The Stein-Weiss inequality
in variable exponent Morrey spaces
October 02, 2025
In this paper we prove the Stein-Weiss inequality in variable exponent Morrey spaces over a bounded domain. Our work extends earlier results in the variable exponent Lebesgue and Morrey settings, and utilizes new proof techniques applicable to Morrey spaces. We build on the foundational paper [1], which introduced Morrey spaces of variable exponents. As an application of our main result, we prove Poincaré-type inequalities using the approach of a recent paper [2] by the first and third authors.
In this paper we are interested in the boundedness of fractional integral operators on various function spaces. Given \(0<\gamma<n\), define the fractional integral \(I_\gamma\) to be the convolution operator \[I_\gamma f(x) = \int_{{\mathbb{R}}^n} \frac{f(y)}{|x-y|^{n-\gamma}}\,dy.\] The boundedness of this operator was first considered on the Lebesgue spaces by Hardy and Littlewood [3], and was later extended by other mathematicians, particularly Sobolev [4], to higher dimensions. The following result is often referred to as the Hardy-Littlewood-Sobolev theorem: for \(0<\gamma<n\) and \(1<p<\frac{n}{\gamma}\), define \(q\) by \(\frac{1}{p}-\frac{1}{q}=\frac{\gamma}{n}\). Then \[\|I_\gamma f\|_{L^q({\mathbb{R}}^n)} \leq C\|f\|_{L^p({\mathbb{R}}^n)}.\]
Stein and Weiss [5] later proved a weighted version of this result.
Theorem 1. Fix \(0<\gamma<n\), \(1<p\leq q <\infty\), and constants \(a\) and \(b\) such that \[-\frac{n}{q} < a \leq b < \frac{n}{p'}.\] Suppose further that \[\frac{1}{p} - \frac{1}{q} = \frac{\gamma}{n} + \frac{a-b}{n}.\] Then there exists a constant \(C>0\) such that for every function \(f\), \[\| |\cdot|^a I_\gamma f\|_{L^q({\mathbb{R}}^n)} \leq C\| |\cdot|^b f\|_{L^p({\mathbb{R}}^n)}.\]
Bounds for the fractional integral operator on the classical Morrey spaces have been studied by a number of authors: see [6]–[10]. Recall that if \(1 \le p < \infty\) and \(0\le \lambda \le n\), the Morrey space \({\mathcal{M}}_{p}^{\lambda}(\mathbb{R}^n)\) is the space of all \(f\in L^p_{loc}({\mathbb{R}}^n)\) such that \[\|f\|_{{\mathcal{M}}_{p}^{\lambda}({\mathbb{R}}^n)} := \sup _{x \in {\mathbb{R}}^n, ~ r>0} \left( r^{-\lambda} \int_{B(x,r)} |f(y)|^{p} dy\right)^{\frac{1}{p}}< \infty,\] where \(B(x, r)= \{ y\in {\mathbb{R}}^n : |x-y|<r\}\). We have the following result due to Spanne (see Peetre [8]).
Theorem 2. Fix \(0 < \gamma < n\), \(1 < p < \frac{n}{\gamma}\), and define \(q\) by \(\frac{1}{p} - \frac{1}{q} = \frac{\gamma}{n}\). Suppose further that we have \(0 < \lambda < \mu< n\) with \[\quad \frac{\lambda}{ p} = \frac{\mu}{ q}.\] Then there exists a constant \(C>0\) such that for every function \(f\), \[\| I_{\gamma} f \|_{M^\mu_q(\mathbb{R}^n)} \leq C \| f \|_{M^\lambda_p(\mathbb{R}^n)}.\]
In the range \(1<p<\frac{n-\lambda}{\gamma}\) Adams [6] proved a sharper result.
Theorem 3. Fix \(0 < \gamma < n\), \(0\leq \lambda \leq n\), \(1 < p < \frac{n-\lambda}{\gamma}\), and define \(q\) by \(\frac{1}{p} - \frac{1}{q} = \frac{\gamma}{n-\lambda}\). Then there exists a constant \(C>0\) such that for every function \(f\), \[\| I_{\gamma} f \|_{ M_{q}^{\lambda}(\mathbb{R}^n)} \leq C \| f \|_{M_{p}^{\lambda}(\mathbb{R}^n)}.\]
Since \(\| f \|_{M^\mu_q(\mathbb{R}^n)} \lesssim \| f \|_{M_{q}^{\lambda}(\mathbb{R}^n)}\), Theorem 3 improves Theorem 2 when \(1 < p < \frac{n - \lambda}{\gamma}\).
A version of Theorem 1 in the Morrey spaces was recently proved by Kassymov, Ragusa, Ruzhansky, and the third author [11].
Theorem 4. Fix \(0 < \gamma < n\), and \(a,\,b\) such that \(0 \leq b-a \leq \gamma\), and \(p\) such that \(1<p<\frac{n}{\gamma-(b-a)}\), and define \(q\) by \(\frac{1}{p}-\frac{1}{q} = \frac{\gamma-(b-a)}{n-\lambda}\). Suppose further that \[- \frac{n-\lambda}{q} < a \leq b < \frac{n}{p'}, \quad 0<\lambda < n- (\gamma-(b-a))p.\] Then there exists a constant \(C>0\) such that for every function \(f\), \[\| |\cdot|^a I_\gamma f \|_{M^\lambda_q({\mathbb{R}}^n)} \leq C\||\cdot|^b f \|_{M^\lambda_p({\mathbb{R}}^n)}.\]
The boundedness of fractional integrals has also been considered on the variable Lebesgue spaces \(L^{p(\cdot)}({\mathbb{R}}^n)\). These are a generalization of the classical \(L^p\) spaces, replacing the constant exponent \(p\) by an exponent function \({p(\cdot)}: {\mathbb{R}}^n \rightarrow [1,\infty]\). For brevity, we defer precise definitions to Section 2 below. It was shown in [12] that if \(1<p_-\leq p_+<n/\gamma\), and if \({p(\cdot)}\) is log-Hölder continuous, then for \({q(\cdot)}\) defined by \[\frac{1}{p(x)} - \frac{1}{q(x)} = \frac{\gamma}{n},\]
the Hardy-Sobolev inequality holds with variable exponents: \[\|I_\gamma f\|_{L^{q(\cdot)}({\mathbb{R}}^n)} \leq C\|f\|_{L^{p(\cdot)}({\mathbb{R}}^n)}.\] For earlier results see [12], [13] and the references they contain.
Samko [14] proved a version of the Stein-Weiss inequality in the variable Lebesgue spaces on a bounded domain. Here we state a somewhat simpler version of his result.
