October 01, 2025
In the setting of a non-complete doubling metric measure space \((\Omega,d,\mu)\), we construct various bounded linear trace and extension operators for homogeneous and inhomogeneous Besov spaces \(B^\alpha_{p,q}\). Equipping the boundary \(\partial\Omega:=\overline{\Omega}\setminus\Omega\) with a measure which is codimension \(\theta\) Ahlfors regular with respect to \(\mu\), these operators take the form \[T:B^\alpha_{p,q}(\Omega)\to B^{\alpha-\theta/p}_{p,q}(\partial\Omega),\quad E:B^\alpha_{p,q}(\partial\Omega)\to B^{\alpha+\theta/p}_{p,q}(\Omega).\] The trace operators are first constructed under the additional assumption that \(\Omega\) is a uniform domain in its completion. We then use such results along with the technique of hyperbolic filling to remove this assumption in the case that \(\Omega\) is bounded. This extends to the doubling setting some earlier results of Marcos and Saksman-Soto proven under the assumption that the ambient measure is Ahlfors regular.
In the setting of a doubling metric measure space \((X,d,\mu)\), we study various trace and extension results for the Besov spaces \(B^\alpha_{p,q}(X,\mu)\), defined by finiteness of the following seminorm: \[\Vert u\Vert_{HB^\alpha_{p,q}(X,\mu )}:= \left(\int_0^\infty\left(\int_X\fint_{B(x,t)}|u(y)-u(x)|^pd\mu(y)d\mu(x)\right)^{q/p}\frac{dt}{t^{\alpha q+1}}\right)^{1/q},\] for \(u\in L^1_\text{loc}(X)\), \(0<\alpha<1\), \(1\le p<\infty\), and \(1\le q\le\infty\). See Section 2.3 for the precise definitions. These nonlocal spaces are closely connected with the study of the fractional Laplacian, nonlocal minimal surfaces, fractional Hardy inequalities, as well as the study of jump processes in connection with Dirichlet forms, for a sampling see [1]–[8].
Relevant to the topic of this paper, the Besov spaces arise naturally as the trace class of Sobolev spaces for sufficiently regular domains in \(\mathbb{R}^n\), when \(p=q\). For example, it was shown by Gagliardo in [9] that there exist bounded linear trace and extension operators between \(W^{1,p}(\Omega)\) and \(B^{1-1/p}_{p,p}(\partial\Omega,\mathcal{H}^{n-1})\) when \(\Omega\subset\mathbb{R}^n\) is a Lipschitz domain. Furthermore, when a closed set \(F\subset\mathbb{R}^n\) is a \(d\)-set, that is, \(\mathcal{H}^d\) is Ahlfors \(d\)-regular on \(F\), then restrictions of functions in \(W^{1,p}(\mathbb{R}^n)\) to \(F\) belong to \(B^{1-(n-d)/p}_{p,p}(F,\mathcal{H}^d)\), as shown by Jonsson-Wallin [10]. Thus, the dimension of the boundary set determines the regularity of the trace class.
In recent decades, a rich theory of analysis on metric measure spaces has been developed, in particular regarding Sobolev space theory in the nonsmooth setting. One natural substitute for the space \(W^{1,p}\) is the Newton-Sobolev class \(N^{1,p}(X)\) introduced in [11]. When \((X,d,\mu)\) is doubling and supports a Poincaré inequality, the Newton-Sobolev class provides a fruitful framework from which to study potential theory, functions of bounded variation, quasiconformal maps, and analogs of PDEs, for example, in the metric setting. For more on these spaces, see the monographs [12]–[14] for example.
In relation to the study of energy minimization and boundary value problems in this context, the trace theory for the Newton-Sobolev class has undergone significant study in recent years. It was shown by Malý in [15] that when \((X,d,\mu)\) is doubling and supports a \(p\)-Poincaré inequality, and when \(\Omega\subset X\) is a uniform domain whose boundary is bounded, then there exist bounded linear trace and extension operators between \(N^{1,p}(\Omega,\mu)\) and \(B^{1-\theta/p}_{p,p}(\partial\Omega,\nu)\). Here, \(\partial\Omega\) is equipped with a measure \(\nu\) which is codimension \(\theta\) Ahlfors regular with respect to \(\mu\) for some \(\theta>0\). When \((X,d,\mu)\) is a doubling metric measure space, this codimensionality condition, see Definition 1, serves as a replacement for the notion of a \(d\)-set, as used in the Euclidean setting in [10]. Furthermore, it was shown by Björn-Björn-Shanmugalingam in [16] that given any compact doubling metric measure space \((Z,d_Z,\nu)\) and \(0<\alpha<1\), there exists a doubling metric measure space \((X,d,\mu)\) supporting a \(1\)-Poincaré inequality such that \(X\) is a uniform domain in its completion, \(Z\) is bi-Lipschitz equivalent to \(\partial X\), and \(B^\alpha_{p,p}(Z,\nu)\) is the trace class for \(N^{1,p}(X,\mu)\). The space \((X,d,\mu)\) is a so-called uniformized hyperbolic filling of \(Z\), see Section 5.1, and is constructed in such a manner so that \(\nu\) is codimension \(p(1-\alpha)\) Ahlfors regular with respect to \(\mu\). Recently, [17] extended these trace and extension results of [15] between the Newton-Sobolev and Besov spaces to the case when the \(\Omega\) is an unbounded uniform domain, equipped with a doubling measure supporting a Poincaré inequality.
In this paper, we are interested in trace and extension results for Besov spaces themselves, rather than between Newton-Sobolev spaces and Besov spaces. As we consider only these nonlocal function spaces, we do not require that our space supports a Poincaré inequality, only assuming that the measure is doubling. In the Euclidean setting, such results between Besov spaces were also established by Jonsson and Wallin in [10], where they proved restriction and extension theorems between \(B^\alpha_{p,q}(\mathbb{R}^n)\) and \(B^{\alpha-(n-d)/p}_{p,q}(F,\mathcal{H}^d)\) for \(d\)-sets \(F\subset\mathbb{R}^n\). In Ahlfors \(Q\)-regular metric measure spaces \((X,d,\mu)\), where for all balls \(B(x,r)\subset X\), one has \[\mu(B(x,r))\simeq r^Q,\] analogous results were established by Marcos [18] via interpolation techniques, and by Saksman-Soto [19], using an equivalent formulation of the Besov seminorm defined through hyperbolic fillings. The trace results in these papers take the form of restriction theorems between globally defined Besov functions and those defined on \(d\)-sets.
A primary goal of this paper is to construct trace operators on Besov spaces in metric measure spaces whose measure is doubling rather than Ahlfors regular, weakening the assumptions of [18], [19]. A particular motivation for this investigation is the recent interest in the construction of fractional \(p\)-Laplace-type operators in the doubling metric measure space setting, see for example [20]–[23], and formulation of Dirichlet-type problems for such operators [24]. Since the Besov spaces \(B^\alpha_{p,p}(\Omega)\) form the natural domain for such operators, the construction of trace operators on Besov spaces is essential to the study of well-posedness of the corresponding Dirichlet problems. The study of well-posedness in [24] required such a trace result for \(B^\alpha_{p,p}(\Omega)\), see [24], which was obtained by leveraging known trace and extension results between Newton-Sobolev and Besov spaces via hyperbolic fillings [16]. In this paper, we also consider the case \(p\ne q\), and so we cannot take advantage of this relationship with Newton-Sobolev spaces.
Mirroring the trace theorems for Newton-Sobolev spaces on domains obtained in [15], [17], we first establish trace theorems under the additional geometric assumption that the domain is uniform in order to obtain the appropriate energy bounds. In contrast to the uniform domains considered in [15], [17], however, we allow for possibly non-rectifiable uniform curves in the definition, and so our approach is slightly different, see Definition 1 and Remark 1. The first of our main results is the following, pertaining to the homogeneous Besov spaces:
Theorem 1. Let \((\Omega,d,\mu)\) be a locally compact, non-complete metric measure space, with \(\mu\) a doubling measure, such that \(\Omega\) is a uniform domain in its completion \(\overline{\Omega}\). Suppose also that \(\partial\Omega:=\overline{\Omega}\setminus\Omega\), the boundary of \(\Omega\), is equipped with a Borel measure \(\nu\) which is codimension \(\theta\) Ahlfors regular with respect to \(\mu\) for some \(\theta>0\). Let \(0<\alpha<1\), \(1\le p<\infty\), and \(1\le q\le\infty\) be such that \(0<\alpha-\theta/p<1\). Then there exists a bounded linear trace operator \[T:HB^\alpha_{p,q}(\Omega,\mu)\to HB^{\alpha-\theta/p}_{p,q}(\partial\Omega,\nu)\] such that for all \(u\in HB^\alpha_{p,q}(\Omega,\mu)\), we have \[\label{eq:intro32trace32condition} \lim_{r\to 0^+}\fint_{B(z,r)}|u-Tu(z)|^pd\mu=0\qquad{(1)}\] for \(\nu\)-a.e.\(z\in\partial\Omega\), and \[\label{eq:hom32trace32energy32bound} \|Tu\|_{HB^{\alpha-\theta/p}_{p,q}(\partial\Omega,\nu)}\le C\|u\|_{HB^\alpha_{p,q}(\Omega,\mu)},\qquad{(2)}\] where the constant \(C\ge 1\) depends only on \(\alpha\), \(p\), \(q\), and \(\theta\), as well as the doubling, codimensional Ahlfors regularity, and uniform domain constants.
When \(\partial\Omega\) is additionally assumed to be bounded, the operator \(T\) obtained in Theorem 1 also acts as a bounded linear trace operator on the inhomogeneous Besov spaces:
Theorem 1. Under the same assumptions as Theorem 1, suppose in addition that \(\partial\Omega\) is bounded. Then there exists a bounded linear trace operator \[T:B^\alpha_{p,q}(\Omega,\mu)\to B^{\alpha-\theta/p}_{p,q}(\partial\Omega,\nu)\] such that for all \(u\in B^\alpha_{p,q}(\Omega,\mu)\), ?? and ?? hold, and \[\label{eq:trace32inhom32Lp} \|Tu\|_{L^p(\partial\Omega,\nu)}\le C\left(\|u\|_{L^p(\Omega,\mu)}+\|u\|_{HB^\alpha_{p,q}(\Omega,\mu)}\right),\qquad{(3)}\] where the constant \(C\ge 1\) depends only on \(\alpha,\,p,\,q,\,\theta\), \(\mathop{\mathrm{diam}}(\partial\Omega)\), as well as the doubling and codimensional Ahlfors regularity constants.
Theorem 1 and Theorem 1 are proved in Sections 3 via Proposition 1 and Proposition 1.
We will later use Theorem 1 and Theorem 1, along with corresponding extension results, to establish a trace theorem on bounded spaces without the uniform domain assumption by using the hyperbolic filling construction, see Theorem 1. However, we believe Theorem 1 and Theorem 1 may be of independent interest, as the operators there are defined intrinsically, without using this auxiliary construction, and these results apply to cases where \(\Omega\) may be unbounded.
A second goal of this paper is to construct corresponding extension operators for Besov spaces in doubling metric measure spaces. In contrast to Theorem 1 and Theorem 1, no uniform domain assumption is needed, and as mentioned above these extension results will be used to construct the trace operator in Theorem 1 without the uniform domain assumption. Our operators are constructed using standard Whitney extension techniques, involving Whitney covers, Lispchitz partitions of unity, and discrete convolution, all of which are available in the doubling metric measure space setting. This general strategy was used in [10] to construct extension operators from \(B^{\alpha-(n-d)/p}_{p,q}(F,\mathcal{H}^d)\) to \(B^\alpha_{p,q}(\mathbb{R}^n)\) for \(d\)-sets \(F\subset\mathbb{R}^n\). There, however, \(C^\infty\) functions were used in the partition of unity, and so to obtain the necessary energy estimates, the authors were able to take advantage of smoothness properties of the discrete convolution along line segments joining certain points in the domain. Such techniques are not available in the metric setting, and so we utilize the Lipschitz partition of unity instead.
In the doubling metric measure space setting, similar Whitney extension operators, via Lipschitz partitions of unity, were constructed in [18] from \(B^{\alpha-\theta/p}_{p,q}(F)\) to \(B^\alpha_{p,q}(X)\) for \(\theta\)-codimensional closed sets \(F\subset X\). A notable difference between our extension results and that of [18] involves the bounds of the extension operator. In both [18] and the analogous Euclidean results in [10], only the inhomogeneous Besov spaces were considered. As such, the bounds on the Besov energy of the extension operator were given in terms of both the Besov energy and the \(L^p\)-norm of the original function. The proof of this bound is included in the preprint version of [18], though omitted from the published version. We first construct an extension operator for the homogeneous Besov spaces, in which we bound the energy of the extension by the energy of the original function alone:
Theorem 1. Let \((\Omega,d,\mu)\) be a locally compact, non-complete metric measure space, with \(\mu\) a doubling measure. Suppose also that \(\partial\Omega:=\overline{\Omega}\setminus\Omega\), the boundary of \(\Omega\), is equipped with a Borel measure \(\nu\) which is codimension \(\theta\) Ahlfors regular with respect to \(\mu\) for some \(\theta>0\). Let \(0<\alpha<1\), \(1\leq p <\infty\), and \(1\le q\le\infty\) be such that \(\alpha<1-\theta/p\). Then there exists a bounded linear extension operator \[\widetilde{E}:HB^\alpha_{p,q}(\partial\Omega,\nu)\to HB^{\alpha+\theta/p}_{p,q}(\Omega,\mu).\] In particular, there exists \(C\ge 1\), depending only on \(\alpha\), \(p\), \(q\), \(\theta\), and the doubling and codimensional Ahlfors regularity constants such that \[\label{eq:hom32extension32energy32bound} \|\widetilde{E}f\|_{HB^{\alpha+\theta /p}_{p,q}(\Omega,\mu)}\le C\|f\|_{HB^\alpha_{p,q}(\partial\Omega,\nu)},\qquad{(4)}\] for all \(f\in HB^\alpha_{p,q}(\partial\Omega,\nu)\).
When \(\Omega\) is bounded, the extension operator \(\widetilde{E}\) constructed for the homogeneous Besov spaces becomes an extension operator for the inhomogeneous spaces:
Corollary 1. Under the assumptions of Theorem 1, suppose in addition that \(\Omega\) is bounded. Then there exists a bounded linear extension operator \[\widetilde{E}:B^\alpha_{p,q}(\partial\Omega,\nu)\to B^{\alpha+\theta/p}_{p,q}(\Omega,\mu)\] such that for all \(f\in B^\alpha_{p,q}(\partial\Omega,\nu)\), ?? holds and \[\label{eq:inhom32Lp32bounds} \|\widetilde{E}f\|_{L^p(\Omega,\mu)}\le C\|f\|_{L^p(\partial\Omega,\nu)},\qquad{(5)}\] where \(C\ge 1\) depends only on \(p\), \(\theta\), \(\mathop{\mathrm{diam}}(\Omega)\), as well as the doubling and codimensional Ahlfors regularity constants.
If \(\Omega\) is not assumed to be bounded, then by applying a Lipschitz cutoff function to the homogeneous extension above, we construct the following extension operator for the inhomogeneous spaces, analogous to that obtained in [18], where the Besov energy of the extension is controlled by both the Besov energy and \(L^p\)-norm of the original function. Our proof is similar to the Euclidean proof in [10], where a smooth cutoff function was used.
Theorem 1. Under the same assumptions as Theorem 1, there exists a bounded linear extension operator \[E:B^\alpha_{p,q}(\partial\Omega,\nu)\to B^{\alpha+\theta/p}_{p,q}(\Omega,\mu).\] In particular, there exists \(C\ge 1\), depending only on \(\alpha\), \(p\), \(q\), \(\theta\), as well as the doubling and codimensional Ahlfors regularity constants, such that \[\begin{align} \|Ef\|_{L^p(\Omega,\mu)}&\le C\|f\|_{L^p(\partial\Omega,\nu)}\quad\text{ and }\\ \|Ef\|_{HB^{\alpha+\theta/p}_{p,q}(\Omega,\mu)}&\le C\left(\|f\|_{HB^\alpha_{p,q}(\partial\Omega,\nu)}+\|f\|_{L^p(\partial\Omega,\nu)}\right) \end{align}\] for all \(f\in B^\alpha_{p,q}(\partial\Omega,\nu).\)
In Corollary 1, we show that under the assumptions of Theorem 1, we have that \(T\circ E\) and \(T\circ\widetilde{E}\) act as identity operators on the Besov spaces on the boundary.
Having established the relevant extension operators, we then obtain the following trace result without the uniform domain assumption:
Theorem 1. Let \((Z,d,\nu)\) be a locally compact, non-complete, bounded metric measure space,with \(\nu\) a doubling measure. Suppose that \(\partial Z:=\overline{Z}\setminus Z\), the boundary of \(Z\), is equipped with a Borel measure \(\pi\), which is codimension \(\theta\) Ahlfors regular with respect to \(\nu\) for some \(\theta >0\). Let \(1\le p<\infty\), \(1\le q\le\infty\) and \(\theta/p<\alpha<1\). Then there exist a bounded, linear trace operator \[T:B^\alpha_{p,q}(Z,\nu)\to B^{\alpha-\theta/p}_{p,q}(\partial Z,\pi),\] such that for all \(u\in B^\alpha_{p,q}(Z,\nu)\), we have \[\label{eq:Intro32gen32Trace32condition} \lim_{r\to 0^+}\fint_{B(z,r)}|u-Tu(z)|^pd\nu=0\qquad{(6)}\] for \(\pi\)-a.e.\(z\in \partial Z\), and
\[\|Tu\|_{HB^{\alpha-\theta/p}_{p,q}(\partial Z,\pi)}\le C\|u\|_{HB^\alpha_{p,q}(Z,\nu)},\qquad\|Tu\|_{L^p(\partial Z,\pi)}\le C\left(\|u\|_{L^p(Z,\nu)}+\|u\|_{HB^\alpha_{p,q}(Z,\nu)}\right),\] where \(C\ge 1\) depends only on \(\alpha\), \(p\), \(q\), \(\theta\), \(\mathop{\mathrm{diam}}(Z)\), as well as the doubling and codimensional Ahlfors regularity constants.
Furthermore, the bounded linear extension operator \(\widetilde{E}:B^{\alpha-\theta/p}_{p,q}(\partial Z,\pi)\to B^\alpha_{p,q}(Z,\nu)\) given by Corollary 1 is a right-inverse of \(T\). That is, for all \(f\in B^{\alpha-\theta/p}_{p,q}(\partial Z,\pi)\), we have that \[T\widetilde{E}f(z)=f(z)\] for \(\pi\)-a.e.\(z\in\partial Z\).
To prove Theorem 1, we utilize the construction given in [16] to realize \((\overline{Z},d,\nu)\) as the boundary of a uniform space \((X,d_\varepsilon,\mu_\beta)\), see Section 5.1, referred to as the uniformized hyperbolic filling of \((\overline{Z},d,\nu)\). This space is constructed such that \(X\) is uniform in its completion, the measure \(\mu_\beta\) is doubling, and \(\nu\) is codimensional Ahlfors regular with respect to \(\mu_\beta\). This allows us to define \(T\) and \(E\) as compositions of operators between the Besov spaces on \(\partial Z\), \(Z\), and \(X\), as given by Theorem 1, Theorem 1, and Theorem 1. We use the notation \((Z,d,\nu)\) in the statement of the above theorems to distinguish this space from its hyperbolic filling.
As mentioned above, Theorem 1 assumes only that \(\nu\) is doubling rather than Ahlfors regular, which is the standing assumption in [18] and [19], used primarily to establish the trace theorems there. In [18], for example, the trace, or restriction, theorem is obtained via interpolation results for Bessel-type potential spaces in Ahlfors regular metric measure spaces. As such, it is proven under the assumption that the measure of the space is infinite and that \(1<p<\infty\), \(1\le q<\infty\), and the regularity parameter \(\alpha\) is sufficiently small. While we assume here that the space has finite measure, Theorem 1 holds for the full range of regularity parameters \(0<\alpha<1-\theta/p\), as well as the cases when \(p=1\) or \(q=\infty\).
The trace result [19], which assumes no restriction on the measure of the space, also utilizes the assumption of Ahlfors \(Q\)-regularity of \(\nu\) to ensure that \(\pi\)-a.e.\(z\in \partial Z\) is a Lebesgue point of a function \(u\in B^\alpha_{p,q}(\overline{Z},\nu)\), where \(\pi\) is an Ahlfors \(d\)-regular measure on \(\partial Z\), with \(0<d<Q\), see [19]. In this manner, the authors show that \(Tu=u|_{\partial Z}\) \(\pi\)-a.e.on \(\partial Z\) for each \(u\in B^\alpha_{p,q}(\overline{Z},\nu)\) [19]. While many of the estimates used in the proof of their result hold under the weaker assumption of doubling and codimension Ahlfors regularity of \(\pi\), it is only shown in this setting that \(Tu=u|_{\partial Z}\) for continuous \(u\in B^\alpha_{p,q}(\overline{Z},\nu)\), as the precise Hausdorff dimension of \(\partial Z\) is not known, see [19]. In our result, however, we show that the analogous property ?? holds for all \(u\in B^\alpha_{p,q}(Z,\nu)\), not just continuous functions.
