In this paper, we obtain sharp remainder terms for the Hardy-Poincaré inequalities with general non-radial weights in the setting of Baouendi-Grushin vector fields (see Theorem [main32thm]). It is worth emphasizing that all of our results are new both in the Baouendi-Grushin and standard Euclidean settings. The method employed allows us to not only unify, but also improve the results of Kombe and
Yener [1] for any \(1<p<\infty\) while holding true for complex-valued functions and providing explicit constants (Corollary
[cor32rec32KY]). As a result, we are able to obtain sharp remainder terms to many known weighted Hardy-type inequalities (see Section sec:sec:sec32sharp32remainders?). Aside from weighted Hardy-type inequalities, we also recover a sharp remainder
formula for the \(L^{p}\)-Poincaré inequality (Corollary [cor32poin]). In the special case of radial weights, we are naturally able to introduce
the notion of Baouendi-Grushin \(p\)-Bessel pairs (see Definition [p-bessel32pair]). Finally, we apply the technique to establish the
sharp remainder term of the Heisenberg-Pauli-Weyl inequality in \(L^{p}\) for \(1<p<\infty\) (Corollary [cor32hpw32sharp]), which includes the sharp constant. This makes it possible to obtain the \(L^{p}\)-analogue for \(2\leq p < n\) (Theorem [stability32hpw]) of a stability result by Cazacu, Flynn, Lam and Lu [2].
The classical Hardy inequality is a cornerstone in the analysis of partial differential equations and functional inequalities. Its multidimensional (in \(L^{p}\)) form is stated as follows: let \(\Omega\subseteq\mathbb{R}^{n}\), \(1 < p < n\), then, for all \(f \in C_0^{\infty}(\Omega)\), the following inequality holds:
\[\label{class95multdim95hardy}
\int_{\Omega} |\nabla f|^p dz \geq \left( \frac{n - p}{p} \right)^p \int_{\Omega} \frac{|f|^p}{|z|^p} dz,\tag{1}\] where the constant \(\left( \frac{n - p}{p} \right)^p\) is sharp but not attained.
One of the well-known applications of the Hardy inequality of the form (1 ) lies in the study of solutions to partial differential equations. For example, in [3], Baras and Goldstein found that the existence of solutions to the heat equation with singular potentials is connected to the extremizers and sharp constants of the Hardy inequality. Later,
Goldstein and Zhang generalized these results to heat equations on the Heisenberg group [4], where they used the Hardy
inequality on the Heisenberg group by Garofalo and Lanconelli [5]. In the meantime, Brezis and Vázques [6] studied blow-up solutions of some nonlinear elliptic problems. There, they obtained a characterization of extremal solutions to a certain semilinear elliptic
equation. However, in order to apply this characterization to standard examples, they needed an improved Hardy inequality, which they successfully derived [6]. Brezis and Vázques also posted an open problem [6], which states whether the improved Hardy inequality
[6] can be further improved in a certain context.
The findings of Baras, Goldstein, Zhang, Brezis and Vázques received enormous attention and led to the development of Hardy inequalities, their improvements and generalizations (e.g. [7]–[14]). We also refer to standard monographs [15]–[21] along with open problems posed by Maz’ya on Hardy inequalities and many other topics [22].
A specific line of development in Hardy inequalities that we aim to focus on originates with the results by Ghoussoub and Moradifam [23], where they gave a sufficient and necessary condition on a pair of positive radial functions \(v(|z|)\) and \(w(|z|)\) so that, we have \[\begin{align}
\int_{B_{R}}v(|z|)|\nabla f|^{2}dz\geq\int_{B_{R}}w(|z|)|f|^{2}dz
\end{align}\] for all \(f\in C^{\infty}_{0}(B_{R})\). The results of Ghoussoub and Moradifam were later extended and further developed (see e.g. [24]–[35]). See also [18] for many
examples and properties of Bessel pairs.
After the contributions of Ghoussoub, Moradifam and others, considerable effort has been devoted to the study of non-radial weights, leading to significant advances in this direction. For example, in [36], Frank and Seiringer developed non-linear and non-local versions of the ground state representation that allow to refine, generalize and unify Hardy inequalities without the requirement on
radial weights: assume that \(\phi\) is a positive weak solution of the weighted \(p\)-Laplace equation \[\begin{align}
\label{frank}
-\nabla \cdot\left(v\left|\nabla \phi\right|^{p-2} \nabla\phi\right) = w\phi^{p-1}.
\end{align}\tag{2}\] Then, one has \[\begin{align}
\int_{\mathbb{R}^n}v|\nabla f|^{p}dz\geq\int_{\mathbb{R}^n}w|f|^{p}dz.
\end{align}\] By utilizing the convexity inequality for \(p\geq2\), one can obtain the same inequality with a positive remainder term [36]. In the same spirit, Goldstein, Kombe and Yener in [37], applied this approach to establish
general weighted Hardy inequalities on a Carnot group. In contrast to the equality in (2 ), they had an inequality, thereby relaxing the condition and giving them a broader class of Hardy inequalities. This idea subsequently found
applications in many different contexts (e.g. [1], [38]–[41]). We also refer to [42] to Hardy
inequalities with homogeneous weights.
However, we are mostly interested in [1]. Before discussing the results of the paper, we first would like to introduce the Baouendi-Grushin setting and its
relation to Hardy inequalities, which will be central to our analysis.
We consider the sub-Riemannian space \(\mathbb{R}^{m+k}=\{z=(x,y):x\in\mathbb{R}^{m}, y\in\mathbb{R}^{k}\}\) defined by the Baouendi-Grushin vector fields \[\begin{align}
\label{grushin32vector32fields}
X_i = \frac{\partial}{\partial x_i}, \quad i = 1, \ldots, m, \qquad
Y_j = |x|^{\gamma} \frac{\partial}{\partial y_j}, \quad j = 1,\ldots,k.
\end{align}\tag{3}\] Here \(\gamma\geq0\) is any nonnegative real number with \(|x|\) representing the usual Euclidean norm of \(x\in\mathbb{R}^{m}\). After being initially introduced by Baouendi [43] and later by Grushin [44], [45], Garofalo was the first to obtain the Hardy inequality in
this setting [46]: \[\begin{align}
\label{Gar}
\int_{\mathbb{R}^n}\left(|\nabla_{x}f|^{2}+|x|^{2\gamma}|\nabla_{y}f|^{2}\right) dz\geq \left(\frac{Q-2}{2}\right)^2 \int_{\mathbb{R}^n}\frac{|x|^{2 \gamma}}{|x|^{2\gamma+2}+(1+\gamma)^{2}|y|^{2}}|f|^2 dz,
\end{align}\tag{4}\] where \(n=m+k\), \(Q=m+(1+\gamma)k\) and \(f \in C_0^{\infty}\left(\mathbb{R}^m \times \mathbb{R}^k
\backslash\{(0,0)\}\right)\), \(\nabla_{x}f\) is the gradient of \(f\) in the variable \(x\) and \(\nabla_{y}f\) is
in the variable \(y\). In the special case of \(\gamma=0\), the inequality (4 ) reduces to the classical multidimensional \(L^{2}\)-Hardy inequality (i.e. \(p=2\) of (1 )).
Let us now give some additional facts on the Baouendi-Grushin framework. The corresponding gradient given by the vector fields (3 ) is defined by \[\begin{align}
\nabla_{\gamma}=\left(X_{1},\ldots,X_{m},Y_{1},\ldots,Y_{k}\right)=(\nabla_{x},|x|^{\gamma}\nabla_{y}).
\end{align}\] The Baouendi-Grushin operator is defined as a differential operator on \(\mathbb{R}^{m+k}\) by \[\begin{align}
\Delta_{\gamma}=\sum_{i=1}^{m}X^{2}_{i}+\sum_{j=1}^{k}Y^{2}_{j}=\Delta_{x}+|x|^{2\gamma}\Delta_{y}=\nabla_{\gamma}\cdot\nabla_{\gamma}.
\end{align}\] The operator in itself has many interesting properties. For example, it is not uniformly elliptic in the space \(\mathbb{R}^{m+k}\) (due to the degeneracy on the subspace \(\{0\}\times\mathbb{R}^{k}\)). It is also sub-elliptic and for non-negative even integers \(\gamma\) satisfies the Hörmander condition, meaning it can be expressed as a sum of squares of vector
fields that form a Lie algebra of full rank at any point in \(\mathbb{R}^{m+k}\). We also have the anisotropic dilation defined by \[\begin{align}
\delta_{a}(x,y)=(ax,a^{1+\gamma}y),
\end{align}\] where \(a>0\). The homogeneous dimension with respect to the dilation is \(Q=m+(1+\gamma)k\). The formula of the change of variables for the Lebesgue measure gives
\[\begin{align}
d \circ \delta_a(x, y)=a^Q d x d y.
\end{align}\] The corresponding distance function \(\rho(z)\) is given by \[\begin{align}
\rho=\rho(z):=\left(|x|^{2(1+\gamma)}+(1+\gamma)^2|y|^2\right)^{\frac{1}{2(1+\gamma)}}.
\end{align}\] Direct computations give us the following formulas \[\begin{align}
\nabla_{\gamma}\rho=\left(\frac{|x|^{2\gamma}x}{\rho^{2\gamma+1}},\frac{(1+\gamma)|x|^{\gamma}y}{\rho^{2\gamma+1}}\right) \quad \text{and} \quad |\nabla_{\gamma}\rho|=\frac{|x|^\gamma}{\rho^\gamma}.
\end{align}\] Also, we have \[\begin{align}
\nabla_{\gamma}\cdot\left(\rho^{c}|x|^{s}\nabla_{\gamma}\rho\right)=(Q+c+s-1)\frac{|x|^{2\gamma+s}}{\rho^{2\gamma+1-c}}
\end{align}\] for \(c,s\in\mathbb{R}\). Additionally, we define the so-called \(p\)-Grushin operator similarly to how the classical Baouendi-Grushin operator is derived from the
Laplacian: \[\begin{align}
\Delta_{\gamma,p}\phi=\nabla_{\gamma}\cdot\left(|\nabla_{\gamma}\phi|^{p-2}\nabla_{\gamma}\phi\right)
\end{align}\] for some complex-valued \(\phi\) on \(\Omega\subseteq\mathbb{R}^{m+k}\), which gives us the following formula \[\begin{align}
\Delta_{\gamma,p} \phi(\rho)
= \frac{|x|^{\gamma p}}{\rho^{\gamma p}} \, \left| \phi'(\rho) \right|^{p-2}
\left[ (p-1)\phi''(\rho) + (Q-1)\frac{\phi'(\rho)}{\rho} \right],
\quad 1<p<\infty
\end{align}\] for all \(z\in\mathbb{R}^{m+k} \backslash\{(0,0)\}\). Lastly, we define \(B_{R}=\{z\in\mathbb{R}^{m+k}:\rho(z)<R\}\) as the \(\rho\)-ball centered at the origin with radius \(R>0\).
Returning to Hardy inequalities, we continue from the result of Garofalo [46]. After that, there has been an intense development. In [47], D’Ambrosio obtained weighted \(L^{p}\)-versions of (4 ) as well as the \(L^{p}\)-Poincaré inequality contained in a slab with a particular assumption on the domain. Then, Niu, Chen and Han [48] employed
the Picone identity from [49] to obtain many different Hardy inequalities on \(B_{R}\), \(\mathbb{R}^{m+k}\backslash B_{R}\) and on \(\mathbb{R}^{m+k}\). Later, D’Ambrosio [50]
further developed and unified the results from [47] by establishing weighted \(L^{p}\)-Hardy inequalities, this time for
general quasilinear second-order differential operators (which include the Baouendi-Grushin operator). Over the next years, Hardy inequalities for Baouendi-Grushin vector fields have remained a huge topic of interest and we refer the reader to many
interesting works in this direction [34],
[51]–[59].
Now the result that we are mostly focused on is from [1], where Kombe and Yener obtained the following general weighted \(L^{p}\)-Hardy type inequality: let \(v \in C^1\left(\mathbb{R}^n\right)\) and \(w \in L_{\text{loc }}^1\left(\mathbb{R}^n\right)\) be nonnegative functions and
\(\phi \in\)\(C^{\infty}\left(\mathbb{R}^n\right)\) be a positive function satisfying the differential inequality \[\begin{align}
\label{condt32kombe}
-\nabla_\gamma \cdot\left(v\left|\nabla_\gamma \phi\right|^{p-2} \nabla_\gamma\phi\right) \geq w\phi^{p-1}
\end{align}\tag{5}\] a.e. in \(\Omega\). There is a positive number \(c_p=c(p)\) such that; for \(p \geq 2\), then
\[\begin{align}
\label{kom1}
\int_{\mathbb{R}^n} v\left|\nabla_\gamma f\right|^p dz \geq \int_{\mathbb{R}^n} w|f|^p dz+c_p \int_{\mathbb{R}^n} v\left|\nabla_\gamma \left(\frac{f}{\phi}\right)\right|^p \phi^p dz
\end{align}\tag{6}\] and for \(1<p<2\), then \[\begin{align}
\label{kom2}
\int_{\mathbb{R}^n} v\left|\nabla_\gamma f\right|^p dz \geq \int_{\mathbb{R}^n} w|f|^p dz+c_p \int_{\mathbb{R}^n} \frac{v\left|\nabla_\gamma \left(\frac{f}{\phi}\right)\right|^2 \phi^2}{\left(\left|\frac{f}{\phi} \nabla_\gamma
\phi\right|+\left|\nabla_\gamma \left(\frac{f}{\phi}\right)\right| \phi\right)^{2-p}} dz
\end{align}\tag{7}\] for all real-valued functions \(f \in C_0^{\infty}\left(\mathbb{R}^n\right)\).
The results, (6 ) and (7 ), in themselves are very interesting as they recover and improve most known Hardy inequalities (e.g. [48], [47], [50]
and etc.) with radial and non-radial weights for all real-valued functions \(f \in C_0^{\infty}\left(\mathbb{R}^n\right)\). Not only that, they also derive new inequalities in the context of the Baouendi-Grushin operator
(e.g. [60], [23]).
The main goal of this paper is to provide a way to obtain sharp remainder terms (identities) and explicit constants in (6 ) and (7 ) without the real-valued assumption. Also, instead of having two different
inequalities for each case of \(p\), it would be good to have one unifying inequality that contains them both. Finally, the condition (5 ) seems to work when \(v=1\) and \(\phi:=\text{the first eigenfunction of -\Delta_{\gamma, p}}\), which should give the \(L^{p}\)-Poincare inequality. However, the first eigenfunction
of \(-\Delta_{\gamma, p}\) is not smooth in general. So, it is interesting to see whether the condition on \(\phi\) can be made less restrictive.
In this paper, we do all of the above. More precisely, we have the following result: let \(1<p<\infty\) and let \(\Omega\subseteq\mathbb{R}^{m+k}\) be an open set such that the
integrals below make sense. Let \(v\in C^{1}(\Omega\backslash\Sigma)\) and \(w\in L^{1}_{loc}(\Omega\backslash\Sigma)\) be nonnegative functions with \(\phi\) being a positive weak supersolution of the weighted \(p\)-Grushin equation, i.e. \(\phi>0\) satisfies (in the weak sense)
\[\begin{align}
\label{intro32main32condition}
-\nabla_{\gamma} \cdot \left( v \left|\nabla_{\gamma} \phi\right|^{p-2} \nabla_{\gamma} \phi \right)\geq w\phi^{p-1}
\end{align}\tag{8}\] a.e. in \(\Omega\backslash\Sigma\). Then, for all complex-valued \(f\in C^{\infty}_{0}(\Omega\backslash\Sigma)\), we have
\[\begin{align}
\label{intro32main32result}
\int_{\Omega}^{}v|\nabla_{\gamma}f|^{p}dz \geq \int_{\Omega}^{}w|f|^{p}dz + \int_{\Omega}^{}vC_{p}\left( \nabla_{\gamma}f,\phi\nabla_{\gamma}\left( \frac{f}{\phi} \right) \right) dz,
\end{align}\tag{9}\] where \(\Sigma\) is the set of singular points of \(v\) and \(w\). Equality, in (9
), holds when \(\phi\) is a positive weak solution of the weighted \(p\)-Grushin equation.
