Megumi Sano, Futoshi Takahashi
We prove a critical Hardy inequality on the half-space by using the harmonic transplantation. Also we give an improvement of the subcritical Hardy inequality on the half-space, which converges to the critical Hardy inequality. Sobolev type inequalities are also discussed.
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The Moser-Trudinger embedding has been generalized in [Adimurthi A.; Sandeep K., A singular Moser-Trudinger embedding and its applications, \textit{NoDEA Nonlinear Differential Equations Appl.}, 13 (2007), no. 5-6, 585--603] to the following weighted version: if $Ω\subset\mathbb{R}^n$ is bounded, $ω_{n-1}$ is the $\mathcal{H}^{n-1}$ measure of the unit sphere, then for $α>0$ and $β\in [0,n)$, $$ \sup_{u\in\mathcal{B}_1}\int_Ω\frac{e^{α|u|^{n/(n-1)}}}{|x|^β}\leq C \ \Leftrightarrow \ \fracα{α_n}+\fracβ{n}\leq1,\qquad $$ where $α_n=n\cnn$ and $\mathcal{B}_1 = \left\{ u \in W_0^{1, n}(Ω) \ | \ \i…
We prove several Sobolev inequalities, which are then used to establish a fractional Hardy-Sobolev- Maz'ya inequality on the upper halfspace.
We give improvements and generalizations of both the classical Hardy inequality and the geometric Hardy inequality based on the divergence theorem. Especially, our improved Hardy type inequality derives both two Hardy type inequalities with best constants. Besides, we improve two Rellich type inequalities by using the improved Hardy type inequality.
We present the best constant and the existence of extremal functions for an Improved Hardy-Sobolev inequality. We prove that, under a proper transformation, this inequality is equivalent to the Sobolev inequality in $\mathbb{R}^N$. We also discuss the connection of the related functional spaces and as a result we obtain some Caffarelli - Kohn - Nirenberg inequalities. Our starting point is the existence of a minimizer for the Bliss' inequality and the indirect dependence of the Hardy inequality at the origin.
We obtain new variants of weighted Gagliardo-Nirenberg interpolation inequalities in Orlicz spaces, as a consequence of weighted Hardy-type inequalities. The weights we consider need not be doubling.
In this article, we derive the existence of positive solutions of a semi-linear, non-local elliptic PDE, involving a singular perturbation of the fractional laplacian, coming from the fractional Hardy-Sobolev-Maz'ya inequality, derived in this paper. We also derive symmetry properties and a precise asymptotic behaviour of solutions.
We prove a characterization of Hardy's inequality in Sobolev-Slobodecki\uı spaces in terms of positive local weak supersolutions of the relevant Euler-Lagrange equation. This extends previous results by Ancona and Kinnunen & Korte for standard Sobolev spaces. The proof is based on variational methods.
For the fractional Laplacian we give Hardy inequality which is optimal in $L^p$ for $1<p<\infty$. As an application, we explicitly characterize the contractivity of the corresponding Feynman-Kac semigroups on $L^p$.
The classical Hardy inequality holds in Sobolev spaces $W_0^{1,p}$ when $1\le p< N$. In the limiting case where $p=N$, it is known that by adding a logarithmic function to the Hardy potential, some inequality which is called the critical Hardy inequality holds in $W_0^{1,N}$. In this note, in order to give an explanation of appearance of the logarithmic function at the potential, we derive the logarithmic function from the classical Hardy inequality with the best constant via some limiting procedure as $p \nearrow N$. And we show that our limiting procedure is also available for the classical …
We show that the sharp constant in the Hardy-Littlewood-Sobolev inequality can be derived using the method that we employed earlier for a similar inequality on the Heisenberg group. The merit of this proof is that it does not rely on rearrangement inequalities; it is the first one to do so for the whole parameter range.
We obtain the sharp constant for the Hardy-Sobolev inequality involving the distance to the origin. This inequality is equivalent to a limiting Caffarelli-Kohn-Nirenberg inequality. In three dimensions, in certain cases the sharp constant coincides with the best Sobolev constant.
We compute the explicit sharp constants of Hardy inequalities in the cone $\mathbb{R}_{k_+}^{n}:=\mathbb{R}^{n-k}\times (\mathbb{R}_{+})^{k}=\{(x_{1},...,x_{n})|x_{n-k+1}>0,...,x_{n}>0\}$ with $1\leq k\leq n$. Furthermore, the spherical harmonic decomposition is given for a function $u\in C^{\infty}_{0}(\mathbb{R}_{k_+}^{n})$. Using this decomposition and following the idea of Tertikas and Zographopoulos, we obtain the Filippas-Tertikas improvement of the Hardy inequality.