In Theorem 1, the boundedness assumption on \((Z,d,\nu)\) is due to our use of the particular construction of the hyperbolic filling from [16]. An alternative hyperbolic filling construction, given via Busemann functions and applicable in the noncompact case, was introduced by Butler in the as-of-yet unpublished papers [25], [26]. In particular, it was shown in [26] that many of the key properties of the uniformized hyperbolic filling from [16] hold with this Busemann function construction. We therefore believe it is likely that Theorem 1 could be proven without the boundedness assumption on \((Z,d,\nu)\) by using this alternate approach. However, this investigation is left for future work.
The structure of the paper is as follows: in Section 2, we discuss the necessary definitions and background notions used throughout the paper. In Section 3, we prove the trace results Theorem 1 and Theorem 1, and Section 4 is devoted to proving the extension results Theorem 1 and Theorem 1. Finally, Section 5 is devoted to introducing the necessary background for the construction of the hyperbolic filling and the proof of Theorem 1.
In this section, we introduce the necessary definitions and notations used throughout the paper. At times, we let \(C>0\) denote a constant which, unless otherwise specified, depends only on the structural constants of the metric measure space, such as the doubling constant, for example. Its precise value is not of interest to us, and may change with each occurrence, even within the same line. Furthermore, given quantities \(A\) and \(B\), we will often use the notation \(A\simeq B\) to mean that there exists a constant \(C\ge 1\) such that \(C^{-1} A\le B\le CA\). Likewise, we use \(A\lesssim B\) and \(A\gtrsim B\) if the left and right inequalities hold, respectively. Given a metric space \((X,d)\), we will at times use the notation \(d(x,A):=\mathop{\mathrm{dist}}(x,A)\) to denote the distance between a point \(x\in X\) and a set \(A\subset X\).
A Borel measure \(\mu\) on a metric space \((X,d)\) is said to be doubling if there exists a constant \(C_\mu\ge 1\) such that \[0<\mu(B(x,2r))\le C_\mu\,\mu(B(x,r))<\infty\] for all \(x\in X\) and \(r>0\). By iteration, there exists constants \(C\ge 1\) and \(Q>0\) depending only on \(C_\mu\) such that \[\frac{\mu(B(y,r))}{\mu(B(x,R))}\ge C^{-1}\left(\frac{r}{R}\right)^Q\] for every \(0<r\le R\), \(x\in X\), and \(y\in B(x,R)\).
Doubling measures play an important role in analysis on metric measure spaces. In particular, many classical results, such as Lebesgue differentiation theorem, Hardy-Littlewood maximal function theorem, and the John-Nirenberg inequality can be generalized to the metric setting when the space is equipped with a doubling measure, see for example [13] and [14]. Another important consequence of the doubling property is the following existence of Whitney covers:
Lemma 1 ([13], Proposition 4.1.15). Let \((X,d,\mu)\) be a doubling metric measure space, and let \(\Omega\subset X\) be an open set such that \(X\setminus\Omega\ne\varnothing\). Then there exists a countable collection \(\mathcal{W}_\Omega=\{B(p_{i,j},r_{i,j})=B_{i,j}\}\) of balls in \(\Omega\) so that
\(\bigcup_{i,j}B_{i,j}=\Omega\)
\(\sum_{i,j}\chi_{B(p_{i,j},2r_{i,j})}\le 2C_\mu^5\)
\(2^{i-1}<r_{i,j}\le 2^i\) for all \(i,\)
\(r_{i,j}=\frac{1}{8}d(p_{i,j}, X\setminus\Omega)\).
For each \(i\in\mathbb{Z}\), \(\{p_{i,j}\}_j\) is a \(2^i/C\)-separated set, and so the collection \(\{ B_{i,j}\}_{j\in\mathbb{Z}}\) has bounded overlap. That is, for each \(K\ge 1\), there exists \(C_K\ge 1\) so that \[\sum_{j}\chi_{KB_{i,j}}\le C_K.\]
Furthermore, there exists a Lipschitz partition of unity \(\{\varphi_{i,j}\}_{i,j}\) subordinate to \(\mathcal{W}_\Omega.\) That is, each \(\varphi_{i,j}\) is \(C/r_{i,j}\)-Lipschitz continuous with \(C\) depending only on \(C_\mu\), such that \(0\le\varphi_{i,j}\le \chi_{2B_{i,j}}\) and \(\sum_{i,j}\varphi_{i,j}\equiv \chi_{\Omega}\).
Definition 1. Let \(\theta>0\). Given an open set \(\Omega\subset X\), we say that a measure \(\nu\) on \(\partial\Omega\) is codimension \(\theta\) Ahlfors regular with respect to \(\mu\) if there exists a constant \(C_\theta\ge 1\) such that for all \(z\in\partial\Omega\) and \(0<r\le 2\mathop{\mathrm{diam}}(\partial\Omega),\) we have that \[\frac{1}{C_\theta}\frac{\mu(B(z,r)\cap\Omega)}{r^\theta}\le\nu(B(z,r)\cap\partial\Omega)\le C_\theta\frac{\mu(B(z,r)\cap\Omega)}{r^\theta}.\]
For a set \(A\subset X\), its codimension \(\theta\) Hausdorff measure with respect to \(\mu\) is defined as \[\mathcal{H}^{-\theta}_{\mu}(A):=\lim_{\varepsilon\to 0^+}\, \inf\bigg\lbrace \sum_{i\in I\subset \mathbb{N}}\frac{\mu(B_i)}{\mathop{\mathrm{rad}}(B_i)^\theta}\, :\, A\subset\bigcup_{i\in I}B_i,\, \mathop{\mathrm{rad}}(B_i)\le \varepsilon\bigg\rbrace.\] Then we have the following result for codimension \(\theta\) Ahlfors regular measures on \(\partial\Omega\) (see also [24]).
Lemma 1. Let \((X,d,\mu )\) be a doubling metric measure space and \(\Omega\subset X\) open and such that \(\mu|_{\Omega}\) is also doubling. If a measure \(\nu\) supported on \(\partial \Omega\) is codimension \(\theta\) Ahlfors regular with respect to \(\mu\) then \(\nu\simeq\mathcal{H}^{-\theta}_{\mu|_{\overline{\Omega}}}|_{\partial\Omega}\). In particular, \(\mathcal{H}^{-\theta}_{\mu|_{\overline{\Omega}}}|_{\partial\Omega}\) is codimension \(\theta\) Ahlfors regular with respect to \(\mu\).
Proof. Let \(\nu\) be codimension \(\theta\) Ahlfors regular with respect to \(\mu\). Notice that by the doubling condition of \(\mu\) and codimensionality we also have \(\nu\) doubling (see Remark 1 below).
Let \(z\in \partial\Omega\), \(r>0\). For \(\varepsilon >0\) consider a covering of \(B(z,r)\) by balls \(\{ B_i\}_{i\in I\subset \mathbb{N}}\) in \(X\) with \(\mathop{\mathrm{rad}}(B_i)<\varepsilon\). Let \(I_1\subset I\) the set of indices such that \(B_i\cap \partial\Omega\neq\varnothing\), and for each \(i\in I\) let \(\hat{B}_i\) with center in \(\partial\Omega\) and \(\mathop{\mathrm{rad}}(B_i)\leq \mathop{\mathrm{rad}}(\hat{B}_i)\leq 2\mathop{\mathrm{rad}}(B_i)\), so that \(B_i\cap\partial\Omega\subset \hat{B}_i\subset 3B_i\). Notice that \(B(z,r)\cap\partial\Omega \subset\bigcup_{i\in I_1} \hat{B}_i\cap\partial\Omega\), and then by the doubling condition and codimensionality we have \[\begin{align} \sum_{i\in I}\frac{\mu|_{\overline{\Omega}} (B_i)}{\mathop{\mathrm{rad}}(B_i)^\theta}\geq C_d^{-2}\sum_{i\in I_1}\frac{\mu (\hat{B}_i\cap\Omega )}{\mathop{\mathrm{rad}}(\hat{B}_i)^\theta}\geq (C_d^2C_\theta)^{-1}\sum_{i\in I_1}\nu (\hat{B}_i \cap\partial\Omega )\geq (C_d^2C_\theta )^{-1}\nu (B(z,r)\cap\partial\Omega ). \end{align}\] Taking infimum over all possible coverings and \(\varepsilon\to 0\) we get \[\nu (B(z,r)\cap\partial\Omega)\lesssim\mathcal{H}^{-\theta}_{\mu|_{\overline{\Omega}}} (B(z,r)\cap\partial\Omega ).\] Now for each \(\eta >0\) there exists \(\varepsilon_\eta >0\) such that for all \(0<\varepsilon <\varepsilon_\eta\) \[\mathcal{H}^{-\theta}_{\mu|_{\overline{\Omega}}} (B(z,r)\cap\partial\Omega )-\eta\leq \inf\bigg\lbrace \sum_{i\in I\subset \mathbb{N}}\frac{\mu|_{\overline{\Omega} }(B_i)}{\mathop{\mathrm{rad}}(B_i)^\theta}\, :\, B(z,r)\cap\partial\Omega \subset\bigcup_{i\in I}B_i,\, \mathop{\mathrm{rad}}(B_i)\le \varepsilon \bigg\rbrace.\] Consider also \(\varepsilon <r\). By the \(5B\)-covering lemma, cover \(B(z,r)\cap\partial\Omega\) by a collection \(\{ B_i\}_{i\in I}\) of balls with radii \(\varepsilon\) and centers in \(B(z,r)\cap\partial\Omega\) so that \(\frac{1}{5}B_i\) are disjoint. Then, since \(B_i\subset B(z,2r)\), \[\begin{align} \mathcal{H}^{-\theta}_{\mu|_{\overline{\Omega}}} (B(z,r)\cap\partial\Omega )-\eta & \leq \sum_{i\in I}\frac{\mu|_{\overline{\Omega}} (B_i)}{\mathop{\mathrm{rad}}(B_i)^\theta}\leq C_d^{3}\sum_{i\in I}\frac{\mu (\frac{1}{5} B_i\cap\Omega )}{5^\theta\mathop{\mathrm{rad}}(\frac{1}{5} B_i)^\theta}\\ &\leq \frac{C^3_dC_\theta}{5^\theta}\sum_{i\in I}\nu (\frac{1}{5}B_i\cap\partial\Omega )\leq \frac{C^3_dC_\theta}{5^\theta}\nu (B(z,2r)\cap\partial\Omega). \end{align}\] Finally, letting \(\eta\) go to \(0\) we have by the above estimate and the doubling condition on \(\nu\) \[\mathcal{H}^{-\theta}_{\mu|_{\overline{\Omega}}} (B(z,r)\cap\partial\Omega )\lesssim \nu (B(z,r)\cap\partial\Omega).\qedhere\] ◻
Remark 1. For a given \(\theta >0\) and open set \(\Omega\), it follows by definition that all measures on \(\partial\Omega\) which are codimension \(\theta\) Ahlfors regular with respect to a fixed ambient measure are equivalent. The above lemma shows that \(\mathcal{H}^{-\theta}_{\mu|_{\overline{\Omega}}}|_{\partial\Omega}\) is the natural \(\theta\)-codimensional measure. Moreover, if this measure is not codimension \(\theta\) Ahlfors regular with respect to \(\mu\), then there is no such measure for the parameter \(\theta\).
In the work of Jonsson and Wallin on \(\mathbb{R}^n\), [10], the assumption that \(\partial\Omega\) is a \(d\)-set implies that the \(d\)-dimensional Hausdorff measure is codimension \((n-d)\) Ahlfors regular with respect to the Lebesgue measure on \(\mathbb{R}^n\). As such, this codimensional assumption on \(\nu\) generalizes the \(d\)-set assumption on \(\mathbb{R}^n\).
Remark 1. If \(\mu|_\Omega\) is doubling, with constant \(C_\mu\), and \(\nu\) is codimension \(\theta\) Ahlfors regular with respect to \(\mu\), then \(\nu\) is doubling. Indeed, for \(z\in\partial\Omega\) and \(0<r\le 2\mathop{\mathrm{diam}}(\partial\Omega)\), we have that \[\nu(B(z,2r)\cap\partial\Omega)\le C_\theta\frac{\mu(B(z,2r)\cap\Omega)}{(2r)^\theta}\le C_\theta C_\mu\frac{\mu(B(z,r)\cap\Omega)}{(2r)^\theta}\le\frac{C_\mu C_\theta^2}{2^\theta}\nu(B(z,r)\cap\partial\Omega).\]
In Sections 3 and 4, we will assume that \((\Omega,d,\mu)\) is a locally compact, non-complete metric measure space, with \(\mu\) a doubling measure. The same setting is considered in Section 5, though with different notation. Under these assumptions, \(\Omega\) is an open set in its completion \(\overline{\Omega}\), and its boundary is given by \(\partial\Omega=\overline{\Omega}\setminus\Omega\). We extend \(\mu\) to \(\overline{\Omega}\) by the zero extension; note that in this setting \(\mu\) is doubling on \(\overline{\Omega}\) with the same constant. Furthermore, we will equip \(\partial\Omega\) with a Borel measure \(\nu\) which is codimension \(\theta\) Ahlfors regular with respect to \(\mu\).
Throughout this paper, a curve is a continuous map \(\gamma:[a,b]\to X\). With a slight abuse of notation, we will also use \(\gamma\) to denote the trajectory of such a curve.
Definition 1. We say that a domain \(\Omega\subset X\) is an \(A\)-uniform domain for some \(A\ge 1\) if for every \(x,y\in\Omega\), there exists a curve \(\gamma\) in \(\Omega\) joining \(x\) to \(y\) such that
\(\mathop{\mathrm{diam}}(\gamma)\le Ad(x,y)\),
for each \(z\in\gamma\), we have that \(\min\{\mathop{\mathrm{diam}}(\gamma_{x,z}),\,\mathop{\mathrm{diam}}(\gamma_{z,y})\}\le Ad(z,X\setminus\Omega)\), where \(\gamma_{x,z}\) denotes a fragment of \(\gamma\) with endpoints \(x\) and \(z\) (and analogously for \(\gamma_{z,y}\)).
Remark 1. In Section 3, we will assume that \(\Omega\) is a uniform domain in its completion in order to obtain the desired energy bounds for our trace operator. In the literature, uniform domains are often defined using the length of curves in place of the diameters used in the above definition, in which case they are sometimes referred to as length uniform domains. Clearly, length uniform domains are uniform domains in the sense of Definition 1. For the study of trace properties of Newton-Sobolev functions, it is often assumed that \(\Omega\) is a length uniform domain, see for example [15], [17], [27], since the standard assumption that the measure is doubling and supports a Poincaré inequality ensures that the space is quasiconvex, hence rectifiably connected.
As our interest is in the nonlocal Besov spaces, we do not assume a Poincaré inequality, and so this general setting may include spaces which possess few or possibly zero rectifiable curves. For this reason, we define uniform domains as above using diameter. With this definition, the uniform domain property is preserved under a quasisymmetric change in the metric, such as snowflaking for example, and for this reason such definitions are commonly used in the study of Dirichlet forms, see [28]–[30].
To obtain the energy bounds for the Newton-Sobolev trace operators in [15], [17], the length uniform domain property was used to obtain a suitable chain of balls inside \(\Omega\) joining pairs of points on \(\partial\Omega\). As the uniform curves in those settings are rectifiable and thus admit arc-length parametrizations, one can use the Arzelà-Ascoli theorem to join each pair of points on \(\partial\Omega\) by a uniform curve. The chain of balls is then obtained by choosing appropriate points along this curve. We similarly use the uniform domain property to obtain a suitable chain of balls, see Lemma 1 below, but as our uniform curves need not be rectifiable, it is not immediate how to extend such curves to the boundary. For this reason, we obtain our chain of balls in a slightly different manner, using Harnack chains and the following Lemmas 1 and 1 from [28].
Definition 1. Let \(\Omega\subset X\) be a domain and \(M\ge 1\). For \(x,y\in\Omega\), an \(M\)-Harnack chain from \(x\) to \(y\) in \(\Omega\) is a sequence of balls \(B_1,\dots,B_n\), each contained in \(\Omega\), such that \(x\in M^{-1}B_{1}\), \(y\in M^{-1}B_n\), and \(M^{-1}B_i\cap M^{-1}B_{i+1}\ne\varnothing\) for each \(i=1,\dots,n-1\). The number \(n\) of balls in a Harnack chain is called the length of the Harnack chain.
Lemma 1 ([28], Lemma 3.4). Let \(\Omega\subset X\) be an \(A\)-uniform domain. For each \(z\in\partial\Omega\) and \(r>0\) such that \(\Omega\setminus B(z,r)\ne\varnothing\), there exists \(z_r\in\Omega\cap\partial B(z,r)\) such that \(d(z_r,X\setminus\Omega)\ge (2A)^{-1}r\).
Lemma 1 ([28], Proposition 2.12). Let \((X,d,\mu)\) be a doubling metric measure space. Let \(\Omega\subset X\) be a domain, and for \(x,y\in\Omega\), let \(\gamma\) be a curve in \(\Omega\) joining \(x\) to \(y\). Assume that \(d(z,X\setminus\Omega)\ge\delta>0\) for all \(z\in\gamma\). Then for any \(M>1\) and any \(0<r<\delta\), there exists an \(M\)-Harnack chain \(\{B_j:=B(x_j,r)\}_j\) from \(x\) to \(y\) in \(\Omega\), with \(x_j\in\gamma\), and with length less than \(C(1+Mr^{-1}\mathop{\mathrm{diam}}(\gamma))^\alpha\), where \(C\) and \(\alpha\) are constants depending only on the doubling constant.
Lemma 1. Let \((X,d,\mu)\) be a doubling metric measure space, and let \(\Omega\subset X\) be an \(A\)-uniform domain. Then for each \(z,w\in\partial\Omega\), there exists a chain of balls \(\{B_k:=B(x_k,r_k)\}_{k\in\mathbb{Z}}\) such that
for each \(k\in\mathbb{Z}\), we have that \(8B_k\subset\Omega\),
\(\lim_{k\to\infty}x_k=z\), \(\lim_{k\to-\infty}x_k=w\),
for each \(k\in\mathbb{Z}\), we have that \(2^{-1}B_k\cap 2^{-1}B_{k+1}\ne\varnothing\),
there exists \(N\ge 1\) such that for each \(k\ge 0\) and each \(x\in B_k\), we have that \[r_k\simeq d(x,X\setminus\Omega)\simeq d(x,z)\lesssim 2^{-|k|/N}d(z,w),\]
there exists \(N\ge 1\) such that for each \(k<0\) and each \(x\in B_k\), we have that \[r_k\simeq d(x,X\setminus\Omega)\simeq d(x,w)\lesssim 2^{-|k|/N}d(z,w),\]
there exists \(C\ge 1\) such that \[\sum_{k}\chi_{4B_k}\le C,\]
The comparison constants, the constant \(N\), and the constant \(C\) above depend only on \(A\) and the doubling constant.
Proof. Let \(z,w\in\partial\Omega\), and let \(r_0:=d(z,w)/4\). By Lemma 1, it follows that for each \(i\in\mathbb{N}\cup\{0\}\), there exists \(x_i\in\Omega\cap\partial B(z,r_0/2^i)\) such that \[\label{eq:xi32distance32to32boundary} d(x_i,X\setminus\Omega)\ge\frac{r_0}{(2A)2^i}.\tag{1}\] For each \(i\in\mathbb{N}\cup\{0\}\), let \(\gamma_i\) be an \(A\)-uniform curve joining \(x_i\) to \(x_{i+1}\) in \(\Omega\). We claim that for all \(\zeta\in\gamma_i\), \[\label{eq:gammai32distance32to32boundary} d(\zeta,X\setminus\Omega)\ge\frac{r_0}{(8A^2)2^i}.\tag{2}\] Indeed, from 1 , we see that if \(\zeta\in\gamma_i\cap B(x_i,r_0/(4A2^i))\), then \(d(\zeta,X\setminus\Omega)\ge\frac{r_0}{(4A)2^i}\). Likewise, if \(\zeta\in B(x_{i+1},r_0/(4A2^{i+1}))\), then \(d(\zeta,X\setminus\Omega)\ge\frac{r_0}{(8A)2^i}\). For the remaining case when \[\zeta\in\gamma_i\setminus\left(B\left(x_i,\frac{r_0}{(4A)2^i}\right)\cup B\left(x_{i+1},\frac{r_0}{(4A)2^{i+1}}\right)\right),\] it follows from the \(A\)-uniform property of \(\gamma_i\) that \[\begin{align} d(\zeta,X\setminus\Omega)\ge A^{-1}\min\{\mathop{\mathrm{diam}}(\gamma_{i,x_i,\zeta}),\,\mathop{\mathrm{diam}}(\gamma_{i,\zeta,x_{i+1}})\}\ge\frac{r_0}{(8A^2)2^i}. \end{align}\] Thus, 2 holds in each case, proving the claim.