Following the approach of [1], one can derive arbitrarily many sharp remainder terms for weighted \(L^{p}\)-Hardy type
inequalities, provided there exists positive \(w\) for some given \(v\) and \(\phi\) such that (8 ) holds with an
equality. To illustrate, we can choose \[\begin{align}
v=|x|^{\beta-\gamma p}\rho^{(1+\gamma)p-\alpha} \quad \text{and} \quad \phi=\rho^{-\frac{Q+\beta -\alpha}{p}},
\end{align}\] which from (8 ) gives \(w=\left(\frac{Q+\beta-\alpha}{p}\right)^{p}\frac{|x|^{\beta}}{\rho^{\alpha}}\) and, hence, the sharp remainder term of [47]: let \(1<p<\infty\), \(m,k\geq1\) and \(\alpha,\beta
\in\mathbb{R}\) such that \(Q\geq\alpha-\beta\). Then, for all complex-valued \(f\in C^{\infty}_{0}(\mathbb{R}^{m+k}\backslash\{(0,0)\})\), we have
\[\begin{gather}
\label{dam32org32eq32intro}
\int_{\mathbb{R}^n}^{}|x|^{\beta-\gamma p}\rho^{(1+\gamma)p-\alpha}|\nabla_{\gamma}f|^{p}dz=\left( \frac{Q+\beta-\alpha}{p} \right) ^{p}\int_{\mathbb{R}^n}^{}\frac{|x|^{\beta}}{\rho^{\alpha}}|f|^{p}dz\\+\int_{\mathbb{R}^n}^{}|x|^{\beta-\gamma
p}\rho^{(1+\gamma)p-\alpha}C_{p}\left( \nabla_{\gamma}f,\rho^{-\frac{Q+\beta-\alpha}{p}}\nabla_{\gamma}\left( \frac{f}{\rho^{-\frac{Q+\beta-\alpha}{p}}}\right) \right)dz.
\end{gather}\tag{10}\] Identity (10 ), under an appropriate choice of parameters, immediately gives the sharp remainder term of [47] and (4 ). Additionally, (10 ), in the Euclidean case, recovers [61].
Here, the result (9 ) is cleanly written as one inequality that (due to [61] and [62]) contains both (6 ) and (7 ) at once with explicit constants.
Moreover, we are able to relax the condition on \(\phi\) in (5 ) by viewing the condition (5 ) as a \(p\)-Grushin equation
with a positive weak supersolution \(\phi\), just like in [36] or (2 ). This allows us to
obtain not only sharp remainders of weighted \(L^{p}\)-Hardy type inequalities, but also of \(L^{p}\)-Poincaré inequalities [63]: let \(1<p<\infty\) and \(\lambda_1\), \(\phi_1\) be the first eigenvalue and
eigenfunction of \(-\Delta_{\gamma,p}\). Then, \[\begin{align}
\int_{\Omega}|\nabla_{\gamma}f|^{p}dz=\lambda_1\int_{\Omega}|f|^{p}dz+\int_{\Omega}C_p\left(\nabla_\gamma f,\phi_{1}\nabla_{\gamma}\left(\frac{f}{\phi_{1}}\right)\right)dz,
\end{align}\] for all complex-valued \(f\in C^{\infty}_{0}(\Omega)\). Here \(\Omega\) is a bounded open subset of \(\mathbb{R}^{m+k}\) where the first
eigenfunction \(\phi_1\) of \(-\Delta_{\gamma,p}\) is positive.
The condition (8 ), in special cases (i.e. for radial weights), makes it possible to introduce the definition of Baouendi-Grushin \(p\)-Bessel pairs, which recovers the \(p\)-Bessel pairs definition introduced by Duy, Lam and Lu [32]. We also refer to [33] for related results. Last but not least, we note that all of our results are new in both Baouendi-Grushin and Euclidean frameworks.
The second aim of this paper is to apply the main result to obtain the sharp remainder formula of the Heisenberg-Pauli-Weyl inequality (in \(L^{p}\)) and consequently derive some stability results. First, let us recall
the original extraordinary paper by Heisenberg [64], where he initially formulated the uncertainty principle.
Essentially, Heisenberg discovered that the position and momentum of a particle cannot be measured at the same time. The idea was revolutionary and naturally led to its rigorous mathematical formulation by Kennard [65] and Weyl [66], [67] (who credited it to Pauli). Nowadays, the following result is referred to as the Heisenberg–Pauli–Weyl (HPW) inequality; for consistency, we state it for
compactly supported functions: let \(f\in C^{\infty}_{0}(\mathbb{R}^{n})\). Then, \[\begin{align}
\label{hpw32classical}
\left(\int_{\mathbb{R}^{n}}|\nabla f|^{2}dz\right)\left(\int_{\mathbb{R}^{n}}|z|^{2}|f|^{2}dz\right)\geq\frac{n^{2}}{4}\left(\int_{\mathbb{R}^{n}}|f|^{2}dz\right)^{2},
\end{align}\tag{11}\] where the constant \(\frac{n^2}{4}\) is optimal. Now the set of all extremizers of (11 ) generally consists of Gaussian profiles of the form \(f = ce^{-\alpha|z|^{2}}\) for some \(c\in\mathbb{R}\) and \(\alpha>0\). However, they are not compactly supported. So, in this context, they should actually
be called extremizers. However, if they describe all possible extremizers, we will still refer them as extremizers without saying that they are .
A natural question to ask is whether (11 ) is stable or not, i.e. if there exists some \(f\) for which (11 ) is close to an equality, does that imply that
\(f\) is close to the set of extremizers? Recently, there has been a lot of progress made in this analysis. McCurdy and Venkatraman, in [68], were the first to address this question. The answer turns out to be positive. However, the paper [68] had some missing gaps, which were filled in the subsequent papers [2], [69], [70]. We also refer to [71] for related results.
In the investigation of stability, one would like to have the explicit form of the so-called \(\delta_{2}(f)\): \[\begin{align}
\delta_{2}(f):=\left(\int_{\mathbb{R}^{n}}|\nabla f|^{2}dz\right)\left(\int_{\mathbb{R}^{n}}|z|^{2}|f|^{2}dz\right)-\frac{n^{2}}{4}\left(\int_{\mathbb{R}^{n}}|f|^{2}dz\right)^{2}.
\end{align}\] For example, in [2], Cazacu, Flynn, Lam and Lu obtained the following sharp remainder formula of a slightly
different version of the HPW inequality: \[\begin{align}
\label{lu32hpw}
\left(\int_{\mathbb{R}^{n}}|\nabla
f|^{2}dz\right)^{\frac{1}{2}}\left(\int_{\mathbb{R}^{n}}|z|^{2}|f|^{2}dz\right)^{\frac{1}{2}}-\frac{n}{2}\int_{\mathbb{R}^{n}}|f|^{2}dz=\frac{\alpha^{2}}{2}\int_{\mathbb{R}^{n}}\left|e^{-\frac{|z|^{2}}{2\alpha^{2}}}\nabla\left(\frac{f}{e^{-\frac{|z|^{2}}{2\alpha^{2}}}}\right)\right|^{2}dz,
\end{align}\tag{12}\] where \(\alpha=\left(\frac{\int_{\mathbb{R}^{n}}|f|^{2}|z|^{2}dz}{\int_{\mathbb{R}^{n}}|\nabla f|^{2}dz}\right)^{\frac{1}{4}}\). The HPW identity (12 ) was shown
to provide valuable insights about the HPW inequality (11 ). For example, the remainder term, in (12 ), vanishes if and only if \(f=ce^{-\frac{|z|^{2}}{2\alpha^{2}}}\) for any \(c\in\mathbb{R}\). In fact, they proved that this implies that all extremizers of the classical HPW inequality (11
) are in the set \(E=\{ce^{-\beta|z|^{2}}:c\in\mathbb{R},\beta>0\}\). Additionally, as shown in [2], the identity (12 ) along with the optimal Poincaré inequality of Gaussian type measure (from [72]) \[\begin{align}
\int_{\mathbb{R}^{n}}|\nabla f|^{2}e^{-\frac{|z|^{2}}{2|\alpha|^{2}}}dz\geq\frac{1}{|\alpha|^{2}}\inf_{c}\int_{\mathbb{R}^{n}}|f-c|^{2}e^{-\frac{|z|^{2}}{2|\alpha|^{2}}}dz
\end{align}\] gives the following sharp stability result of the altered HPW inequality: \[\begin{align}
\label{stability32ineq}
\left(\int_{\mathbb{R}^{n}}|\nabla f|^{2}dz\right)^{\frac{1}{2}}\left(\int_{\mathbb{R}^{n}}|z|^{2}|f|^{2}dz\right)^{\frac{1}{2}}-\frac{n}{2}\int_{\mathbb{R}^{n}}|f|^{2}dz\geq d(f,E)^{2},
\end{align}\tag{13}\] where \(d(f,E)^{2}\) is the \(L^{2}\) distance function from \(f\) to the set \(E\).
Moreover, the inequality (13 ) is sharp and the equality can be obtained by nontrivial functions \(f\notin E\). In [2], the authors demonstrated that the stability properties of the altered HPW inequality actually imply and improve the corresponding stability properties of the classical HPW inequality (11 ).
In this paper, we establish the sharp remainder formula and stability of the \(L^{p}\)-HPW inequality. Here, due to the novelty of the results, we will state them in the standard Euclidean framework (see Section 3.2 for its generalization to the Baouendi-Grushin setting, and [1] for related results): let \(1<p<\infty\) and \(p'=\frac{p}{p-1}\). Then, for all non-zero complex-valued \(f\in C^{\infty}_{0}(\mathbb{R}^{n})\), we have
\[\begin{gather}
\label{p32hpw32identity}
\left(\int_{\mathbb{R}^{n}}|\nabla f|^{p}dz\right)\left(\int_{\mathbb{R}^{n}}|z|^{p'}|f|^{p}dz\right)-\left(\frac{n}{p}\right)^{p}\left(\int_{\mathbb{R}^{n}}|f|^{p}dz\right)^{p}
\\=\left(\int_{\mathbb{R}^{n}}|z|^{p'}|f|^{p}dz\right)^{p-1}\int_{\mathbb{R}^{n}}C_{p}\left(\nabla f,e^{-\alpha_{p,n}|z|^{p'}}\nabla\left(\frac{f}{e^{-\alpha_{p,n}|z|^{p'}}}\right)\right)dz,
\end{gather}\tag{14}\] where \(\alpha_{p,n}=\frac{n}{p}\frac{p-1}{p}\frac{\int_{\mathbb{R}^{n}}|f|^{p}dz}{\int_{\mathbb{R}^{n}}|z|^{p'}|f|^{p}dz}\).
As in (12 ), the identity (14 ) automatically gives the fact that all extremizers of the corresponding \(L^{p}\)-HPW inequality are of the form \(f=ce^{-\alpha_{p,n}|z|^{p'}}\) for any \(c\in\mathbb{C}\). We refer to [61]
and [62] for the estimates of the \(C_p\)-functional that allow us to make such claim. On top of that, since for any \(f=ce^{\beta|z|^{p'}}\) (\(\beta>0\)) the formula for \(\alpha_{p,n}\) gives \(\alpha_{p,n}=\beta\), we have that the
manifold of extremizers of the \(L^{p}\)-HPW inequality is \(E_{p}=\{ce^{-\beta|z|^{p'}} : c\in\mathbb{C},\beta>0\}\).
When \(p=2\), in (14 ), we derive the following sharp remainder of the now classical HPW inequality (11 ): for all non-zero complex-valued \(f\in C^{\infty}_{0}(\mathbb{R}^{n})\)\[\begin{gather}
\left(\int_{\mathbb{R}^{n}}|\nabla f|^{2}dz\right)\left(\int_{\mathbb{R}^{n}}|z|^{2}|f|^{2}dz\right)-\frac{n^2}{4}\left(\int_{\mathbb{R}^{n}}|f|^{2}dz\right)^{2}
\\=\left(\int_{\mathbb{R}^{n}}|z|^{2}|f|^{2}dz\right)\int_{\mathbb{R}^{n}}C_{2}\left(\nabla f,e^{-\alpha_{2,n}|z|^{2}}\nabla\left(\frac{f}{e^{-\alpha_{2,n}|z|^{2}}}\right)\right)dz
\\=\left(\int_{\mathbb{R}^{n}}|z|^{2}|f|^{2}dz\right)\int_{\mathbb{R}^{n}}\left|e^{-\alpha_{2,n}|z|^{2}}\nabla\left(\frac{f}{e^{-\alpha_{2,n}|z|^{2}}}\right)\right|^{2}dz,
\end{gather}\] where \(\alpha_{2,n}=\frac{n}{4}\frac{\int_{\mathbb{R}^{n}}|f|^{2}dz}{\int_{\mathbb{R}^{n}}|z|^{2}|f|^{2}dz}\). Similarly, here we have \(E=\{ce^{-\beta|z|^{2}} :
c\in\mathbb{C}, \beta>0\}\).
Similarly as in [2], our \(L^{p}\)-HPW identity (14 ) along
with weighted \(L^{p}\)-Poincaré inequality for the log-concave probability measure [72] gives us a quantitative stability
result of the \(L^{p}\)-HPW inequality. First, let us define the \(L^{p}\)-Heisenberg deficit term: \[\begin{align}
\delta_{p}(f):=\left(\int_{\mathbb{R}^{n}}|\nabla f|^{p}dz\right)\left(\int_{\mathbb{R}^{n}}|z|^{p'}|f|^{p}dz\right)-\left(\frac{n}{p}\right)^{p}\left(\int_{\mathbb{R}^{n}}|f|^{p}dz\right)^{p}.
\end{align}\] Let \(2\leq p<n\) and \(p'=\frac{p}{p-1}\). Then, for all non-zero complex-valued \(f\in C^{\infty}_{0}(\mathbb{R}^n)\), we have
\[\begin{align}
\delta_{p}(f)\geq \widetilde{C}(n,p)\left(\int_{\mathbb{R}^{n}}|z|^{p'}|f|^{p}dz\right)^{p-1}d(f,E_{p})^{p},
\end{align}\] where \(\widetilde{C}(n,p)>0\) and \(d(f,E_{p})\) is the \(L^{p}\) distance to the manifold of extremizers, i.e. \[\begin{align}
d(f,E_{p})^p:=\inf_{c\in\mathbb{C},\beta>0}\left\{\left\lVert f-ce^{-\beta|z|^{p'}}\right\rVert^{p}_{L^{p}(\mathbb{R}^n)}\right\}.
\end{align}\]
The paper is organized as follows. Section 2 presents our main results and their consequences for general weights. In Section 3, we apply Theorem 5 to recover the sharp remainders of most known \(L^{p}\)-Hardy-type inequalities (Section 3.1), covering both radial
and non-radial weights. We also establish new identities for non-radial weights and derive the sharp remainder term for the \(L^{p}\)-Poincaré inequality. Section 3.2 shows the sharp
remainder and stability of the \(L^{p}\)-HPW inequality. Section 4 contains the proof of the main result, and Section 5 provides the proofs of
all applications of the main result.
In this section, we present the main results of the paper. Before stating the main results, we recall the following definition and estimates of the \(C_p\)-functional that will be used later.
Definition 1. Let \(1<p<\infty\). Then, for \(\xi,\eta\in\mathbb{C}^{n}\), we define \[\begin{align}
C_p(\xi,\eta):=|\xi|^p-|\xi-\eta|^p-p|\xi-\eta|^{p-2}\textrm{Re}(\xi-\eta)\cdot\overline{\eta}\geq0.
\end{align}\]
Lemma 2 (). Let \(p\geq2\). Then, for \(\xi,\eta\in\mathbb{C}^n\), we have \[\begin{align}
C_{p}(\xi,\eta)\geq c_1(p)|\eta|^{p},
\end{align}\] where \[\begin{align}
\label{c132const}
c_1(p)
= \inf_{(s,t)\in\mathbb{R}^2\setminus\{(0,0)\}}
\frac{\bigl[t^2 + s^2 + 2s + 1\bigr]^{\frac{p}{2}} -1-ps}{\bigl[t^2 + s^2\bigr]^{\frac{p}{2}}}\in(0,1].
\end{align}\qquad{(1)}\]
Theorem 5. Let \(1<p<\infty\) and let \(\Omega\subseteq\mathbb{R}^{m+k}\) be an open set such that the integrals below make sense. Let \(v\in C^{1}(\Omega\backslash\Sigma)\) and \(w\in L^{1}_{loc}(\Omega\backslash\Sigma)\) be nonnegative functions with \(\phi\) being a positive weak supersolution
of the weighted \(p\)-Grushin equation, i.e. \(\phi>0\) satisfies (in the weak sense) \[\begin{align}
\label{main32condition}
-\nabla_{\gamma} \cdot \left( v \left|\nabla_{\gamma} \phi\right|^{p-2} \nabla_{\gamma} \phi \right)\geq w\phi^{p-1}
\end{align}\qquad{(4)}\] a.e. in \(\Omega\backslash\Sigma\). Then, for all complex-valued functions \(f\in C_{0}^{\infty}(\Omega\backslash\Sigma)\), we have, \[\begin{align}
\label{main32result}
\int_{\Omega}^{}v|\nabla_{\gamma}f|^{p}dz \geq \int_{\Omega}^{}w|f|^{p}dz + \int_{\Omega}^{}vC_{p}\left( \nabla_{\gamma}f,\phi\nabla_{\gamma}\left( \frac{f}{\phi} \right) \right) dz,
\end{align}\qquad{(5)}\] where \(\Sigma\) is the set of singular points of \(v\) and \(w\). Equality, in (?? ), holds when \(\phi\) is a positive weak solution of the weighted \(p\)-Grushin equation.