Theorem 5. Given \(0<\gamma<n\), and a bounded open set \(\Omega \subset \mathbb{R}^n\), fix \(x_0 \in \overline{\Omega}\). Suppose that \({p(\cdot)}\in \mathcal{P}(\Omega)\) is such that \(1<p_-\leq p_+< \frac{n}{\gamma}\) and \({p(\cdot)}\in LH_0(\Omega)\). Define \({q(\cdot)}\) by \(\frac{1}{q(x)} = \frac{1}{p(x)} - \frac{\gamma}{n}\). Fix constants \(\mu\) and \(\nu\) such that \[\gamma p(x_0) - n < \nu < n \big(p(x_0) - 1\big), \quad \mu = \frac{q(x_0)}{p(x_0)} \nu.\] Then there exists a constant \(C>0\) such that for every function \(f\), \[\| |\cdot-x_0|^{\frac{\mu}{{q(\cdot)}}}I_{\gamma} f \|_{L^{q(\cdot)}(\Omega)} \le C \||x - x_0|^{\frac{\nu}{{p(\cdot)}}} f \|_{L^{{p(\cdot)}}(\Omega)}.\]
More recently, the first and third authors [2], building on the work of Melchiori and Pradolini [15], proved a version of the Stein-Weiss inequality in variable Lebesgue spaces.
Theorem 6. Fix \(0<\gamma<n\). Given \({p(\cdot)},\,{q(\cdot)}\in \mathcal{P}({\mathbb{R}}^n)\), suppose that \({p(\cdot)},\,{q(\cdot)}\in LH({\mathbb{R}}^n)\), \(1<p_-\leq p_+<\infty\), \(1<q_-\leq q_+<\infty\), and \(p(x)\leq q(x)\) for all \(x\in {\mathbb{R}}^n\). Given constants \(a,\, b\) that satisfy \[\label{eqn:vsw1} -\frac{n}{q_+} < a\leq b < \frac{n}{(p_-)'},\tag{1}\] suppose further that for all \(x\in {\mathbb{R}}^n\), \[\label{eqn:vsw2} \frac{1}{p(x)}-\frac{1}{q(x)} = \frac{\gamma}{n} + \frac{a-b}{n}.\tag{2}\] Then for all \(f\) there exists a constant \(C>0\) such that \[\||\cdot|^a I_\gamma f\|_{{q(\cdot)}} \leq C \||\cdot|^b f\|_{{p(\cdot)}}.\]
In this paper we generalize Theorems 4, 5, and 6 by proving a version of the Stein-Weiss inequality in variable exponent Morrey spaces defined on a bounded set \(\Omega\). The variable exponent Morrey spaces were introduced by Almeida, Hasanov, and Samko [1]; they proved bounds for the Hardy-Littlewood maximal operator, fractional maximal operators, and fractional integrals on bounded subsets of \({\mathbb{R}}^n\). Additional results on bounded domains can be found in [16]–[22]. Our main result is the following.
Theorem 7. Fix \(0 < \gamma < n\). Let \(\Omega\subset {\mathbb{R}}^n\) be a bounded domain and fix \(x_0 \in \Omega\). Given \({p(\cdot)}, q(\cdot) \in \mathcal{P}(\Omega)\), suppose that \({p(\cdot)}, q(\cdot) \in LH_0(\Omega)\), \(1 < p_{-} \le p_{+} < \infty\), \(1 < q_{-} \le q_{+} < \infty\), and \(p(x) \le q(x)\) for all \(x \in \Omega\). Let \({\lambda(\cdot)}: \Omega \to [0,n]\) be such that \(0 < \lambda_- \leq \lambda_+ <n\) and \({\lambda(\cdot)}\in LH_0(\Omega)\). Let \(a,\,b\) be constants such that \[\begin{gather} 0 \leq b -a \le \gamma, \tag{3} \\ \frac{\lambda_{+}-n}{q_+} < a \le b < \frac{n}{ (p')_{+}}, \tag{4} \\ 0 < \lambda_{-} \leq \lambda(x) < n - (\gamma - b + a)p(x), \tag{5} \end{gather}\] and \[\label{eqn:condD} \frac{\gamma + a- b}{n - \lambda(x)}= \frac{1}{p(x)} - \frac{1}{q(x)}.\tag{6}\] Then, there exists a constant \(C>0\) such that for every function \(f\), \[\label{SWI} \| |x-x_0|^{a} I_{\gamma} f \|_{M_{q(\cdot)}^{{\lambda(\cdot)}}(\Omega)} \leq C \| |x-x_0|^{b} f \|_{M_{{p(\cdot)}}^{{\lambda(\cdot)}}(\Omega)}.\tag{7}\]
Remark 8. Our proof, while based on that of Theorem 4, does not seem to extend to unbounded domains. It is an open problem to prove a Stein-Weiss type inequality on variable exponent Morrey spaces defined on all of \({\mathbb{R}}^n\). For unweighted bounds for fractional integrals and fractional maximal operators in this setting, see Ho [23].
The remainder of the paper is organized as follows. In Section 2 we gather some basic results about variable Lebesgue spaces, variable exponent Morrey spaces, fractional integral operators and the associated fractional maximal operators. In Section 3 we prove Theorem 7. Finally, in Section 4 we give an application of our main result and use it to prove Sobolev and Poincaré inequalities in the variable exponent Morrey spaces.
In this section we establish some basic notation and preliminary results. Throughout, \(n\) will denote the dimension of the underlying space \({\mathbb{R}}^n\), and \(\Omega \subseteq {\mathbb{R}}^n\) will be a bounded, open set. For \(x\in \Omega\) and \(r>0\), we define \(B(x,r)=\{ y \in {\mathbb{R}}^n : |x-y|<r\}\) and \(\tilde{B}(x,r) = B(x,r) \cap \Omega\).
By \(C\) we will mean a constant whose values depend on the underlying parameters and whose value may change from line to line. If we write \(A\lesssim B\), we mean that there exists a constant \(c\) such that \(A\leq cB\).
Here we define the variable Lebesgue spaces and give some basic properties. For more detailed information and proofs of these properties, we refer the reader to [13].
A variable exponent is a measurable function \({p(\cdot)}: \Omega \rightarrow [1,\infty]\). We denote the set of exponent functions by \(\mathcal{P}(\Omega)\). Let \[p^{+}= \mathop{\mathrm{ess \, sup}}_{x \in \Omega} p(x) \quad \text{and} \quad p^{-}= \mathop{\mathrm{ess \, inf}}_{x \in \Omega} p(x).\]
A function \(r(\cdot) : \Omega \to {\mathbb{R}}\) is said to be locally Log-Hölder continuous, denoted by \(r(\cdot) \in LH_0(\Omega)\), if there exists a constant \(C_0\) such that \[|r(x) - r(y)| \leq \frac{C_0}{-\log(|x - y|)}\] for all \(x, y \in \Omega\) with \(|x - y| < \frac{1}{2}\).
Given an exponent \({p(\cdot)}\in \mathcal{P}(\Omega)\), define the associated modular by \[\rho_{p(\cdot)}(f) = \int_{\Omega\setminus \Omega_\infty} |f(x)|^{p(x)}\,dx + \|f\|_{L^\infty(\Omega_\infty)},\] where \(\Omega_\infty=\{ x\in \Omega : p(x)=\infty\}\).
Remark 9. When \(p_+<\infty\), the set \(\Omega_\infty\) is empty and the modular reduces to the first integral.
Define \(L^{p(\cdot)}(\Omega)\) to be the collection of measurable functions with domain \(\Omega\) such that \[\|f\|_{p(\cdot)}= \|f\|_{L^{p(\cdot)}(\Omega)} = \inf\{ \lambda >0 : \rho(f/\lambda) \leq 1 \}.\] Then \(\|\cdot\|_{p(\cdot)}\) is a norm and \(L^{p(\cdot)}(\Omega)\) is a Banach function space.