Letting \[r_i:=\frac{r_0}{(64A^2)2^i},\] we then apply Lemma 1 to obtain a \(2\)-Harnack chain \(\{B_{i,j}:=B(x_{i,j},r_i)\}_{j=1}^{n_i}\) from \(x_i\) to \(x_{i+1}\) in \(\Omega\), with \(x_{i,j}\in\gamma_i\), and such that \[\label{eq:Harnack32chain32length} n_i\le C(1+2r_i^{-1}\mathop{\mathrm{diam}}(\gamma_i))^\alpha\le C(1+192 A^3)^\alpha=:N.\tag{3}\] Here the last inequality follows from the coarse estimate that \(\mathop{\mathrm{diam}}(\gamma_i)\le(3A/2)r_0/2^i\). We claim that for each \(j=1,\dots,n_i\), the following holds for all \(x\in B_{i,j}\), with comparison constants depending only on \(A\) and the doubling constant: \[\begin{align} \label{eq:Bij32distance32to32boundary} r_i\simeq d(x,X\setminus\Omega)\simeq d(x,z)\simeq 2^{-i}d(z,w). \end{align}\tag{4}\] Indeed, \(r_i\simeq 2^{-i}d(z,w)\) by the definition of \(r_i\) and \(r_0\). Moreover, we have that \[\begin{align} \label{eq:ri32comparison} d(x,X\setminus\Omega)\le d(x,z)\le d(x,x_{i,j})+d(x_{i,j},x_{i+1})+d(x_{i+1},z)\le r_i+\mathop{\mathrm{diam}}(\gamma_i)+r_0/2^{i+1}\lesssim r_i, \end{align}\tag{5}\] and by 2 , we have that \[d(x,X\setminus\Omega)\ge d(x_{i,j},X\setminus\Omega)-d(x_{i,j},x)\gtrsim r_i.\] Thus, 4 holds, proving the claim.
By relabeling the collection \(\bigcup_{i=0}^\infty\{B_{i,j}\}_{j=1}^{n_i}=\{B_k:=B(x_k,r_k)\}_{k\ge 0}\) with the appropriate ordering, it follows from the construction, the definition of \(2\)-Harnack chains, and 2 , that claims \((i)\), \((ii)\), and \((iii)\) of the lemma hold (when \(k\ge 0\)). Furthermore, if \(B_k=B(x_{i,j},r_i)\) for some \(i,j\), then from 4 and 3 , it follows that \(k\le Ni\). Thus, \(-i\le -k/N\), and so claim \((iv)\) holds.
To define the collection of balls \(\{B_k\}_{k<0}\), we repeat the argument above, this time using Lemma 1 to obtain \(y_i\in\Omega\cap\partial B(w,r_0/2^i)\) such that \(d(y,X\setminus\Omega)\ge r_0/(2A2^i)\) for each \(i\in\mathbb{N}\cup\{0\}\). We then use uniform curves joining \(y_i\) to \(y_{i+1}\) and Lemma 1 to obtain the analogous collection of balls \(\{B(y_{i,j},r_i)\}_{j=1}^{n_i}\) via \(2\)-Harnack chains, with length bounded by \(N\), along the uniform curves. We also join \(x_0\) to \(y_0\) by an \(A\)-uniform curve, and then apply Lemma 1 to likewise obtain a \(2\)-Harnack chain \(\{B(z_{0,j},r_0/(64A^2))\}_j\) from \(x_0\) to \(y_0\) of length bounded by \(N\). We then relabel the collection \[\{B(z_{0,j},r_0/(64A^2))\}_j\cup\bigcup_{i=0}^\infty\{B(y_{i,j},r_i)\}_{j=1}^{n_i}=\{B_k:=B(x_k,r_k)\}_{k<0},\] and by the above arguments, it follows that the collection \(\{B_k\}_{k\in\mathbb{Z}}\) satisfies claims \((i)\)-\((v)\) of the Lemma.
It remains to prove claim \((vi)\). To this end, we denote \[\mathcal{B}_i^+:=\{B(x_{i,j},r_i)\}_{j},\quad\mathcal{B}_i^-:=\{B(y_{i,j},r_i)\}_j,\quad\mathcal{B}_0:=\{B(z_{i,j},r_0/(64A^2))\}_j.\] Then \(\{B_k\}_{k\in\mathbb{Z}}\) is just a relabeling of \(\mathcal{B}_0\cup\bigcup_{i=0}^\infty(\mathcal{B}_i^+\cup\mathcal{B}_i^-)\). Let \(B:=B(x_{i,j},r_i)\) be a ball in the collection \(\mathcal{B}_i^+\), and let \(x\in 4B\). Then, from 2 , it follows that \[\begin{align} d(x,X\setminus\Omega)\ge d(x_{i,j},X\setminus\Omega)-d(x,x_{i,j})\ge\frac{r_0}{(8A^2)2^i}-\frac{r_0}{(16A^2)2^i}=\frac{r_0}{(16A^2)2^i}. \end{align}\] Likewise, similar to 5 , we have that \[\begin{align} d(x,X\setminus\Omega)\le d(x,z)\le\frac{r_0}{(16A^2)2^i}+(3A/2)\frac{r_0}{2^i}+\frac{r_0}{2^{i+1}}=\left((16A^2)^{-1}+3A/2+1/2\right)\frac{r_0}{2^i}. \end{align}\] This shows that there exists \(C\ge1\) depending only on \(A\) such that if \(i,i'\in\mathbb{N}\cup\{0\}\) with \(|i-i'|\ge C\), then \[4B\cap 4B'=\varnothing\] for all \(B\in\mathcal{B}^+_i\) and \(B'\in \mathcal{B}^+_{i'}\), and by symmetry for all \(B\in\mathcal{B}^-_{i}\) and \(B'\in\mathcal{B}^-_{i'}\). Similar arguments show that there exists (a possibly different) \(C\ge 1\), depending only on \(A\), such that if \(i,i'\in\mathbb{N}\cup\{0\}\) are such that \(i,i'\ge C\), then for all \(B\in\mathcal{B}_i^+\), \(B'\in\mathcal{B}^-_{i'}\), \(B''\in\mathcal{B}_0\), we have \[4B\cap 4B'=4B\cap 4B''=4B'\cap4B''=\varnothing.\] Thus, for a given \(B_k\), if \[4B_{k'}\cap 4B_k\ne\varnothing,\] then there are at most \(C\) of the collections \(\mathcal{B}_0\), \(\mathcal{B}_i^{+}\), \(\mathcal{B}_{i}^-\), \(i=0,1,2,\dots\) in which \(B_{k'}\) may belong. As there are at most \(N\) balls in each of these collections, with \(N\) depending only on \(A\) and the doubling constant, it follows that claim \((vi)\) holds. ◻
Let \(1\leq p,q<\infty\) and \(0<\alpha<1\). Given a function \(f\in L^1_\text{loc}(X,\mu)\), we define its Besov energy by \[\Vert f\Vert_{HB^\alpha_{p,q}(X,\mu )}:= \left(\int_0^\infty\left(\int_X\fint_{B(x,t)}|f(y)-f(x)|^pd\mu(y)d\mu(x)\right)^{q/p}\frac{dt}{t^{\alpha q+1}}\right)^{1/q}.\] As usual, we define the homogeneous Besov space \(HB^\alpha_{p,q}(X,\mu)\) as the class of functions in \(L^1_\text{loc}(X,\mu)\) for which this energy is finite, while the inhomogeneous Besov space is defined by \(B^\alpha_{p,q}(X,\mu ):=HB^\alpha_{p,q}(X,\mu )\cap L^p(X,\mu )\), and is equipped with the norm \[\|u\|_{B^\alpha_{p,q}(X,\mu)}:=\|u\|_{HB^\alpha_{p,q}(X,\mu)}+\|u\|_{L^p(X,\mu)}.\] We follow the ideas in the proof of [31] in order to obtain the following equivalent seminorm for the Besov energy.
Lemma 1. Let \((X,d,\mu )\) be a metric measure space with \(\mu\) a doubling measure. Let \(1\leq p,q<\infty\) and \(0<\alpha<1\). Given a function \(f\in L^1_\text{loc}(X,\mu)\), then \[\label{eq:SumForm} \|f\|^q_{HB^\alpha_{p,q}( X ,\mu)}\simeq \sum_{k\in\mathbb{Z}}2^{-k\alpha q}\left(\int_ X \fint_{B(x,2^k)}|f(y)-f(x)|^pd\mu(y)d\mu(x)\right)^{q/p},\qquad{(7)}\] where the comparison constants depend only on \(\alpha\), \(p\), \(q\) and \(C_\mu\). Moreover, if \(f\in L^p( X,\mu )\) then \[\label{eq:SumFormLp} \|f\|^q_{HB^\alpha_{p,q}( X ,\mu)}\lesssim \| f\|^q_{L^p( X ,\mu )}+\sum_{k\in\mathbb{N}}2^{k\alpha q}\left(\int_ X \fint_{B(x,2^{-k})}|f(y)-f(x)|^pd\mu(y)d\mu(x)\right)^{q/p},\qquad{(8)}\] where the comparison constants depend only on \(\alpha\), \(p\), \(q\) and \(C_\mu\).
Proof. By the doubling property we obtain ?? following the steps in [31]. To prove ?? , consider \(k\in \mathbb{N}\). Then \[\int_ X\fint_{B(x,2^k)}|f(x)-f(y)|^pd\mu (y)d\mu (x) \lesssim \int_ X |f(x)|^pd\mu (x)+\int_ X \fint_{B(x,2^k)}|f(y)|^pd\mu (y)d\mu(x).\]
By Tonelli’s theorem and doubling, we have that \[\begin{align} \int_X\fint_{B(x,2^k)}|f(y)|^pd\mu(y)d\mu(x)&=\int_X\int_{X}|f(y)|^p\frac{\chi_{B(x,2^k)}(y)}{\mu(B(x,2^k))}d\mu(y)d\mu(x)\\ &=\int_X|f(y)|^p\int_{X}\frac{\chi_{B(y,2^k)}(x)}{\mu(B(x,2^k))}d\mu(x)d\mu(y)\lesssim\int_X|f(y)|^pd\mu(y). \end{align}\] Since \(\sum_{k\in\mathbb{N}}2^{-k\alpha q}\) converges, ?? holds by splitting the sum in ?? into \(k>0\) and \(k\le 0\). ◻
We now record the following sum-rearrangement lemma from [32], which we will use frequently in establishing the energy bounds for our trace and extension operators.
Lemma 1 ([32], Lemma 3.1). Let \(1<a<\infty\), \(0<b<\infty\), and \(c_i\ge 0\), \(i\in\mathbb{Z}\). Then there is a constant \(C=C(a,b)\) such that \[\sum_{i\in\mathbb{Z}}\left(\sum_{j\in\mathbb{Z}}a^{-|j-i|}c_j\right)^b\le C\sum_{j\in \mathbb{Z}}c_j^b.\]
To close this section, we provide the definition of the Besov class for \(q=\infty\), which will be treated separately within the proofs of the main results.
\[\begin{align} \Vert f\Vert_{HB^\alpha_{p,\infty}(X,\mu )}&:= \sup_{t>0}\frac{1}{t^{\alpha}}\left(\int_X\fint_{B(x,t) }|f(y)-f(x)|^pd\mu(y)d\mu(x)\right)^{1/p}\\ &\simeq \sup_{k\in\mathbb{Z}}2^{-k\alpha }\left(\int_X \fint_{B(x,2^k) }|f(y)-f(x)|^pd\mu(y)d\mu(x)\right)^{1/p}, \end{align}\] Moreover, if \(f\in L^\infty (X ,\mu )\) then \[\Vert f\Vert_{HB^\alpha_{p,\infty}(X,\mu )}\lesssim \Vert f\Vert_{L^\infty (X ,\mu )}+\sup_{k\in\mathbb{N}}2^{k\alpha }\left(\int_X \fint_{B(x,2^{-k}) }|f(y)-f(x)|^pd\mu(y)d\mu(x)\right)^{1/p},\] The seminorm equivalences above are obtained analogously to those in ?? and ?? .
In this section, we assume that \((\Omega,d,\mu)\) is a locally compact, non-complete metric measure space, with \(\mu\) a doubling measure, such that \(\Omega\) is a uniform domain in its completion \(\overline{\Omega}\), see Definition 1. We extend \(\mu\) to \(\overline{\Omega}\) by the zero extension, and note that the metric measure space \((\overline{\Omega},d,\mu)\) is still doubling with the same constant. Due to this zero extension, it follows that \(HB^\alpha_{p,q}(\Omega,\mu)=HB^\alpha_{p,q}(\overline{\Omega},\mu)\); by a slight abuse of notation, we will at times use the spaces interchangeably. We equip \(\partial\Omega:=\overline{\Omega}\setminus\Omega\), the boundary of \(\Omega\), with a Borel measure \(\nu\) which is codimension \(\theta\) Ahlfors regular with respect to \(\mu\) for some \(\theta>0\). Recall that this implies that \(\nu\) is doubling, see Remark 1. The arguments in this section are inspired by those of [15], [17], where analogous trace results were proven for the Newton-Sobolev class.
Fix \(R>0\) and let \(\theta>0\) be as above. We define the following restricted fractional maximal function following [15], \[M^R_\theta f(z)=\sup_{0<r<R}r^\theta\fint_{B(z,r)}|f|d\mu,\] which maps functions in \(L^1_{\text{loc}}(\Omega)\) to the space of lower semicontinuous functions on \(\partial\Omega\). The following lemma from [15] was proven in greater generality; for the reader’s convenience, we include the proof of the case relevant to us, that \(M^R_\theta\) is bounded from \(L^1(\Omega)\) to weak-\(L^1(\partial\Omega)\):
Lemma 1 ([15], Lemma 4.2). There exists a constant \(C\), depending only on \(\theta\), \(C_\mu\) and \(C_\theta\), such that for all \(\lambda>0\) and \(f\in L^1(\Omega,\mu)\), \[\nu(\{z\in\partial\Omega:M^R_\theta f(z)>\lambda\})\le \frac{C}{\lambda}\int_\Omega|f|d\mu.\]
Proof. Let \(E_\lambda:=\{z\in\partial\Omega:M^R_\theta f(z)>\lambda\}\). Then for each \(z\in E_\lambda\), there exists \(B_z:=B(z,r_z)\) such that \(r_z^\theta\fint_{B_z}|f|d\mu>\lambda\), and so we have that \[\frac{\mu(B_z)}{r_z^\theta}\le\frac{1}{\lambda}\int_{B_z}|f|d\mu.\] Since \(r_z<R\) for all \(z\in E_\lambda\), we use the 5-covering lemma to obtain a disjoint countable subcollection \(\{B_i:=B(z_i,r_{z_i})\}_i\) such that \[\bigcup_{z\in E_\lambda} B_z\subset\bigcup_i5B_i.\] Therefore, it follows from doubling, codimensionality of \(\nu\), and the disjointness of \(\{B_i\}_i\) that \[\begin{align} \nu(E_\lambda)\le\nu\left(\bigcup_i 5B_i\cap\partial\Omega\right)&\le\sum_i\nu(5B_i\cap\partial\Omega)\\ &\lesssim\sum_i\frac{\mu(B_i)}{r_{z_i}^\theta}\le\frac{1}{\lambda}\sum_i\int_{B_i}|f|d\mu\le\frac{1}{\lambda}\int_{\Omega}|f|d\mu.\qedhere \end{align}\] ◻
Before studying the trace operator we need the following lemma that will ensure the trace is well defined.
Lemma 1. Let \(1\le p<\infty\), \(1\leq q\leq\infty\), \(0<\alpha<1\), and let \(u\in HB^\alpha_{p,q}(\Omega,\mu)\). Then for each \(z\in\partial\Omega\) and \(r>0\), we have that \[\int_{B(z,r)}|u|d\mu<\infty.\]
Proof. Since \(\|u\|_{HB^\alpha_{p,q}(\Omega,\mu)}<\infty\), it follows that \[\begin{align} \infty>\int_{B(z,r) }\fint_{B(x,t) }|u(x)-u(y)|^pd\mu(y)d\mu(x) \end{align}\] for \(\mathcal{L}\)-a.e.\(0<t<\infty\). Since \(u\in L^1_\text{loc}(\Omega,\mu)\), we then have that \(|u(x)|<\infty\) and \[\infty>\fint_{B(x,t) }|u(x)-u(y)|^pd\mu(y)\] for \(\mu\)-a.e.\(x\in B(z,r)\). Choosing such \(t\ge 2r\) and \(x\in B(z,r)\), the conclusion follows, as \[\begin{align} \infty>\left(\fint_{B(x,t) }|u(x)-u(y)|^pd\mu(y)\right)^{1/p}&\ge\fint_{B(x,t) }|u(x)-u(y)|d\mu(y)\\ &\ge\frac{1}{\mu(B(x,t) )}\left(\int_{B(z,r) }|u(y)|d\mu(y)-|u(x)|\right).\qedhere \end{align}\] ◻
We note that the assumption that \(\Omega\) is uniform in its completion is not needed to prove the following result. However, this assumption will be used in the following subsection to attain the appropriate energy bounds.
Proposition 1. Let \(1\le p<\infty\), \(1\leq q\leq\infty\) and \(0<\alpha<1\) be such that \(\alpha-\theta/p>0\). Then there exists a linear operator \(T: HB^\alpha_{p,q}(\Omega,\mu)\rightarrow L^p(\partial\Omega,\nu)\) such that \[\label{eq:TraceCondition} \lim_{r\to 0^+}\fint_{B(z,r) }|u-Tu(z)|^pd\mu=0\text{ for }\nu\text{-a.e\;}z\in\partial\Omega.\qquad{(9)}\] If in addition \(\partial\Omega\) is bounded, then \[\label{eq:Trace32Lp32bound} \|Tu\|_{L^p(\partial\Omega,\nu)}\lesssim \mathop{\mathrm{diam}}(\partial\Omega)^{-\theta/p}\|u\|_{L^p(\Omega,\mu)}+\mathop{\mathrm{diam}}(\partial\Omega)^{\alpha -\theta/p}\|u\|_{HB^{\alpha}_{p,q}(\Omega,\mu)}.\qquad{(10)}\]
Proof. We first consider the case \(1\le q<\infty\). The case \(q=\infty\) holds with natural modifications briefly described at the end of the proof. For each \(r>0\), choose a maximal \(r\)-separated subset \(\{z_i^r\}_{i\in I_r}\) of \(\partial\Omega\). For each \(i\in I_r\), set \(B_i^r:=B(z_i^r,r)\), and let \(U_i^r:=B(z_i^r,r)\cap\partial\Omega\). Since \(\mu\) is doubling, there exists a Lipschitz partition of unity \(\{\varphi_i^r\}_{i\in I_r}\) subordinated to \(\{B_i^r\}_{i\in I_r}\). That is, each \(\varphi_i^r\) is \(C/r\)-Lipschitz, \(0\le \varphi_i^r\le\chi_{2B_i^r}\), and for all \(x\in\bigcup_{i\in I_r} B_i^r\), we have that \(\sum_{i\in I_r}\varphi_i^r(x)=1\). For proof of these facts, see [33], for example.
For each \(u\in HB_{p,q}^\alpha(\Omega,\mu)\), we then set \[u_{B^r_i}:=\fint_{B_i^r }u\,d\mu,\] and define \[T_ru:=\sum_{i\in I_r}u_{B_i^r}\varphi_i^r|_{\partial\Omega}.\] Note that \(T_ru\) well-defined by Lemma 1 and locally Lipschitz, hence \(\nu\)-measurable. We claim that for any sequence \(r_k\to 0^+\), the sequence \(\{T_{r_k}u\}_k\) is Cauchy in \(L^p(\partial\Omega,\nu)\). Toward this, consider \(r,R>0\) such that \(0<R/2<r\le R\). Since \(\mu\) is doubling, we have \[\begin{align} \label{eq:intersected32comparable32measure} \mu(B(z,r))\simeq \mu(B(z,R)). \end{align}\tag{6}\] Setting \[\beta:=\frac{\alpha p+\theta}{2},\quad\sigma:=\alpha-\beta/p,\] we note that \(\sigma>0\) since \(\alpha>\theta/p\). Using the properties of the partition of unity, we then have that
\[\begin{align} \|T_Ru-T_ru\|^p_{L^p(\partial\Omega,\nu)}&\le\sum_{i\in I_r}\int_{U_i^r}\left|\sum_{j\in I_r}u_{B_j^r}\varphi_j^r(z)-\sum_{k\in I_R}u_{B_k^R}\varphi_k^R(z)\right|^pd\nu(z)\\ &\le\sum_{i\in I_r}\int_{U_i^r}\left(\sum_{j\in I_r}|u_{B_j^r}-u_{B_i^r}|\varphi_j^r(z)+\sum_{k\in I_R}|u_{B_i^r}-u_{B_k^R}|\varphi_k^R(z)\right)^pd\nu(z)\\ &\lesssim\sum_{i\in I_r}\int_{U_i^r}\fint_{B(z,3R)}\fint_{B(z,3R)}|u(x)-u(y)|^pd\mu(y)d\mu(x)d\nu(z)\\ &\lesssim\int_{\partial\Omega}\fint_{B(z,3R)}\fint_{B(z,3R)}|u(x)-u(y)|^pd\mu(y)d\mu(x)d\nu(z). \end{align}\] To obtain the last two inequalities, we have used the fact that if \(z\in U_i^r\), and \(\varphi_k^R(z)\ne 0\), then \(B_k^R\subset B(z,3R)\) (and similarly if \(\varphi_j^r(z)\ne 0\)), along with 6 and bounded overlap of the collection of balls, due to the doubling property.