When \(\gamma=0\), in Theorem 5, the result is also new in the Euclidean setting.
Corollary 6. Let \(1<p<\infty\) and let \(\Omega\subset\mathbb{R}^{n}\) be an open set such that the integrals below make sense. Let \(v\in C^{1}(\Omega\backslash\Sigma)\) and \(w\in L^{1}_{loc}(\Omega\backslash\Sigma)\) be nonnegative functions with \(\phi\) being a positive weak supersolution
of the weighted \(p\)-Laplace equation, i.e. \(\phi>0\) satisfies (in the weak sense) \[\begin{align}
-\nabla \cdot \left( v \left|\nabla \phi\right|^{p-2} \nabla \phi \right)\geq w\phi^{p-1}
\end{align}\] a.e. in \(\Omega\backslash\Sigma\). Then, for all complex-valued functions \(f\in C_{0}^{\infty}(\Omega\backslash\Sigma)\), we have, \[\begin{align}
\label{euc32main32result}
\int_{\Omega}^{}v|\nabla f|^{p}dz \geq \int_{\Omega}^{}w|f|^{p}dz + \int_{\Omega}^{}vC_{p}\left( \nabla f,\phi\nabla \left( \frac{f}{\phi} \right) \right) dz,
\end{align}\qquad{(6)}\] where \(\Sigma\) is the set of singular points of \(v\) and \(w\). Equality, in (?? ), holds when \(\phi\) is a positive weak solution of the weighted \(p\)-Laplace equation.
The inequality (?? ) not only improves the results of Kombe and Yener (6 ) and (7 ), but also the two weighted Hardy type inequalities, in the sense that it contains the two inequalities, (6 ) and (7 ), at once. Moreover, the inequalities are with explicit constants.
Corollary 7. Let \(\Omega\), \(v, w\) and \(\phi\) be from Theorem 5. Then, for \(p\geq2\), we have \[\begin{align}
\int_{\Omega} v\left|\nabla_\gamma f\right|^p dz \geq \int_{\Omega} w|f|^p dz+c_1(p) \int_{\Omega} v\left|\nabla_\gamma \left(\frac{f}{\phi}\right)\right|^p \phi^p dz,
\end{align}\] where the constant \(c_1(p)\) is given in (?? ). In addition, for \(1<p<2\leq n\), we have \[\begin{align}
\int_{\Omega}v\left|\nabla_\gamma f\right|^p dz \geq \int_{\Omega} w|f|^p dz+\widetilde{c_2}(p) \int_{\Omega} \frac{v\left|\nabla_\gamma \left(\frac{f}{\phi}\right)\right|^2 \phi^2}{\left(\left|\frac{f}{\phi} \nabla_\gamma \phi\right|+\left|\nabla_\gamma
\left(\frac{f}{\phi}\right)\right| \phi\right)^{2-p}} dz,
\end{align}\] where \(\widetilde{c_2}(p)=2^{p-2}c_2(p)\) with \(c_2(p)\) given in (?? ).
Remark 8. If \(v, w\) and \(\phi\) are radial (i.e. depend only on \(\rho\)), then the condition (?? ) can be reduced to the \(p\)-Bessel pair (in the setting of Baouendi-Grushin vector fields) first introduced in [32] in the standard Euclidean setting.
Here, we show this implication: suppose we have \[\begin{align}
\label{bess1}
-\nabla_{\gamma} \cdot \left( v \left|\nabla_{\gamma} \phi\right|^{p-2} \nabla_{\gamma} \phi \right) = w \phi^{p-1}
\end{align}\tag{15}\] with \(v=v(\rho)\), \(w=w(\rho)\) and \(\phi=\phi(\rho)\). Then, we have \[\begin{align}
\label{bess2}
\nabla_{\gamma}\phi=\phi'(\rho)\nabla_{\gamma}\rho \implies |\nabla_{\gamma}\phi|=|\phi'(\rho)||\nabla_{\gamma}\rho|.
\end{align}\tag{16}\] Using (16 ), we can rewrite (15 ) as \[\begin{align}
\label{bess3}
-\nabla_{\gamma}\cdot\left(v(\rho)|\phi'(\rho)|^{p-2}\phi'(\rho)|\nabla_{\gamma}\rho|^{p-2}\nabla_{\gamma}\rho\right)=w(\rho)\phi^{p-1}.
\end{align}\tag{17}\] To simplify the calculations, let us denote \[\begin{align}
A(\rho):=v|\phi'(\rho)|^{p-2}\phi'(\rho).
\end{align}\] Now we calculate \[\begin{align}
\label{bess4}
\nabla_{\gamma}\cdot\left(A(\rho)|\nabla_{\gamma}\rho|^{p-2}\nabla_{\gamma}\rho\right)&=A'(\rho)|\nabla_{\gamma}\rho|^{p}+A(\rho)\Delta_{\gamma,p}\rho \nonumber
\\&=A'(\rho)|\nabla_{\gamma}\rho|^{p}+\frac{Q-1}{\rho}A(\rho)|\nabla_{\gamma}\rho|^{p} \nonumber
\\&=\frac{|\nabla_{\gamma}\rho|^p}{\rho^{Q-1}}\left(\rho^{Q-1} A(\rho)\right)'.
\end{align}\tag{18}\] Substituting (18 ) into (17 ), we get \[\begin{gather}
-\nabla_{\gamma}\cdot\left(v(\rho)|\phi'(\rho)|^{p-2}\phi'(\rho)|\nabla_{\gamma}\rho|^{p-2}\nabla_{\gamma}\rho\right)=-\frac{|\nabla_{\gamma}\rho|^p}{\rho^{Q-1}}\left(\rho^{Q-1}
v|\phi'(\rho)|^{p-2}\phi'(\rho)\right)'\\=w(\rho)\phi^{p-1},
\end{gather}\] which gives us \[\begin{align}
\left(\rho^{Q-1} v|\phi'(\rho)|^{p-2}\phi'(\rho)\right)'+\frac{\rho^{Q-1}}{|\nabla_{\gamma}\rho|^p}w(\rho)\phi^{p-1}=0.
\end{align}\] If we do not assume that \(\phi\) is positive, then (15 ) becomes \[\begin{align}
-\nabla_{\gamma} \cdot \left( v \left|\nabla_{\gamma} \phi\right|^{p-2} \nabla_{\gamma} \phi \right) = w |\phi|^{p-2}\phi.
\end{align}\] By the same process, we obtain \[\begin{align}
\label{final32final32bessel}
\left(\rho^{Q-1} v(\rho)|\phi'(\rho)|^{p-2}\phi'(\rho)\right)'+\frac{\rho^{Q-1}}{|\nabla_{\gamma}\rho|^p}w(\rho)|\phi|^{p-2}\phi=0.
\end{align}\tag{19}\] If \(\gamma=0\) in (19 ), we recover the \(p\)-Bessel pair definition from [32] by Duy, Lam and Lu. Therefore, we can also introduce the following definition of \(p\)-Bessel pairs, now, in the setting of Baouendi-Grushin
vector fields.
Definition 9. The pair \((v(\rho),w(\rho))\) is called a Baouendi-Grushin \(p\)-Bessel pair on \((0,R)\) if \[\left(\rho^{Q-1} v|\phi'(\rho)|^{p-2}\phi'(\rho)\right)' + \frac{\rho^{Q-1}}{|\nabla_{\gamma}\rho|^p}w(\rho)|\phi|^{p-2}\phi = 0\] has a positive solution on \((0,R)\).
In this section, we apply the Theorem 5. However, before we start applying the Theorem 5, we note
that to make the proofs rigorous, one should (for any given \(\epsilon>0\)) replace the function \(\rho\) by its regularization \[\begin{align}
\rho_{\epsilon}:=\left(|x|^{2(1+\gamma)}_{\epsilon}+(1+\gamma)^{2}|y|^{2}\right)^{\frac{1}{2(1+\gamma)}}
\end{align}\] with \(|x|_{\epsilon}:=\left(\epsilon^{2}+\sum_{i=1}^{m}x^{2}_{i}\right)^{\frac{1}{2}}\) and then, after calculations take limit \(\epsilon \rightarrow 0\). However, for
the simplicity, we proceed formally.
Lastly, to clarify, we note that the results below provide new identities to known inequalities. When the Baouendi-Grushin vector fields are reduced to the standard Euclidean gradient, all of the results below are also new.
3.1 Sharp remainders of weighted Hardy-type, Poincaré and Hardy-Poin-caré inequalities↩︎
Now as previously mentioned, we can employ Theorem 5 to obtain sharp remainder terms of known weighted Hardy type, Poincaré and Hardy-Poincaré inequalities associated to the
Baouendi-Grushin operator. We begin with \[\begin{align}
v = 1 \quad \text{and} \quad \phi=(R-\rho)^{\frac{p-1}{p}},
\end{align}\] which gives us the sharp remainder terms of the Hardy type inequality obtained by Niu, Chen and Han [48]:
Corollary 10. Let \(1<p<\infty\) and \(R>0\). Then, for all complex-valued \(f\in
C^{\infty}_{0}(B_{R}\backslash\{(0,0)\})\), we have \[\begin{gather}
\int_{B_{R}}^{}|\nabla_{\gamma}f|^{p}dz=\left( \frac{p-1}{p} \right) ^{p}\int_{B_{R}}^{}\frac{|x|^{\gamma p}}{(R-\rho)^{p}\rho^{\gamma p}}|f|^{p}dz\\+\int_{B_{R}}^{}C_{p}\left( \nabla_{\gamma}f,(R-\rho)^{\frac{p-1}{p}}\nabla_{\gamma}\left(
\frac{f}{(R-\rho)^{\frac{p-1}{p}}}\right) \right)dz\\+(Q-1)\left( \frac{p-1}{p} \right) ^{p-1}\int_{B_{R}}^{}\frac{|x|^{\gamma p}}{(R-\rho)^{p-1}\rho^{\gamma p+1}}|f|^{p}dz.
\end{gather}\]
We can also obtain the sharp remainder term of the result by D’Ambrosio [47]. In particular, if we set \[\begin{align}
v=|x|^{\beta-\gamma p}\rho^{(1+\gamma)p-\alpha} \quad \text{and} \quad \phi=\rho^{-\frac{Q+\beta -\alpha}{p}},
\end{align}\] then we get \(w=\left( \frac{Q+\beta-\alpha}{p} \right)^{p}\frac{|x|^{\beta}}{\rho^{\alpha}}\), and therefore the following result:
Corollary 11. Let \(1<p<\infty\), \(m,k\geq1\) and \(\alpha,\beta \in\mathbb{R}\) such that \(Q\geq\alpha-\beta\). Then, for all complex-valued \(f\in C^{\infty}_{0}(\mathbb{R}^{m+k}\backslash\{(0,0)\})\), we have \[\begin{gather}
\label{dam32org32eq}
\int_{\mathbb{R}^n}^{}|x|^{\beta-\gamma p}\rho^{(1+\gamma)p-\alpha}|\nabla_{\gamma}f|^{p}dz=\left( \frac{Q+\beta-\alpha}{p} \right) ^{p}\int_{\mathbb{R}^n}^{}\frac{|x|^{\beta}}{\rho^{\alpha}}|f|^{p}dz\\+\int_{\mathbb{R}^n}^{}|x|^{\beta-\gamma
p}\rho^{(1+\gamma)p-\alpha}C_{p}\left( \nabla_{\gamma}f,\rho^{-\frac{Q+\beta-\alpha}{p}}\nabla_{\gamma}\left( \frac{f}{\rho^{-\frac{Q+\beta-\alpha}{p}}}\right) \right)dz.
\end{gather}\qquad{(7)}\]
Remark 12. Due to the importance of [47], a lot of consequent results emerge out of (?? ). For instance, when \(\alpha=(1+\gamma)p\) and \(\beta=\gamma p\) in (?? ), we can obtain the sharp remainder formula of [47]. That is, we have \[\begin{align}
\label{new32res}
\int_{\mathbb{R}^n}|\nabla_{\gamma}f|^{p}dz=\left(\frac{Q-p}{p}\right)^{p}\int_{\mathbb{R}^n}\frac{|x|^{\gamma p}}{\rho^{\gamma
p+p}}|f|^{p}dz+\int_{\mathbb{R}^n}C_{p}\left(\nabla_{\gamma}f,\rho^{-\frac{Q-p}{p}}\nabla_{\gamma}\left(\frac{f}{\rho^{-\frac{Q-p}{p}}}\right)\right)dz,
\end{align}\tag{20}\] which for \(p=2\), in (20 ), gives the sharp remainder term of (4 ). Furthermore, when \(\gamma=0\), we
recover the sharp remainder term of the classical \(L^{p}\)-Hardy inequality (1 ) from [61].
We also obtain a particular case of the result from [50]. That is, picking \[\begin{align}
v=|x|^{\alpha+p} \quad \text{and} \quad \phi=|x|^{\frac{|m+\alpha|}{p}}
\end{align}\] gives exactly \(w=\left(\frac{|m+\alpha|}{p}\right)^{p}|x|^{\alpha}\). Hence, the next corollary holds.
Corollary 13. Let \(1<p<\infty\), \(\alpha<-m\) and \(\Omega\subset(\mathbb{R}^{m}\backslash\{0\})\times\mathbb{R}^k\) be an
open set. Then, we have \[\begin{gather}
\label{dam32124x12432eq}
\int_{\Omega} |x|^{\alpha+p}\left|\nabla_{\gamma}f\right|^p dz =\left(\frac{|m+\alpha|}{p}\right)^{p} \int_{\Omega}|x|^{\alpha}|f|^pdz
\\+\int_{\Omega} |x|^{\alpha+p}C_p\left(\nabla_{\gamma}f, |x|^{\frac{|m+\alpha|}{p}} \nabla_{\gamma}\left(\frac{f}{|x|^{\frac{|m+\alpha|}{p}}}\right)\right)dz
\end{gather}\qquad{(8)}\] for all complex-valued \(f\in C^{\infty}_{0}(\Omega)\).
Apart from Hardy type inequalities, we also obtain the sharp remainder formula of the \(L^{p}\)-Poincaré inequality from [63] by specifying \[\begin{align}
v=1 \quad \text{and} \quad \phi=\phi_1:=\text{the first eigenfunction of } -\Delta_{\gamma,p},
\end{align}\] we get precisely \(w=\lambda_{1}\). In this context, \(\lambda_{1}\) corresponds to the first eigenvalue of the minus Dirichlet \(p\)-Grushin operator \(-\Delta_{\gamma,p}\) on the open bounded subset \(\Omega\) of \(\mathbb{R}^{m+k}\) where the first
eigenfunction is positive.
Corollary 14. Let \(1<p<\infty\) and \(\Omega\subset\mathbb{R}^{m+k}\) . Then, \[\begin{align}
\int_{\Omega}|\nabla_{\gamma}f|^{p}dz=\lambda_1\int_{\Omega}|f|^{p}dz+\int_{\Omega}C_p\left(\nabla_\gamma f,\phi_{1}\nabla_{\gamma}\left(\frac{f}{\phi_{1}}\right)\right)dz,
\end{align}\] for all complex-valued \(f\in C^{\infty}_{0}(\Omega)\). Here, \(\Omega\) is a bounded open subset of \(\mathbb{R}^{m+k}\) where the
first eigenfunction \(\phi_1\) of \(-\Delta_{\gamma,p}\) is positive.