Given \({p(\cdot)}\in \mathcal{P}(\Omega)\), define the conjugate exponent \(p'(\cdot)\)pointwise by \[\frac{1}{p(x)}+\frac{1}{p'(x)} = 1,\] with the conventions that \(\frac{1}{\infty}=0\). Hölder’s inequality extends to the variable Lebesgue spaces: \[\label{Holder} \int_\Omega |f(x) g(x)| \, dx \leq 4 \|f\|_{{p(\cdot)}} \|g\|_{p'(\cdot)}.\tag{8}\] The following generalized Hölder’s inequality [13] also holds: if \({p(\cdot)}, q(\cdot), r(\cdot) \in \mathcal{P} (\Omega)\) satisfy \[\frac{1}{r(x)} = \frac{1}{p(x)} + \frac{1}{q(x)}, \quad \text{for all } x \in \Omega,\] then there exists a constant \(K=K({p(\cdot)},{q(\cdot)})\geq 1\) such that for any \(f \in L^{{p(\cdot)}}(\Omega)\) and \(g \in L^{q(\cdot)}(\Omega)\), \[\label{Gholder} \|fg\|_{L^{r(\cdot)}(\Omega)} \leq K \|f\|_{L^{{p(\cdot)}}(\Omega)} \|g\|_{L^{q(\cdot)}(\Omega)}.\tag{9}\]
The following lemmas will be used repeatedly in our proofs. Most of them rely heavily on the fact that \(\Omega\) is bounded. The first is from [1].
Lemma 1. Given a bounded open set \(\Omega\subset {\mathbb{R}}^n\), suppose \({\lambda(\cdot)}\in LH_0(\Omega)\). Then there exists a constant \(C=C(\lambda,\Omega)\) such that for any \(r>0\) and all \(x, y \in \Omega\) such that \(|x - y| \leq r\), \[C^{-1} r^{-\lambda(y)} \leq r^{-\lambda(x)} \leq C r^{-\lambda(y)}.\]
The second is from [1].
Lemma 2. Given a bounded open set \(\Omega\subset {\mathbb{R}}^n\), suppose \({p(\cdot)}\in \mathcal{P}(\Omega)\) satisfies \({p(\cdot)}\in LH_0(\Omega)\). Then there exists a constant \(C=C({p(\cdot)},\Omega)\) such that for all \(x\in \Omega\) and \(r>0\), \[\|\chi_{B(x, r)}\|_{{{p(\cdot)}}} \leq C |B(x, r)|^{\frac{1}{p(x)}} .\]
The third is a consequence of well-known properties of the variable exponent norm.
Lemma 3. Given a bounded open set \(\Omega \subset {\mathbb{R}}^n\), and \(s(\cdot) \in \mathcal{P}(\Omega)\), \(\|\chi_\Omega\|_{L^{s(\cdot)}(\Omega)} \leq 1+|\Omega|\).
Proof. If \(\|\chi_\Omega\|_{L^{s(\cdot)}(\Omega)} \leq 1\), then we are done, so assume the opposite inequality holds. Then by [13], \[\|\chi_\Omega\|_{L^{s(\cdot)}(\Omega)} \leq \rho_{s(\cdot)}(\chi_\Omega) = \int_{\Omega\setminus\Omega_\infty} \,dx + \|\chi_\Omega\|_{L^\infty(\Omega_\infty)} \leq |\Omega|+1.\] ◻
Finally, we will need the following integral estimate.
Lemma 4. Let \({\lambda(\cdot)}:\Omega \to [0,n]\) be such that \(0<\lambda_-\leq \lambda_+<n\). Then for every \(x_0 \in \Omega\) and \(r>0\). \[\label{Bound} \int_{\tilde{B}(x_0,r)} |x-x_0|^{\lambda(x) - n} dx \le C r^{\lambda(x_0)}.\tag{10}\]
Proof. We prove this by considering two cases. First, suppose that \(r\le 1\). Then for all \(x\in \tilde{B}(x_0,r)\), \(|x-x_0|\le 1\), and so (integrating in polar coordinates), \[\int_{\tilde{B}(x_0,r)} |x-x_0|^{\lambda(x) - n} dx \le \int_{B(x_0,r)} |x-x_0|^{\lambda_{+} - n} dx = C(n,\lambda_+) r^{\lambda_{+}} \leq C(n,\lambda_+) r^{\lambda(x_0)}.\]
Now suppose \(r>1\). We decompose \[\tilde{B}(x_0,r) = \tilde{B}(x_0,r)\cap B(x_0,1) \cup \tilde{B}(x_0,r)\setminus B(x_0,1),\] and evaluate the corresponding parts of the integral separately. For the first set we have that \[\begin{gather} \int_{ \tilde{B}(x_0,r)\cap B(x_0,1)} |x-x_0|^{\lambda(x) - n} dx \le \int_{\tilde{B}(x_0,r)\cap B(x_0,1)} |x-x_0|^{\lambda_{+} - n} dx \\ \le \int_{B(x_0,1) } |x-x_0|^{\lambda_{+} - n} dx = C(n, \lambda_+) \le C(n, \lambda_+) r^{\lambda(x_0)}. \end{gather}\] For the second set, we have that \[\int_{\tilde{B}(x_0,r)\setminus B(x_0,1)} |x-x_0|^{\lambda(x) - n} dx \le \int_{B(x_0,r)} |x-x_0|^{\lambda_{-} - n} dx = C(n,\lambda_-) r^{\lambda_{-}} \le C(n,\lambda_-)r^{\lambda(x_0)}.\] If we combine these two estimates, we see that 10 holds. ◻
Our definition of the variable exponent Morrey spaces is taken from [1] and we refer the reader there for more information. Given \({\lambda(\cdot)}: \Omega \to [0,n]\) measurable and \({p(\cdot)}\in \mathcal{P}(\Omega)\), define the modular \[I_{{p(\cdot)}, {\lambda(\cdot)}}(f) := \sup_{x \in \Omega, \, r > 0} r^{-\lambda(x)} \int_{\tilde{B}(x, r)} |f(y)|^{p(y)} \, dy.\] Then the variable exponent Morrey space \(M_{{p(\cdot)}}^{{\lambda(\cdot)}}(\Omega)\) is defined as the set of all measurable functions \(f\) on \(\Omega\) such that \[\|f\|_{M_{{p(\cdot)}}^{{\lambda(\cdot)}}(\Omega)} = \inf \left\{ \eta > 0 : I_{{p(\cdot)}, {\lambda(\cdot)}}\left( \frac{f}{\eta} \right) \le 1 \right\} < \infty.\]
To prove norm inequalities in the variable exponent Morrey spaces, we need the following lemma. This property is well known in the variable Lebesgue spaces (cf. [13]).
Lemma 5. Given a function \(g\), \(\|g\|_{M^{{\lambda(\cdot)}}_{p(\cdot)}(\Omega)} \leq C_1\) if and only if \(I_{{p(\cdot)},{\lambda(\cdot)}}(g) \leq C_2\) for some constants \(C_1,\,C_2>0\).