Furthermore, we have \[\begin{align} \|T_Ru-&T_ru\|_{L^p(\partial\Omega,\nu)}^p\\ &\lesssim R^{\beta}\int_{\partial\Omega}\fint_{B(z,3R) }\int_{B(z,3R) }\frac{|u(x)-u(y)|^p}{d(x,y)^{\beta}\mu(B(x,d(x,y)) )}d\mu(y)d\mu(x)d\nu(z)\\ &\lesssim R^{\beta-\theta}\int_{\partial\Omega}\int_{B(z,3R) }\int_{B(z,3R) }\frac{|u(x)-u(y)|^p\,d\mu(y)d\mu(x)}{d(x,y)^{\beta}\mu(B(x,d(x,y)))\nu(B(z,R)\cap\partial\Omega)}d\nu(z). \end{align}\]
Letting \(\Omega_{3R}:=\{x\in\Omega:d(x,\partial\Omega )<3R\}\), we have by Tonelli’s theorem that \[\begin{align} \|T_Ru&-T_ru\|_{L^p(\partial\Omega,\nu)}^p\nonumber\\ &\lesssim R^{\beta-\theta}\int_{\Omega_{3R}}\int_{B(x,6R) }\int_{B(x,3R)\cap\partial\Omega}\frac{|u(x)-u(y)|^pd\nu(z)d\mu(y)d\mu(x)}{d(x,y)^{\beta}\mu(B(x,d(x,y)) )\nu(B(z,R)\cap\partial\Omega)}\nonumber\\ &\lesssim R^{\beta-\theta}\int_{\Omega}\int_{B(x,6R) }\frac{|u(x)-u(y)|^p}{d(x,y)^{\beta}\mu(B(x,d(x,y)) )}d\mu(y)d\mu(x). \end{align}\] Let \(N_R\in\mathbb{Z}\) be such that \(2^{N_R-1}\le 6R<2^{N_R}\). For each \(x\in \Omega\) and \(m\in\mathbb{Z}\), consider the annulus \(A_m(x):= B(x,2^{-m})\backslash B(x,2^{-m-1})\). We then have that \[\begin{align} \|T_Ru-T_ru\|_{L^p(\partial\Omega,\nu)}\nonumber &\lesssim \left( R^{\beta-\theta}\int_{\Omega}\sum^\infty_{m=-N_R}\int_{A_m(x)}\frac{|u(x)-u(y)|^p}{d(x,y)^{\beta}\mu(B(x,d(x,y)) )}d\mu(y)d\mu(x)\right)^{1/p}\nonumber\\ &\lesssim\left( R^{\beta-\theta}\sum^\infty_{m=-N_R}2^{m\beta}\int_{\Omega}\int_{A_m(x)}\frac{|u(x)-u(y)|^p}{\mu(B(x,2^{-m}) )}d\mu(y)d\mu(x)\right)^{1/p}\nonumber\\ &\le\left( R^{\beta-\theta}\sum^\infty_{m=-N_R}2^{m\beta}\int_{\Omega}\fint_{B(x,2^{-m}) }|u(x)-u(y)|^pd\mu(y)d\mu(x)\right)^{1/p}\nonumber\\ &\le R^{(\beta-\theta)/p}\;\sum^\infty_{m=-N_R}2^{m\beta/p}\left(\int_{\Omega}\fint_{B(x,2^{-m}) }|u(x)-u(y)|^pd\mu(y)d\mu(x)\right)^{1/p}\nonumber. \end{align}\] Here in the last inequality, we have used the fact that for any \(a_i>0\), \(i\in\mathbb{N}\), and \(0<b\leq 1\), it follows that \[\left(\sum_{i\in\mathbb{N}}a_i\right)^b\leq \sum_{i\in\mathbb{N}}a_i^b.\] By Hölder’s inequality, and our choices of \(\beta\), \(\sigma\), and \(N_R\), we then obtain \[\begin{align} \|T&_Ru-T_ru\|_{L^p(\partial\Omega,\nu)}\\ &\lesssim R^{\frac{\beta-\theta}{p}}\left(\sum_{m=-N_R}^\infty 2^{\frac{-m\sigma q}{q-1}}\right)^{\frac{q-1}{q}}\left(\sum^\infty_{m=-N_R}2^{m\left(\frac{\beta}{p}+\sigma\right)q}\left(\int_{\Omega}\fint_{B(x,2^{-m}) }|u(x)-u(y)|^pd\mu(y)d\mu(x)\right)^{\frac{q}{p}}\right)^{\frac{1}{q}}\\ &\simeq R^{(\beta-\theta)/p+\sigma}\left(\sum^\infty_{m=-N_R}2^{m\alpha q}\left(\int_{\Omega}\fint_{B(x,2^{-m}) }|u(x)-u(y)|^pd\mu(y)d\mu(x)\right)^{q/p}\right)^{1/q}\\ &\le R^{\alpha-\theta/p}\|u\|_{HB^\alpha_{p,q}(\Omega,\mu)}. \end{align}\] Now, for any \(0<r\le R\), there exists \(N\in\mathbb{N}\) such that \(2^{-N}R<r\le 2^{-N+1}R\). Thus, we have that \[\begin{align} \label{eqn:L94pCauchy} \|T_Ru-T_ru\|_{L^p(\partial\Omega,\nu)}&\le\sum_{k=1}^N\|T_{2^{-k+1}R}u-T_{2^{-k}R}u\|_{L^p(\partial\Omega,\nu)}\nonumber\\ &\lesssim\|u\|_{HB^\alpha_{p,q}(\Omega,\mu)}\sum_{k=1}^N(2^{-k}R)^{\alpha-\theta/p}\lesssim R^{\alpha-\theta/p}\|u\|_{HB^\alpha_{p,q}(\Omega,\mu)}\to 0 \end{align}\tag{7}\] as \(R\to 0^+\), since \(u\in HB^\alpha_{p,q}(\Omega,\mu)\). Hence, for any sequence \(r_k\to 0^+\), \(\{T_{r_k}u\}_k\) is Cauchy in \(L^p(\partial\Omega,\nu)\), and so there exists \(Tu\in L^p(\partial\Omega,\nu)\) such that \(T_ru\to Tu\) in \(L^p(\partial\Omega,\nu)\) as \(r\to 0^+\). Passing to a subsequence if necessary, we have that \(T_ru(z)\to Tu(z)\) for \(\nu\)-a.e.\(z\in\partial\Omega\).
Now, fix \(0<R\leq 1\) and define \[F(x):=\int_{B(x,R) }\frac{|u(x)-u(y)|^p}{d(x,y)^{\beta}\mu(B(x,d(x,y)))}d\mu(y).\] Then, using a similar argument via Hölder’s inequality and doubling as above, we have that \[\begin{align} \left(\int_\Omega |F|d\mu\right)^{1/p}& = \left(\int_\Omega \int_{B(x,R) }\frac{|u(x)-u(y)|^p}{d(x,y)^{\beta}\mu(B(x,d(x,y)))}d\mu(y)d\mu(x)\right)^{1/p}\\ &\le\left(\sum_{m\geq 0}\int_\Omega\int_{B(x,2^{-m+1})\setminus B(x,2^{-m}) }\frac{|u(x)-u(y)|^p}{d(x,y)^{\beta}\mu(B(x,d(x,y)))}d\mu(y)d\mu(x)\right)^{1/p}\\ &\lesssim\left(\sum_{m\geq 0}2^{m\beta}\int_\Omega\fint_{B(x,2^{-m}) }|u(x)-u(y)|^pd\mu(y)d\mu(x)\right)^{1/p}\\ &\le\sum_{m\geq 0}2^{m\beta/p}\left(\int_\Omega\fint_{B(x,2^{-m}) }|u(x)-u(y)|^pd\mu(y)d\mu(x)\right)^{1/p}\\ &\lesssim \left(\sum_{m\geq 0}2^{m(\beta/p+\sigma)q}\left(\int_\Omega\fint_{B(x,2^{-m}) }|u(x)-u(y)|^pd\mu (y)d\mu (x)\right)^{q/p}\right)^{1/q}\\ &\lesssim\| u\|_{HB^\alpha_{p,q}(\Omega ,\mu)}. \end{align}\] Since \(u\in HB^\alpha_{p,q}(\Omega,\mu)\), it follows that \(F\in L^1(\Omega,\mu)\), and so by Lemma 1, we have that \(M^R_\theta F(z)<\infty\) for \(\nu\)-a.e.\(z\in\partial\Omega\).
Let \(z\in\partial\Omega\) such that \(M^R_{\theta}F(z)<\infty\) and \(T_ru(z)\to Tu(z)\) as \(r\to 0^+\). This holds for \(\nu\)-a.e.\(z\in\partial\Omega\). Let \(0<r<R\). Then, by properties of the partition of unity, 6 , our definition of \(\beta\), and since \(\alpha>\theta/p\), we have that \[\begin{align} \fint_{B(z,r) }|u-&T_ru(z)|^pd\mu=\fint_{B(z,r) }\left|u(x)-\sum_{i\in I_r}u_{B_i^r}\varphi_i^r(z)\right|^pd\mu(x)\\ &\lesssim\fint_{B(z,r) }\fint_{B(z,3r) }|u(x)-u(y)|^pd\mu(y)d\mu(x)\\ &\lesssim r^{\beta}\fint_{B(z,r) }\int_{B(z,3r) }\frac{|u(x)-u(y)|^p}{d(x,y)^{\beta}\mu(B(x,d(x,y)) )}d\mu(y)d\mu(x)\\ &\le r^{\beta-\theta}r^\theta\fint_{B(z,r) }\int_{B(z,R) }\frac{|u(x)-u(y)|^p}{d(x,y)^{\beta}\mu(B(x,d(x,y)) )}d\mu(y)d\mu(x)\\ &= r^{(\alpha p-\theta)/2}\left(r^\theta\fint_{B(z,r) }|F(x)|d\mu(x)\right)\le r^{(\alpha p-\theta)/2}M^R_{\theta}F(z)\to 0, \end{align}\] as \(r\to 0^+\). Hence, we have that \[\begin{align} \fint_{B(z,r) }|u-Tu(z)|^pd\mu\lesssim\fint_{B(z,r) }|u-T_ru(z)|^pd\mu+|T_ru(z)-Tu(z)|^p\to 0 \end{align}\] as \(r\to 0^+\), which gives us ?? .
Assume now that \(\partial\Omega\) is bounded, and let \(R=2\mathop{\mathrm{diam}}(\partial\Omega)\). Then, \[\begin{align} \|Tu\|_{L^p(\partial\Omega,\nu)}&\le\lim_{r\to 0^+}\|T_ru-T_Ru\|_{L^p(\partial\Omega,\nu)}+\|T_Ru\|_{L^p(\partial\Omega,\nu)}, \end{align}\] By 7 , we have that \[\|T_ru-T_Ru\|_{L^p(\partial\Omega,\nu)}\lesssim \mathop{\mathrm{diam}}(\partial\Omega)^{\alpha -\theta/p}\|u\|_{HB^\alpha_{p,q}(\Omega,\mu)}.\] By properties of the partition of unity, as well as bounded overlap of \(\{B_i^R\}_{i\in I_R}\), we also have that \[\begin{align} \|T_Ru\|^p_{L^p(\partial\Omega,\nu)}&\lesssim\int_{\partial\Omega}\fint_{B(z,3R) }|u|^pd\mu d\nu(z)\\ &\le\|u\|^p_{L^p(\Omega,\mu)}\int_{\partial\Omega}\frac{1}{\mu(B(z,3R) )}d\nu(z)\\ &\lesssim\|u\|^p_{L^p(\Omega,\mu)}\int_{\partial\Omega}\frac{1}{R^\theta\nu(B(z,R)\cap\partial\Omega)}d\nu(z)=(2\mathop{\mathrm{diam}}(\partial\Omega))^{-\theta}\|u\|^p_{L^p(\Omega,\mu)}. \end{align}\] Hence, it follows that \[\|Tu\|_{L^p(\partial\Omega,\nu)}\lesssim \mathop{\mathrm{diam}}(\partial\Omega)^{-\theta/p}\|u\|_{L^p(\Omega,\mu)}+\mathop{\mathrm{diam}}(\partial\Omega)^{\alpha -\theta/p}\|u\|_{HB^{\alpha}_{p,q}(\Omega,\mu)},\] which gives us ?? .
For the case \(q=\infty\) we notice that \[\begin{align} \sum_{m=-N_R}^\infty 2^{m\beta/p}\left(\int_\Omega\fint_{B(x,2^{-m}) }|u(x)-u(y)|^pd\mu (y)d\mu (x)\right)^{1/p} &\lesssim 2^{-N_R(\beta /p-\alpha )}\Vert u\Vert_{B^\alpha_{p,\infty}(\Omega ,\mu )} \\ &\lesssim R^\sigma \Vert u\Vert_{B^\alpha_{p,\infty}(\Omega ,\mu )} \end{align}\] and a similar estimate holds replacing \(-N_R\) by \(1\) (getting then \(2^\sigma\) instead of \(R^\sigma\)). Replacing the use of the Hölder inequality in the proof of the case \(q<\infty\) with these estimates yields \[\lim_{R\to 0^+}\|T_Ru-T_ru\|_{L^p(\partial\Omega,\nu)} =0\quad and \quad F\in L^1(\Omega ,\mu ),\] while the rest of the proof is exactly as when \(q<\infty\). ◻
Equipped with the chain of balls given by Lemma 1, we are now able to prove the energy bounds for the trace operator defined in Proposition 1.
Proposition 1. Let \((\Omega,d,\mu)\) be a locally compact, non-complete metric measure space, with \(\mu\) a doubling measure, such that \(\Omega\) is an \(A\)-uniform domain in its completion \(\overline{\Omega}\). Let \(\partial\Omega:=\overline{\Omega}\setminus\Omega\), the boundary of \(\Omega\), be equipped with a Borel measure \(\nu\) which is codimension \(\theta\) Ahlfors regular with respect to \(\mu\). Let \(1\le p<\infty\), \(1\leq q\leq\infty\) and let \(0<\alpha<1\) be such that \(0<\alpha-\theta/p<1\). Then there exists a bounded linear trace operator \[T:HB^\alpha_{p,q}(\Omega,\mu)\to HB^{\alpha-\theta/p}_{p,q}(\partial\Omega,\nu).\] That is, there exists a constant \(C\ge 1\), depending only on \(\alpha\), \(p\), \(q\), \(\theta\), \(C_\mu\), \(C_\theta\), and \(A\), such that \[\| Tu\|_{HB^{\alpha-\theta/p}_{p,q}(\partial\Omega,\nu)}\le C\|u\|_{HB^{\alpha}_{p,q}(\Omega,\mu)}\] for all \(u\in HB^{\alpha}_{p,q}(\Omega,\mu)\).
Proof. As before, we first consider the case \(1\le q<\infty\). Consider the operator \(T:HB^\alpha_{p,q}(\Omega ,\mu)\rightarrow L^p(\partial\Omega ,\nu )\) given by Proposition 1. Then for \(\nu\)-a.e. \(z\in \partial\Omega\) we have \[\label{eq:tracelimit} Tu(z):=\lim_{r\to0^+}\fint_{B(z,r) }u\,d\mu.\tag{8}\] Let \(z,w\in\partial\Omega\) such that 8 holds for both of them, and consider the corresponding chain of balls \(\{ B_k:=B(x_k,r_k) \}_{k\in\mathbb{Z}}\) provided by Lemma 1. For the reader’s convenience we recall here the properties of the collection \(\{ B_k\}_k\):
For each \(k\in\mathbb{Z}\), we have that \(8B_k\subset\Omega\),
\(\lim_{k\to\infty}x_k=z\), \(\lim_{k\to-\infty}x_k=w\),
for each \(k\in\mathbb{Z}\), we have that \(2^{-1}B_k\cap 2^{-1}B_{k+1}\ne\varnothing\),
there exists \(N\ge 1\), such that for each \(k\ge 0\), we have that \(2^{-1}r_k \leq r_{k+1}\leq r_k\), and for each \(x\in B_k\), we have that \[r_k\simeq d(x,\partial\Omega)\simeq d(x,z)\lesssim 2^{-|k|/N}d(z,w),\]
there exists \(N\ge 1\) such that for each \(k<0\), we have that \(2^{-1}r_k \leq r_{k-1}\leq r_k\), and for each \(x\in B_k\), we have that \[r_k\simeq d(x,\partial\Omega)\simeq d(x,w)\lesssim 2^{-|k|/N}d(z,w),\]
there exists \(C\ge 1\) such that \[\sum_{k}\chi_{4B_k}\le C,\]
Here the constants \(C\) and \(N\) and the comparison constants depend only on \(A\) and \(C_\mu\). Note that when \(k\ge 0\), it follows from these properties that \(2B_k\supset B_{k+1}\), and \(2B_k\supset B_{k-1}\) when \(k<0\). Since \[\lim_{k\to\infty}|Tu(z)-u_{B_k}|=0=\lim_{k\to-\infty}|Tu(w)-u_{B_k}|,\] it then follows from induction and doubling that \[\begin{align} |Tu&(z)-Tu(w)|^p\le\left(\sum_{k\ge 0}|u_{B_{k+1}}-u_{B_k}|+\sum_{k<0}|u_{B_k}-u_{B_{k-1}}|\right)^p\\ &\lesssim\left(\sum_{k\ge 0}\fint_{2B_k}\fint_{2B_k}|u(x)-u(y)|d\mu(y)d\mu(x)\right)^p+ \left(\sum_{k<0}\fint_{2B_k}\fint_{2B_k}|u(x)-u(y)|d\mu(y)d\mu(x)\right)^p. \end{align}\] Since \(\alpha p-\theta>0\) by hypothesis, we choose \(\beta>0\) sufficiently small so that \(\alpha p-\theta-\beta p>0\). It then follows from Hölder’s inequality, property \((iv)\) of the chain of balls, and doubling that \[\begin{align} \Bigg(\sum_{k\ge 0}&\fint_{2B_k}\fint_{2B_k}|u(x)-u(y)|d\mu(y)d\mu(x)\Bigg)^p =\left(\sum_{k\ge 0}\frac{r_k^\beta}{r_k^\beta}\fint_{2B_k}\fint_{2B_k}|u(x)-u(y)|d\mu(y)d\mu(x)\right)^p \\ &\lesssim \left(\sum_{k\ge 0}\frac{(2^{-k/N}d(z,w))^\beta}{d(x,\partial\Omega )^\beta}\fint_{2B_k}\fint_{2B_k}|u(x)-u(y)|d\mu(y)d\mu(x)\right)^p\\ &\le\left(\sum_{k\ge 0}(2^{-k/N}d(z,w))^{\beta p'}\right)^{p/p'}\sum_{k\ge 0}d(x,\partial\Omega )^{-\beta p}\fint_{2B_k}\fint_{2B_k}|u(x)-u(y)|^p d\mu(y)d\mu(x)\\ &\lesssim d(z,w)^{\beta p}\sum_{k\ge 0}\int_{2B_k}\int_{2B_k}\frac{|u(x)-u(y)|^p}{d(x,\partial\Omega)^{\beta p}\mu(B(x, d(x,\partial\Omega)))^2}d\mu(y)d\mu(x)\\ &\lesssim d(z,w)^{\beta p}\int_{C_{z,w}^1}\int_{B(x,d(x,\partial\Omega))}\frac{|u(x)-u(y)|^p}{d(x,\partial\Omega)^{\beta p}\mu(B(x, d(x,\partial\Omega)))^2}d\mu(y)d\mu(x). \end{align}\] Here, we use the notation \(C_{z,w}^1:=\bigcup_{k\ge 0}2B_k\). In the above, we have also used property \((iv)\), \((i)\), and \((vi)\) of the chain of balls to obtain the last two inequalities. Similarly, letting \(C_{z,w}^2:=\bigcup_{k<0} 2B_k\), we have that \[\begin{align} \Bigg(\sum_{k<0}&\fint_{2B_k}\fint_{2B_k}|u(x)-u(y)|d\mu(y)d\mu(x)\Bigg)^p\\ &\lesssim d(z,w)^{\beta p}\int_{C_{z,w}^2}\int_{B(x,d(x,\partial\Omega))}\frac{|u(x)-u(y)|^p}{d(x,\partial\Omega)^{\beta p}\mu(B(x, d(x,\partial\Omega))^2}d\mu(y)d\mu(x). \end{align}\] Hence, using ?? , it follows that \[\begin{align} \label{eq:trace32I43II} \|Tu\|^q_{HB^{\alpha-\theta/p}_{p,q}(\partial\Omega,\nu)}&\simeq\sum_{i\in\mathbb{Z}}2^{-i(\alpha-\theta/p)q}\left(\int_{\partial \Omega}\fint_{B(z,2^i)}|Tu(z)-Tu(w)|^pd\nu(w)d\nu(w)\right)^{q/p}\nonumber\\ &\lesssim I+II, \end{align}\tag{9}\] where \[\begin{align} I:=\sum_{i\in\mathbb{Z}}2^{-i(\alpha-\theta/p)q}\left(\int_{\partial\Omega}\int_{B(z,2^i)}\int_{C_{z,w}^1}\int_{B(x,d(x,\partial\Omega))}\frac{|u(x)-u(y)|^pd(z,w)^{\beta p}d\mu(y)d\mu(x)d\nu(w)d\nu(z)}{d(x,\partial\Omega)^{\beta p}\mu(B(x,d(x,\partial\Omega)))^2\nu(B(z,2^i))}\right)^{q/p} \end{align}\] and \[\begin{align} II:=\sum_{i\in\mathbb{Z}}2^{-i(\alpha-\theta/p)q}\left(\int_{\partial\Omega}\int_{B(z,2^i)}\int_{C_{z,w}^2}\int_{B(x,d(x,\partial\Omega))}\frac{|u(x)-u(y)|^pd(z,w)^{\beta p}d\mu(y)d\mu(x)d\nu(w)d\nu(z)}{d(x,\partial\Omega)^{\beta p}\mu(B(x,d(x,\partial\Omega)))^2\nu(B(z,2^i))}\right)^{q/p}. \end{align}\] Here the comparison constants depend only on \(\alpha\), \(p\), \(q\), \(C_\mu\), and \(A\).