Returning to Hardy inequalities, we observe that choosing \[\begin{align}
v=\left(\log\frac{R}{\rho}\right)^{\alpha+p} \quad \text{and} \quad \phi=\left(\log \frac{R}{\rho}\right)^{\frac{|\alpha+1|}{p}}
\end{align}\] produces the Hardy weight and an extra remainder term. That is, we get \[\begin{align}
w=\left(\frac{|\alpha+1|}{p}\right)^p\left(\log\frac{R}{\rho}\right)^{\alpha}\frac{|x|^{\gamma p}}{\rho^{\gamma p + p}}+\left(\frac{|\alpha+p|}{p}\right)^{p-1}(Q-p)\left(\log\frac{R}{\rho}\right)^{\alpha+1}\frac{|x|^{\gamma p}}{\rho^{\gamma p +p}}
\end{align}\] and, thus, the following sharp remainder formula of the power logarithmic \(L^{p}\)-Hardy type inequality by D’Ambrosio [50]:
Corollary 15. Let \(1<p<\infty\), \(\alpha<-1\) and \(Q\geq p\). Then, we have \[\begin{gather}
\int_{B_{R}}^{}\left(\log\frac{R}{\rho}\right)^{\alpha+p}|\nabla_{\gamma}f|^{p}dz = \left( \frac{|\alpha+1|}{p} \right) ^{p}\int_{B_{R}}^{}\left( \log \frac{R}{\rho} \right)^{\alpha}\frac{|x|^{\gamma p}}{\rho^{\gamma p+p}}|f|^{p}dz\\+\left(
\frac{|\alpha+1|}{p} \right) ^{p-1}(Q-p)\int_{B_{R}}^{}\left( \log \frac{R}{\rho}\right)^{\alpha+1}\frac{|x|^{\gamma p}}{\rho^{\gamma p+p}}|f|^{p}dz
\\+\int_{B_{R}}^{} \left(\log\frac{R}{\rho}\right)^{\alpha+p}C_{p}\left( \nabla_{\gamma}f,\left( \log \frac{R}{\rho}\right) ^{\frac{|\alpha+1|}{p}}\nabla_{\gamma}\left( \frac{f}{\left( \log \frac{R}{\rho}\right) ^{\frac{|\alpha+1|}{p}}} \right) \right)dz
\end{gather}\] for all complex-valued \(f\in C^{\infty}_{0}(B_{R}\backslash\{(0,0)\})\).
A similar identity can be obtained, but now with different logarithmic weights, which are from [50]. So, if we take \[\begin{align}
v=\left( \log \frac{R}{|x|} \right)^{\alpha+p} \quad \text{and} \quad \phi=\left( \log \frac{R}{|x|} \right)^{\frac{|\alpha+1|}{p}},
\end{align}\] we get \(w=\left(\frac{|\alpha+1|}{p}\right)^{p}\frac{\left(\log \frac{R}{|x|}\right)^{\alpha}}{|x|^{p}}
+\left(\frac{|\alpha+1|}{p}\right)^{p-1}(m-p)\frac{\left(\log \frac{R}{|x|}\right)^{\alpha+1}}{|x|^{p}}\).
Corollary 16. Let \(1<p<\infty\), \(\alpha<-1\) and \(m\geq p\). Then, for all complex-valued \(f\in C^{\infty}_{0}(\Omega\backslash\{x=0\})\), we have \[\begin{gather}
\int_{\Omega} \left(\log\frac{R}{|x|}\right)^{\alpha+p} |\nabla_{\gamma} f|^{p} dz
= \left( \frac{|\alpha+1|}{p} \right)^{p} \int_{\Omega} \left(\log\frac{R}{|x|}\right)^{\alpha} \frac{|f|^{p}}{|x|^{p}} dz \\ + \left( \frac{|\alpha+1|}{p} \right)^{p-1} (m-p) \int_{\Omega} \left(\log\frac{R}{|x|}\right)^{\alpha+1} \frac{|f|^{p}}{|x|^{p}}
dz \\+ \int_{\Omega} \left(\log\frac{R}{|x|}\right)^{\alpha+p}
C_{p} \left( \nabla_{\gamma} f, \left( \log\frac{R}{|x|} \right)^{\frac{|\alpha+1|}{p}}
\nabla_{\gamma} \left( \frac{f}{\left( \log\frac{R}{|x|} \right)^{\frac{|\alpha+1|}{p}}} \right) \right) dz,
\end{gather}\] where \(\Omega=\{z\in\mathbb{R}^{m+k}:|x|<R\}\).
By choosing \[\begin{align}
v=\left( 1+\rho^{\frac{p}{p-1}} \right)^{\alpha(p-1)} \quad \text{and} \quad \phi=\left( 1+\rho^{\frac{p}{p-1}} \right)^{1-\alpha},
\end{align}\] we get \(w=Q\left(\frac{(\alpha-1)p}{p-1}\right)^{p-1}\left(1+\rho^{\frac{p}{p-1}}\right)^{(\alpha-1)(p-1)}\frac{|x|^{\gamma p}}{\rho^{\gamma p}}\), which allows us to derive the sharp remainder formula
of the inequality proved in [1] by Kombe and Yener.
Corollary 17. Let \(1<p<\infty\) and \(\alpha>1\). Then, for all complex-valued \(f\in
C^{\infty}_{0}(\mathbb{R}^{m+k}\backslash\{(0,0)\})\), we have \[\begin{gather}
\label{rho32alpha32eq}
\int_{\mathbb{R}^{n}}\left(1+\rho^{\frac{p}{p-1}}\right)^{\alpha(p-1)}|\nabla_{\gamma}f|^{p}dz\\=Q\left(\frac{(\alpha-1)p}{p-1}\right)^{p-1}\int_{\mathbb{R}^{n}}\left(1+\rho^{\frac{p}{p-1}}\right)^{(\alpha-1)(p-1)}\frac{|x|^{\gamma p}}{\rho^{\gamma
p}}|f|^{p}dz\\+\int_{\mathbb{R}^{n}}\left(1+\rho^{\frac{p}{p-1}}\right)^{\alpha(p-1)}C_{p}\left(\nabla_{\gamma}f,\left( 1+\rho^{\frac{p}{p-1}} \right)^{1-\alpha}\nabla_{\gamma}\left(\frac{f}{\left( 1+\rho^{\frac{p}{p-1}}
\right)^{1-\alpha}}\right)\right)dz.
\end{gather}\qquad{(9)}\]
Remark 18. Under \(\gamma=0\), in (?? ), we obtain the sharp remainder formula of the Hardy-Poincaré inequalities by Skrzypczak from [60] and [25]. We note that the inequality form of (?? ) has many
applications in the theory of nonlinear diffusions [73]–[76]. Also, we refer the reader to [25] for the comparisons with the
classical Hardy inequality and to [25] for the connections between (?? ) and the results from [23], [74], [77].
We also establish sharp remainder terms of Hardy-type inequalities by Yener [40]. That is, considering \[\begin{align}
v=\frac{(a+b\rho^{\alpha})^{\beta}}{\rho^{\ell p}} \quad \text{and} \quad \phi=\rho^{-\frac{Q-p\ell-p}{p}}
\end{align}\] we derive that \[\begin{gather}
w=\left(\frac{Q-p\ell-p}{p}\right)^{p}(a+b\rho^{\alpha})^{\beta}\frac{|x|^{\gamma p}}{\rho^{\gamma p+\ell p+p}}
\\+\left(\frac{Q-p\ell-p}{p}\right)^{p-1}\beta\alpha b(a+b\rho^{\alpha})^{\beta-1}\frac{|x|^{\gamma p}}{\rho^{\gamma p+\ell p+p-\alpha}},
\end{gather}\] which, as a result, gives:
Corollary 19. Let \(a,b>0\) and \(\alpha,\beta, \ell\) be real numbers such that \(\alpha\beta>0\) and \(Q\geq pl+p\). Then, for all complex-valued \(f\in C^{\infty}_{0}(\mathbb{R}^{m+k}\backslash\{(0,0)\})\), we have \[\begin{gather}
\label{super32eq}
\int_{\mathbb{R}^{n}}\frac{(a+b\rho^{\alpha})^{\beta}}{\rho^{\ell p}}
|\nabla_{\gamma}f|^{p}dz
=\left(\frac{Q-p\ell-p}{p}\right)^{p}\int_{\mathbb{R}^{n}}
\frac{(a+b\rho^{\alpha})^{\beta}}{\rho^{\ell p+p}}\frac{|x|^{\gamma p}}{\rho^{\gamma p}}|f|^{p}dz
\\+\left(\frac{Q-p\ell-p}{p}\right)^{p-1}\beta\alpha b\int_{\mathbb{R}^{n}}
\frac{(a+b\rho^{\alpha})^{\beta-1}}{\rho^{\ell p+p-\alpha}}\frac{|x|^{\gamma p}}{\rho^{\gamma p}}|f|^{p}dz
\\+\int_{\mathbb{R}^{n}}\frac{(a+b\rho^{\alpha})^{\beta}}{\rho^{\ell p}}
C_{p}\left(\nabla_{\gamma}f,\rho^{-\frac{Q-p\ell-p}{p}}\nabla_{\gamma}\left(\frac{f}{\rho^{-\frac{Q-p\ell-p}{p}}}\right)\right)dz.
\end{gather}\qquad{(10)}\]
Remark 20. In the \(L^{2}\) Euclidean case (\(\gamma=0\)), the identity (?? ) gives sharp remainder terms of the weighted Hardy inequalities by Ghoussoub and Moradifam
[23]. For the applications of (?? ) in the theory of ODE, we refer to [23].
In the spirit of [40], we also prove several sharp remainder formulas of \(L^{p}\)-Hardy inequalities
related to the Baouendi-Grushin operator with other non-radial weights. For example, setting \[\begin{align}
v=\left(\frac{y_1}{|x|^{\gamma}}\right)^{p-2}\log x_{1} \quad \text{and} \quad \phi=\log y_{1},
\end{align}\] gives the following sharp remainder formula of [40]:
Corollary 21. Let \(1<p<\infty\). Then, for all complex-valued \(f\in C^{\infty}_{0}(\Omega)\), we have \[\begin{gather}
\int_{\Omega}\left(\frac{y_1}{|x|^{\gamma}}\right)^{p-2}\log x_{1}|\nabla_{\gamma}f|^{p}dz=\int_{\Omega}\frac{|x|^{2\gamma}\log x_{1}}{y^{2}_{1}\log^{p-1}y_{1}}|f|^{p}dz
\\+\int_{\Omega}\left(\frac{y_1}{|x|^{\gamma}}\right)^{p-2}\log x_{1}C_{p}\left(\nabla_{\gamma}f,\log y_{1}\nabla_{\gamma}\left(\frac{f}{\log y_{1}}\right)\right)dz,
\end{gather}\] where \(\Omega=\{(x,y)\in\mathbb{R}^{m+k}: x_{1},y_{1}>1\}\).
The method can be applied analogously to obtain sharp remainder terms of inequalities [40].
3.2 Sharp remainder and stability of the \(L^{p}\)-Heisenberg-Pauli-Weyl inequality↩︎
As noted earlier, the motivation for obtaining the sharp remainder formula of the HPW inequality comes from the question of stability. That is, as shown in [2], if we know the explicit form of the \[\begin{align}
\delta_{2}(f):=\left(\int_{\mathbb{R}^{n}}|\nabla f|^{2}dz\right)\left(\int_{\mathbb{R}^{n}}|z|^{2}|f|^{2}dz\right)-\frac{n^{2}}{4}\left(\int_{\mathbb{R}^{n}}|f|^{2}dz\right)^{2},
\end{align}\] which (for a slightly different deficit) turns out to be \[\begin{align}
\left(\int_{\mathbb{R}^{n}}|\nabla
f|^{2}dz\right)^{\frac{1}{2}}\left(\int_{\mathbb{R}^{n}}|z|^{2}|f|^{2}dz\right)^{\frac{1}{2}}-\frac{n}{2}\int_{\mathbb{R}^{n}}|f|^{2}dz=\frac{\alpha^{2}}{2}\int_{\mathbb{R}^{n}}\left|e^{-\frac{|z|^{2}}{2\alpha^{2}}}\nabla\left(\frac{f}{e^{-\frac{|z|^{2}}{2\alpha^{2}}}}\right)\right|^{2}dz,
\end{align}\] where \(\alpha=\left(\frac{\int_{\mathbb{R}^{n}}|f|^{2}|z|^{2}dz}{\int_{\mathbb{R}^{n}}|\nabla f|^{2}dz}\right)^{\frac{1}{4}}\), then with the combination of the optimal Poincaré inequality of Gaussian
type measure, one can have a certain result relating to the stability of the HPW inequality.
In this context, we apply the main result (Theorem 5) to obtain the sharp remainder term and stability of the \(L^{p}\)-HPW inequality. In
particular, if we set \[\begin{align}
v=\frac{\rho^{\gamma p}}{|x|^{\gamma p}}, \quad
\phi=e^{-\alpha \rho^{p'}}, \quad
p'=\frac{p}{p-1} \quad \text{with} \quad \alpha>0,
\end{align}\] we obtain the following identity: \[\begin{gather}
\int_{\mathbb{R}^n}\frac{\rho^{\gamma p}}{|x|^{\gamma p}}|\nabla_{\gamma}f|^{p}dz = \left(\frac{\alpha p}{p-1}\right)^{p-1}\int_{\mathbb{R}^n}\left(Q-\alpha p\rho^{\frac{p}{p-1}}\right)|f|^{p}dz\\+\int_{\mathbb{R}^n}\frac{\rho^{\gamma p}}{|x|^{\gamma
p}}C_{p}\left(\nabla_{\gamma}f,e^{-\alpha \rho^{\frac{p}{p-1}}}\nabla_{\gamma}\left(\frac{f}{e^{-\alpha \rho^{\frac{p}{p-1}}}}\right)\right)dz,
\end{gather}\] which after maximization with respect to \(\alpha\) implies the next result.
Corollary 22. Let \(1<p<\infty\) and \(p'=\frac{p}{p-1}\). Then, for all complex-valued \(f\in
C_{0}^{\infty}(\mathbb{R}^{m+k}\backslash\{x=0\})\), we have \[\begin{gather}
\label{hpw32sharp32eq}
\left(\int_{\mathbb{R}^{n}}\frac{\rho^{\gamma p}}{|x|^{\gamma
p}}|\nabla_{\gamma}f|^{p}dz\right)\left(\int_{\mathbb{R}^{n}}\rho^{p'}|f|^{p}dz\right)^{p-1}=\left(\frac{Q}{p}\right)^{p}\left(\int_{\mathbb{R}^{n}}|f|^{p}dz\right)^{p}\\+\left(\int_{\mathbb{R}^{n}}\rho^{p'}|f|^{p}dz\right)^{p-1}\int_{\mathbb{R}^{n}}\frac{\rho^{\gamma
p}}{|x|^{\gamma p}}C_{p}\left(\nabla_{\gamma}f,e^{-\alpha_{p,Q,\gamma} \rho^{p'}}\nabla_{\gamma}\left(\frac{f}{e^{-\alpha_{p,Q,\gamma} \rho^{p'}}}\right)\right)dz,
\end{gather}\qquad{(11)}\] where \(\alpha_{p,Q,\gamma}=\frac{Q}{p}\frac{p-1}{p}\frac{\int_{\mathbb{R}^{n}}|f|^{p}dz}{\int_{\mathbb{R}^{n}}\rho^{p'}|f|^{p}dz}\).
Remark 23. We note that (due to Lemma 2, 3 and 4) the identity (?? ) implies that all extremizers of the \(L^{p}\)-HPW inequality (related to the Baouendi-Grushin operator) are in the set \(E_{p,\gamma}=\{ce^{-\beta\rho^{p'}}:c\in\mathbb{C},\beta>0\}\) and similar remarks can be made for the \(\gamma=0\) and \(p=2\) cases.
When \(\gamma=0\), in (?? ), we obtain the sharp remainder formula of the \(L^{p}\)-HPW inequality for standard Euclidean gradient.
Corollary 24. Let \(1<p<\infty\) and \(p'=\frac{p}{p-1}\). Then, for all non-zero complex-valued \(f\in
C_{0}^{\infty}(\mathbb{R}^{n})\), we have \[\begin{gather}
\label{in32sec32p32hpw32identity}
\left(\int_{\mathbb{R}^{n}}|\nabla f|^{p}dz\right)\left(\int_{\mathbb{R}^{n}}|z|^{p'}|f|^{p}dz\right)-\left(\frac{n}{p}\right)^{p}\left(\int_{\mathbb{R}^{n}}|f|^{p}dz\right)^{p}
\\=\left(\int_{\mathbb{R}^{n}}|z|^{p'}|f|^{p}dz\right)^{p-1}\int_{\mathbb{R}^{n}}C_{p}\left(\nabla f,e^{-\alpha_{p,n}|z|^{p'}}\nabla\left(\frac{f}{e^{-\alpha_{p,n}|z|^{p'}}}\right)\right)dz,
\end{gather}\qquad{(12)}\] where \(\alpha_{p,n}=\frac{n}{p}\frac{p-1}{p}\frac{\int_{\mathbb{R}^{n}}|f|^{p}dz}{\int_{\mathbb{R}^{n}}|z|^{p'}|f|^{p}dz}\).