Proof. First, suppose that \(I_{{p(\cdot)},{\lambda(\cdot)}}(g) \leq C_2\). We may assume \(C_2\geq 1\), (for \(0 < C_{2} < 1\), one can choose \(C_{1} = 1\)). Then we have that \[1 \geq C_2^{-1} \sup_{x \in \Omega, \, r > 0} r^{-\lambda(x)} \int_{\tilde{B}(x, r)} |g(y)|^{p(y)} \, dy \geq \sup_{x \in \Omega, \, r > 0} r^{-\lambda(x)} \int_{\tilde{B}(x, r)} |C_2^{-\frac{1}{p_-}}g(y)|^{p(y)} \, dy.\] Thus, by the definition of the variable exponent Morrey space norm, \(\|C_2^{-\frac{1}{p_-}} g\|_{M^{\lambda(\cdot)}_{p(\cdot)}(\Omega)} \leq 1\), which gives us the desired estimate with \(C_1=C_2^{\frac{1}{p_-}}\).
Now suppose that \(\|g\|_{M^{{\lambda(\cdot)}}_{p(\cdot)}(\Omega)} \leq C_1\). Without loss of generality, we may assume \(C_1\geq 1\). Then, again by the definition, we have that \[1 \geq \sup_{x \in \Omega, \, r > 0} r^{-\lambda(x)} \int_{\tilde{B}(x, r)} |C_1^{-1}g(y)|^{p(y)} \, dy \geq C_1^{- p_+} \sup_{x \in \Omega, \, r > 0} r^{-\lambda(x)} \int_{\tilde{B}(x, r)} |g(y)|^{p(y)} \, dy.\] This gives us the desired inequality with \(C_2=C_1^{p_+}\). ◻
We will need to show that with our hypotheses \(\frac{{\lambda(\cdot)}}{{p(\cdot)}}\) is log-Hölder continuous.
Lemma 6. Given \({p(\cdot)}\in \mathcal{P}(\Omega)\) such that \(p_+<\infty\) and \({p(\cdot)}\in LH_0(\Omega)\), suppose that \({\lambda(\cdot)}: \Omega \to [0,n]\) satisfies \({\lambda(\cdot)}\in LH_0(\Omega)\). Then \(\frac{{\lambda(\cdot)}}{{p(\cdot)}}\in LH_0(\Omega)\).
Proof. Fix \(x,\,y \in \Omega\) with \(|x-y|<\frac{1}{2}\). Then \[\left|\frac{\lambda(x)}{p(x)}-\frac{\lambda(y)}{p(y)}\right| \leq \left|\frac{\lambda(x)}{p(x)}-\frac{\lambda(y)}{p(x)}\right| + \left|\frac{\lambda(y)}{p(x)}-\frac{\lambda(y)}{p(y)}\right| \leq \frac{1}{p_-}|\lambda(x)-\lambda(y)| + \frac{\lambda_+}{p_-^2}|p(x)-p(y)|.\] It follows immediately that \(\frac{{\lambda(\cdot)}}{{p(\cdot)}}\in LH_0(\Omega)\). ◻
We will also need an equivalent form for the norm, which was essentially shown in [1].
Lemma 7. If \({\lambda(\cdot)}, \, {p(\cdot)}\in LH_0(\Omega)\), and \(\lambda_{+}<\infty\), \(p_+<\infty\), then \[\|f\|_{M_{{p(\cdot)}}^{{\lambda(\cdot)}}(\Omega)} \approx \sup_{x \in \Omega, \, r > 0} r^{-\frac{\lambda(x)}{p(x)}} \left\| f \, \chi_{\tilde{B}(x, r)} \right\|_{{p(\cdot)}}.\]
Proof. By [1], we have that \[\|f\|_{M_{{p(\cdot)}}^{{\lambda(\cdot)}}(\Omega)} \approx \sup_{x \in \Omega, \, r > 0} \left\| r^{-\frac{{\lambda(\cdot)}}{{p(\cdot)}}} f \, \chi_{\tilde{B}(x, r)} \right\|_{{p(\cdot)}}.\] By Lemma 6, \(\frac{{\lambda(\cdot)}}{{p(\cdot)}}\in LH_0(\Omega)\), and so by Lemma 1, we have that for all \(y\in \tilde{B}(x,r)\), \[r^{-\frac{\lambda(y)}{p(y)}} \approx r^{-\frac{\lambda(x)}{p(x)}}.\] The desired equivalence follows at once. ◻
Finally, we need bounds for maximal operators on variable exponent Morrey spaces. For \(0\leq \sigma <n\), define \[M_\sigma f(x) = \sup_{\substack{z\in \Omega \\r > 0}} \frac{1}{|B(z, r)|^{1-\frac{\sigma}{n}}} \int_{B(z, r)} |f(y)| \, dy \cdot \chi_{B(z,r)}(x).\] When \(\sigma=0\) this is the Hardy-Littlewood maximal operator, and when \(\sigma>0\) the fractional maximal operator. The following result was proved in [1].
Theorem 10. Given a bounded, open set \(\Omega\subset \mathbb{R}^n\), let \({p(\cdot)}\in \mathcal{P}(\Omega)\) be such that \(1 < p_{-} \leq p_{+} < \infty\) and \({p(\cdot)}\in LH_0(\Omega)\). Let \({\lambda(\cdot)}\) be a measurable function on \(\Omega\) such that \(0 \leq \lambda_- \leq \lambda_{+} < n\) and \[\sup_{x \in \Omega} [\lambda(x) + \sigma p(x)] < n.\] Fix \(\sigma\), \(0\leq \sigma < n\), and define \({q(\cdot)}\in \mathcal{P}(\Omega)\) by \[\frac{1}{q(x)} = \frac{1}{p(x)} - \frac{\sigma}{n - \lambda(x)}.\] Then the maximal operator \(M_\sigma\) is bounded from \(M^{\lambda(\cdot)}_{p(\cdot)}(\Omega)\) to \(M^{\lambda(\cdot)}_{q(\cdot)}(\Omega)\). The same is also true for the fractional integral operator \(I_\sigma\).
To prove this result, we make some reductions. Fix \(x_0 \in \Omega\). First, since the \(M_{{p(\cdot)}}^{{\lambda(\cdot)}}(\Omega)\) norm is homogeneous, it will suffice to prove this result for \(f\) such that \(\||\cdot-x_0|^b f\|_{M_{{p(\cdot)}}^{{\lambda(\cdot)}}(\Omega)}=1\), and so we need to show that for some \(C\geq 1\), \(\||\cdot-x_0|^a I_\gamma f\|_{M_{{q(\cdot)}}^{{\lambda(\cdot)}}(\Omega)}\leq C\). (If \(C<1\), there is nothing to prove.) Second, since \(I_\gamma\) is a positive operator, we may also assume that \(f\) is nonnegative.