We provide sufficient conditions for boundary Hardy inequality to hold in bounded Lipschitz domains, complement of a point (the so-called point Hardy inequality), domain above the graph of a Lipschitz function, the complement of a bounded Lipschitz domain in fractional Orlicz-Sobolev setting. As a consequence, we get sufficient conditions for regional fractional Orlicz Poincaré inequality in bounded Lipschitz domains. Necessary conditions for fractional Orlicz Hardy and regional fractional Orlicz Poincaré inequalities are also given for bounded Lipschitz domains. Various sufficient conditions …
We give a new proof of certain cases of the sharp HLS inequality. Instead of symmetric decreasing rearrangement it uses the reflection positivity of inversions in spheres. In doing this we extend a characterization of the minimizing functions due to Li and Zhu.
We prove a fractional Hardy-type inequality for vector fields over the half space based on a modified fractional semi-norm. A priori, the modified semi-norm is not known to be equivalent to the standard fractional semi-norm and in fact gives a smaller norm, in general. As such, the inequality we prove improves the classical fractional Hardy inequality for vector fields. We will use the inequality to establish the equivalence of a space of functions (of interest) defined over the half space with the classical fractional Sobolev spaces, which amounts to proving a fractional version of the classi…
The main aim of this paper to provide several scales of equivalent conditions for the bilinear Hardy inequalities in the case $1< q, p_1, p_2<\infty$ with $q \geq \max(p_1,p_2)$.
In this paper we establish improved Hardy and Rellich type inequalities on Riemannian manifold $M$. Furthermore, we also obtain sharp constant for the improved Hardy inequality and explicit constant for the Rellich inequality on hyperbolic space $\mathbb{H}^n$.
We consider the optimizers $u$ in the Hardy-Sobolev inequality for the space $\dot{W}^{s,p}({\mathbb R}^N)$ with order of differentiability $s\in ]0,1[$. After proving existence through concentration-compactness, we derive the pointwise asymptotic $u(x)\simeq |x|^{-\frac{N-ps}{p-1}}$ for large $|x|$ and the summability estimate $u\in \dot{W}^{s,γ}({\mathbb R}^N)$ for all $γ>\frac{N(p-1)}{N-s}$. These estimates are optimal in the limit $s\to 1^-$, in which case optimizers are explicitly known.
We prove some sharp Hardy inequalities for domains with a spherical symmetry. In particular, we prove an inequality for domains of the unit $n$-dimensional sphere with a point singularity, and an inequality for functions defined on the half-space $\R_+^{n+1}$} vanishing on the hyperplane $\{x_{n+1}=0\}$, with singularity along the $x_{n+1}$-axis. The proofs rely on a one-dimensional Hardy inequality involving a weight function related to the volume element on the sphere, as well as on symmetrization arguments. The one-dimensional inequality is derived in a general form.
Intrinsic Hardy inequality for fractional Kolmogorov operator
本文研究了当漂移项对原点的吸引力强度取临界值时，非局部扩散算子的Hardy不等式。
A Stein-Weiss type inequality on Carnot groups is established by proving the boundedness of an integral operator and the Hardy-Littlewood-Sobolev type inequality on Carnot groups is also derived.
We establish a sharp Adams-type inequality invoking a Hardy inequality for any even dimension. This leads to a non compact Sobolev embedding in some Orlicz space. We also give a description of the lack of compactness of this embedding in the spirit of \cite{Bahouri}.
Multilinear trace restriction inequalities are obtained for Hardy's inequality. More generally, detailed development is given for new multilinear forms for Young's convolution inequality, and a new proof for the multilinear Hardy-Littlewood-Sobolev inequality, and its realization on hyperbolic space.
In this note, we obtain a reverse version of the integral Hardy inequality on metric measure spaces. Moreover, we give necessary and sufficient conditions for the weighted reverse Hardy inequality to be true. The main tool in our proof is a continuous version of the reverse Minkowski inequality. Also, we present some consequences of the obtained reverse Hardy inequality on the homogeneous groups, hyperbolic spaces and Cartan-Hadamard manifolds.
We give a simple proof of Hardy's inequality, based on the logarithmic Caccioppoli estimate for p-superharmonic functions in several variables.
Some Hardy and Rellich type inequalities for affine connections
该文研究完备非紧黎曼流形上仿射联络的多种Hardy型及相关不等式，所得结果改进并包含了诸多已有结果作为特例。
We prove a one-dimensional Hardy inequality on the halfline with sharp constant, which improves the classical form of this inequality. As a consequence of this new inequality we can rederive known doubly weighted Hardy inequalities. Our motivation comes from the theory of Schrödinger operators and we explain the use of Hardy inequalities in that context.