We first estimate \(I\). For each \(j\in\mathbb{Z}\), let \[\Omega_j:=\{x\in\Omega:2^{j-1}\le d(x,\partial\Omega)<2^{j}\}.\] For each \(i\in\mathbb{Z}\), we note that if \(z\in\partial\Omega\) and \(w\in B(z,2^i)\cap\partial\Omega\), then by property \((iv)\) of the chain of balls, we have that \(C_{z,w}^1\subset\bigcup_{j=-\infty}^{i+C}\Omega_j\) for some constant \(C\) depending only on \(A\) and \(C_\mu\). Hence, from this fact and Tonelli’s theorem, we have that \[\begin{align} \int_{\partial\Omega}&\int_{B(z,2^i)}\int_{C_{z,w}^1}\int_{B(x,d(x,\partial\Omega)}\frac{|u(x)-u(y)|^pd(z,w)^{\beta p}d\mu(y)d\mu(x)d\nu(w)d\nu(z)}{d(x,\partial\Omega)^{\beta p}\mu(B(x,d(x,\partial\Omega)))^2\nu(B(z,2^i))}\\ &\lesssim\sum_{j=-\infty}^{i+C}\int_{\partial\Omega}\int_{B(z,2^i)}\int_{C_{z,w}^1\cap\Omega_j}\int_{B(x,2^j)}\frac{|u(x)-u(y)|^pd(z,w)^{\beta p}d\mu(y)d\mu(x)d\nu(w)d\nu(z)}{2^{j\beta p}\mu(B(x,2^j))^2\nu(B(z,2^i))}\\ &=\sum_{j=-\infty}^{i+C}\int_{\partial\Omega}\int_{B(z,2^i)}\int_{\Omega}\int_{B(x,2^j)}\frac{|u(x)-u(y)|^pd(z,w)^{\beta p}\chi_{C_{z,w}^1\cap\Omega_j}(x)d\mu(y)d\mu(x)}{2^{j\beta p}\mu(B(x,2^j))^2\nu(B(z,2^i))}d\nu(w)d\nu(z)\\ &=\sum_{j=-\infty}^{i+C}2^{-j\beta p}\int_{\Omega}\fint_{B(x,2^j)}\frac{|u(x)-u(y)|^p}{\mu(B(x,2^j))}\int_{\partial\Omega}\int_{B(z,2^i)}\frac{d(z,w)^{\beta p}\chi_{C_{z,w}^1\cap\Omega_j}(x)}{\nu(B(z,2^i))}d\nu(w)d\nu(z)d\mu(y)d\mu(x). \end{align}\] We note that if \(x\in C_{z,w}^1\cap\Omega_j\), then by property \((iv)\) of the chain of balls, there exists \(C'\ge 1\), depending only on \(A\) and \(C_\mu\), such that \(z\in\partial\Omega\cap B(x,C'2^j)\). Thus, using the \(\theta\)-codimensional relationship between \(\nu\) and \(\mu\), we have that \[\begin{align} \int_{\partial\Omega}\int_{B(z,2^i)}\frac{d(z,w)^{\beta p}\chi_{C_{z,w}^1\cap\Omega_j}(x)}{\nu(B(z,2^i))}d\nu(w)d\nu(z)&=\int_{\partial\Omega\cap B(x,C'2^j)}\int_{B(z,2^i)}\frac{d(z,w)^{\beta p}}{\nu(B(z,2^i))}d\nu(w)d\nu(z)\\ &\le 2^{i\beta p}\nu(B(x,C'2^j)\cap\partial\Omega)\\ &\lesssim 2^{i\beta p-j\theta}\mu(B(x,2^j)). \end{align}\] Substituting this into the previous expression, we then obtain the following estimate for \(I\): \[\begin{align} I&\lesssim\sum_{i\in\mathbb{Z}}\left(\sum_{j=-\infty}^{i+C}2^{-i(\alpha-\theta/p)p-j\beta p+i\beta p-j\theta}\int_\Omega\fint_{B(x,2^j)}|u(x)-u(y)|^pd\mu(y)d\mu(x)\right)^{q/p}. \end{align}\]
For \(i\in\mathbb{Z}\) and \(j\in\mathbb{Z}\) with \(j\le i+C\), we have that \[\begin{align} 2^{-i(\alpha-\theta/p)p-j\beta p+i\beta p-j\theta}&=\left(2^{\alpha p-\theta-\beta p}\right)^C\left(2^{\alpha p-\theta-\beta p}\right)^{-|j-(i+C)|}2^{-j\alpha p} \end{align}\] Setting \(a:=2^{\alpha p-\theta-\beta p}\), we have from our choice of \(\beta\) that \(1<a<\infty\). Setting \(b:=q/p\) and \[c_j:=2^{-j\alpha p}\int_\Omega\fint_{B(x,2^j)}|u(x)-u(y)|^pd\mu(y)d\mu(x),\] we then have that \[I\lesssim a^{Cb}\sum_{i\in\mathbb{Z}}\left(\sum_{j=-\infty}^{i+C}a^{-|j-(i+C)|}c_j\right)^b\le a^{Cb}\sum_{i\in\mathbb{Z}}\left(\sum_{j\in\mathbb{Z}}a^{-|j-i|}c_j\right)^b.\] It then follows from Lemma 1 and ?? that \[\begin{align} \label{eq:I32estimate} I\lesssim\sum_{j\in\mathbb{Z}}c_j^b=\sum_{j\in\mathbb{Z}}2^{-j\alpha q}\left(\int_\Omega\fint_{B(x,2^j)}|u(x)-u(y)|^pd\mu(y)d\mu(x)\right)^{q/p}\simeq \|u\|_{HB^\alpha_{p,q}(\Omega,\mu)}^q, \end{align}\tag{10}\] with comparison constants depending only on \(\alpha\), \(p\), \(q\), \(\theta\), \(C_\mu\), \(C_\theta\) and \(A\).
By a similar argument, with the roles of \(w\) and \(z\) reversed, we also obtain the estimate \[II\lesssim\|u\|^q_{HB^\alpha_{p,q}(\Omega,\mu)},\] with the same dependencies for the comparison constant. Combining these two estimates with 9 , we have that \[\|Tu\|_{B^{\alpha-\theta/p}_{p,q}(\partial\Omega,\nu)}\lesssim\|u\|_{B^\alpha_{p,q}(\Omega,\mu)}.\]
For the case \(q=\infty\) set, for each \(i\in\mathbb{Z}\), \[\begin{align} I_i&:=\left(\sum_{j=-\infty}^{i+C}2^{-i(\alpha-\theta/p)p-j\beta p+i\beta p-j\theta}\int_\Omega\fint_{B(x,2^j)}|u(x)-u(y)|^pd\mu(y)d\mu(x)\right)^{1/p}. \end{align}\] By the definition of the \(HB^\alpha_{p,\infty}\)-seminorm and the above arguments, it suffices to prove \(I_i\lesssim \Vert u\Vert_{B^\alpha_{p,\infty}(\Omega ,\mu )}\) for each \(i\in\mathbb{Z}\). Defining \(a\) and \(c_j\) as before, it is clear that for any \(j\in \mathbb{Z}\) one has \(c_j\leq \Vert u\Vert^p_{B^\alpha_{p,\infty}(\Omega ,\mu )}\) and so estimating \(I_i\) analogously as it was done for \(I\), we have \[I_i\lesssim a^{C/p}\left(\sum_{j=-\infty}^{i+C}a^{-|j-(i+C)|}c_j\right)^{\frac{1}{p}}\leq a^{C/p} \Vert u\Vert_{B^\alpha_{p,\infty}(\Omega ,\mu )}\left(\sum_{j=-\infty}^{i+C}a^{-|j-(i+C)|}\right)^{\frac{1}{p}}\lesssim \Vert u\Vert_{B^\alpha_{p,\infty}(\Omega ,\mu )}.\qedhere\] ◻
Theorem 1 and Theorem 1 are now proved by combining Proposition 1 and Proposition 1.
In this section, we assume that \((\Omega,d,\mu)\) is a locally compact, non-complete metric measure space, with \(\mu\) a doubling measure. We extend \(\mu\) to \(\overline{\Omega}\) by the zero extension, and note that the metric measure space \((\overline{\Omega},d,\mu)\) is still doubling with the same constant. We equip \(\partial\Omega:=\overline{\Omega}\setminus\Omega\), the boundary of \(\Omega\), with a Borel measure \(\nu\) which is codimension \(\theta\) Ahlfors regular with respect to \(\mu\) for some \(\theta>0\). Recall that this implies that \(\nu\) is doubling, see Remark 1. Unlike the previous section, we do not assume that \(\Omega\) is a uniform domain.
We first prove Theorem 1, constructing an extension operator for the homogeneous Besov spaces using standard Whitney extension techniques. Let \(\{ B_{i,j}\}_{i,j}\) be the Whitney cover of \(\Omega\) given by Lemma 1, and for each \(B_{i,j}=B(p_{i,j},r_{i,j})\), we consider its corresponding “shadow” on the boundary, given by \[\label{eq:Uij} U_{i,j}:=B(q_{i,j},r_{i,j})\cap\partial\Omega ,\tag{11}\] where \(q_{i,j}\in\partial\Omega\) is a closest point to \(p_{i,j}\). By Lemma 1 (v), the collection \(\{B_{i,j}\}_j\) has bounded overlap for each \(i\), and so this implies bounded overlap of the collection \(\{U_{i,j}\}_j\), as shown by the following lemma:
Lemma 1. Fix \(i\in\mathbb{Z}\). For each \(K\ge 1\), there exists \(C_K'\ge 1\) such that \[\sum_j\chi_{KU_{i,j}}\le C_K',\] where \(KU_{i,j}=B(q_{i,j},Kr_{i,j})\cap\partial\Omega\).
Proof. If \(z\in KU_{i,j}\), then we have that \[d(z,p_{i,j})\le d(z,q_{i,j})+d(q_{i,j},p_{i,j})<Kr_{i,j}+8r_{i,j},\] and so \(KU_{i,j}\subset (K+8)B_{i,j}\). Therefore by Lemma 1 (v), it follows that \[\sum_j\chi_{KU_{i,j}}\le\sum_{j}\chi_{(K+8)B_{i,j}}\le C_{K+8}=:C_K'.\qedhere\] ◻
We now prove Theorem 1:
Proof of Theorem 1. We first consider the case \(1\le q<\infty\). Let \(f\in HB^{\alpha}_{p,q}(\partial\Omega,\nu)\), and let \(\mathcal{W}_\Omega=\{B(p_{i,j},r_{i,j})=:B_{i,j}\}\) and \(\{\varphi_{i,j}\}\) be the Whitney cover and partition of unity given by Lemma 1. For each \((i,j)\) let \(U_{i,j}\) as in 11 and \(a_{i,j}:=\fint_{U_{i,j}}fd\nu.\) Then, for each \(x\in\Omega,\) let \[\label{eq:homExt} \widetilde{E}f(x):=\sum_{i,j}a_{i,j}\varphi_{i,j}(x).\tag{12}\] By this construction, the map \(f\mapsto \widetilde{E}f\) is linear. By ?? , it follows that \[\begin{align} \label{eq:First} \|\widetilde{E}f\| ^q_{HB^{\alpha+\theta /p}_{p,q}(\Omega,\mu)}&\simeq\sum_{k\in\mathbb{Z}}2^{-k(\alpha+\theta/p)q}\left(\int_{\Omega}\fint_{B(x,2^k)}|\widetilde{E}f(x)-\widetilde{E}f(y)|^pd\mu(y)d\mu(x)\right)^{q/p}\nonumber\\ &\le\sum_{k\in\mathbb{Z}}2^{-k(\alpha+\theta/p)q}\left(\sum_{i,j}\int_{B_{i,j}}\fint_{B(x,2^k)}|\widetilde{E}f(x)-\widetilde{E}f(y)|^pd\mu(y)d\mu(x)\right)^{q/p}. \end{align}\tag{13}\] Fixing \(k\in\mathbb{Z}\), we then have \[\begin{align} \label{eq:Inside32parenth32I43II} \sum_{i,j}\int_{B_{i,j}}\fint_{B(x,2^k) }&|\widetilde{E}f(x)-\widetilde{E}f(y)|^pd\mu(y)d\mu(x)\nonumber\\ &=\sum_{\substack{i,j\text{ s.t. }\\ i\ge k+1}}\int_{B_{i,j}}\fint_{B(x,2^k) }|\widetilde{E}f(x)-\widetilde{E}f(y)|^pd\mu(y)d\mu(x)\nonumber\\ &\qquad+\sum_{\substack{i,j\text{ s.t. }\\i<k+1}}\int_{B_{i,j}}\fint_{B(x,2^k) }|\widetilde{E}f(x)-\widetilde{E}f(y)|^pd\mu(y)d\mu(x)=:I+II. \end{align}\tag{14}\]
To estimate \(I\), let \((i,j)\) be such that \(i\ge k+1.\) Then for \(x\in B_{i,j}\) and \(y\in B(x,2^k)\), we have that \(B(x,2^k)\subset 2B_{i,j}\), and so by the partition of unity, it follows that \[\begin{align} |\widetilde{E}f(x)-\widetilde{E}f(y)|&=\left|\sum_{l,m} a_{l,m}\varphi_{l,m}(x)-\sum_{l,m}a_{l,m}\varphi_{l,m} (y)\right|\\ &= \left| \sum_{l,m}a_{l,m}\varphi_{l,m}(x)-a_{i,j}\sum_{l,m}\varphi_{l,m}(x)+a_{i,j}\sum_{l,m}\varphi_{l,m} (y)-\sum_{l,m}a_{l,m}\varphi_{l,m}(y)\right| \\ &\le \sum_{l,m}|a_{l,m}-a_{i,j}||\varphi_{l,m}(x)-\varphi_{l,m}(y)|=\sum_{\substack{l,m\text{ s.t. }\\2B_{i,j}\cap 2B_{l,m}\ne\varnothing}}|a_{l,m}-a_{i,j}||\varphi_{l,m}(x)-\varphi_{l,m}(y)|. \end{align}\] The last equality follows since \(\varphi_{l,m}\) is supported in \(2B_{l,m}\).
If \((l,m)\) is such that \(2B_{i,j}\cap 2B_{l,m}\ne\varnothing\), then since \(r_{i,j}=\frac{1}{8}d(p_{i,j},\partial\Omega)\), it follows that \(2^{-1}r_{i,j}\le r_{l,m}\le 2r_{i,j}\), and so \(i-2\le l\le i+2\) due to (iii) in Lemma 1. By the triangle inequality, we then have that \[\label{eq:U9442} U_{i,j}^*:=B(q_{i,j},32r_{i,j})\cap\partial\Omega\supset U_{l,m}.\tag{15}\] Furthermore, it follows that \(B_{i,j}\subset 8B_{l,m}\), and so by Lemma 1 (v), it follows that there are at most \(C\ge 1\) indices \((l,m)\) such that \(2B_{l,m}\cap 2B_{i,j}\ne\varnothing\), with \(C\) depending only on \(C_\mu\). Finally, since \(r_{i,j}\simeq r_{l,m}\), we have that \(\varphi_{l,m}\) is \(C/r_{i,j}\)-Lipschitz, with \(C\) depending only on the doubling constant. Using these facts, the doubling property of \(\nu\), and Jensen’s inequality, it follows that \[\begin{align} |\widetilde{E}f(x)-\widetilde{E}f(y)|^p&\lesssim \left(\frac{d(x,y)}{r_{i,j}}\sum_{\substack{l,m\text{ s.t. }\\2B_{i,j}\cap 2B_{l,m}\ne\varnothing}}|a_{l,m}-a_{i,j}|\right)^p\\ &\lesssim \left( \frac{d(x,y)}{r_{i,j}}\sum_{\substack{l,m\text{ s.t. }\\2B_{i,j}\cap 2B_{l,m}\ne\varnothing}}\fint_{U_{l,m}}\fint_{U_{i,j}}|f(w)-f(z)|d\nu(w)d\nu(z)\right)^p \\ &\lesssim \left( \frac{d(x,y)}{r_{i,j}}\fint_{U_{i,j}^*}\fint_{U_{i,j}^*}|f(w)-f(z)|d\nu(w)d\nu(z)\right)^p\\ &\le\frac{d(x,y)^p}{r^p_{i,j}}\fint_{U_{i,j}^*}\fint_{U_{i,j}^*}|f(w)-f(z)|^pd\nu(w)d\nu(z), \end{align}\] with comparison constant depending only on \(\theta\), \(C_\mu\) and \(C_\theta\). Hence, we have that \[\begin{align} \int_{B_{i,j}}\fint_{B(x,2^k)}|\widetilde{E}f(x)&-\widetilde{E}f(y)|^pd\mu(y)d\mu(x)\\ &\lesssim\frac{1}{r^p_{i,j}}\fint_{U_{i,j}^*}\fint_{U_{i,j}^*}|f(w)-f(z)|^pd\nu(w)d\nu(z)\int_{B_{i,j}}\fint_{B(x,2^k)}d(x,y)^pd\mu(y)d\mu(x)\\ &\le\frac{1}{r^p_{i,j}}\fint_{U_{i,j}^*}\fint_{U_{i,j}^*}|f(w)-f(z)|^pd\nu(w)d\nu(z)\int_{B_{i,j}}\fint_{B(x,2^k)}2^{pk}d\mu(y)d\mu(x)\\ &\lesssim 2^{p(k-i)}\mu(B_{i,j})\fint_{U_{i,j}^*}\fint_{U_{i,j}^*}|f(w)-f(z)|^pd\nu(w)d\nu(z)\\ &\le2^{p(k-i)}\frac{\mu(B(q_{i,j},32r_{i,j}))}{\nu(U_{i,j}^*)}\int_{U_{i,j}^*}\fint_{U_{i,j}^*}|f(w)-f(z)|^pd\nu(w)d\nu(z)\\ &\lesssim 2^{p(k-i)} r_{i,j}^\theta\int_{U_{i,j}^*}\fint_{U_{i,j}^*}|f(w)-f(z)|^pd\nu(w)d\nu(z)\\ &\le2^{p(k-i)+i\theta}\int_{U_{i,j}^*}\fint_{U_{i,j}^*}|f(w)-f(z)|^pd\nu(w)d\nu(z), \end{align}\] with the comparison constant depending only on \(\theta\), \(C_\mu\), and \(C_\theta\). By Lemma 1, we then have that \[\begin{align} \label{eq:Inside32Parenth32I3240141} I\lesssim\sum_{i\ge k+1}2^{p(k-i)+i\theta}\int_{\partial\Omega}\fint_{B(z,2^{i+6})\cap\partial\Omega}|f(w)-f(z)|^pd\nu(w)d\nu(z), \end{align}\tag{16}\]
To estimate \(II\) in 14 , we consider \(i,j\) such that \(i<k+1\).