Third, for each \(x\in \Omega\) we decompose \(I_\gamma f(x)\) as follows: \[\begin{align} I_\gamma f(x) &= \int_{\Omega} \frac{f(y)}{|x - y|^{n - \gamma}} \, dy \\ &= \int_{\tilde{B}(x_0,\frac{|x-x_0|}{2})} \frac{f(y)}{|x - y|^{n - \gamma}} \, dy \\ & \qquad + \int_{\tilde{B}(x_0, 2|x-x_0|)\setminus \tilde{B}(x_0,\frac{|x-x_0|}{2})} \frac{f(y)}{|x - y|^{n - \gamma}} \, dy \\ & \qquad + \int_{\Omega \setminus \tilde{B}(x_0,2|x-x_0|)} \frac{f(y)}{|x - y|^{n - \gamma}} \, dy \\ &= J_1(x) + J_2(x) + J_3(x). \end{align}\] Thus, it will suffice to show that for \(1\leq i \leq 3\), \(\||\cdot-x_0|^a J_i\|_{M_{{q(\cdot)}}^{{\lambda(\cdot)}}(\Omega)}\leq C\).
Finally, by Lemma 5 it will suffice to prove that there exists \(C\geq 1\) such that for every \(z\in \Omega\) and every \(r>0\), \[\label{eqn:Ji-estimate} r^{-\lambda(z)}\int_{\tilde{B}(z,r)} \big[|x-x_0|^a J_i(x)\big]^{q(x)}\,dx \leq C.\tag{11}\]
We estimate each of these in turn. We first consider the \(J_1\) term. In \(\tilde{B}(x_0, \frac{|x-x_0|}{2})\) we have that \(|y-x_0| < \frac{|x-x_0|}{2}\), which in turn implies that \(|x - y| \geq |x-x_0| - |y-x_0| > \frac{|x-x_0|}{2}\). Hence, \[\begin{align} J_1(x) &\leq \int_{\tilde{B}(x_0,\frac{|x-x_0|}{2})} \frac{f(y)}{(\frac{|x-x_0|}{2})^{n - \gamma}} \, dy \\ &= \frac{2^{n-\gamma}}{|x-x_0|^{n - \gamma}} \int_{\tilde{B}(x_0,\frac{|x-x_0|}{2})} f(y) \, dy \\ & \lesssim \sum_{k=1}^{\infty} \int_{\tilde{B}(x_0,2^{-k}|x-x_0|)\setminus \tilde{B}(x_0, 2^{-k-1}|x-x_0|)} f(y) \, dy \\ & \lesssim \sum_{k=1}^{\infty} (2^{-k}|x-x_0|)^{-b} \int_{\tilde{B}(x_0,2^{-k}|x-x_0|)} |y-x_0|^{b} |f(y)| \, dy. \intertext{By H\"older’s inequality \eqref{Holder} and Lemma \ref{Lemma326}, } &\lesssim \sum_{k=1}^{\infty} (2^{-k}|x-x_0|)^{-b} \| \chi_{\tilde{B}(x_0,2^{-k}|x-x_0|)} |\cdot-x_0|^{b } f \|_{L^{p(\cdot)}(\Omega)} \| \chi_{B(x_0,2^{-k}|x-x_0|)} \|_{L^{p'(\cdot)}(\Omega)} \\ &\lesssim \sum_{k=1}^{\infty} (2^{-k}|x-x_0|)^{-b+\frac{n}{p'(x_0)}} \| \chi_{\tilde{B}(x_0,2^{-k}|x-x_0|)} |\cdot-x_0|^{b } f \|_{L^{p(\cdot)}(\Omega)}. \\ \intertext{If we now apply the equivalent norm from Lemma~\ref{lemma:equiv-norm} and our norm assumption on f we get} &= \sum_{k=1}^{\infty} (2^{-k}|x-x_0|)^{-b+\frac{n}{p'(x_0)}+\frac{\lambda(x_0)}{p(x_0)}} \\ & \qquad \qquad \times (2^{-k}|x-x_0|)^{\frac{-\lambda(x_0)}{p(x_0)}} \| \chi_{\tilde{B}(x_0,2^{-k}|x-x_0|)} |\cdot-x_0|^{b } f \|_{L^{p(\cdot)}(\Omega)} \\ &\lesssim \sum_{k=1}^{\infty} (2^{-k}|x-x_0|)^{-b+\frac{n}{p'(x_0)}+\frac{\lambda(x_0)}{p(x_0)}} \||\cdot-x_0|^{b }f \|_{M_{{p(\cdot)}}^{{\lambda(\cdot)}}(\Omega)} \\ & = \sum_{k=1}^{\infty} (2^{-k}|x-x_0|)^{-b+\frac{n}{p'(x_0)}+\frac{\lambda(x_0)}{p(x_0)}}; \\ \intertext{by the second inequality in \eqref{eqn:condB}, -b+\frac{n}{p'(x_0)}\geq -b+\frac{n}{(p')_+}>0, so the series converges and we get} & \lesssim |x-x_0|^{-b+\frac{n}{p'(x_0)}+\frac{\lambda(x_0)}{p(x_0)}} \\ & \lesssim |x-x_0|^{-b + n -\frac{n-\lambda(x)}{p(x)}}. \end{align}\] The last inequality follows from Lemma 1, with \(r=|x-x_0|\) and Lemma 6.
We can now estimate 11 for \(J_1\). Fix \(z\in \Omega\) and \(r>0\). Then by the above estimate, the identity 6 , and Lemma 4, we get \[\begin{align} r^{-\lambda(z)}\int_{\tilde{B}(z,r)} \big[|x-x_0|^a J_1(x)\big]^{q(x)}\,dx & \lesssim r^{-\lambda(z)}\int_{\tilde{B}(z,r)} \big[|x-x_0|^a |x-x_0|^{-b + n -\frac{n-\lambda(x)}{p(x)}}\big]^{q(x)}\,dx \\ & \leq r^{-\lambda(z)}\int_{B(z,r)} |x-x_0|^{\lambda(x)-n}\,dx \\ & \lesssim 1. \end{align}\] Since the implicit constant is independent of \(z\) and \(r\), we have proved 11 for \(J_1\).