Remark 25. In Corollary 22 we excluded the singular point \(x=0\) from the domain. However, when \(\gamma=0\) the singular weight \(\tfrac{1}{|x|^{\gamma p}}\) vanishes, and consequently the point \(x=0\) can be included in the domain of \(f\) without restriction.
Now let us define the \(L^{p}\)-Heisenberg deficit: \[\begin{align}
\delta_{p}(f):=\left(\int_{\mathbb{R}^{n}}|\nabla f|^{p}dz\right)\left(\int_{\mathbb{R}^{n}}|z|^{p'}|f|^{p}dz\right)-\left(\frac{n}{p}\right)^{p}\left(\int_{\mathbb{R}^{n}}|f|^{p}dz\right)^{p}.
\end{align}\]
Utilizing the sharp remainder formula of the \(L^{p}\)-HPW inequality (?? ) and Lemma 2 with the weighted \(L^{p}\)-Poincaré inequality for the log-concave probability measure by Do, Flynn, Lam and Lu [72], we are able to obtain the
quantitative stability result of the \(L^{p}\)-HPW inequality.
Theorem 26. Let \(2\leq p<n\) and \(p'=\frac{p}{p-1}\). Then, for all non-zero complex-valued \(f\in
C^{\infty}_{0}(\mathbb{R}^n)\), we have \[\begin{align}
\label{stab32eq}
\delta_{p}(f)\geq \widetilde{C}(n,p)\left(\int_{\mathbb{R}^{n}}|z|^{p'}|f|^{p}dz\right)^{p-1}d(f,E_{p})^{p},
\end{align}\qquad{(13)}\] where \(\widetilde{C}(n,p)>0\) and \(d(f,E_{p})\) is the \(L^{p}\) distance to the manifold of extremizers,
i.e. \[\begin{align}
d(f,E_{p})^p:=\inf_{c\in\mathbb{C},\beta>0}\left\{\left\lVert f-ce^{-\beta|z|^{p'}}\right\rVert^{p}_{L^{p}(\mathbb{R}^n)}\right\}.
\end{align}\]
We note that the \(p=2\) case of (?? ) gives the sharp remainder formula for the classical \(L^{2}\)-HPW inequality.
Corollary 27. For all non-zero complex-valued \(f\in C_{0}^{\infty}(\mathbb{R}^{n})\), we have \[\begin{gather}
\left(\int_{\mathbb{R}^{n}}|\nabla f|^{2}dz\right)\left(\int_{\mathbb{R}^{n}}|z|^{2}|f|^{2}dz\right)-\frac{n^2}{4}\left(\int_{\mathbb{R}^{n}}|f|^{2}dz\right)^{2}
\\=\left(\int_{\mathbb{R}^{n}}|z|^{2}|f|^{2}dz\right)\int_{\mathbb{R}^{n}}C_{2}\left(\nabla f,e^{-\alpha_{2,n}|z|^{2}}\nabla\left(\frac{f}{e^{-\alpha_{2,n}|z|^{2}}}\right)\right)dz
\\=\left(\int_{\mathbb{R}^{n}}|z|^{2}|f|^{2}dz\right)\int_{\mathbb{R}^{n}}\left|e^{-\alpha_{2,n}|z|^{2}}\nabla\left(\frac{f}{e^{-\alpha_{2,n}|z|^{2}}}\right)\right|^{2}dz,
\end{gather}\] where \(\alpha_{2,n}=\frac{n}{4}\frac{\int_{\mathbb{R}^{n}}|f|^{2}dz}{\int_{\mathbb{R}^{n}}|z|^{2}|f|^{2}dz}\).
In a similar way, from (?? ), we obtain the stability of the classical \(L^{2}\)-HPW inequality:
Corollary 28. For all non-zero complex-valued \(f\in C^{\infty}_{0}(\mathbb{R}^n)\), we have \[\begin{align}
\delta_{2}(f)\geq \widetilde{C}(n)\left(\int_{\mathbb{R}^n}|z|^{2}|f|^{2}dz\right)d(f,E)^{2},
\end{align}\] where \(\widetilde{C}(n)>0\) and \(d(f,E)\) is the \(L^{2}\) distance to the manifold of extremizers, i.e \[\begin{align}
d(f,E)^{2}:=\inf_{c\in\mathbb{C},\beta>0}\left\{\left\lVert f-ce^{\beta|z|^{2}}\right\rVert^{2}_{L^{2}(\mathbb{R}^n)}\right\}.
\end{align}\]
4 Sharp Remainder of the Hardy-Poincaré Inequality With General Weights–Proof of Theorem 5↩︎
Proof of Theorem 5. For any complex-valued \(f \in C_0^{\infty}(\Omega)\), we set \[\psi
:= \frac{f}{\phi},\] where \(\phi\) is a positive weak supersolution of the weighted \(p\)-Grushin equation. Applying the formula for the \(C_p\)-functional with the vectors \[\xi = \psi \nabla_{\gamma}\phi + \phi\nabla_{\gamma}\psi, \quad
\eta = \phi \nabla_{\gamma} \psi,\] we obtain \[\begin{align}
\label{step1}
C_{p}(\psi \nabla_{\gamma}\phi + \phi\nabla_{\gamma}\psi, \phi\nabla_{\gamma}\psi) \nonumber
&= \left| \psi \nabla_\gamma \phi + \phi \nabla_\gamma \psi \right|^p
- \left| \psi \nabla_\gamma \phi \right|^p
\\& \quad- p \left| \psi \nabla_\gamma \phi \right|^{p-2}
\text{Re}\!\left( (\psi \nabla_\gamma \phi) \cdot \overline{(\phi \nabla_\gamma \psi)} \right) \nonumber
\\
&= |\nabla_{\gamma}f|^{p}
- |\nabla_{\gamma}\phi|^{p} |\psi|^{p} \nonumber
\\& \quad- p \text{Re} |\psi|^{p-2} \psi \overline{\nabla_{\gamma}\psi}
|\nabla_{\gamma}\phi|^{p-2}\phi\nabla_{\gamma}\phi \nonumber
\\
&= |\nabla_{\gamma}f|^{p}
- |\nabla_{\gamma}\phi|^{p} |\psi|^{p} \nonumber
\\& \quad- \nabla_{\gamma}(|\psi|^{p})
|\nabla_{\gamma}\phi|^{p-2} \phi \nabla_{\gamma}\phi.
\end{align}\tag{21}\] Integrating both sides of (21 ) and multiplying by \(v\), we have \[\begin{align}
\int_{\Omega} v |\nabla_{\gamma}f|^{p} dz
&= \int_{\Omega}
C_{p}(\psi \nabla_{\gamma}\phi + \phi\nabla_{\gamma}\psi, \phi\nabla_{\gamma}\psi) v dz
+ \int_{\Omega} |\nabla_{\gamma}\phi|^{p} |\psi|^{p} v dz
\\
&\quad + \int_{\Omega}
\nabla_{\gamma}(|\psi|^{p})
|\nabla_{\gamma}\phi|^{p-2}
\phi \nabla_{\gamma}\phi v dz.
\end{align}\] By integrating by parts, we get \[\begin{align}
\int_{\Omega} v |\nabla_{\gamma}f|^{p} dz
&= \int_{\Omega}
C_{p}(\psi \nabla_{\gamma}\phi + \phi\nabla_{\gamma}\psi, \phi\nabla_{\gamma}\psi) v dz
+ \int_{\Omega} |\nabla_{\gamma}\phi|^{p} |\psi|^{p} v dz
\\
&\quad - \int_{\Omega}
|\psi|^{p} \nabla_{\gamma} \cdot
\left( \phi v |\nabla_{\gamma}\phi|^{p-2} \nabla_{\gamma}\phi \right) dz.
\end{align}\] Expanding the divergence term yields \[\begin{align}
\int_{\Omega} v |\nabla_{\gamma}f|^{p} dz
&= \int_{\Omega}
C_{p}(\psi \nabla_{\gamma}\phi + \phi\nabla_{\gamma}\psi, \phi\nabla_{\gamma}\psi) v dz
+ \int_{\Omega} |\nabla_{\gamma}\phi|^{p} |\psi|^{p} v dz
\\
&\quad - \int_{\Omega}
|\psi|^{p} \nabla_{\gamma}\phi
\left( v |\nabla_{\gamma}\phi|^{p-2}\nabla_{\gamma}\phi \right) dz
\\
&\quad - \int_{\Omega}
|\psi|^{p} \phi
\nabla_{\gamma} \cdot \left( v |\nabla_{\gamma}\phi|^{p-2} \nabla_{\gamma}\phi \right) dz.
\end{align}\] Simplifying, we get \[\begin{align}
\int_{\Omega} v |\nabla_{\gamma}f|^{p} dz
&= \int_{\Omega}
C_{p}(\psi \nabla_{\gamma}\phi + \phi\nabla_{\gamma}\psi, \phi\nabla_{\gamma}\psi) v dz
+ \int_{\Omega} |\nabla_{\gamma}\phi|^{p} |\psi|^{p} v dz
\\
&\quad - \int_{\Omega}
|\nabla_{\gamma}\phi|^{p} |\psi|^{p} v dz
- \int_{\Omega}
|\psi|^{p} \phi
\nabla_{\gamma} \cdot \left( v |\nabla_{\gamma}\phi|^{p-2} \nabla_{\gamma}\phi \right) dz
\\
&= \int_{\Omega}
C_{p}(\psi \nabla_{\gamma}\phi + \phi\nabla_{\gamma}\psi, \phi\nabla_{\gamma}\psi) v dz
\\& \quad - \int_{\Omega}
|\psi|^{p} \phi
\nabla_{\gamma} \cdot \left( v |\nabla_{\gamma}\phi|^{p-2} \nabla_{\gamma}\phi \right) dz.
\end{align}\] Since \[-\nabla_{\gamma} \cdot \left( v \left|\nabla_{\gamma} \phi\right|^{p-2} \nabla_{\gamma} \phi \right) \geq w \phi^{p-1},\] we deduce \[\begin{align}
\int_{\Omega} v |\nabla_{\gamma}f|^{p} dz
&\geq \int_{\Omega}
C_{p}(\psi \nabla_{\gamma}\phi + \phi\nabla_{\gamma}\psi, \phi\nabla_{\gamma}\psi) v dz
+ \int_{\Omega} |\psi|^{p} w |\phi|^{p} dz.
\end{align}\] After performing the change of variables \(\psi=\frac{f}{\phi}\), we get \[\begin{align}
\int_{\Omega} v |\nabla_{\gamma}f|^{p} dz
\geq \int_{\Omega}
C_{p}\left( \nabla_{\gamma}f, \phi \nabla_{\gamma} \left( \frac{f}{\phi} \right) \right) v dz
+ \int_{\Omega} w|f|^{p} dz.
\end{align}\] The proof is complete. ◻
5 Proofs of Corollary 7 and of Applications of the Main Result↩︎
Proof of Corollary 7. Using Lemma 2, in (?? ), we immediately get (6 ) with the explicit constant \(c_1(p)\): \[\begin{align}
\int_{\Omega} v\left|\nabla_\gamma f\right|^p dz \geq \int_{\Omega} w|f|^p dz+c_1(p) \int_{\Omega} v\left|\nabla_\gamma \left(\frac{f}{\phi}\right)\right|^p \phi^p dz.
\end{align}\] To get (7 ), we set \[\begin{align}
\label{variables}
\xi=\nabla_{\gamma}f, \quad \eta=\phi\nabla_{\gamma}\left(\frac{f}{\phi}\right) \quad \text{and} \quad \xi-\eta=\frac{f}{\phi}\nabla_{\gamma}\phi
\end{align}\tag{22}\] so that \[\begin{align}
|\xi|+|\xi-\eta|=|\eta+(\xi-\eta)|+|\xi-\eta|\leq 2|\xi-\eta|+|\eta|\leq 2\left(|\xi-\eta|+|\eta|\right).
\end{align}\] Thus, we have \[\begin{align}
\left(|\xi| + |\xi - \eta|\right)^{2-p}
\leq
\left[2\left(|\xi - \eta| + |\eta|\right)\right]^{2-p}
=
2^{2-p}\left(|\xi - \eta| + |\eta|\right)^{2-p},
\end{align}\] which gives us \[\begin{align}
\label{xi32eta32est}
\frac{|\eta|^2}{\left(|\xi| + |\xi - \eta|\right)^{2-p}}\geq
\frac{|\eta|^2}{2^{2-p}\left(|\xi - \eta| + |\eta|\right)^{2-p}}
=
2^{p-2}
\frac{|\eta|^2}{\left(\left|\frac{f}{\phi}\nabla\phi\right| + \left|\phi\nabla \left(\frac{f}{\phi}\right)\right|\right)^{2-p}}.
\end{align}\tag{23}\] Substituting (22 ) back to (23 ), we derive \[\begin{align}
\label{xi32eta32last}
\frac{\left| \phi \nabla_{\gamma} \left( \frac{f}{\phi} \right) \right|^2}{\left( |\nabla_{\gamma}f| + \left| \nabla_{\gamma}f - \phi \nabla_{\gamma} \left( \frac{f}{\phi} \right) \right| \right)^{2-p}} \geq
2^{p-2}\frac{\left|\phi\nabla_{\gamma}\left(\frac{f}{\phi}\right)\right|^{2}}{\left(\left|\frac{f}{\phi}\nabla\phi\right| + \left|\phi\nabla \left(\frac{f}{\phi}\right)\right|\right)^{2-p}}.
\end{align}\tag{24}\] Using Lemma 3, in (?? ), we get \[\begin{gather}
\int_{\Omega}^{}v|\nabla_{\gamma}f|^{p}dz \geq \int_{\Omega}^{}w|f|^{p}dz + \int_{\Omega}^{}vC_{p}\left( \nabla_{\gamma}f,\phi\nabla_{\gamma}\left( \frac{f}{\phi} \right) \right) dz
\\ \geq \int_{\Omega}^{}w|f|^{p}dz+c_{2}(p)\int_{\Omega}^{}\frac{v\left| \phi \nabla_{\gamma} \left( \frac{f}{\phi} \right) \right|^2}{\left( |\nabla_{\gamma}f| + \left| \nabla_{\gamma}f - \phi \nabla_{\gamma} \left( \frac{f}{\phi} \right) \right|
\right)^{2-p}}dz.
\end{gather}\] Now we utilize the obtained estimate (24 ): \[\begin{align}
\int_{\Omega}^{}v|\nabla_{\gamma}f|^{p}dz\geq \int_{\Omega}^{}w|f|^{p}dz+\widetilde{c_{2}}(p)\int_{\Omega}^{}\frac{v\left| \phi \nabla_{\gamma} \left( \frac{f}{\phi} \right) \right|^2}{\left(\left|\frac{f}{\phi}\nabla\phi\right| + \left|\phi\nabla
\left(\frac{f}{\phi}\right)\right|\right)^{2-p}}dz,
\end{align}\] where \(\widetilde{c_{2}}(p)=2^{p-2}c_2(p)\). ◻
5.1 Proofs of sharp remainder formulas of weighted
Hardy-type and Poincaré inequalities–Proof of Corollary 10-21↩︎
Proof of Corollary 10. Let \[\begin{align}
v=1, \quad \phi=(R-\rho)^{\frac{p-1}{p}}.
\end{align}\] We need to calculate \[\begin{align}
\label{w1}
w=\frac{-\nabla_{\gamma}\cdot \left( v|\nabla_{\gamma}\phi|^{p-2}\nabla_{\gamma}\phi \right) }{\phi^{p-1}}=\frac{-\nabla_{\gamma}\cdot T}{\phi^{p-1}}=\frac{\Phi}{\phi^{p-1}},
\end{align}\tag{25}\] where \[\begin{align}
\label{Phi1}
T=v|\nabla_{\gamma}\phi|^{p-2}\nabla_{\gamma}\phi \quad \text{ and } \quad \Phi=-\nabla_{\gamma}\cdot T.
\end{align}\tag{26}\] Now since \(\phi=\phi(\rho)\), we have \[\begin{align}
\nabla_{\gamma}\phi=\phi'(\rho)\nabla_{\gamma}\rho \quad \text{and} \quad |\nabla_{\gamma}\phi|=|\phi'(\rho)||\nabla_{\gamma}\rho|,
\end{align}\] which gives us \[\begin{align}
\nabla_{\gamma}\phi=-c(R-\rho)^{-c-1}\nabla_{\gamma}\rho \quad \text{and} \quad |\nabla_{\gamma}\phi|=c\frac{|\nabla_{\gamma}\rho|}{\left( R-\rho \right)^{\frac{1}{p}} },
\end{align}\] where \(c=\frac{p-1}{p}\). Therefore, we have \[\begin{align}
\label{T1}
T=-c^{p-1}\frac{|\nabla_{\gamma}\rho|^{p-2}\nabla_{\gamma}\rho}{\left( R-\rho \right) ^{c}}.