For each \(x\in B_{i,j}\), we have that \(B(x,2^k)\subset B(p_{i,j},2^{k+1})\). Letting \[I_{i,j,k}:=\{l,m: B_{l,m}\cap B(p_{i,j},2^{k+1})\ne\varnothing\},\] we have that \[\begin{align} \int_{B_{i,j}}\fint_{B(x,2^k) }|\widetilde{E}f(x)-\widetilde{E}f(y)|^pd\mu(y)d\mu(x)\nonumber&\le\sum_{l,m\in I_{i,j,k}}\int_{B_{i,j}}\int_{B_{l,m}}\frac{|\widetilde{E}f(x)-\widetilde{E}f(y)|^p}{\mu(B(x,2^k) )}d\mu(y)d\mu(x). \end{align}\] For \(x\in B_{i,j}\) and \(y\in B_{l,m}\) for \((l,m)\in I_{i,j,k},\) it follows from the partition of unity, Lemma 1 (ii) and Jensen’s inequality that \[\begin{align} |\widetilde{E}f(x)&-\widetilde{E}f(y)|^p \le\left( \sum_{s,t}|a_{s,t}-a_{i,j}||\varphi_{s,t}(x)-\varphi_{s,t}(y)|\right)^p\\ &\le \left( \sum_{\substack{s,t\text{ s.t. }\\x\in 2B_{s,t}}}|a_{s,t}-a_{i,j}|+\sum_{\substack{s,t\text{ s.t. }\\y\in 2B_{s,t}}}|a_{s,t}-a_{i,j}|\right)^p\\ &\lesssim \left( \fint_{U_{i,j}^*}\fint_{U_{i,j}^*}|f(w)-f(z)|d\nu(w)d\nu(z)+\fint_{U_{i,j}^*}\fint_{U_{l,m}^*}|f(w)-f(z)|d\nu(w)d\nu(z)\right)^p\\ &\lesssim\fint_{U_{i,j}^*}\fint_{U_{i,j}^*}|f(w)-f(z)|^pd\nu(w)d\nu(z)+\fint_{U_{i,j}^*}\fint_{U_{l,m}^*}|f(w)-f(z)|^pd\nu(w)d\nu(z), \end{align}\] where the comparison constant depends only on \(\theta\), \(p\), \(C_\mu\) and \(C_\theta\). Substituting this into the above expression, we have that \[\begin{align} \label{eq:Second} \int_{B_{i,j}}&\fint_{B(x,2^k) }|\widetilde{E}f(x)-\widetilde{E}f(y)|^pd\mu(y)d\mu(x)\nonumber\\ &\lesssim \fint_{U_{i,j}^*}\fint_{U_{i,j}^*}|f(w)-f(z)|^pd\nu(w)d\nu(z)\sum_{l,m\in I_{i,j,k}}\int_{B_{i,j}}\int_{B_{l,m}}\frac{1}{\mu(B(x,2^k) )}d\mu(y)d\mu(x)\nonumber\\ &+\sum_{l,m\in I_{i,j,k}}\left(\fint_{U_{i,j}^*}\fint_{U_{l,m}^*}|f(w)-f(z)|^pd\nu(w)d\nu(z)\int_{B_{i,j}}\int_{B_{l,m}}\frac{1}{\mu(B(x,2^k) )}d\mu(y)d\mu(x)\right)\nonumber\\ &=:III+IV. \end{align}\tag{17}\]
For \((l,m)\in I_{i,j,k}\), we have by Lemma 1 and the assumption that \(i<k+1\) that \[\begin{align} 8r_{l,m}=d(p_{l,m},\partial\Omega)\le r_{l,m}+2^{k+1}+d(p_{i,j},\partial\Omega)\le r_{l,m}+2^{k+1}+8r_{i,j}\le r_{l,m}+10\cdot 2^{k}, \end{align}\] and so it follows that \[3\cdot 2^{l}\le 10\cdot 2^k.\] Hence, for \((l,m)\in I_{i,j,k}\), we have that \(l\le k+2\), and so if \(x\in B_{i,j}\) and \(y\in B_{l,m}\) for \((l,m)\in I_{i,j,k}\), it follows that \[d(x,y)\le 2r_{l,m}+r_{i,j}+2^{k+1}\le 2^{k+4}.\] Therefore, by bounded overlap, we can sum over \((l,m)\in I_{i,j,k}\) to estimate \(III\) as follows:
\[III\lesssim\fint_{U_{i,j}^*}\fint_{U_{i,j}^*}|f(w)-f(z)|^pd\nu(w)d\nu(z)\int_{B_{i,j}}\int_{B(x,2^{k+4}) }\frac{1}{\mu(B(x,2^k) )}d\mu(y)d\mu(x).\] By doubling of \(\mu\) and the codimensionality of \(\nu\), it follows that \[\begin{align} \label{eq:III} III&\lesssim\frac{\mu(B_{i,j})}{\nu(U_{i,j}^*)}\int_{U_{i,j}^*}\fint_{U_{i,j}^*}|f(w)-f(z)|^pd\nu(w)d\nu(z)\nonumber\\ &\simeq\frac{\mu(B(q_{i,j}, 32r_{i,j}))}{\nu(U_{i,j}^*)}\int_{U_{i,j}^*}\fint_{U_{i,j}^*}|f(w)-f(z)|^pd\nu(w)d\nu(z)\nonumber\\ &\simeq r_{i,j}^\theta\int_{U_{i,j}^*}\fint_{U_{i,j}^*}|f(w)-f(z)|^pd\nu(w)d\nu(z)\le 2^{i\theta}\int_{U_{i,j}^*}\fint_{U_{i,j}^*}|f(w)-f(z)|^pd\nu(w)d\nu(z) \end{align}\tag{18}\] with comparison constant depending only on \(\theta\), \(C_\mu\) and \(C_\theta\).
Now, for \(x\in B_{i,j},\) we have that \(\mu(B(x,2^k) )\simeq\mu(B(q_{i,j},2^k) )\) by the doubling condition of \(\mu\). Using this, we have that \[\begin{align} IV&\simeq \frac{1}{\mu(B(q_{i,j},2^k) )}\sum_{l,m\in I_{i,j,k}}\frac{\mu(B_{i,j})\mu(B_{l,m})}{\nu(U_{i,j}^*)\nu(U_{l,m}^*)}\int_{U_{i,j}^*}\int_{U_{l,m}^*}|f(w)-f(z)|^pd\nu(w)d\nu(z)\nonumber\\ &\lesssim\frac{2^{i\theta}}{\mu(B(q_{i,j},2^k) )}\sum_{l,m\in I_{i,j,k}}2^{l\theta}\int_{U_{i,j}^*}\int_{U_{l,m}^*}|f(w)-f(z)|^pd\nu(w)d\nu(z),\nonumber\\ \end{align}\] with comparison constant again depending only on \(\theta\), \(C_\mu\) and \(C_\theta\).
Recall that if \((l,m)\in I_{i,j,k}\), by definition we have \(d(p_{i,j},p_{l,m})\leq 2^{k+1}+r_{l,m}\). Moreover, we also saw that \(l\le k+2\), and so we have that
\[d(q_{i,j},q_{l,m})\leq 8(r_{i,j}+r_{l,m})+d(p_{i,j},p_{l,m})<8\cdot 2^{k}+9\cdot 2^{k+2}+2^{k+1}\leq 2^{k+6},\] Therefore, for \(z\in U^*_{i,j}\) and \(w\in U^*_{l,m}\) it follows that \(d(z,w)\leq 32(r_{i,j}+r_{l,m})+d(q_{i,j},q_{l,m})\leq 2^{k+8}\) and so \(U^*_{l,m}\subset B(z,2^{k+8})\cap\partial\Omega\). Using this fact, Lemma 1, and the codimensionality and doubling property of \(\nu\), we have that \[\begin{align} \label{eq:IV} IV&\lesssim \frac{2^{i\theta}}{\mu(B(q_{i,j},2^k) )}\sum_{l=-\infty}^{k+2}\sum_{\substack{m\text{ s.t. }\\l,m\in I_{i,j,k}}}2^{l\theta}\int_{U_{i,j}^*}\int_{U_{l,m}^*}|f(w)-f(z)|^pd\nu(w)d\nu(z)\nonumber\\ &\lesssim \frac{2^{i\theta}}{\mu(B(q_{i,j},2^k) )}\int_{U_{i,j}^*}\int_{B(z,2^{k+ 8})\cap\partial\Omega}|f(w)-f(z)|^pd\nu(w)d\nu(z)\sum_{l=-\infty}^{k+2}2^{l\theta}\nonumber\\ &\simeq \frac{2^{i\theta+k\theta}}{\mu(B(q_{i,j},2^k) )}\int_{U_{i,j}^*}\int_{B(z,2^{k+ 8})\cap\partial\Omega}|f(w)-f(z)|^pd\nu(w)d\nu(z)\nonumber\\ &\lesssim \frac{2^{i\theta}}{\nu(B(q_{i,j},2^k)\cap\partial\Omega)}\int_{U_{i,j}^*}\int_{B(z,2^{k+ 8})\cap\partial\Omega}|f(w)-f(z)|^pd\nu(w)d\nu(z)\nonumber\\ &\lesssim2^{i\theta}\int_{U_{i,j}^*}\fint_{B(z,2^{k+ 8})}|f(w)-f(z)|^pd\nu(w)d\nu(z), \end{align}\tag{19}\] with comparison constant depending only on \(\theta\), \(C_\mu\) and \(C_\theta\). In particular the constant depends on the convergence of \(\sum_{l=-\infty}^0 2^{l\theta}\).
Combining 14 , 17 , 18 , and 19 , and using Lemma 1, we have that \[\begin{align} \label{eq:IIdecomp} II\lesssim \sum_{i<k+1}&\sum_j 2^{i\theta}\int_{U_{i,j}^*}\fint_{U_{i,j}^*}|f(w)-f(z)|^pd\nu(w)d\nu(z)\nonumber\\ &+\sum_{i<k+1} 2^{i\theta}\int_{\partial\Omega}\fint_{B(z,2^{k+ 8})\cap\partial\Omega}|f(w)-f(z)|^pd\nu(w)d\nu(z). \end{align}\tag{20}\] By 13 , 14 , 16 , and 20 , we then have that \[\begin{align} \label{eq:V43VI43VII} \|\widetilde{E}f\|&^q_{HB^{\alpha+\theta /p}_{p,q}(\Omega,\mu)}\nonumber\\ &\lesssim\sum_{k\in\mathbb{Z}}2^{-k(\alpha+\theta/p)q}\left(\sum_{i\ge k+1}2^{p(k-i)+i\theta}\int_{\partial\Omega}\fint_{B(z,2^{i+ 6})\cap\partial\Omega}|f(w)-f(z)|^pd\nu(w)d\nu(z)\right)^{q/p}\nonumber\\ &+\sum_{k\in\mathbb{Z}}2^{-k(\alpha+\theta/p)q}\left(\sum_{i<k+1}\sum_j 2^{i\theta}\int_{U_{i,j}^*}\fint_{U_{i,j}^*}|f(w)-f(z)|^pd\nu(w)d\nu(z)\right)^{q/p}\nonumber\\ &+\sum_{k\in\mathbb{Z}}2^{-k(\alpha+\theta/p)q}\left(\sum_{i<k+1} 2^{i\theta}\int_{\partial\Omega}\fint_{B(z,2^{k+ 8})\cap\partial\Omega}|f(w)-f(z)|^pd\nu(w)d\nu(z)\right)^{q/p}\nonumber\\ &=:V+VI+VII. \end{align}\tag{21}\]
We note that for \(i\ge k+1\), we have that \[\begin{align} 2^{-k(\alpha p+\theta)+p(k-i)+i\theta}=2^{-k\alpha p+i\alpha p-i\alpha p}\left(2^{p-\theta}\right)^{-|i-k|}=\left(2^{p-\theta-\alpha p}\right)^{-|i-k|}2^{-i\alpha p}. \end{align}\]
Let \(a=2^{p-\theta-\alpha p}\), \(b=q/p\), and let \[c_i=2^{-i\alpha p}\int_{\partial\Omega}\fint_{B(z, 2^{i+ 6})\cap\partial\Omega}|f(w)-f(z)|^pd\nu(w)d\nu(z).\] Note that \(1<a<\infty\) by our assumption that \(\alpha<1-\theta/p\), we then use Lemma 1 and ?? to estimate \(V\) by \[\begin{align} \label{eq:V} V&=\sum_{k\in\mathbb{Z}}\left(\sum_{i\ge k+1}2^{-k(\alpha p+\theta)}2^{p(k-i)+i\theta}\int_{\partial\Omega}\fint_{B(z,2^{i+ 6})\cap\partial\Omega}|f(w)-f(z)|^pd\nu(w)d\nu(z)\right)^{q/p}\nonumber\\ &\le\sum_{k\in\mathbb{Z}}\left(\sum_{i\in\mathbb{Z}}\left(2^{p-\theta-\alpha p}\right)^{-|i-k|}2^{-i\alpha p}\int_{\partial\Omega}\fint_{B(z,2^{i+ 6})\cap\partial\Omega}|f(w)-f(z)|^pd\nu(w)d\nu(z)\right)^{q/p}\nonumber\\ &=\sum_{k\in\mathbb{Z}}\left(\sum_{i\in\mathbb{Z}}a^{-|i-k|}c_i\right)^b\nonumber\\ &\lesssim \sum_{i\in\mathbb{Z}}c_i^b\lesssim\sum_{i\in\mathbb{Z}}2^{-i\alpha q}\left(\int_{\partial\Omega}\fint_{B(z,2^{i+6})\cap\partial\Omega}|f(w)-f(z)|^pd\nu(w)d\nu(z)\right)^{q/p}\simeq\|f\|_{HB^\alpha_{p,q}(\partial\Omega,\nu)}^q. \end{align}\tag{22}\]
To estimate \(VI\), we note that for \(i<k+1\), \[2^{-k(\alpha p+\theta)+i\theta}=\left(2^{\alpha p+\theta}\right)^{-|i-k|}2^{-i\alpha p}.\] Setting \(a=2^{\alpha p+\theta}\), \(b=q/p\), and \[c_i=2^{-i\alpha p}\int_{\partial\Omega}\fint_{B(z,2^{i+ 6})\cap\partial\Omega}|f(w)-f(z)|^pd\nu(w)d\nu(z),\] we have by Lemma 1, Lemma 1, and ?? , \[\begin{align} \label{eq:VI} VI&=\sum_{k\in\mathbb{Z}}\left(\sum_{i<k+1}2^{-k(\alpha p+\theta)+i\theta}\sum_j \int_{U_{i,j}^*}\fint_{U_{i,j}^*}|f(w)-f(z)|^pd\nu(w)d\nu(z)\right)^{q/p}\nonumber\\ &\lesssim\sum_{k\in\mathbb{Z}}\left(\sum_{i\in\mathbb{Z}}\left(2^{\alpha p+\theta}\right)^{-|i-k|}2^{-i\alpha p} \int_{\partial\Omega}\fint_{B(z,2^{i+ 6})}|f(w)-f(z)|^pd\nu(w)d\nu(z)\right)^{q/p}\nonumber\\ &=\sum_{k\in\mathbb{Z}}\left(\sum_{i\in\mathbb{Z}}a^{-|i-k|}c_i\right)^{b}\nonumber\\ &\lesssim \sum_{i\in\mathbb{Z}}c_i^b\lesssim\sum_{i\in\mathbb{Z}}2^{-i\alpha q}\left(\int_{\partial\Omega}\fint_{B(z,2^{i+6})}|f(w)-f(z)|^pd\nu(w)d\nu(z)\right)^{q/p}\simeq\|f\|_{HB^\alpha_{p,q}(\partial\Omega,\nu)}^q. \end{align}\tag{23}\]
Finally, since \(\sum_{i<k+1}2^{i\theta}\simeq 2^{k\theta}\), it follows that \[\begin{align} VII\lesssim\sum_{k\in\mathbb{Z}}2^{-k\alpha q}\left(\int_{\partial\Omega}\fint_{B(z,2^{k+ 8})\cap\partial\Omega}|f(w)-f(z)|^pd\nu(w)d\nu(z)\right)^{q/p}\simeq\|f\|_{HB^\alpha_{p,q}(\partial\Omega,\nu)}^q. \end{align}\] Notice that the estimates of \(V\) and \(VI\) add the dependency on \(p\), \(q\), \(\alpha\) and \(\theta\) to the comparison constant due to the use of Lemma 1 with our choice of \(a\) and \(b\). Combining this estimate with 22 , 23 , and 21 completes the proof for the case \(1\leq q<\infty\).
For the case \(q=\infty\), we follow the same arguments as above until reaching 21 , and now instead set \[\begin{align} &V_k:=2^{-k(\alpha+\theta/p)}\left(\sum_{i\ge k+1}2^{p(k-i)+i\theta}\int_{\partial\Omega}\fint_{B(z,2^{i+ 6})\cap\partial\Omega}|f(w)-f(z)|^pd\nu(w)d\nu(z)\right)^{1/p}\nonumber\\ &VI_k:=2^{-k(\alpha+\theta/p)}\left(\sum_{i<k+1}\sum_j 2^{i\theta}\int_{U_{i,j}^*}\fint_{U_{i,j}^*}|f(w)-f(z)|^pd\nu(w)d\nu(z)\right)^{1/p}\nonumber\\ &VII_k:=2^{-k(\alpha+\theta/p)}\left(\sum_{i<k+1} 2^{i\theta}\int_{\partial\Omega}\fint_{B(z,2^{k+ 8})\cap\partial\Omega}|f(w)-f(z)|^pd\nu(w)d\nu(z)\right)^{1/p} \end{align}\] for each \(k\in \mathbb{Z}\). We then have to prove \(V_k+VI_k+VII_k\lesssim \|f\|_{HB^\alpha_{p,\infty}(\partial\Omega,\nu)}\) for every \(k\in \mathbb{Z}\). Notice that given any \(i\in\mathbb{Z}\) then \(c_i\lesssim \|f\|_{HB^\alpha_{p,\infty}(\partial\Omega,\nu)}^p\) by definition, and thus, estimating analogously to the case \(q<\infty\) we get \[V_k\lesssim \left(\sum_{i\in\mathbb{Z}}a^{-|i-k|}c_i\right)^{\frac{1}{p}} \lesssim \|f\|_{HB^\alpha_{p,\infty}(\partial\Omega,\nu)}\left(\sum_{i\in\mathbb{Z}}a^{-|i-k|}\right)^{\frac{1}{p}}\simeq \|f\|_{HB^\alpha_{p,\infty}(\partial\Omega,\nu)},\] while an analogous argument yields the estimate for \(VI_k\). Furthermore, \[VII_k\lesssim 2^{-k\alpha }\left(\int_{\partial\Omega}\fint_{B(z,2^{k+ 8})\cap\partial\Omega}|f(w)-f(z)|^pd\nu(w)d\nu(z)\right)^{1/p}\lesssim \|f\|_{HB^\alpha_{p,\infty}(\partial\Omega,\nu)}.\qedhere\] ◻
We now prove Corollary 1, showing that \(\widetilde{E}\), in fact, gives a bounded linear extension operator for the inhomogeneous spaces \(B^\alpha_{p,q}(\partial\Omega\,\nu)\) when \(\Omega\) is bounded:
Proof of Corollary 1. For each \(f\in B^\alpha_{p,q}(\partial\Omega,\nu)\), let \(\widetilde{E}f\) be defined by 12 . The linearity of \(\widetilde{E}\) and energy bounds ?? follow from Theorem 1. It remains to show ?? . To this end, we note that as \(\Omega\) is bounded, there exists \(N_\Omega\in\mathbb{Z}\), depending only on \(\mathop{\mathrm{diam}}(\Omega)\), such that \(\mathcal{W}_\Omega=\bigcup_{i=-\infty}^{N_\Omega}\{B_{i,j}\}_j\). That is, the Whitney cover of \(\Omega\) given by Lemma 1 contains balls of radius no greater than \(2^{N_\Omega}\). Therefore, by the definition of \(\widetilde{E}f\), we have that \[\begin{align} \label{eq:layer32estimate32part321} \int_\Omega|\widetilde{E}f|^pd\mu\le\sum_{i=\infty}^{N_\Omega}\sum_j\int_{B_{i,j}}|\widetilde{E}f|^pd\mu &\le\sum_{i=-\infty}^{N_\Omega}\sum_{j}\int_{B_{i,j}}\Bigg|\sum_{l,m}\left(\fint_{U_{l,m}}fd\nu\right)\varphi_{l,m}(x)\Bigg|^pd\mu(x)\nonumber\\ &\le\sum_{i=-\infty}^{N_\Omega}\sum_{j}\int_{B_{i,j}}\Bigg|\sum_{\substack{l,m\text{ s.t. }\\2B_{l,m}\cap B_{i,j}\ne\varnothing}}\left(\fint_{U_{l,m}}fd\nu\right)\Bigg|^pd\mu(x). \end{align}\tag{24}\] By bounded overlap, there are at most \(C\) indices \((l,m)\) such that \(2B_{l,m}\cap B_{i,j}\ne\varnothing\). Furthermore, for such indices, we have that \(U_{l,m}\subset U^*_{i,j}\), where \(U_{i,j}^*\) is given by 15 , and so \(\nu(U_{l,m})\simeq\nu(U^*_{i,j})\) by the doubling property of \(\nu\). From these facts, Jensen’s inequality, the doubling property of \(\mu\), the codimension \(\theta\) Ahlfors regularity of \(\nu\), and Lemma 1, we have \[\begin{align} \label{eq:layer32estimate32part322} \int_\Omega|\widetilde{E}f|^pd\mu&\lesssim\sum_{i=-\infty}^{N_\Omega}\sum_{j}\int_{B_{i,j}}\left(\fint_{U_{i,j}^*}|f|d\nu\right)^pd\mu(x)\nonumber\\ &\le\sum_{i=-\infty}^{N_\Omega}\sum_j\int_{B_{i,j}}\fint_{U_{i,j}^*}|f|^pd\nu d\mu(x)\nonumber\\ &\le\sum_{i=-\infty}^{N_\Omega}\sum_{j}\frac{\mu(B(q_{i,j},32r_{i,j}))}{\nu(U_{i,j}^*)}\int_{U_{i,j}^*}|f|^pd\nu\nonumber\\ &\lesssim \sum_{i=-\infty}^{N_\Omega}2^{i\theta}\sum_{j}\int_{U_{i,j}^*}|f|^pd\nu\lesssim \sum_{i=-\infty}^{N_\Omega}2^{i\theta}\|f\|^p_{L^p(\partial\Omega,\nu)}\lesssim\|f\|^p_{L^p(\partial\Omega,\nu)}, \end{align}\tag{25}\] where the comparison constant depends only on \(p\), \(\theta\), \(C_\mu\), \(C_\theta\) and \(\mathop{\mathrm{diam}}(\Omega)\). ◻
If \(\Omega\) is not assumed to be bounded, then an extension result for the inhomogeneous Besov spaces can still be obtained by applying a Lipschitz cutoff function to the extension operator obtained in the previous theorem. We now prove Theorem 1, obtaining the desired extension operator.