Next we will consider 11 for \(J_3\), \[J_3(x) = \int_{\Omega \setminus\tilde{B}(x_0,2|x-x_0|)} \frac{|f(y)|}{|y - x|^{n-\gamma}} \, dy.\] For all \(y\) such that \(2|x-x_0| \leq |y-x_0|\) we have that \[|y-x_0| = |y -x_0- x + x| \leq |y - x| + |x-x_0| \leq |y - x| + \frac{|y-x_0|}{2};\] hence, \[\label{eqn:J3-est} \frac{|y-x_0|}{2} \leq |y - x|.\tag{12}\]
Moreover, by the first inequality in 4 we have that \[\frac{n-\lambda(x)}{q(x)} \geq \frac{n-\lambda(x)}{q_+} \geq \frac{n-\lambda_+}{q_+} > - a.\] Hence, \[\frac{1}{q(x)} = \frac{1}{p(x)} - \frac{\gamma - b + a}{n - \lambda(x)} < \frac{1}{p(x)} + \frac{1}{q(x)} - \frac{\gamma - b}{n - \lambda(x)},\] which in turn implies \[\label{lazim} \frac{n - \lambda(x)}{p(x)} - \gamma + b > 0.\tag{13}\]
We can now argue much as we did in our estimate for \(J_1\): by inequality 12 \[\begin{align} J_3(x) & \lesssim \int_{\Omega \setminus \tilde{B}(x_0,2|x-x_0|)} \frac{|f(y)|}{|y - x_0|^{n-\gamma}} \, dy \\ & = \sum_{k=1}^\infty \int_{\tilde{B}(x_0,2^{k+1}|x-x_0|)\setminus \tilde{B}(x_0,2^k|x-x_0|)} \frac{|y-x_0|^b|f(y)|}{|y - x_0|^{n-\gamma+b}} \, dy \\ & \leq \sum_{k=1}^\infty (2^k|x-x_0|)^{\gamma-n-b} \int_{\tilde{B}(x_0,2^{k+1}|x-x_0|)} |y-x_0|^b|f(y)| \, dy. \\ \intertext{By H\"older's inequality \eqref{Holder} and Lemma~\ref{Lemma326}, we get } & \leq \sum_{k=1}^\infty (2^k|x-x_0|)^{\gamma-n-b} \|\chi_{\tilde{B}(x_0,2^{k+1}|x-x_0|)}\|_{L^{p'(\cdot)}(\Omega)} \||\cdot-x_0|^b f\chi_{\tilde{B}(x_0,2^{k+1}|x-x_0|)}\|_{L^{p(\cdot)}(\Omega)} \\ & \lesssim \sum_{k=1}^\infty (2^k|x-x_0|)^{\gamma-n-b+\frac{n}{p'(x_0)}+\frac{\lambda(x_0)}{p(x_0)}} \\ & \qquad \qquad \times (2^{k+1}|x-x_0|)^{-\frac{\lambda(x_0)}{p(x_0)}} \||\cdot-x_0|^b f\chi_{\tilde{B}(x_0,2^{k+1}|x-x_0|)}\|_{L^{p(\cdot)}(\Omega)}\\ & = \sum_{k=1}^\infty (2^k|x-x_0|)^{\gamma-b-\frac{n-\lambda(x_0)}{p(x_0)}} (2^{k+1}|x-x_0|)^{-\frac{\lambda(x_0)}{p(x_0)}} \||\cdot-x_0|^b f\chi_{\tilde{B}(x_0,2^{k+1}|x-x_0|)}\|_{L^{p(\cdot)}(\Omega)}. \\ \intertext{By Lemma~\ref{lemma:l47p-LH0}, \frac{{\lambda(\cdot)}}{{p(\cdot)}}\in LH_0(\Omega); hence, by Lemmas~\ref{Lemma324} and~\ref{lemma:equiv-norm},} & \lesssim \sum_{k=1}^\infty (2^k|x-x_0|)^{\gamma-b-\frac{n-\lambda(x)}{p(x)}} \||\cdot-x_0|^b f\|_{M^{\lambda(\cdot)}_{p(\cdot)}(\Omega)}. \end{align}\] By inequality 13 , the series converges, and so by our assumption on the norm of \(f\) we have shown that \[J_3(x) \lesssim (|x-x_0|)^{\gamma-b-\frac{n-\lambda(x)}{p(x)}}.\]
We can now estimate 11 for \(J_3\). Fix \(z\in \Omega\) and \(r>0\); then, arguing exactly as we did for \(J_1\), we have that \[\begin{gather} r^{-\lambda(z)}\int_{\tilde{B}(z,r)} \big[|x-x_0|^a J_3(x)\big]^{q(x)}\,dx \\ \lesssim r^{-\lambda(z)}\int_{\tilde{B}(z,r)} \big[|x-x_0|^a |x-x_0|^{-b + n -\frac{n-\lambda(x)}{p(x)}}\big]^{q(x)}\,dx \lesssim 1. \end{gather}\]
Finally, we consider 11 for \(J_2\), \[J_2(x) = \int_{\tilde{B}(x_0, 2|x-x_0|)\setminus \tilde{B}(x_0,\frac{|x-x_0|}{2})} \frac{f(y)}{|x - y|^{n - \gamma}} \, dy.\] Temporarily, for brevity let \(A=\tilde{B}(x_0, 2|x-x_0|)\setminus \tilde{B}(x_0,\frac{|x-x_0|}{2})\). We will consider two cases. Suppose first that \(b>a\). If \(y\in A\), then \[|x-y| \leq |x-x_0| + |x_0-y| \leq 3|x-x_0|.\] Therefore, we can cover \(A\) by dyadic annuli. More precisely, \[A = \bigcup_{k=0}^\infty D_k,\] where \(D_k = \{ y \in A : 3\cdot2^{-k-1} |x-x_0| \leq |x-y| < 3 \cdot 2^{-k}|x-x_0|\}\). By 3 , \(\gamma \geq b-a>0\), so we can define \(\sigma\geq 0\) by \(\gamma=\sigma+b-a\). We can now estimate as follows: \[\begin{align} J_2(x) & \le \sum_{k=0}^\infty \int_{D_k} \frac{f(y)}{|x - y|^{n - \gamma}} \, dy \\ & \lesssim \sum_{k=0}^\infty \int_{D_k} (2^{-k}|x-x_0|)^{\gamma-n} f(y)\, dy \\ & = \sum_{k=0}^\infty 2^{-k(b-a)} |x-x_0|^{-a} (2^{-k}|x-x_0|)^{\sigma-n} \int_{D_k} |x-x_0|^{b} f(y)\, dy \\ & \lesssim \sum_{k=0}^\infty 2^{-k(b-a)} |x-x_0|^{-a} (2^{-k}|x-x_0|)^{\sigma-n} \int_{D_k} |y-x_0|^{b} f(y)\, dy. \\ \intertext{Since D_k \subset \tilde{B}(x,3\cdot 2^{-k}|x-x_0|), by the definition of the fractional maximal operator (or the maximal operator if \sigma=0), } & \lesssim \sum_{k=0}^\infty 2^{-k(b-a)} |x-x_0|^{-a} M_\sigma (|\cdot-x_0|^b f)(x) \\ & \lesssim |x-x_0|^{-a} M_\sigma (|\cdot-x_0|^b f)(x). \end{align}\] The last inequality holds since \(b-a>0\) and so the series converges.
We can now estimate 11 for \(J_2\): for any \(z\in \Omega\) and \(r>0\), \[\begin{gather} r^{-\lambda(z)} \int_{\tilde{B}(z,r)} [|x-x_0|^a J_2(x)]^{q(x)} \,dx \\ \lesssim r^{-\lambda(z)} \int_{\tilde{B}(z,r)} M_\sigma(|\cdot-x_0|^b f)(x)^{q(x)} \,dx \leq I_{{p(\cdot)},{\lambda(\cdot)}}(M_\sigma(|\cdot-x_0|^b f). \end{gather}\] By our normalization assumption on \(f\), \(\||\cdot-x_0|^b f\|_{M^{\lambda(\cdot)}_{p(\cdot)}(\Omega)}=1\), and so by Theorem 10, \(\|M_\sigma(|\cdot-x_0|^b f)\|_{M^{\lambda(\cdot)}_{q(\cdot)}(\Omega)}\leq C\). Therefore, by Lemma 5, \(I_{{p(\cdot)},{\lambda(\cdot)}}(M_\sigma(|\cdot-x_0|^b f)\leq C\). This completes the estimate for \(J_2\) when \(b>a\).