\end{align}\tag{27}\] Substituting (27 ) to (26 ), we get \[\begin{align}
\Phi&=c^{p-1}\nabla_{\gamma}\cdot \left( \frac{|\nabla_{\gamma}\rho|^{p-2}\nabla_{\gamma}\rho}{\left( R-\rho \right) ^{c}} \right)
\\&= c^{p-1}\left[ \nabla_{\gamma}\left( \left( R-\rho \right) ^{-c} \right)|\nabla_{\gamma}\rho|^{p-2}\nabla_{\gamma}\rho + \left( R-\rho \right) ^{-c}\Delta_{\gamma}\rho \right]
\\&=c^{p-1}\left[ c\left( R-\rho \right) ^{-c-1}|\nabla_{\gamma}\rho|^{p}+\frac{Q-1}{\rho}\frac{|\nabla_{\gamma}\rho|^{p}}{(R-\rho)^{c}} \right]
\\&=c^{p}\frac{|\nabla_{\gamma}\rho|^{p}}{(R-\rho)^{c+1}}+(Q-1)c^{p-1}\frac{|\nabla_{\gamma}\rho|^{p}}{\rho(R-\rho)^{c}}.
\end{align}\] Putting (26 ) to (25 ), we derive \[\begin{align}
w&=c^{p}\frac{|\nabla_{\gamma}\rho|^{p}}{(R-\rho)^{pc+1}}+\frac{|\nabla_{\gamma}\rho|^{p}}{\rho(R-\rho)^{pc}}
\\&=\left( \frac{p-1}{p} \right) ^{p}\frac{|x|^{\gamma p}}{(R-\rho)^{p}\rho^{\gamma p}}+(Q-1)\left( \frac{p-1}{p} \right) ^{p-1}\frac{|x|^{\gamma p}}{(R-\rho)^{p-1}\rho^{\gamma p + 1}}
\end{align}\] and the proof is complete. ◻
Proof of Corollary 11. Let \[\begin{align}
v=|x|^{\beta-\gamma p}\rho^{(1+\gamma)p-\alpha}, \qquad \phi=\rho^{-\frac{Q+\beta-\alpha}{p}}.
\end{align}\] We need to calculate \[\begin{align}
\label{w2}
w=\frac{-\nabla_{\gamma}\cdot\left(v|\nabla_{\gamma}\phi|^{p-2}\nabla_{\gamma}\phi\right)}{\phi^{p-1}}
=\frac{-\nabla_{\gamma}\cdot T}{\phi^{p-1}}
=\frac{\Phi}{\phi^{p-1}},
\end{align}\tag{28}\] where \[\begin{align}
\label{Phi2}
T=v|\nabla_{\gamma}\phi|^{p-2}\nabla_{\gamma}\phi \quad \text{and} \quad \Phi=-\nabla_{\gamma}\cdot T.
\end{align}\tag{29}\] Since \(\phi=\phi(\rho)\), we have \[\begin{align}
\nabla_{\gamma}\phi=\phi'(\rho)\nabla_{\gamma}\rho \quad \text{and} \quad |\nabla_{\gamma}\phi|=|\phi'(\rho)||\nabla_{\gamma}\rho|.
\end{align}\] Let \(c=\dfrac{Q+\beta-\alpha}{p}\), then, this yields \[\begin{align}
\nabla_{\gamma}\phi=-c\rho^{-c-1}\nabla_{\gamma}\rho \quad \text{and} \quad
|\nabla_{\gamma}\phi|=c\frac{|\nabla_{\gamma}\rho|}{\rho^{c+1}}.
\end{align}\] Therefore, \[\begin{align}
T&=-c^{p-1}|x|^{\beta-\gamma p}\rho^{(1+\gamma)p-\alpha}
\frac{|\nabla_{\gamma}\rho|^{p-2}}{\rho^{(c+1)(p-2)}}
\rho^{-c-1}\nabla_{\gamma}\rho\notag
\\&=-c^{p-1}\frac{|x|^{\beta-\gamma p}}{\rho^{(c+1)(p-1)+\alpha-(1+\gamma)p}}|\nabla_{\gamma}\rho|^{p-2}\nabla_{\gamma}\rho\notag
\\&=-c^{p-1}\frac{|x|^{\beta-\gamma p}}{\rho^{(c+1)(p-1)+\alpha-(1+\gamma)p}}
\frac{|x|^{\gamma(p-2)}}{\rho^{\gamma(p-2)}}\nabla_{\gamma}\rho\notag
\\&=-c^{p-1}\frac{|x|^{\beta-2\gamma}}{\rho^{(c+1)(p-1)+\alpha-p-2\gamma}}\nabla_{\gamma}\rho.\label{T2}
\end{align}\tag{30}\] Substituting (30 ) into (29 ), we get \[\begin{align}
\Phi
&=c^{p-1}\nabla_{\gamma}\cdot\left(|x|^{\beta-2\gamma}\rho^{-(c+1)(p-1)-\alpha+p+2\gamma}\nabla_{\gamma}\rho\right)
\\&=c^{p-1}\left(Q+\beta-2\gamma-(c+1)(p-1)-\alpha+p+2\gamma-1\right)\frac{|x|^{\beta}}{\rho^{1+(c+1)(p-1)+\alpha-p}}
\\&=c^{p-1}\left(Q+\beta-\alpha-cp+c\right)
\frac{|x|^{\beta}}{\rho^{c(p-1)+\alpha}}
\\&=c^{p}\frac{|x|^{\beta}}{\rho^{c(p-1)+\alpha}}.
\end{align}\] Putting (29 ) into (28 ), we obtain \[\begin{align}
w
&=c^{p}\frac{|x|^{\beta}}{\rho^{\alpha}}
=\left(\frac{Q+\beta-\alpha}{p}\right)^{p}\frac{|x|^{\beta}}{\rho^{\alpha}},
\end{align}\] completing the proof. ◻
Proof of Corollary 13. Let \[\begin{align}
v=|x|^{\alpha+p}, \qquad \phi=|x|^{s}, \qquad s=\frac{|m+\alpha|}{p}.
\end{align}\] We need to calculate \[\begin{align}
\label{w3}
w=\frac{-\nabla_{\gamma}\cdot\left(v|\nabla_{\gamma}\phi|^{p-2}\nabla_{\gamma}\phi\right)}{\phi^{p-1}}
=\frac{-\nabla_{\gamma}\cdot T}{\phi^{p-1}}
=\frac{\Phi}{\phi^{p-1}},
\end{align}\tag{31}\] where \[\begin{align}
T=v|\nabla_{\gamma}\phi|^{p-2}\nabla_{\gamma}\phi, \qquad \Phi=-\nabla_{\gamma}\cdot T.
\end{align}\] Since \(\phi=\phi(x)\) depends only on the \(x\) variables, we have \[\begin{align}
\nabla_{\gamma}\phi=(\nabla_{x}\phi,0), \quad \nabla_{x}\phi=s|x|^{s-2}x, \quad |\nabla_{\gamma}\phi|=s|x|^{s-1}.
\end{align}\] Therefore, \[\begin{align}
|\nabla_{\gamma}\phi|^{p-2}=s^{p-2}|x|^{(s-1)(p-2)}.
\end{align}\] Hence \[\begin{align}
T&=|x|^{\alpha+p}s^{p-2}|x|^{(s-1)(p-2)}\left(s|x|^{s-2}x,0\right)
\\&=s^{p-1}|x|^{\alpha+p+(s-1)(p-2)+(s-2)}(x,0).
\end{align}\] A simple computation gives \[\begin{align}
\alpha+p+(s-1)(p-2)+(s-2)=\alpha+s(p-1),
\end{align}\] so \[\begin{align}
T=s^{p-1}|x|^{\alpha+s(p-1)}(x,0).
\end{align}\] Taking the divergence, we get \[\begin{align}
\label{Phi3}
\Phi=-\sum_{i=1}^{m}\partial_{x_{i}}\left(s^{p-1}|x|^{\alpha+s(p-1)}x_{i}\right)
&=-s^{p-1}\left(m+\alpha+s(p-1)\right)|x|^{\alpha+s(p-1)}.
\end{align}\tag{32}\] Since \(\phi^{p-1}=|x|^{s(p-1)}\), from (31 ) and (32 ), we obtain \[\begin{align}
w&=-s^{p-1}\left(m+\alpha+s(p-1)\right)\frac{|x|^{\alpha+s(p-1)}}{|x|^{s(p-1)}}.
\end{align}\] Using \(-(m+\alpha)=|m+\alpha|\), we deduce \[\begin{align}
w=s^{p}|x|^{\alpha}=\left(\frac{|m+\alpha|}{p}\right)^{p}|x|^{\alpha}.
\end{align}\] The proof is complete. ◻
Proof of Corollary 14. Take \[\begin{align}
\label{pair}
v=1 \quad \text{and} \quad \phi=\phi_1:=\text{the first eigenfunction of } -\Delta_{\gamma,p}.
\end{align}\tag{33}\] Assuming that \(\phi_1>0\), (33 ) immediately gives us \[\begin{align}
w=\frac{-\nabla_{\gamma}\cdot\left(|\nabla_{\gamma}\phi_{1}|^{p-2}\nabla_{\gamma}\phi_{1}\right)}{\phi_{1}^{p-1}}=\frac{- \Delta_{\gamma,p} \phi_{1}}{\phi^{p-1}_{1}}=\lambda_{1},
\end{align}\] where \(\lambda_{1}\) is a number (eigenvalue) such that \[\begin{align}
- \Delta_{\gamma,p} \phi_{1} = \lambda_{1} \phi_{1}^{p-1}.
\end{align}\] The proof is complete. ◻
Proof of Corollary 15. Let \[\begin{align}
v=\left( \log \frac{R}{\rho} \right)^{\alpha+p}, \quad
\phi=\left( \log \frac{R}{\rho} \right)^{s}, \quad
s=\frac{|\alpha+1|}{p}.
\end{align}\] We need to calculate \[\begin{align}
\label{wlog1}
w=\frac{-\nabla_{\gamma}\cdot\left(v|\nabla_{\gamma}\phi|^{p-2}\nabla_{\gamma}\phi\right)}{\phi^{p-1}}
=\frac{-\nabla_{\gamma}\cdot T}{\phi^{p-1}}
=\frac{\Phi}{\phi^{p-1}},
\end{align}\tag{34}\] where \[\begin{align}
\label{Philog1}
T=v|\nabla_{\gamma}\phi|^{p-2}\nabla_{\gamma}\phi, \quad
\Phi=-\nabla_{\gamma}\cdot T.
\end{align}\tag{35}\] Since \(\phi=\phi(\rho)\), we have \[\begin{align}
\nabla_{\gamma}\phi=\phi'(\rho)\nabla_{\gamma}\rho, \quad
|\nabla_{\gamma}\phi|=|\phi'(\rho)||\nabla_{\gamma}\rho|.
\end{align}\] Now \[\begin{align}
\phi'(\rho)
= \frac{d}{d\rho}\left( \log \frac{R}{\rho} \right)^{s}
= -s\frac{\left(\log \frac{R}{\rho}\right)^{s-1}}{\rho},
\end{align}\] so that \[\begin{align}
\nabla_{\gamma}\phi
=-s\frac{\left(\log \frac{R}{\rho}\right)^{s-1}}{\rho}\nabla_{\gamma}\rho, \quad
|\nabla_{\gamma}\phi|=s\frac{\left(\log \frac{R}{\rho}\right)^{s-1}}{\rho}|\nabla_{\gamma}\rho|.
\end{align}\] Therefore \[\begin{align}
T&=-\left(\log \frac{R}{\rho}\right)^{\alpha+p}
\left(s\frac{\left(\log \frac{R}{\rho}\right)^{s-1}}{\rho}|\nabla_{\gamma}\rho|\right)^{p-2}
s\frac{\left(\log \frac{R}{\rho}\right)^{s-1}}{\rho}\nabla_{\gamma}\rho\notag
\\&=-s^{p-1}\left(\log \frac{R}{\rho}\right)^{\alpha+p+(s-1)(p-1)}
\frac{|\nabla_{\gamma}\rho|^{p-2}\nabla_{\gamma}\rho}{\rho^{p-1}}\notag
\\&=-s^{p-1}L^{\alpha+1+s(p-1)}\rho^{-(p-1)}|\nabla_{\gamma}\rho|^{p-2}\nabla_{\gamma}\rho,\label{Tlog1}
\end{align}\tag{36}\] where \[\begin{align}
L=\log \frac{R}{\rho}.
\end{align}\] From (35 ) and (36 ), we get \[\begin{align}
\Phi&=-\nabla_{\gamma}\cdot T
=s^{p-1}\nabla_{\gamma}\cdot\left(L^{\alpha+1+s(p-1)}\rho^{-(p-1)}|\nabla_{\gamma}\rho|^{p-2}\nabla_{\gamma}\rho\right)
\\&=s^{p-1}\nabla_{\gamma}A(\rho)|\nabla_{\gamma}\rho|^{p-2}\nabla_{\gamma}\rho
+s^{p-1}A(\rho)\frac{Q-1}{\rho}|\nabla_{\gamma}\rho|^{p},
\end{align}\] where \[\begin{align}
A(\rho)=L^{\alpha+1+s(p-1)}\rho^{-(p-1)}.
\end{align}\] We compute \[\begin{align}
\nabla_{\gamma}A(\rho)
&=\nabla_{\gamma}\left(L^{\alpha+1+s(p-1)}\right)\rho^{-(p-1)}
+L^{\alpha+1+s(p-1)}\nabla_{\gamma}\rho^{-(p-1)}
\\&=-(\alpha+1+s(p-1))L^{\alpha+s(p-1)}\rho^{-p}\nabla_{\gamma}\rho
-(p-1)L^{\alpha+1+s(p-1)}\rho^{-p}\nabla_{\gamma}\rho
\\&=-L^{\alpha+s(p-1)}\rho^{-p}\nabla_{\gamma}\rho\left[(\alpha+1+s(p-1))+(p-1)L\right].
\end{align}\] Since \(\alpha+1=-|\alpha+1|\), we obtain \[\begin{align}
\nabla_{\gamma}A(\rho)
=L^{\alpha+s(p-1)}\rho^{-p}\nabla_{\gamma}\rho\left[s-(p-1)L\right].
\end{align}\] Thus, we get \[\begin{align}
\Phi&=s^{p-1}L^{\alpha+s(p-1)}\rho^{-p}\left[s-(p-1)L\right]|\nabla_{\gamma}\rho|^{p}
+s^{p-1}(Q-1)L^{\alpha+1+s(p-1)}\frac{|\nabla_{\gamma}\rho|^{p}}{\rho^{p}}\notag
\\&=s^{p}L^{\alpha+s(p-1)}\frac{|\nabla_{\gamma}\rho|^{p}}{\rho^{p}}
+s^{p-1}(Q-p)L^{\alpha+1+s(p-1)}\frac{|\nabla_{\gamma}\rho|^{p}}{\rho^{p}}.\label{Philog2}
\end{align}\tag{37}\] Dividing (37 ) by \(\phi^{p-1}=L^{s(p-1)}\) and using (34 ), we obtain \[\begin{align}
w=s^{p}L^{\alpha}\frac{|\nabla_{\gamma}\rho|^{p}}{\rho^{p}}
+s^{p-1}(Q-p)L^{\alpha+1}\frac{|\nabla_{\gamma}\rho|^{p}}{\rho^{p}}.
\end{align}\] Putting everything back, we get exactly \[\begin{align}
w=\left(\frac{|\alpha+1|}{p}\right)^p\left(\log\frac{R}{\rho}\right)^{\alpha}\frac{|x|^{\gamma p}}{\rho^{\gamma p + p}}+\left(\frac{|\alpha+p|}{p}\right)^{p-1}(Q-p)\left(\log\frac{R}{\rho}\right)^{\alpha+1}\frac{|x|^{\gamma p}}{\rho^{\gamma p +p}}
\end{align}\] and the proof is complete. ◻
Proof of Corollary 16. Let \[\begin{align}
v=\left(\log\frac{R}{|x|}\right)^{\alpha+p}, \quad
\phi=\left(\log\frac{R}{|x|}\right)^{s}, \quad
s=\frac{|\alpha+1|}{p},
\end{align}\] and set \[\begin{align}
L=\log\frac{R}{|x|}.