Proof of Theorem 1. Let \(f\in B^{\alpha}_{p,q}(\partial\Omega,\nu)\). Let \(\Phi:\Omega\to[0,1]\) be a \(1\)-Lipschitz function such that \(\Phi\equiv 1\) in \(\{x\in\Omega:d(x,\partial\Omega)\le 1\}\) and \(\Phi\equiv 0\) in \(\{x\in\Omega:d(x, \partial\Omega)>2\}\). For \(x\in\Omega\), we then set \[Ef(x)=\Phi(x)\widetilde{E}f(x),\] where \(\widetilde{E}f\) is given by 12 .
Let \(\Omega_2:=\{x\in\Omega:d(x,\partial\Omega)\le 2\}\). If \(i,j\) is such that \(B_{i,j}\cap\Omega_2\ne\varnothing\), then \[4\cdot 2^i\le 8r_{i,j}=d(p_{i,j},\partial\Omega)\le r_{i,j}+2\le 2^i+2,\] and so it follows that \(2^i\le 2/3\). Hence \(i<0\). Here \(\mathcal{W}_\Omega=\{B_{i,j}\}_{i,j}\) is the Whitney cover used in Theorem 1. Since \(0\le\Phi\le 1\), we then have that \[\begin{align} \label{eq:i600} \|Ef\|^p_{L^p(\Omega,\mu)}\le\int_{\Omega_2}|\widetilde{E}f|^pd\mu\le\sum_{i=-\infty}^0\sum_j\int_{B_{i,j}}|\widetilde{E}f|^pd\mu. \end{align}\tag{26}\] For \(i<0\), we see from estimates 24 and 25 that \[\begin{align} \sum_j\int_{B_{i,j}}|\widetilde{E}f|^pd\mu\lesssim 2^{i\theta}\|f\|^p_{L^p(\partial\Omega,\nu)}. \end{align}\]
Substituting this estimate into 26 , we obtain \[\label{eq:ExtL94pBound} \|Ef\|_{L^p(\Omega,\mu)}\le\|\widetilde{E}f\|_{L^p(\Omega_2,\mu)}\le C\|f\|_{L^p(\partial\Omega,\nu)},\tag{27}\] where \(C\ge 1\) depends only on \(p\), \(\theta\), \(C_\mu\) and \(C_\theta\).
It remains to estimate the Besov energy of \(Ef\). Let \(1\le q<\infty\) and recall by ?? and 27 that \[\begin{align} \label{eq:InhomExt32Energy32bound} \|Ef\|^q_{HB^{\alpha+\theta/p}_{p,q}(\Omega,\mu)}&\lesssim\|Ef\|_{L^p(\Omega,\mu)}^q+\sum_{i\in\mathbb{N}}2^{i(\alpha+\theta/p)q}\left(\int_{\Omega}\fint_{B(x,2^{-i}) }|Ef(x)-Ef(y)|^pd\mu(y)d\mu(x)\right)^{q/p}\nonumber\\ &\lesssim\|f\|_{L^p(\partial\Omega,\nu)}^q+I+II, \end{align}\tag{28}\] where \[I:=\sum_{i\in\mathbb{N}}2^{i(\alpha+\theta/p)q}\left(\int_{\Omega_2}\fint_{B(x,2^{-i}) }|Ef(x)-Ef(y)|^pd\mu(y)d\mu(x)\right)^{q/p}\] and \[II:=\sum_{i\in\mathbb{N}}2^{i(\alpha+\theta/p)q}\left(\int_{\Omega\setminus\Omega_2}\int_{B(x,2^{-i})\cap \Omega_2}\frac{|Ef(x)-Ef(y)|^p}{\mu(B(x,2^{-i}) )}d\mu(y)d\mu(x)\right)^{q/p}.\] By the triangle inequality, we have \[|Ef(x)-Ef(y)|\le|\Phi(y)||\widetilde{E}f(x)-\widetilde{E}f(y)|+|\widetilde{E}f(x)||\Phi(x)-\Phi(y)|,\] and since \(\Phi\) is \(1\)-Lipschitz and \(0\le\Phi\le 1\), it follows that \[\begin{align} I&\lesssim \sum_{i\in\mathbb{N}}2^{i(\alpha+\theta/p)q}\left(\int_{\Omega_2}\fint_{B(x,2^{-i}) }|\widetilde{E}f(x)-\widetilde{E}f(y)|^pd\mu(y)d\mu(x)\right)^{q/p}\\ &\qquad+\sum_{i\in\mathbb{N}}2^{i(\alpha +\theta/p)q}\left(\int_{\Omega_2}|\widetilde{E}f(x)|^p\fint_{B(x,2^{-i}) }|\Phi (x)-\Phi (y)|^pd\mu(y)d\mu(x)\right)^{q/p}\\ &\lesssim \|\widetilde{E}f\|_{HB^{\alpha +\theta /p}_{p,q}(\Omega,\mu)}^q+\sum_{i\in\mathbb{N}}2^{i(\alpha+ \theta/p)q}\left(\int_{\Omega_2}|\widetilde{E}f(x)|^p\fint_{B(x,2^{-i}) }d(x,y)^p d\mu(y)d\mu(x)\right)^{q/p}\\ &\leq \|\widetilde{E}f\|_{HB^{\alpha +\theta /p}_{p,q}(\Omega,\mu)}^q+\sum_{i\in\mathbb{N}}2^{i(\alpha+ \theta/p)q}\left(\int_{\Omega_2} 2^{-ip}|\widetilde{E}f(x)|^pd\mu(x)\right)^{q/p}\\ &\leq \|\widetilde{E}f\|_{HB^{\alpha +\theta /p}_{p,q}(\Omega,\mu)}^q+\sum_{i\in\mathbb{N}}2^{i(\alpha+\theta/p-1)q}\left(\int_{\Omega_2}|\widetilde{E}f(x)|^pd\mu(x)\right)^{q/p}\\ &\lesssim \|f\|_{HB^\alpha_{p,q}(\partial\Omega,\nu)}^q+\|f\|_{L^p(\partial\Omega,\nu)}^q, \end{align}\] where the last step follows from the assumption that \(\alpha+\theta /p<1\) for the convergence of the series, as well as estimate 27 and Theorem 1.
In order to estimate \(II\), we note that for \(i\in\mathbb{N}\), \(x\in\Omega\setminus\Omega_2\), and \(y\in B(x,2^{-i})\cap \Omega_2\), we have by the triangle inequality that \[\begin{align} |Ef(x)-Ef(y)|\le|\Phi(x)||\widetilde{E}f(x)-\widetilde{E}f(y)|+|\widetilde{E}f(y)||\Phi(x)-\Phi(y)|=|\widetilde{E}f(y)||\Phi(x)-\Phi(y)|, \end{align}\] since \(\Phi (x)=0\). By the doubling property of \(\mu\), it also follows that for such \(x\) and \(y\), \[\label{eq:doubB40x44294i41} \mu( B(y,2^{-i}) )\leq \mu (B(x,2^{-i+1}) )\leq C_\mu \mu (B(x,2^{-i}) ).\tag{29}\] Hence by 29 , Tonelli’s theorem, and the \(1\)-Lipschitz condition of \(\Phi\), we have \[\begin{align} II&\lesssim\sum_{i\in\mathbb{N}}2^{i(\alpha+\theta/p)q}\left(\int_{\Omega\backslash\Omega_2}\int_{B(x,2^{-i})\cap \Omega_2}\frac{|\widetilde{E}f(y)|^p|\Phi(x)-\Phi(y)|^p}{\mu (B(y,2^{-i}) )}d\mu(y)d\mu(x)\right)^{q/p}\\ &= \sum_{i\in\mathbb{N}}2^{i(\alpha+\theta/p)q}\left(\int_{\Omega_2}\int_{B(y,2^{-i})\cap(\Omega\backslash\Omega_2)}\frac{|\widetilde{E}f(y)|^p|\Phi(x)-\Phi(y)|^p}{\mu (B(y,2^{-i}) )}d\mu(x)d\mu(y)\right)^{q/p}\\ &\le \sum_{i\in\mathbb{N}}2^{i(\alpha+\theta/p)q}\left(\int_{\Omega_2}|\widetilde{E}f(y)|^p\int_{B(y,2^{-i}) }\frac{d(x,y)^p}{\mu (B(y,2^{-i}) )}d\mu(x)d\mu(y)\right)^{q/p}\\ &\lesssim \sum_{i\in\mathbb{N}}2^{i(\alpha+\theta/p-1)q}\left(\int_{\Omega_2}|\widetilde{E}f(y)|^pd\mu(y)\right)^{q/p} \lesssim \|f\|_{L^p(\partial\Omega,\nu)}^q. \end{align}\] It then follows by the estimates of \(I\), \(II\), and 28 that \[\|Ef\|_{HB^{\alpha+\theta/p}_{p,q}(\Omega,\mu)}\le C\left(\|f\|_{HB^\alpha_{p,q}(\partial\Omega,\nu)}+\|f\|_{L^p(\partial\Omega,\nu)}\right),\] where \(C\) depends on \(\alpha\), \(p\), \(q\), \(\theta\), \(C_\mu\) and \(C_\theta\). This completes the proof.
Now let \(q=\infty\). In this case, set \[I:=\sup_{i\in\mathbb{N}}2^{i(\alpha+\theta/p)}\left(\int_{\Omega_2}\fint_{B(x,2^{-i}) }|Ef(x)-Ef(y)|^pd\mu(y)d\mu(x)\right)^{1/p}\] and \[II:=\sup_{i\in\mathbb{N}}2^{i(\alpha+\theta/p)}\left(\int_{\Omega\setminus\Omega_2}\int_{B(x,2^{-i})\cap \Omega_2}\frac{|Ef(x)-Ef(y)|^p}{\mu(B(x,2^{-i}) )}d\mu(y)d\mu(x)\right)^{1/p}.\] By the same triangle inequality arguments as before, \[\begin{align} I&\lesssim \|\widetilde{E}f\|_{HB^{\alpha +\theta /p}_{p,\infty}(\Omega,\mu)}+\sup_{i\in\mathbb{N}}2^{i(\alpha+ \theta/p)}\left(\int_{\Omega}|\widetilde{E}f(x)|^p\fint_{B(x,2^{-i}) }d(x,y)^p d\mu(y)d\mu(x)\right)^{1/p}\\ &\lesssim \|f\|_{HB^\alpha_{p,\infty}(\partial\Omega,\nu)}+\sup_{i\in\mathbb{N}}2^{i(\alpha +\theta/p -1)} \|f\|_{L^p(\partial\Omega,\nu)}= \|f\|_{HB^\alpha_{p,\infty}(\partial\Omega,\nu)}+2^{\alpha +\theta/p -1} \|f\|_{L^p(\partial\Omega,\nu)}. \end{align}\] Similar adaptations from the case \(1\le q<\infty\) yield the corresponding estimate for \(II\). ◻
Finally, we show that the trace of this extension operator recovers the original function.
Lemma 1. Let \(f\in B^\alpha_{p,q}(\partial\Omega,\nu)\) and let \(Ef\) be defined as in Theorem 1. Then for \(\nu\)-a.e.\(z\in\partial\Omega\), we have that \[\lim_{r\to 0^+}\fint_{B(z,r) }|Ef-f(z)|d\mu=0.\]
Proof. As \(f\in B^\alpha_{p,q}(\partial\Omega,\nu)\subset L^p(\partial\Omega,\nu)\), \(f\) is locally integrable on \(\partial\Omega\). Thus, by the Lebesgue differentiation theorem, \(\nu\)-a.e.\(z\in\partial\Omega\) is a Lebesgue point of \(f\). Fix such a point \(z\in\partial\Omega\) and \(0<r<1\). Notice that for such \(r\) we always have \(Ef(x) = \widetilde{E}f (x)\) for each \(x\in B(z,r)\). By the partition of unity, we then have that \[\begin{align} \int_{B(z,r) }&|Ef-f(z)|d\mu=\int_{B(z,r) }\Bigg|\sum_{i,j}\left(\fint_{U_{i,j}}fd\nu\right)\varphi_{i,j}(x)-f(z)\Bigg|d\mu(x)\\ &\le\int_{B(z,r) }\sum_{i,j}\left(\fint_{U_{i,j}}|f-f(z)|d\nu\right)\varphi_{i,j}(x)d\mu(x)\\ &\le\sum_{\substack{l,m\text{ s.t. }\\B_{l,m}\cap B(z,r)\ne\varnothing}}\int_{B_{l,m} }\sum_{i,j}\left(\fint_{U_{i,j}}|f-f(z)|d\nu\right)\varphi_{i,j}(x)d\mu(x)\\ &=\sum_{\substack{l,m\text{ s.t. }\\B_{l,m}\cap B(z,r)\ne\varnothing}}\int_{B_{l,m} }\sum_{\substack{i,j\text{ s.t. }\\2B_{i,j}\cap B_{l,m}\ne\varnothing}}\left(\fint_{U_{i,j}}|f-f(z)|d\nu\right)d\mu(x). \end{align}\] By bounded overlap, there are at most \(C\) indices \((i,j)\) such that \(2B_{i,j}\cap B_{l,m}\ne\varnothing\), with \(C\) depending only on the doubling constant. Furthermore, for such \((i,j)\), we have that \(U_{i,j}\subset U_{l,m}^*\), with \(U_{l,m}^*\) defined as in 15 , and so by doubling, it follows that \(\nu(U_{i,j})\simeq\nu(U_{l,m}^*)\). Hence, we have that \[\begin{align} \int_{B(z,r) }|Ef-f(z)|d\mu&\lesssim\sum_{\substack{l,m\text{ s.t. }\\B_{l,m}\cap B(z,r)\ne\varnothing}}\int_{B_{l,m} }\fint_{U_{l,m}^*}|f-f(z)|d\nu d\mu(x)\\ &=\sum_{\substack{l,m\text{ s.t. }\\B_{l,m}\cap B(z,r)\ne\varnothing}}\frac{\mu(B_{l,m} )}{\nu(U^*_{l,m})}\int_{U_{l,m}^*}|f-f(z)|d\nu\\ &\lesssim\sum_{\substack{l,m\text{ s.t. }\\B_{l,m}\cap B(z,r)\ne\varnothing}}r_{l,m}^\theta\int_{U_{l,m}^*}|f-f(z)|d\nu\le\sum_{\substack{l,m\text{ s.t. }\\B_{l,m}\cap B(z,r)\ne\varnothing}}2^{\theta l}\int_{U_{l,m}^*}|f-f(z)|d\nu. \end{align}\] If \((l,m)\) is such that \(B_{l,m}\cap B(z,r)\ne\varnothing\), then \[r\ge d(p_{l,m},z)-r_{l,m}\ge d(p_{l,m},\partial\Omega )-r_{l,m}\geq 7r_{l,m}\ge 7(2^{l-1}),\] and so we have that \(2^l<r\). Let \(l_r\in\mathbb{Z}\) be such that \(2^{l_r-1}\le r<2^{l_r}\). Also, if \(\zeta\in U^*_{l,m}\) for such \((l,m)\), it follows that \[\begin{align} d(\zeta,z)&\le d(\zeta,q_{l,m})+d(q_{l,m},p_{l,m})+d(p_{l,m},z)\\ &<32r_{l,m}+8r_{l,m}+r_{l,m}+r\le 42 r, \end{align}\] and so \(U^*_{l,m}\subset B(z,42r)\). We then have by Lemma 1 that \[\begin{align} \int_{B(z,r) }|Ef-f(z)|d\mu&\lesssim\sum_{l=-\infty}^{l_r-1}2^{\theta l}\sum_{\substack{m\text{ s.t. }\\B_{l,m}\cap B(z,r)\ne\varnothing}}\int_{U_{l,m}^*}|f-f(z)|d\nu\\ &\lesssim\sum_{l=-\infty}^{l_r-1}2^{\theta l}\int_{B(z,42r)\cap\partial\Omega}|f-f(z)|d\nu\\ &\lesssim r^\theta\int_{B(z,42r)\cap\partial\Omega}|f-f(z)|d\nu. \end{align}\] Therefore by the \(\theta\) codimensional Ahlfors regularity and doubling property of \(\nu\), and since \(z\) is a Lebesgue point of \(f\), we have that \[\begin{align} \fint_{B(z,r) }|Ef-f(z)|d\mu&\lesssim\frac{r^\theta}{\mu(B(z,r) )}\int_{B(z,42r)\cap\partial\Omega}|f-f(z)|d\nu\\ &\lesssim\fint_{B(z,42r)\cap\partial\Omega}|f-f(z)|d\nu\to 0 \end{align}\] as \(r\to 0^+\). ◻
Notice that the above lemma also holds when \(Ef\) is replaced by \(\widetilde{E}f\), since for sufficiently small \(r\), we have that \(Ef(x)= \widetilde{E}f(x)\) for all \(x\in B(z,r)\). Likewise the following result also holds for \(\widetilde{E}\), although we state it for \(E\). In both cases, if \(\widetilde{E}\) is considered, then we can consider \(f\) to be in \(HB_{p,q}^\alpha (\Omega ,\mu )\).
Corollary 1. Under the assumptions of Theorem 1, we have that \(T\circ E=\mathop{\mathrm{Id}}\). That is, for each \(f\in B^{\alpha}_{p,q}(\partial\Omega,\nu)\), we have that \(TEf(z)=f(z)\) for \(\nu\)-a.e.\(z\in\partial\Omega\).
Proof. By ?? and Lemma 1, we have that \[\lim_{r\to 0^+}\fint_{B(z,r) }|Ef-TEf(z)|d\mu=0=\lim_{r\to 0^+}\fint_{B(z,r) }|Ef-f(z)|d\mu\] for \(\nu\)-a.e.\(z\in\partial\Omega\). Therefore, it follows that \(TEf(z)=f(z)\) for \(\nu\)-a.e.\(z\in\partial\Omega\). ◻
In Section 3, we established trace theorems under the geometric assumption that \(\Omega\) is a uniform domain in its completion. In this section, we establish the corresponding result, Theorem 1, without this assumption. To do so, we consider uniformized hyperbolic fillings of a metric measure space, as constructed in [16].
Let \((Z,d,\nu)\) be a compact metric measure space, with \(\nu\) a doubling measure. By rescaling the metric if necessary, we may assume that \(\mathop{\mathrm{diam}}(Z)<1\). We will realize \((Z,d,\nu)\), up to a bi-Lipschitz change in the metric, as the boundary of a uniform domain, which is a metric graph obtained through the following construction.
Fix \(z_0\in Z\), and let \(A_0:=\{z_0\}\). For each \(n\in\mathbb{N}\), let \(A_n\subset Z\) be a maximal \(2^{-n}\)-separated subset of \(Z\), chosen so that \(A_n\subset A_{n+1}\). We obtain the vertex set \(V\) of the desired graph by associating to each point in \(A_n\) its corresponding level. That is, \[V:=\bigcup_{n=0}^\infty\bigcup_{z\in A_{n}}(z,n).\] The edge relationship \(\sim\) between vertices is defined as follows. Two vertices \((z,n)\) and \((y,m)\) are joined by an edge, i.e.\((z,n)\sim (y,m)\), if and only if \(n=m\) and \(B(z, 2^{-n+1})\cap B(y,2^{-n+1})\ne\varnothing\) or \(m=n\pm 1\) and \(B(z,2^{-n})\cap B(z,2^{-m})\ne\varnothing\). We obtain a metric graph \(X\) by associating a copy of the unit interval to each edge, and we equip \(X\) with the path metric \(d_X\). We denote by \([v,w]\) the interval associated to the edge joining vertices \(v\) and \(w\). The metric space \((X,d_X)\) is the hyperbolic filling of \((Z,d)\).