Now suppose that \(b=a\). Then for every \(y\in A\), we have that \[1 = |x-x_0|^{b-a} =|x-x_0|^{-a} |x-x_0|^b \approx |x-x_0|^{-a}|y-x_0|^b.\] Therefore, \[\begin{gather} J_2(x) \lesssim |x-x_0|^{-a} \int_{A} \frac{|y-x_0|^bf(y)}{|x - y|^{n - \gamma}} \, dy \\ \leq |x-x_0|^{-a} \int_{A} \frac{|y-x_0|^bf(y)}{|x - y|^{n - \gamma}} \, dy \leq |x-x_0|^{-a} I_\gamma( |\cdot-x_0|^bf)(x). \end{gather}\] We can now repeat the above argument for 11 , except that we use the bound for \(I_\gamma\) in Theorem 10 to estimate the modular. This completes the estimate for \(J_2\) and so completes the proof.
In this section, as an application of Theorem 7, we show that Poincaré and Sobolev inequalities in the variable exponent Morrey spaces are a consequence of the Stein-Weiss inequality. Given a set \(\Omega\) such that \(0 < |\Omega| < \infty\) and a locally integrable function \(f\), define \[f_\Omega = \frac{1}{|\Omega|} \int_\Omega f(x) \, dx.\]
We first prove a Poincaré inequality in variable exponent Morrey spaces.
Theorem 11. Let \(\Omega\) be a bounded, open, convex set containing the origin. Let \({p(\cdot)},\, q(\cdot), r(\cdot) \in P(\Omega)\) be such that \({p(\cdot)}, r(\cdot) \in LH_{0}(\Omega)\), \(1 < p_- \leq p_+ < \infty\), \(1 < r_- \leq r_+ < \infty\), and \(p(x) \leq q(x)\leq r(x)\) for all \(x \in \Omega\). Let \({\lambda(\cdot)}: \Omega \to [0,n]\) be such that \(0< \lambda_-\leq \lambda_+ <n\) and \({\lambda(\cdot)}\in LH_0(\Omega)\). Let \(a\) and \(b\) be constants such that conditions 3 and 5 hold for \(\gamma = 1\), and 4 and 6 with \(\gamma=1\) hold for \({p(\cdot)}\) and \(r(\cdot)\). Then, for all \(f\in C^1(\Omega)\) there exists a constant \(C>0\) such that \[\| |\cdot|^{a} [f - f_\Omega] \|_{M_{q(\cdot)}^{{\lambda(\cdot)}}(\Omega)} \leq C \| |\cdot|^{b} \nabla f \|_{M_{{p(\cdot)}}^{{\lambda(\cdot)}}(\Omega)}.\]
Proof. Fix a bounded, open, convex set \(\Omega\) such that \(0 \in \Omega\). Then, for every \(x \in \Omega\) and \(f\in C^1(\Omega)\), \[\label{WPI} |f(x) - f_{\Omega}| \leq I_1(|\nabla f| \chi_{\Omega})(x)\tag{14}\] holds. (See [13].) Since \(q(x) \leq r(x)\), we can define the exponent \(s(\cdot)\) by \[\frac{1}{q(x)} = \frac{1}{s(x)} + \frac{1}{r(x)}.\] By the generalized Hölder’s inequality 9 , Lemma 3, and inequality 14 , we get \[\begin{align} \label{yar} \begin{aligned} \| | \cdot |^{a} [f - f_{\Omega}] \|_{L^{q(\cdot)}(\Omega)} &\leq K \| | \cdot |^{a} [f - f_{\Omega}] \|_{L^{r(\cdot)}(\Omega)} \| \chi_{\Omega} \|_{L^{s(\cdot)}(\Omega)} \\ &\le K (|\Omega| +1 ) \| | \cdot |^{a} [f - f_{\Omega}] \|_{L^{r(\cdot)}(\Omega)} \\ &\lesssim \| | \cdot |^{a} I_1(|\nabla f| \chi_{\Omega}) \|_{L^{r(\cdot)}(\Omega)} . \end{aligned} \end{align}\tag{15}\]
Hence, by Lemma 7 (applied twice), inequality 15 , and inequality 7 , we have that \[\begin{align} \| |\cdot|^{a} [f - f_\Omega] \|_{M_{q(\cdot)}^{\lambda(\cdot)}(\Omega)} &\lesssim \sup_{x \in \Omega, \, r > 0} r^{-\frac{\lambda(x)}{q(x)}} \left\| |\cdot|^{a} [f - f_\Omega] \chi_{\tilde{B}(x, r)} \right\|_{L^{q(\cdot)}(\Omega)} \\ & \lesssim \sup_{x \in \Omega, \, r > 0} r^{-\frac{\lambda(x)}{q(x)}} \| | \cdot |^{a} I_1(|\nabla f| \chi_{\tilde{B}(x, r)}) \|_{L^{r(\cdot)}(\Omega)} \\ & \lesssim \| | \cdot |^{a} I_1(|\nabla f| \chi_{\tilde{B}(x, r)}) \|_{M_{r(\cdot)}^{\lambda(\cdot)}(\Omega)} \\ & \lesssim \| | \cdot |^{b} \nabla f \|_{M_{p(\cdot)}^{\lambda(\cdot)}(\Omega)}, \end{align}\] which is the desired result. ◻
We next prove a weighted Hardy-Sobolev inequality in variable exponent Morrey spaces.
Theorem 12. Let \(\Omega \subset \mathbb{R}^n\) be a bounded open set containing the origin. Let \(p(\cdot), q(\cdot) \in \mathcal{P}(\Omega)\) be such that \(p(\cdot), q(\cdot) \in LH_{0}(\Omega)\), \(1 < p_{-} \le p_{+} < \infty\), \(1 < q_{-} \le q_{+} < \infty\), and \(p(x) \le q(x)\) for all \(x \in \Omega\). Let \({\lambda(\cdot)}: \Omega \to [0,n]\) be such that \(0< \lambda_-\leq \lambda_+ <n\) and \({\lambda(\cdot)}\in LH_0(\Omega)\). Let \(a\) and \(b\) be constants such that conditions 3 , 4 , 5 , and 6 hold for \(\gamma = 1\). Then, for all \(f\in C_c^1(\Omega)\) there exists a constant \(C>0\) such that \[\label{SWI-2} \| |\cdot|^{a} f \|_{M_{q(\cdot)}^{\lambda(\cdot)}(\Omega)} \leq C \| |\cdot|^{b} \nabla f \|_{M_{p(\cdot)}^{\lambda(\cdot)}(\Omega)}.\tag{16}\]
Proof. For all \(f\in C_c^1(\Omega)\), we have that \[|f(x)| \lesssim I_1(|\nabla f|)(x)\] holds. (See [13].) Therefore, by inequality 7 , \[\| |\cdot|^{a} f \|_{M_{q(\cdot)}^{\lambda(\cdot)}(\Omega)} \lesssim \left\| |\cdot|^{a} I_1(|\nabla f|) \right\|_{M_{q(\cdot)}^{\lambda(\cdot)}(\Omega)} \le \| |\cdot|^b \nabla f \|_{M_{p(\cdot)}^{\lambda(\cdot)}(\Omega)},\] which is the desired result. ◻
As a corollary to Theorem 12 we prove a Gagliardo-Nirenberg inequality in variable exponent Morrey spaces.