\end{align}\] We need to calculate \[\begin{align}
\label{wxlog1}
w=\frac{-\nabla_{\gamma}\cdot\left(v|\nabla_{\gamma}\phi|^{p-2}\nabla_{\gamma}\phi\right)}{\phi^{p-1}}
=\frac{-\nabla_{\gamma}\cdot T}{\phi^{p-1}}
=\frac{\Phi}{\phi^{p-1}},
\end{align}\tag{38}\] where \[\begin{align}
\label{Phixlog1}
T=v|\nabla_{\gamma}\phi|^{p-2}\nabla_{\gamma}\phi, \quad
\Phi=-\nabla_{\gamma}\cdot T.
\end{align}\tag{39}\] Since \(\phi=\phi(|x|)\) depends only on the \(x\) variables, we have \[\begin{align}
\nabla_{\gamma}\phi=(\nabla_x\phi,0), \quad
\nabla_x\phi=sL^{s-1}\nabla_x\left(\log\frac{R}{|x|}\right)
=-sL^{s-1}\frac{x}{|x|^{2}}.
\end{align}\] Thus, we have \[\begin{align}
|\nabla_{\gamma}\phi|=s\frac{L^{s-1}}{|x|}, \quad
|\nabla_{\gamma}\phi|^{p-2}=s^{p-2}L^{(s-1)(p-2)}|x|^{-(p-2)}.
\end{align}\] Therefore, we get \[\begin{align}
T&=v|\nabla_{\gamma}\phi|^{p-2}\nabla_{\gamma}\phi\notag
\\&=-s^{p-1}L^{\alpha+p+(s-1)(p-2)+(s-1)}|x|^{-(p-2)}\frac{x}{|x|^{2}}\notag
\\&=-s^{p-1}L^{\alpha+1+s(p-1)}|x|^{-p}(x,0).\label{Txlog1}
\end{align}\tag{40}\] Let \[\begin{align}
A(|x|)=L^{\alpha+1+s(p-1)}|x|^{-p}.
\end{align}\] Since \(\nabla_x\cdot(Ax)=mA+|x|A'(|x|)\), from (39 ) and (40 ) we obtain \[\begin{align}
\Phi=-\nabla_{\gamma}\cdot T=s^{p-1}\left(mA+|x|A'(|x|)\right).
\end{align}\] A direct computation gives \[\begin{align}
|x|A'(|x|)=|x|^{-p}\left(-pL^{k}-kL^{k-1}\right), \quad
k=\alpha+1+s(p-1),
\end{align}\] and hence \[\begin{align}
\Phi=s^{p-1}|x|^{-p}\left((m-p)L^{k}-kL^{k-1}\right).\label{Phixlog2}
\end{align}\tag{41}\] Dividing (41 ) by \(\phi^{p-1}=L^{s(p-1)}\) and using (38 ), we get \[\begin{align}
w=s^{p-1}|x|^{-p}\left((m-p)L^{\alpha+1}-(\alpha+1+s(p-1))L^{\alpha}\right).
\end{align}\] Since \(\alpha+1=-|\alpha+1|\) we have \(s=\frac{|\alpha+1|}{p}\), which yields \[\begin{align}
w&=\left(\frac{|\alpha+1|}{p}\right)^{p}\frac{L^{\alpha}}{|x|^{p}}
+\left(\frac{|\alpha+1|}{p}\right)^{p-1}(m-p)\frac{L^{\alpha+1}}{|x|^{p}}
\\&=\left(\frac{|\alpha+1|}{p}\right)^{p}\frac{\left(\log \frac{R}{|x|}\right)^{\alpha}}{|x|^{p}}+\left(\frac{|\alpha+1|}{p}\right)^{p-1}(m-p)\frac{\left(\log \frac{R}{|x|}\right)^{\alpha+1}}{|x|^{p}}
\end{align}\] and the proof is complete. ◻
Proof of Corollary 17. Let \[\begin{align}
v=\left( 1+\rho^{a} \right)^{\alpha(p-1)}, \quad
\phi=\left( 1+\rho^{a} \right)^{1-\alpha}, \quad
a=\frac{p}{p-1}, \quad \alpha>1.
\end{align}\] Set \(b=\alpha-1>0\), then \[\begin{align}
\phi=\left( 1+\rho^{a} \right)^{-b}.
\end{align}\] We need to calculate \[\begin{align}
\label{w32new}
w=\frac{-\nabla_{\gamma}\cdot \left( v|\nabla_{\gamma}\phi|^{p-2}\nabla_{\gamma}\phi \right)}{\phi^{p-1}}
=\frac{-\nabla_{\gamma}\cdot T}{\phi^{p-1}}
=\frac{\Phi}{\phi^{p-1}},
\end{align}\tag{42}\] where \[\begin{align}
\label{Phi32new}
T=v|\nabla_{\gamma}\phi|^{p-2}\nabla_{\gamma}\phi, \quad \Phi=-\nabla_{\gamma}\cdot T.
\end{align}\tag{43}\] Since \(\phi=\phi(\rho)\), we have \[\begin{align}
\phi'(\rho)=-b\,a\,\rho^{a-1}(1+\rho^{a})^{-b-1}, \quad
\nabla_{\gamma}\phi=\phi'(\rho)\nabla_{\gamma}\rho, \quad
|\nabla_{\gamma}\phi|=|\phi'(\rho)||\nabla_{\gamma}\rho|.
\end{align}\] Hence \[\begin{align}
|\phi'|^{p-2}\phi'=-(ba)^{p-1}\rho^{(a-1)(p-1)}(1+\rho^{a})^{-(b+1)(p-1)}.
\end{align}\] Multiplying by \(v=(1+\rho^{a})^{(b+1)(p-1)}\) gives \[\begin{align}
T&=-(ba)^{p-1}\rho^{(a-1)(p-1)}|\nabla_{\gamma}\rho|^{p-2}\nabla_{\gamma}\rho.
\end{align}\] Since \((a-1)(p-1)=1\), we obtain \[\begin{align}
\label{T32new}
T=-(ba)^{p-1}\rho\,|\nabla_{\gamma}\rho|^{p-2}\nabla_{\gamma}\rho.
\end{align}\tag{44}\] Using the radial identity \[\begin{align}
\nabla_{\gamma}\cdot\left(A(\rho)|\nabla_{\gamma}\rho|^{p-2}\nabla_{\gamma}\rho\right)
=\left(A'(\rho)+\frac{Q-1}{\rho}A(\rho)\right)|\nabla_{\gamma}\rho|^{p}
\end{align}\] with \(A(\rho)=\rho\) and \(A'=1\), we get from (43 ) and (44 ) \[\begin{align}
\Phi=(ba)^{p-1}Q|\nabla_{\gamma}\rho|^{p}.
\end{align}\] Dividing by \(\phi^{p-1}=(1+\rho^{a})^{-b(p-1)}\) and using (42 ), we have \[\begin{align}
w=(ba)^{p-1}Q(1+\rho^{a})^{b(p-1)}|\nabla_{\gamma}\rho|^{p}.
\end{align}\] Recalling that \(|\nabla_{\gamma}\rho|^{p}=\dfrac{|x|^{\gamma p}}{\rho^{\gamma p}}\) and \(ba=\dfrac{(\alpha-1)p}{p-1}\), we arrive at \[\begin{align}
w=Q\left(\frac{(\alpha-1)p}{p-1}\right)^{p-1}
\left(1+\rho^{\frac{p}{p-1}}\right)^{(\alpha-1)(p-1)}
\frac{|x|^{\gamma p}}{\rho^{\gamma p}}.
\end{align}\] The proof is complete. ◻
Proof of Corollary 19. Let \[\begin{align}
v=\frac{(a+b\rho^{\alpha})^{\beta}}{\rho^{\ell p}}, \quad
\phi=\rho^{-s}, \quad s=\frac{Q-p\ell-p}{p}.
\end{align}\] We need to calculate \[\begin{align}
\label{wrad1}
w=\frac{-\nabla_{\gamma}\cdot\left(v|\nabla_{\gamma}\phi|^{p-2}\nabla_{\gamma}\phi\right)}{\phi^{p-1}}
=\frac{-\nabla_{\gamma}\cdot T}{\phi^{p-1}}
=\frac{\Phi}{\phi^{p-1}},
\end{align}\tag{45}\] where \[\begin{align}
\label{Phirad1}
T=v|\nabla_{\gamma}\phi|^{p-2}\nabla_{\gamma}\phi, \quad \Phi=-\nabla_{\gamma}\cdot T.
\end{align}\tag{46}\] Since \(\phi=\phi(\rho)\), we have \[\begin{align}
\nabla_{\gamma}\phi=\phi'(\rho)\nabla_{\gamma}\rho=-s\rho^{-s-1}\nabla_{\gamma}\rho, \quad
|\nabla_{\gamma}\phi|=s\rho^{-s-1}|\nabla_{\gamma}\rho|.
\end{align}\] Therefore \[\begin{align}
T=-s^{p-1}(a+b\rho^{\alpha})^{\beta}\rho^{-\ell p-(s+1)(p-1)}|\nabla_{\gamma}\rho|^{p-2}\nabla_{\gamma}\rho.\label{Trad1}
\end{align}\tag{47}\] Let \[\begin{align}
A(\rho)=(a+b\rho^{\alpha})^{\beta}\rho^{-\ell p-(s+1)(p-1)}.
\end{align}\] Using the radial identity \[\begin{align}
\nabla_{\gamma}\cdot\left(A(\rho)|\nabla_{\gamma}\rho|^{p-2}\nabla_{\gamma}\rho\right)
=\left(A'(\rho)+\frac{Q-1}{\rho}A(\rho)\right)|\nabla_{\gamma}\rho|^{p},
\end{align}\] we obtain from (46 ) and (47 ) \[\begin{align}
\Phi=s^{p-1}\left(A'(\rho)+\frac{Q-1}{\rho}A(\rho)\right)|\nabla_{\gamma}\rho|^{p}.
\end{align}\] A direct computation gives \[\begin{align}
\frac{A'(\rho)}{A(\rho)}
=\frac{\beta\alpha b\,\rho^{\alpha-1}}{a+b\rho^{\alpha}}
-\frac{\ell p+(s+1)(p-1)}{\rho},
\end{align}\] and thus \[\begin{align}
A'(\rho)+\frac{Q-1}{\rho}A(\rho)
=A(\rho)\left[\frac{\beta\alpha b\,\rho^{\alpha-1}}{a+b\rho^{\alpha}}
+\frac{Q-1-\ell p-(s+1)(p-1)}{\rho}\right].
\end{align}\] Since \(s=\frac{Q-p\ell-p}{p}\), we have \[\begin{align}
Q-1-\ell p-(s+1)(p-1)=s.
\end{align}\] Therefore \[\begin{align}
\Phi=s^{p-1}A(\rho)|\nabla_{\gamma}\rho|^{p}
\left[\frac{\beta\alpha b\,\rho^{\alpha-1}}{a+b\rho^{\alpha}}+\frac{s}{\rho}\right].\label{Phirad2}
\end{align}\tag{48}\] Dividing (48 ) by \(\phi^{p-1}=\rho^{-s(p-1)}\) and using (45 ), we get \[\begin{align}
w=s^{p-1}A(\rho)\rho^{s(p-1)}|\nabla_{\gamma}\rho|^{p}
\left[\frac{\beta\alpha b\,\rho^{\alpha-1}}{a+b\rho^{\alpha}}+\frac{s}{\rho}\right].
\end{align}\] Since \(A(\rho)\rho^{s(p-1)}=(a+b\rho^{\alpha})^{\beta}\rho^{-\ell p-(p-1)}\) and \(|\nabla_{\gamma}\rho|^{p}=\frac{|x|^{\gamma p}}{\rho^{\gamma p}}\), it follows that
\[\begin{align}
w&=s^{p}(a+b\rho^{\alpha})^{\beta}\frac{|x|^{\gamma p}}{\rho^{\gamma p+\ell p+p}}
+s^{p-1}\beta\alpha b(a+b\rho^{\alpha})^{\beta-1}\frac{|x|^{\gamma p}}{\rho^{\gamma p+\ell p+p-\alpha}}
\\&=\left(\frac{Q-p\ell-p}{p}\right)^{p}(a+b\rho^{\alpha})^{\beta}\frac{|x|^{\gamma p}}{\rho^{\gamma p+\ell p+p}}
\\&\quad+\left(\frac{Q-p\ell-p}{p}\right)^{p-1}\beta\alpha b(a+b\rho^{\alpha})^{\beta-1}\frac{|x|^{\gamma p}}{\rho^{\gamma p+\ell p+p-\alpha}}.
\end{align}\] The proof is complete. ◻
Proof of Corollary 21. Let \[\begin{align}
v=\left(\frac{y_{1}}{|x|^{\gamma}}\right)^{p-2}\log x_{1}, \qquad \phi=\log y_{1},
\end{align}\] with \(x_{1},y_{1}>1\). We need to calculate \[\begin{align}
\label{w32newlogy}
w=\frac{-\nabla_{\gamma}\cdot\left(v|\nabla_{\gamma}\phi|^{p-2}\nabla_{\gamma}\phi\right)}{\phi^{p-1}}
=\frac{-\nabla_{\gamma}\cdot T}{\phi^{p-1}}
=\frac{\Phi}{\phi^{p-1}},
\end{align}\tag{49}\] where \[\begin{align}
\label{Phi32newlogy}
T=v|\nabla_{\gamma}\phi|^{p-2}\nabla_{\gamma}\phi, \qquad \Phi=-\nabla_{\gamma}\cdot T.
\end{align}\tag{50}\] We have \[\begin{align}
X_{i}(\log y_{1})=\frac{\partial}{\partial x_{i}}\log y_{1}=0, \quad i=1,\dots,n,
\end{align}\] and \[\begin{align}
Y_{j}(\log y_{1})=\left(|x|^{\gamma}\frac{\partial}{\partial y_{j}}\right)(\log y_{1})
=\begin{cases}
\dfrac{|x|^{\gamma}}{y_{1}}, & j=1,\\
0, & j\neq 1.
\end{cases}
\end{align}\] Hence \[\begin{align}
\nabla_{\gamma}\log y_{1}=\left(0,\dots,0,\frac{|x|^{\gamma}}{y_{1}},0,\dots,0\right),
\end{align}\] and therefore \[\begin{align}
|\nabla_{\gamma}\log y_{1}|^{p-2}=\left(\frac{|x|^{\gamma}}{y_{1}}\right)^{p-2}.
\end{align}\] It follows from (50 ) that \[\begin{align}
T&=\left(\frac{y_{1}}{|x|^{\gamma}}\right)^{p-2}\log x_{1}\left(\frac{|x|^{\gamma}}{y_{1}}\right)^{p-2}\left(0,\dots,0,\frac{|x|^{\gamma}}{y_{1}},0,\dots,0\right)\\
&=\left(0,\dots,0,\frac{|x|^{\gamma}\log x_{1}}{y_{1}},0,\dots,0\right).
\end{align}\] Taking the divergence, we obtain \[\begin{align}
\Phi&=-Y_{1}\left(\frac{|x|^{\gamma}\log x_{1}}{y_{1}}\right)
=-|x|^{2\gamma}\log x_{1}\frac{\partial}{\partial y_{1}}\left(\frac{1}{y_{1}}\right)
=\frac{|x|^{2\gamma}\log x_{1}}{y_{1}^{2}}.
\end{align}\] Using (49 ) and \(\phi^{p-1}=\log^{p-1}y_{1}\), we get \[\begin{align}
w=\frac{|x|^{2\gamma}\log x_{1}}{y_{1}^{2}\log^{p-1}y_{1}},
\end{align}\] which completes the proof. ◻
5.2 Proof of the sharp remainder formula of the Heisenberg-Pauli-Weyl
inequality–Proof of Corollary 22↩︎
Proof of Corollary 22. Let \[\begin{align}
v=\frac{\rho^{\gamma p}}{|x|^{\gamma p}}, \quad
\phi=e^{-\alpha \rho^{p'}}, \quad
p'=\frac{p}{p-1}, \quad \alpha>0.
\end{align}\] We need to calculate \[\begin{align}
\label{w32hpw}
w=\frac{-\nabla_{\gamma}\cdot\left(v|\nabla_{\gamma}\phi|^{p-2}\nabla_{\gamma}\phi\right)}{\phi^{p-1}}
=\frac{-\nabla_{\gamma}\cdot T}{\phi^{p-1}}
=\frac{\Phi}{\phi^{p-1}},
\end{align}\tag{51}\] where \[\begin{align}
\label{Phi32hpw}
T=v|\nabla_{\gamma}\phi|^{p-2}\nabla_{\gamma}\phi, \qquad \Phi=-\nabla_{\gamma}\cdot T.