Letting \(\varepsilon:=\log 2\) and \(v_0:=(z_0,0)\), we define the uniformized metric \(d_\varepsilon\) on \(X\) by \[d_\varepsilon(x,y):=\inf_{\gamma}\int_{\gamma}e^{-\varepsilon d_X(\cdot,v_0)}ds\] where the infimum is over all paths \(\gamma\) in \(X\) with endpoints \(x\) and \(y\). We denote by \(\overline{X}_\varepsilon\) the completion of \(X\) with respect to the metric \(d_\varepsilon\), and we denote \(\partial_\varepsilon X:=\overline{X}_\varepsilon\setminus X\). With this choice of \(\varepsilon\), \((\overline{X}_\varepsilon,d_\varepsilon)\) is bounded.
For each \(\beta>0\), we also lift up the measure \(\nu\) to essentially a weighted measure \(\mu_\beta\) on \(X\) as follows. We first define the vertex weights \[\hat{\mu}_\beta((z,n)):=e^{-\beta n}\nu(B(z,2^{-n})),\] and then for each \(A\subset X\), we define \[\mu_\beta(A):=\sum_{v\in V}\sum_{w\sim v}(\hat{\mu}_\beta(v)+\hat{\mu}_\beta(w))\mathcal{H}^1([v,w]\cap A).\] The following properties of \((\overline{X}_\varepsilon,d_\varepsilon,\mu_\beta)\) were obtained in [16]:
Theorem 1. Let \((Z,d,\nu)\) be a compact metric measure space, with \(\nu\) a doubling measure, and let \(\varepsilon:=\log 2\). Then for each \(\beta>0\), the metric measure space \((\overline{X}_\varepsilon,d_\varepsilon,\mu_\beta)\) satisfies the following:
\((Z,d)\) is bi-Lipschitz equivalent with \((\partial_\varepsilon X,d_\varepsilon)\),
\((\overline{X}_\varepsilon,d_\varepsilon)\) is geodesic,
\(X\) is a length uniform domain in \(\overline{X}_\varepsilon\), see Remark 1,
\(\mu_\beta\) is doubling,
For all \(z\in Z\), \(0<r<2\mathop{\mathrm{diam}}(Z)\), we have \[\nu(B(z,r)\cap Z)\simeq\frac{\mu_\beta(B(z,r))}{r^{\beta/\varepsilon}}.\]
In each of the above, the comparison constants depend only on \(\varepsilon\), \(\beta\), and the doubling constant of \(\nu\).
In a slight abuse of notation, we do not distinguish between balls defined with respect to \(d\) and \(d_\varepsilon\) due to the bi-Lipschitz equivalence between \((Z,d)\) and \((\partial_\varepsilon X,d_\varepsilon)\). Furthermore, for \(A\subset\overline{X}_\varepsilon\), we interpret \(\nu(A)\) to mean \(\nu(A\cap Z)\), for ease of notation.
In [16], it was further shown that \((\overline{X}_\varepsilon,d_\varepsilon,\mu_\beta)\) supports a \(1\)-Poincaré inequality and that bounded linear trace and extension operators exist between Besov spaces on \(Z\) and Newton-Sobolev spaces on \(\overline{X}_\varepsilon\). As these properties are not relevant to us, we have not included them in the statement of the above theorem.
We now prove Theorem 1, restated below as Theorem 1. To do so, we will use the following lemma from [34]:
Lemma 1 ([34], Lemma 3.10). Let \((X,d,\mu)\) be a doubling metric measure space. Let \(f\in L^p(X)\), \(0<t<p\), and \(M>0\). Then \(\mathcal{H}^{-t}(E_M)=0\), where \[E_M=\left\{x\in U:\limsup_{r\to 0^+}r^t\fint_{B(x,r)}|f|^pd\mu>M^p\right\}.\]
Theorem 1. Let \((Z,d,\nu)\) be a locally compact, non-complete, bounded metric measure space, with \(\nu\) a doubling measure. Suppose that \(\partial Z:=\overline{Z}\setminus Z\), the boundary of \(Z\), is equipped with a Borel measure \(\pi\), which is codimension \(\theta\) Ahlfors regular with respect to \(\nu\) for some \(\theta >0\). Let \(1\le p<\infty\), \(1\le q\le\infty\), and \(\theta/p<\alpha<1\). Then there exists a bounded, linear trace operator \[T:B^\alpha_{p,q}(Z,\nu)\to B^{\alpha-\theta/p}_{p,q}(\partial Z,\pi),\] such that for all \(u\in B^\alpha_{p,q}(Z,\nu)\), we have \[\label{eq:general32trace32condition} \lim_{r\to 0^+}\fint_{B(z,r)}|u-Tu(z)|^pd\nu=0\qquad{(11)}\] for \(\pi\)-a.e.\(z\in \partial Z\), and
\[\label{eq:gen32trace32energy32and32Lp} \|Tu\|_{HB^{\alpha-\theta/p}_{p,q}(\partial Z,\pi)}\le C\|u\|_{HB^\alpha_{p,q}(Z,\nu)},\qquad\|Tu\|_{L^p(\partial Z,\pi)}\le C\left(\|u\|_{L^p(Z,\nu)}+\|u\|_{HB^\alpha_{p,q}(Z,\nu)}\right),\qquad{(12)}\] where \(C\ge 1\) depends only on \(\alpha\), \(p\), \(q\), \(\theta\), \(C_\nu\), \(C_\theta\), and \(\mathop{\mathrm{diam}}(Z)\).
Furthermore, the bounded linear extension operator \(\widetilde{E}:B^{\alpha-\theta/p}_{p,q}(\partial Z,\pi)\to B^\alpha_{p,q}(Z,\nu)\), given by Corollary 1, is a right-inverse of \(T\). That is, for all \(f\in B^{\alpha-\theta/p}_{p,q}(\partial Z,\pi)\), we have that \[\label{eq:gen32TEf61f} T\widetilde{E}f(z)=f(z)\qquad{(13)}\] for \(\pi\)-a.e.\(z\in\partial Z\).
Proof. Consider the uniformized hyperbolic filling \((\overline{X}_\varepsilon,d_\varepsilon,\mu_\beta)\) of \((\overline{Z},d,\nu)\), constructed with parameters \(\varepsilon=\log 2\) and \(\beta>0\) chosen so that \(\sigma:=\beta/\varepsilon=p(1-\alpha)/2\), hence \(\sigma<p(1-\alpha)\). Note that by Theorem 1, \(\mu:=\mu_\beta\) is doubling, and the measure \(\nu\) is codimension \(\sigma\) Ahlfors regular with respect to \(\mu\), and \((\overline{X}_\varepsilon,d_\varepsilon)\) is bounded. By Corollary 1, there exists a bounded, linear extension operator \[E_Z:B^\alpha_{p,q}(\overline{Z},\nu)\to B^{\alpha+\sigma/p}_{p,q}(\overline{X}_\varepsilon,\mu).\] Note that the codimensionality between \(\mu\) and \(\nu\) implies that \(\mu(\overline{Z})=0\), and so the above holds since \(B^{\alpha+\sigma/p}_{p,q}(\overline{X}_\varepsilon,\mu)=B^{\alpha+\sigma/p}_{p,q}(X_\varepsilon,\mu)\).
Since \(X\) is a length uniform domain in \(\overline{X}_\varepsilon\), the corresponding uniform curves can be extended to the boundary \(\overline{Z}\) by the Arzelà-Ascoli theorem, see Remark 1. Thus, it follows that \(\overline{X}_\varepsilon\setminus\partial Z\) is an \(A\)-uniform domain in \(\overline{X}_\varepsilon\), with \(A\) depending only on \(\alpha\), \(p\), and \(C_\nu\). Furthermore, the measure \(\pi\) is codimension \((\sigma+\theta)\) Ahlfors regular with respect to \(\mu_\beta\). Thus, by Theorem 1, there exists a bounded, linear trace operator \[T_{\partial Z}:B^{\alpha+\sigma/p}_{p,q}(\overline{X}_\varepsilon,\mu)\to B^{\alpha-\theta/p}_{p,q}(\partial Z,\pi),\] again using the fact that \(\mu(\partial Z)=0\). Define \(T:=T_{\partial Z}\circ E_Z\), and let \(u\in B^\alpha_{p,q}(Z,\nu)\). The estimates ?? follow immediately from ?? , ?? , ?? , and ?? .
We now show ?? . To this end, let \(0<r<1\). By the definition of \(T\) and Proposition 1, it follows that for \(\pi\)-a.e.\(z\in \partial Z\), \[\label{eq:Tu40z41} Tu(z)=\lim_{\rho\to 0^+}\fint_{B(z,\rho)}E_Zu\,d\mu.\tag{30}\] Fix such a \(z\in \partial Z\). By Theorem 1 and Corollary 1, there exists a bounded, linear trace operator \[\; T_Z:HB^{\alpha+\sigma/p}_{p,q}(\overline{X}_\varepsilon,\mu)\to HB^\alpha_{p,q}(\overline{Z},\nu).\] such that \(T_Z\circ E_Z=\mathop{\mathrm{Id}}\). Again using Proposition 1, it follows that \[\label{eq:u40w41} u(w)=\lim_{\rho\to 0^+}\fint_{B(w,\rho)}E_Z u\,d\mu\tag{31}\] for \(\nu\)-a.e.\(w\in B(z,r)\cap \overline{Z}\). As \(\overline{X}_\varepsilon\setminus \overline{Z}\) is an \(A\)-uniform domain, with \(A\) depending only on \(\alpha\), \(p\), and \(C_\nu\), we can join \(z\) and \(w\) by a chain of balls \(\{B_k:=B(x_k,r_k)\}_{k\in\mathbb{Z}}\) in \(\overline{X}_\varepsilon\), as described in Lemma 1. Similar to the proof of Proposition 1, we have by 30 , 31 , and the properties of this chain of balls that \[\begin{align} |u(w)-Tu(z)|^p&\le\left(\sum_{k\ge 0}|(E_Zu)_{B_{k+1}}-(E_Zu)_{B_k}|+\sum_{k<0}|(E_Zu)_{B_{k+1}}-(E_Zu)_{B_k}|\right)^p\\ &\lesssim\left(\sum_{k\ge 0}\fint_{2B_k}\fint_{2B_k}|E_Zu(x)-E_Zu(y)|d\mu(y)d\mu(x)\right)^p\\ &\qquad+\left(\sum_{k< 0}\fint_{2B_k}\fint_{2B_k}|E_Zu(x)-E_Zu(y)|d\mu(y)d\mu(x)\right)^p. \end{align}\]
Set \(\delta:=\theta/p\). As in the proof of Proposition 1 (with \(\delta\) in place of \(\beta\) there), we then have \[\begin{align} \Bigg(\sum_{k\ge 0}\fint_{2B_k}\fint_{2B_k}&|E_Zu(x)-E_Zu(y)|d\mu(y)d\mu(x)\Bigg)^p\\ &\lesssim d(z,w)^{\delta p}\int_{C_{z,w}^1}\int_{B(x,d(x,\overline{Z}))}\frac{|E_Zu(x)-E_Zu(y)|^p}{d(x,\overline{Z})^{\delta p}\mu(B(x,d(x,\overline{Z})))^2}d\mu(y)d\mu(x) \end{align}\] and \[\begin{align} \Bigg(\sum_{k< 0}\fint_{2B_k}\fint_{2B_k}&|E_Zu(x)-E_Zu(y)|d\mu(y)d\mu(x)\Bigg)^p\\ &\lesssim d(z,w)^{\delta p}\int_{C_{z,w}^2}\int_{B(x,d(x,\overline{Z}))}\frac{|E_Zu(x)-E_Zu(y)|^p}{d(x,\overline{Z})^{\delta p}\mu(B(x,d(x,\overline{Z})))^2}d\mu(y)d\mu(x), \end{align}\] where \(C_{z,w}^1:=\bigcup_{k\ge 0}2B_k\) and \(C_{z,w}^2:=\bigcup_{k<0}2B_k\). Therefore, it follows that \[\begin{align} \label{eq:General32Trace32I32and32II} \fint_{B(z,r)}|u&-Tu(z)|^pd\nu\nonumber\\ &\lesssim\fint_{B(z,r)}d(z,w)^{\delta p}\int_{C_{z,w}^1}\int_{B(x,d(x,\overline{Z}))}\frac{|E_Zu(x)-E_Zu(y)|^p}{d(x,\overline{Z})^{\delta p}\mu(B(x,d(x,\overline{Z})))^2}d\mu(y)d\mu(x)d\nu(w)\nonumber\\ &+\fint_{B(z,r)}d(z,w)^{\delta p}\int_{C_{z,w}^2}\int_{B(x,d(x,\overline{Z}))}\frac{|E_Zu(x)-E_Zu(y)|^p}{d(x,\overline{Z})^{\delta p}\mu(B(x,d(x,\overline{Z})))^2}d\mu(y)d\mu(x)d\nu(w)\nonumber\\ &=:I+II. \end{align}\tag{32}\] Using the \(\sigma\)-codimensionality between \(\nu\) and \(\mu\), as well as Tonelli’s theorem, we have \[\begin{align} I&\lesssim\frac{r^\sigma}{\mu(B(z,r))}\int_{B(z,r)}\int_{C_{z,w}^1}\int_{B(x,d(x,\overline{Z}))}\frac{|E_Zu(x)-E_Zu(y)|^pd(z,w)^{\delta p}}{d(x,\overline{Z})^{\delta p}\mu(B(x,d(x,\overline{Z})))^2}d\mu(y)d\mu(x)d\nu(w)\\ &\lesssim r^\sigma\fint_{B(z,Cr)}\int_{B(x,d(x,\overline{Z}))}\frac{|E_Zu(x)-E_Zu(y)|^p}{d(x,\overline{Z})^{\delta p}\mu(B(x,d(x,\overline{Z})))^2}\int_{B(z,r)}d(z,w)^{\delta p}\chi_{C_{z,w}^1}(x)d\nu(w)d\mu(y)d\mu(x). \end{align}\] For each \(i\in\mathbb{Z}\), let \(X_i:=\{x\in\overline{X}_\varepsilon:2^{i-1}\le d(x,\overline{Z})<2^{i}\}\). Recalling that \(0<r<1\), we then have \[\begin{align} I&\lesssim r^{\sigma}\fint_{B(z,Cr)}\sum_{i=-\infty}^{N_C}\chi_{X_i}(x)\fint_{B(x,2^i)}\frac{|E_Zu(x)-E_Zu(y)|^p}{2^{i\delta p}\mu(B(x,2^i))}\int_{B(z,r)}d(z,w)^{\delta p}\chi_{C_{z,w}^1}(x)d\nu(w)d\mu(y)d\mu(x)\\ &\lesssim r^{\sigma+\delta p}\fint_{B(z,Cr)}\sum_{i=-\infty}^{N_C}2^{-i\delta p}\fint_{B(x,2^i)}|E_Zu(x)-E_Zu(y)|^p\frac{\nu(B(x,2^i)\cap \overline{Z})}{\mu(B(x,2^i))}d\mu(y)d\mu(x)\\ &\lesssim r^{\sigma+\delta p}\fint_{B(z,Cr)}\sum_{i=-\infty}^{N_C}2^{-i(\delta p+\sigma)}\fint_{B(x,2^i)}|E_Zu(x)-E_Zu(y)|^pd\mu(y)d\mu(x). \end{align}\] Similarly, we obtain \[\begin{align} II\lesssim r^{\sigma+\delta p}\fint_{B(z,Cr)}\sum_{i=-\infty}^{N_C}2^{-i(\delta p+\sigma)}\fint_{B(x,2^i)}|E_Zu(x)-E_Zu(y)|^pd\mu(y)d\mu(x), \end{align}\] By our choice of \(\delta\) and 32 , we then have \[\begin{align} \label{eq:Haus32null32ineq} \fint_{B(z,r)}|u-Tu(z)|^pd\nu&\lesssim r^{\sigma+\theta}\fint_{B(z,Cr)} \sum_{i=-\infty}^{N_C}2^{-i(\sigma+\theta)}\fint_{B(x,2^i)}|E_Zu(x)-E_Zu(y)|^pd\mu(y)d\mu(x)\nonumber\\ &=:r^{\sigma+\theta}\fint_{B(z,Cr)}g(x)^pd\mu(x), \end{align}\tag{33}\] where we have defined the function \[g(x):=\left(\sum_{i=-\infty}^{N_C}2^{-i(\sigma+\theta)}\fint_{B(x,2^i)}|E_Zu(x)-E_Zu(y)|^pd\mu(y)\right)^{1/p}.\]
We claim that \(g\in L^p(\overline{X}_\varepsilon,\mu)\). To show this, let \(\tau:=\alpha-\theta/p\) and note that \(\tau>0\) by our assumptions. We then have by Hölder’s inequality and the boundedness of the operator \(E_Z\) that \[\begin{align} \|g\|_{L^p(\overline{X}_\varepsilon,\mu)}&=\left(\int_{\overline{X}_\varepsilon}\sum_{i=-\infty}^{N_C}2^{-i(\sigma+\theta)}\fint_{B(x,2^i)}|E_Zu(x)-E_Zu(y)|^pd\mu(y)d\mu(x)\right)^{1/p}\\ &\le\sum_{i=-\infty}^{N_C}2^{-i(\sigma+\theta)/p}\left(\int_{\overline{X}_\varepsilon}\fint_{B(x,2^i)}|E_Zu(x)-E_Zu(y)|^pd\mu(y)d\mu(x)\right)^{1/p}\\ &\le\sum_{i=-\infty}^{N_C}2^{-i((\sigma+\theta)/p+\tau)}2^{i\tau}\left(\int_{\overline{X}_\varepsilon}\fint_{B(x,2^i)}|E_Zu(x)-E_Zu(y)|^pd\mu(y)d\mu(x)\right)^{1/p}\\ &\lesssim\left(\sum_{i=-\infty}^{N_C}2^{-i((\sigma+\theta)/p+\tau)q}\left(\int_{\overline{X}_\varepsilon}\fint_{B(x,2^i)}|E_Zu(x)-E_Zu(y)|^pd\mu(y)d\mu(x)\right)^{q/p}\right)^{1/q}\\ &=\left(\sum_{i=-\infty}^{N_C}2^{-i(\alpha+\sigma/p)q}\left(\int_{\overline{X}_\varepsilon}\fint_{B(x,2^i)}|E_Zu(x)-E_Zu(y)|^pd\mu(y)d\mu(x)\right)^{q/p}\right)^{1/q}\\ &\le\|E_Zu\|_{HB^{\alpha+\sigma/p}_{p,q}(\overline{X}_\varepsilon,\mu)}<\infty. \end{align}\] Though the above estimate is written for \(1\le q<\infty\), it holds with the standard modifications for \(q=\infty\).
Since \(\theta<\alpha p\) and \(\sigma<p(1-\alpha)\), we have that \(\sigma+\theta<p\). Hence, from Lemma 1 and 33 , it follows that \[\lim_{r\to 0^+}\fint_{B(z,r)}|u-Tu(z)|^pd\nu=0\] for \(\mathcal{H}^{-(\sigma+\theta)}_\mu\)-a.e.\(z\in\partial Z\). As \(\pi\) is codimension \((\sigma+\theta)\) Ahlfors regular with respect to \(\mu\), we have by Lemma 1 that \(\pi\simeq\mathcal{H}^{-(\sigma+\theta)}_\mu|_{\partial Z}\), which completes the proof of ?? .
It remains to show ?? . To this end, let \(f\in B^{\alpha-\theta/p}_{p,q}(\partial Z,\pi)\). Then from Corollary 1, it follows that \(\widetilde{E}f\in B^\alpha_{p,q}(\overline{Z},\nu)\), and so from ?? , we have that for \(\pi\)-a.e.\(z\in\partial Z\), \[\lim_{r\to 0^+}\fint_{B(z,r)}|\widetilde{E}f-T\widetilde{E}f(z)|^pd\nu=0.\] Moreover, from Lemma 1, with \(\nu\) and \(\pi\) playing the role of \(\mu\) and \(\nu\) there, respectively, it follows that for \(\pi\)-a.e.\(z\in\partial Z\), \[\lim_{r\to 0^+}\fint_{B(z,r)}|\widetilde{E}f-f(z)|d\nu=0.\] By these limits and the triangle inequality, we have that for \(\pi\)-a.e.\(z\in\partial Z\), \[\begin{align} |f(z)-T\widetilde{E}f(z)|\le\lim_{r\to 0^+}\fint_{B(z,r)}|\widetilde{E}f-T\widetilde{E}f(z)|^pd\nu+\lim_{r\to 0^+}\fint_{B(z,r)}|\widetilde{E}f-f(z)|d\nu=0, \end{align}\] which gives us ?? . ◻
Remark 1. As mentioned in the introduction, it may be possible to remove the boundedness assumption in Theorem 1 by using the hyperbolic filling construction given in [25], [26] instead of the construction from [16]. However, as these papers are as-of-yet unpublished, we leave this investigation for future work.
I.C.: Institute of Mathematics of the Polish Academy of Sciences, Jędrzeja Śniadeckich 8, 00-656 Warsaw, Poland.
E-mail: icaamanoaldemunde@impan.pl J.K.: Dept. of Mathematical Sciences, P.O. Box 210025, University of Cincinnati, Cincinnati, OH 45221-0025, U.S.A.
E-mail: klinejp@ucmail.uc.edu