Theorem 13. Let \(\Omega \subset \mathbb{R}^n\) be a bounded open set containing the origin. Let \(p(\cdot), p^*(\cdot) \in \mathcal{P}(\Omega)\) be such that \(p(\cdot), p^*(\cdot) \in LH_{0}(\Omega)\), \(1 < p_{-} \le p_{+} < \infty\), \(1 < p^*_{-} \le p^*_{+} < \infty\), and \(p(x) \le p^*(x)\) for all \(x \in \Omega\). Let \({\lambda(\cdot)}: \Omega \to [0,n]\) be such that \(0 < \lambda_-\leq \lambda_+ <n\) and \({\lambda(\cdot)}\in LH_0(\Omega)\). Let \(a\) be a constant such that conditions 3 , 4 , 5 , and 6 hold for \(\gamma = 1\), \(b=a\), and \(q(\cdot)\) replaced by \(p^*(\cdot)\). Fix \({q(\cdot)}\in \mathcal{P}(\Omega)\) and \(0\leq \theta\leq 1\), and define the exponent \(r(\cdot)\) by \[\frac{1}{r(x)} = \frac{\theta}{p^*(x)}+\frac{1-\theta}{q(x)}.\] Then, for all \(f\in C_c^1(\Omega)\) there exists a constant \(C>0\) such that for all \(f\in C_c^1(\Omega)\), \[\| |\cdot|^{a} f \|_{M_{r(\cdot)}^{\lambda(\cdot)}(\Omega)} \leq C \| |\cdot|^{a} \nabla f \|_{M_{p(\cdot)}^{\lambda(\cdot)}(\Omega)}^{\theta} \| |\cdot|^{a} f \|_{M_{q(\cdot)}^{\lambda(\cdot)}(\Omega)}^{1-\theta}.\]
Proof. By the generalized Hölder’s inequality 9 , the rescaling property for variable Lebesgue space norms [13], and Lemma 7, we have that for every \(z\in \Omega\) and \(r>0\), \[\begin{align} \| |\cdot|^{a} &f \chi_{B(z,r)}\|_{L_{r(\cdot)}} \\ &\le K \|\left(|\cdot|^{a}f\right)^{\theta} \chi_{B(z,r)}\|_{L_{\frac{1}{\theta}p^*(\cdot)}} \|\left(|\cdot|^{a}f\right)^{1-\theta} \chi_{B(z,r)}\|_{L_{\frac{1}{(1-\theta)}q(\cdot)}} \\ &= K \| |\cdot|^{a} f \chi_{B(z,r)}\|^{\theta}_{L_{ p^*(\cdot)}} \||\cdot|^{a} f \chi_{B(z,r)}\|^{1-\theta}_{L_{q(\cdot)}} \\ &= K r^{\theta \frac{\lambda(z)}{p^{*}(z)}+ (1- \theta) \frac{\lambda(z)}{q(z)}} \left( r^{- \frac{\lambda(z)}{p^{*}(z)}} \| |\cdot|^{a} f \chi_{B(z,r)}\|_{L_{ p^*(\cdot)}}\right)^{\theta} \left( r^{- \frac{\lambda(z)}{q(z)}} \| |\cdot|^{a} f \chi_{B(z,r)}\|_{L_{q(\cdot)}}\right)^{1-\theta} \\ &\lesssim r^{ \frac{\lambda(z)}{r(z)}} \| |\cdot|^{a} f \|_{M_{p^*(\cdot)}^{\lambda(\cdot)}\Omega)}^{\theta} \||\cdot|^{a} f \|_{M_{q(\cdot)}^{\lambda(\cdot)}(\Omega)}^{1-\theta}. \end{align}\] If we now apply Theorem 12 with \(b=a\) and \({q(\cdot)}\) replaced by \(p^*(\cdot)\), we get \[r^{- \frac{\lambda(z)}{r(z)}}\||\cdot|^{a} f \chi_{B(z,r)}\|_{L_{r(\cdot)}} \lesssim \| |\cdot|^{a} f \|_{M_{p^*(\cdot)}^{\lambda(\cdot)}(\Omega)}^{\theta} \| |\cdot|^{a}f \|_{M_{q(\cdot)}^{\lambda(\cdot)}(\Omega)}^{1-\theta} \lesssim \| |\cdot|^{a} \nabla f \|_{M_{p(\cdot)}^{\lambda(\cdot)}(\Omega)}^{\theta} \| |\cdot|^{a} f \|_{M_{q(\cdot)}^{\lambda(\cdot)}(\Omega)}^{1-\theta}.\] Since the constant is independent of \(z\) and \(r\), by Lemma 7 we get the desired result. ◻
Finally, we prove a fractional Hardy-Sobolev inequality in variable exponent Morrey spaces.
Theorem 14. Let \(\Omega \subset \mathbb{R}^n\) be a bounded open set containing the origin. Let \(s \in [0,1]\) and \(p(\cdot), q(\cdot) \in \mathcal{P}(\Omega)\). Assume that \(p(\cdot), q(\cdot) \in LH_{0}(\Omega)\), \(1 < p_{-} \le p_{+} < \infty\), \(1 < q_{-} \le q_{+} < \infty\), and \(p(x) \le q(x)\) for all \(x \in \Omega\). Let \({\lambda(\cdot)}: \Omega \to [0,n]\) be such that \(0 < \lambda_-\leq \lambda_+ <n\) and \({\lambda(\cdot)}\in LH_0(\Omega)\). Let \(a\) and \(b\) be constants such that 3 , 4 , 5 , and 6 hold with \(\gamma=2s\). Then there exists a constant \(C>0\) such that for all function \(f\in C^2(\Omega)\), \[\| |\cdot|^{a} f \|_{M_{q(\cdot)}^{\lambda(\cdot)}(\Omega)} \leq C \| |\cdot|^{b} \left(-\Delta\right)^{s} f \|_{M_{p(\cdot)}^{\lambda(\cdot)}(\Omega)}.\]
Proof. By the definition of the fractional Laplacian and the properties of the Fourier transform, it follows that for \(s \in [0, 1]\), \[I_{2s}\big((-\Delta)^s f\big)(x) = f(x).\] The desired result follows now follows immediately from Theorem 7. ◻
Remark 15. In Theorem 14, if fix \(0\leq s \leq 1\), \({p(\cdot)}\) such that \(p_+<\frac{n-\lambda_+}{2s}\), and let \({q(\cdot)}={p(\cdot)}\), \(b = 0\) and \(a=-2s\), we get a fractional Hardy-Rellich inequality, \[\| |\cdot|^{-2s} f \|_{M_{p(\cdot)}^{\lambda(\cdot)}(\Omega)} \leq C \| \left(-\Delta\right)^{s} f \|_{M_{p(\cdot)}^{\lambda(\cdot)}(\Omega)}.\]
The first author is partially supported by a Simons Foundation Travel Support for Mathematicians Grant and by NSF grant DMS-2349550. The second two authors were funded by Nazarbayev University under Collaborative Research Program Grant 20122022CRP1601.↩︎