\end{align}\tag{52}\] Since \(\phi=\phi(\rho)\), we have \[\begin{align}
\phi'(\rho)=-\alpha p'\,\rho^{p'-1}e^{-\alpha\rho^{p'}}, \quad
\nabla_{\gamma}\phi=\phi'(\rho)\nabla_{\gamma}\rho, \quad
|\nabla_{\gamma}\phi|=|\phi'(\rho)||\nabla_{\gamma}\rho|.
\end{align}\] Thus \[\begin{align}
|\nabla_{\gamma}\phi|^{p-2}\nabla_{\gamma}\phi
=-(\alpha p')^{p-1}\rho^{(p'-1)(p-1)}e^{-\alpha(p-1)\rho^{p'}}
|\nabla_{\gamma}\rho|^{p-2}\nabla_{\gamma}\rho,
\end{align}\] and therefore \[\begin{align}
\label{T32hpw}
T=-(\alpha p')^{p-1}A(\rho)v|\nabla_{\gamma}\rho|^{p-2}\nabla_{\gamma}\rho,
\end{align}\tag{53}\] where \[\begin{align}
A(\rho)=\rho^{(p'-1)(p-1)}e^{-\alpha(p-1)\rho^{p'}}.
\end{align}\] Using the product rule together with the radial identity \[\begin{align}
\nabla_{\gamma}\cdot\big(|\nabla_{\gamma}\rho|^{p-2}\nabla_{\gamma}\rho\big)
=\frac{Q-1}{\rho}|\nabla_{\gamma}\rho|^{p},
\end{align}\] we get from (53 ) and (52 ) \[\begin{align}
\Phi&=(\alpha p')^{p-1}\nabla_{\gamma}\cdot\Big[A(\rho)v|\nabla_{\gamma}\rho|^{p-2}\nabla_{\gamma}\rho\Big]
\\&=(\alpha p')^{p-1}\Big[\nabla_{\gamma}\left(A(\rho) v\right)|\nabla_{\gamma}\rho|^{p-2}\nabla_{\gamma}\rho+A(\rho)v\Delta_{\gamma,p}\rho\Big]
\\&=(\alpha p')^{p-1}\Big[A'(\rho)v|\nabla_{\gamma}\rho|^{p}+A(\rho)|\nabla_{\gamma}\rho|^{p-2}\nabla_{\gamma}v\cdot\nabla_{\gamma}\rho+A(\rho)v\Delta_{\gamma,p}\rho\Big]
\\&=(\alpha p')^{p-1}\left[ A(\rho)|\nabla_{\gamma}\rho|^{p-2}\nabla_{\gamma}v\cdot\nabla_{\gamma}\rho
+v\left(A'(\rho)+\tfrac{Q-1}{\rho}A(\rho)\right)|\nabla_{\gamma}\rho|^{p}\right]
\\&=(\alpha p')^{p-1}v\left(A'(\rho)+\tfrac{Q-1}{\rho}A(\rho)\right)|\nabla_{\gamma}\rho|^{p}.
\end{align}\] Therefore, we have \[\begin{align}
\label{phi32hpw322}
\Phi=(\alpha p')^{p-1}\left(A'(\rho)+\frac{Q-1}{\rho}A(\rho)\right).
\end{align}\tag{54}\] Since \[\begin{align}
A'(\rho)=\rho^{(p'-1)(p-1)-1}e^{-\alpha(p-1)\rho^{p'}}
\Big((p'-1)(p-1)-\alpha(p-1)p'\rho^{p'}\Big),
\end{align}\] we have \[\begin{gather}
A'(\rho)+\frac{Q-1}{\rho}A(\rho)
\\=\rho^{(p'-1)(p-1)-1}e^{-\alpha(p-1)\rho^{p'}}
\Big((p'-1)(p-1)+Q-1-\alpha(p-1)p'\rho^{p'}\Big).
\end{gather}\] Dividing by \(\phi^{p-1}=e^{-\alpha(p-1)\rho^{p'}}\) and using (54 ) with (51 ), we get \[\begin{align}
w=(\alpha p')^{p-1}\rho^{(p'-1)(p-1)-1}
\Big((p'-1)(p-1)+Q-1-\alpha(p-1)p'\rho^{p'}\Big).
\end{align}\] With \(p'=\frac{p}{p-1}\) we have \((p'-1)(p-1)=1\), hence \[\begin{align}
w=(\alpha p')^{p-1}\Big(Q-\alpha(p-1)p'\rho^{p'}\Big)
=\left(\frac{\alpha p}{p-1}\right)^{p-1}\left(Q-\alpha p\rho^{\frac{p}{p-1}}\right).
\end{align}\] By Theorem 5, this gives us \[\begin{gather}
\label{hpw32eq}
\int_{\mathbb{R}^n}\frac{\rho^{\gamma p}}{|x|^{\gamma p}}|\nabla_{\gamma}f|^{p}dz = \left(\frac{\alpha p}{p-1}\right)^{p-1}\int_{\mathbb{R}^n}\left(Q-\alpha p\rho^{\frac{p}{p-1}}\right)|f|^{p}dz\\+\int_{\mathbb{R}^n}\frac{\rho^{\gamma p}}{|x|^{\gamma
p}}C_{p}\left(\nabla_{\gamma}f,e^{-\alpha \rho^{\frac{p}{p-1}}}\nabla_{\gamma}\left(\frac{f}{e^{-\alpha \rho^{\frac{p}{p-1}}}}\right)\right)dz.
\end{gather}\tag{55}\] Now set \[\begin{align}
\label{hpw32change}
K:=\int_{\mathbb{R}^n}\frac{\rho^{\gamma p}}{|x|^{\gamma p}}|\nabla_{\gamma}f|^{p}dz, \quad L:=\int_{\mathbb{R}^n}\rho^{p'}|f|^{p}dz, \quad M:=\int_{\mathbb{R}^n}|f|^{p}dz.
\end{align}\tag{56}\] Taking (56 ) and (55 ) into consideration, we get \[\begin{align}
\label{k61}
K=\left(\frac{\alpha p}{p-1}\right)^{p-1}(QM-\alpha pL)+R(\alpha),
\end{align}\tag{57}\] where \[\begin{align}
R(\alpha)=\int_{\mathbb{R}^n}\frac{\rho^{\gamma p}}{|x|^{\gamma p}}C_{p}\left(\nabla_{\gamma}f,e^{-\alpha \rho^{p'}}\nabla_{\gamma}\left(\frac{f}{e^{-\alpha \rho^{p'}}}\right)\right)dz.
\end{align}\] Now let \[\begin{align}
\label{s61alp32p}
s:=\alpha p
\end{align}\tag{58}\] and with (58 ) set \[\begin{align}
\label{Gs}
G(s):=\left(\frac{s}{p-1}\right)^{p-1}(QM-sL)=(p-1)^{-(p-1)}(s^{p-1}QM-s^{p}L).
\end{align}\tag{59}\] Next, we maximize (59 ) with respect to \(s\): \[\begin{gather}
G'(s)=(p-1)^{-(p-1)}s^{p-2}((p-1)QM-psL)=0 \\ \iff s=0 \quad \text{or} \quad s=\frac{(p-1)QM}{pL}.
\end{gather}\] Since \(1<p<\infty\) and \(\alpha>0\), the only possible critical point is \[\begin{align}
\label{s32crit}
s=\frac{(p-1)QM}{pL} \iff \alpha=\frac{(p-1)QM}{p^{2}L}:=\alpha_{p,Q,\gamma}
\end{align}\tag{60}\] for all non-zero complex-valued \(f\in C^{\infty}_{0}(\mathbb{R}^n)\). Substituting (60 ) to (59 ), we get \[\begin{align}
G(s)=\left(\frac{QM}{pL}\right)^{p-1}\frac{QM}{p}=\left(\frac{Q}{p}\right)^{p}\frac{M^{p}}{L^{p-1}}.
\end{align}\] Therefore, from (57 ), we have \[\begin{align}
\label{pre32last32hpw}
K=\left(\frac{Q}{p}\right)^{p}\frac{M^{p}}{L^{p-1}}+R(\alpha_{p,Q,\gamma}).
\end{align}\tag{61}\] Multiplying both sides of (61 ) by \(L^{p-1}\), we derive \[\begin{align}
KL^{p-1}=\left(\frac{Q}{p}\right)^{p}M^{p}+L^{p-1}R(\alpha_{p,Q,\gamma}).
\end{align}\] Thus, from (56 ), we conclude \[\begin{gather}
\left(\int_{\mathbb{R}^n}\frac{\rho^{\gamma p}}{|x|^{\gamma
p}}|\nabla_{\gamma}f|^{p}dz\right)\left(\int_{\mathbb{R}^n}\rho^{p'}|f|^{p}dz\right)^{p-1}=\left(\frac{Q}{p}\right)^{p}\left(\int_{\mathbb{R}^n}|f|^{p}dz\right)^{p}\\+\left(\int_{\mathbb{R}^n}\rho^{p'}|f|^{p}dz\right)^{p-1}\int_{\mathbb{R}^n}\frac{\rho^{\gamma
p}}{|x|^{\gamma p}}C_{p}\left(\nabla_{\gamma}f,e^{-\alpha_{p,Q,\gamma} \rho^{p'}}\nabla_{\gamma}\left(\frac{f}{e^{-\alpha_{p,Q,\gamma} \rho^{p'}}}\right)\right)dz
\end{gather}\] and the proof is complete. ◻
5.3 Proof of the stability of the Heisenberg-Pauli-Weyl inequality–Proof of Theorem 26↩︎
Proof of Theorem 26. Let \(0\neq f\in C^{\infty}_{0}(\mathbb{R}^n)\). From Corollary 24, we have \[\begin{gather}
\label{step132stab}
\left(\int_{\mathbb{R}^{n}}|\nabla f|^{p}dz\right)\left(\int_{\mathbb{R}^{n}}|z|^{p'}|f|^{p}dz\right)-\left(\frac{n}{p}\right)^{p}\left(\int_{\mathbb{R}^{n}}|f|^{p}dz\right)^{p}
\\=\left(\int_{\mathbb{R}^{n}}|z|^{p'}|f|^{p}dz\right)^{p-1}\int_{\mathbb{R}^{n}}C_{p}\left(\nabla f,e^{-\alpha_{p,n}|z|^{p'}}\nabla\left(\frac{f}{e^{-\alpha_{p,n}|z|^{p'}}}\right)\right)dz,
\end{gather}\tag{62}\] where \(1<p<\infty\) and \(p'=\frac{p}{p-1}>0\) and \(\alpha_{p,n}=\frac{n}{p}\frac{p-1}{p}\frac{\int_{\mathbb{R}^{n}}|f|^{p}dz}{\int_{\mathbb{R}^{n}}|z|^{p'}|f|^{p}dz}\). Using Lemma \(\ref{lem1}\) to (62 ), we
get \[\begin{gather}
\label{stab32main32part}
\left(\int_{\mathbb{R}^{n}}|\nabla f|^{p}dz\right)\left(\int_{\mathbb{R}^{n}}|z|^{p'}|f|^{p}dz\right)-\left(\frac{n}{p}\right)^{p}\left(\int_{\mathbb{R}^{n}}|f|^{p}dz\right)^{p}\\\geq
c_{1}(p)\left(\int_{\mathbb{R}^{n}}|z|^{p'}|f|^{p}dz\right)^{p-1}\int_{\mathbb{R}^n}\left|e^{-\alpha_{p,n}|z|^{p'}}\nabla\left(\frac{f}{e^{-\alpha_{p,n}|z|^{p'}}}\right)\right|^{p}dz
\\=c_{1}(p)\left(\int_{\mathbb{R}^{n}}|z|^{p'}|f|^{p}dz\right)^{p-1}\int_{\mathbb{R}^n}\left|\nabla\left(\frac{f}{e^{-\alpha_{p,n}|z|^{p'}}}\right)\right|^{p}e^{-p\alpha_{p,n}|z|^{p'}}dz
\end{gather}\tag{63}\] for \(2 \leq p < \infty\). Let us define \[\begin{align}
u:=\frac{f}{e^{-\alpha_{p,n}|z|^{p'}}}.
\end{align}\] Then, we get that \[\begin{align}
\label{almost32stability}
\int_{\mathbb{R}^n}\left|\nabla\left(\frac{f}{e^{-\alpha_{p,n}|z|^{p'}}}\right)\right|^{p}e^{-p\alpha_{p,n}|z|^{p'}}dz=\int_{\mathbb{R}^n}^{}\left|\nabla u\right|^{p}e^{-p\alpha_{p,n}|z|^{p'}}dz,
\end{align}\tag{64}\] where \(2\leq p<\infty\), \(\alpha_{p,n}>0\) and \(p'=\frac{p}{p-1}>1\). Now let us recall the weighted
\(L^{p}\)-Poincaré inequality for the log-concave probability measure by Do, Flynn, Lam and Lu [72]: for some \(\delta>0\), \(n-p>\mu\geq0\) and \(\alpha\geq\frac{n-p-\mu}{n-p}\), we have for \(u\in
C^{\infty}_{0}(\mathbb{R}^n\backslash\{0\})\) that \[\begin{align}
\label{lu32poincare}
\int_{\mathbb{R}^n}\frac{|\nabla u|^{p}}{|z|^{\mu}}e^{-\delta|z|^{\alpha}}dz\geq C(n,p,\alpha,\delta,\mu)\inf_{c}\int_{\mathbb{R}^n}\frac{|u-c|^{p}}{|z|^{\frac{n\mu}{n-p}}}e^{-\delta|z|^{\alpha}}dz,
\end{align}\tag{65}\] where \(C(N,p,\alpha,\delta,\mu)>0\) is some universal positive constant. Setting \(\mu=0\), \(\delta=p
\alpha_{p,n}>0\), \(\alpha=p'>1\), in (65 ), we get that for \(u\in C^{\infty}_{0}(\mathbb{R}^n)\)\[\begin{align}
\int_{\mathbb{R}^n}|\nabla u|^{p}e^{-p \alpha_{p,n}|z|^{p'}}dz\geq C(n,p)\inf_{c}\int_{\mathbb{R}^n}|u-c|^{p}e^{-p \alpha_{p,n}|z|^{p'}}dz,
\end{align}\] which when applied to (64 ) gives \[\begin{align}
\label{stab32last}
\int_{\mathbb{R}^n}\left|\nabla\left(\frac{f}{e^{-\alpha_{p,n}|z|^{p'}}}\right)\right|^{p}e^{-p\alpha_{p,n}|z|^{p'}}dz \nonumber
&=\int_{\mathbb{R}^n}^{}\left|\nabla u\right|^{p}e^{-p\alpha_{p,n}|z|^{p'}}dz \nonumber
\\&\geq C(n,p)\inf_{c}\int_{\mathbb{R}^n}|u-c|^{p}e^{-p \alpha_{p,n}|z|^{p'}}dz \nonumber
\\&=C(n,p)\inf_{c}\int_{\mathbb{R}^n}\left|\frac{f}{e^{-\alpha_{p,n}|z|^{p'}}}-c\right|^{p}e^{-p \alpha_{p,n}|z|^{p'}}dz \nonumber
\\&=C(n,p)\inf_{c}\int_{\mathbb{R}^n}\left|f-ce^{- \alpha_{p,n}|z|^{p'}}\right|^{p}dz.
\end{align}\tag{66}\] Using (66 ) in (63 ), we finally obtain \[\begin{gather}
\left(\int_{\mathbb{R}^{n}}|\nabla f|^{p}dz\right)\left(\int_{\mathbb{R}^{n}}|z|^{p'}|f|^{p}dz\right)-\left(\frac{n}{p}\right)^{p}\left(\int_{\mathbb{R}^{n}}|f|^{p}dz\right)^{p}\\\geq
c_{1}(p)C(n,p)\left(\int_{\mathbb{R}^{n}}|z|^{p'}|f|^{p}dz\right)^{p-1}\inf_{c}\int_{\mathbb{R}^n}\left|f-ce^{- \alpha_{p,n}|z|^{p'}}\right|^{p}dz
\\=\widetilde{C}(n,p)\left(\int_{\mathbb{R}^{n}}|z|^{p'}|f|^{p}dz\right)^{p-1}\inf_{c}\int_{\mathbb{R}^n}\left|f-ce^{- \alpha_{p,n}|z|^{p'}}\right|^{p}dz.
\end{gather}\] ◻
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This research is funded by the Committee of Science of the Ministry of Science and Higher Education of the Republic of Kazakhstan (Grant No. AP23490970).↩︎