The Hardy-Sobolev trace inequality can be obtained via Harmonic extensions on the half-space of the Stein and Weiss weighted Hardy-Littlewood-Sobolev inequality. In this paper we consider a bounded domain and study the influence of the boundary mean curvature in the Hardy-Sobolev trace inequality on the underlying domain. We prove existence of minimizers when the mean curvature is negative at the singular point of the Hardy potential.
We compute the optimal constant for a generalized Hardy-Sobolev inequality, and using the product of two symmetrizations we present an elementary proof of the symmetries of some optimal functions. This inequality was motivated by a nonlinear elliptic equation arising in astrophysics.
In this work we improve the sharp Hardy inequality in the case $p>n$ by adding an optimal weighted Hoelder semi-norm. To achieve this we first obtain a local improvement. We also obtain a refinement of both the Sobolev inequality for $p>n$ and the Hardy inequality, the latter having the best constant.
We show existence of minimizers for the Hardy-Sobolev-Maz'ya inequality in $R^{m+n}\setminus\R^n$ for $m=1$ and $n>2$ or for $m>2$ and $n>0$.
We use the Morrey norm estimate for the imaginary power of the Laplacian to prove an interpolation inequality for the fractional power of the Laplacian on Morrey spaces. We then prove a Hardy-type inequality and use it together with the interpolation inequality to obtain a Heisenberg-type inequality in Morrey spaces.
We prove an improved version of the trace-Hardy inequality, so-called Kato's inequality, on the half-space in Finsler context. The resulting inequality extends the former one obtained by \cite{AFV} in Euclidean context. Also we discuss the validity of the same type of inequalities on open cones.
We prove Hardy's inequality for the fractional powers of the generalized sublaplacian and the fractional powers of the Grushin operator. We also find an integral representation and a ground state representation for the fractional powers of generalized sublaplacian.
In this paper we prove sharp Hardy inequalities by using Maximal function theory. Our results improve and extend the well-known results of G.Hardy \cite{Ha04}, T.Cazenave \cite {Ca03}, J.-Y.Chemin\cite {Ch06} and T.Tao\cite {TT06}.
In this paper, we establish several new anisotropic Hardy-Sobolev inequalities in mixed Lebesgue spaces and mixed Lorentz spaces, which covers many known corresponding results. As an application, this type of inequalities allows us to generalize some regularity criteria of the 3D axisymmetric Navier-Stokes equations.
We prove the attainability of the best constant in the fractional Hardy--Sobolev inequality with boundary singularity for the Spectral Dirichlet Laplacian. The main assumption is the average concavity of the boundary at the origin.
We prove, using elementary methods of complex analysis, the following generalization of the isoperimetric inequality: if $p\in\re$, $Ω\subset\re^2$ then the inequality $$ \left(\frac{|Ω|}π\right)^{\frac{p+1}{2}}\leq\frac{1}{2π}\int_{\delomega}|x|^pdσ(x) $$ holds true under appropriate assumptions on $Ω$ and $p.$ This solves an open problem arising in the context of isoperimetric problems with density and poses some new ones (for instance generalizations to $\re^n$). We prove the equivalence with a Hardy-Sobolev inequality, giving the best constant, and generalize thereby the equivalence betwee…
Let $Ø$ be a smooth bounded domain in $\R^N$ with $N\ge 1$. In this paper we study the Hardy-Poincaré inequality with weight function singular at the boundary of $Ø$. In particular we provide sufficient and necessary conditions on the existence of minimizers.
In this paper, we prove Hardy-Leray inequality for three-dimensional solenoidal (i.e., divergence-free) fields with the best constant. To derive the best constant, we impose the axisymmetric condition only on the swirl components. This partially complements the former work by O. Costin and V. Maz'ya \cite{Costin-Mazya} on the sharp Hardy-Leray inequality for axisymmetric divergence-free fields.
In this paper, we study the asymptotic behavior of radial extremal functions to an inequality involving Hardy potential and critical Sobolev exponent. Based on the asymptotic behavior at the origin and the infinity, we shall deduce a strict inequality between two best constants. Finally, as an application of this strict inequality, we consider the existence of nontrivial solution of a quasilinear Brezis-Nirenberg type problem with Hardy potential and critical Sobolev exponent.
We study linear and non-linear equations related to the fractional Hardy--Sobolev inequality. We prove nondegeneracy of ground state solutions to the basic equation and investigate existence and qualitative properties, including symmetry of solutions to some perturbed equations.
We prove an optimal Hardy inequality for the fractional Laplacian on the half-space.
We give a~new proof of the known criteria for the inequality \begin{equation*} \left(\int_{0}^{\infty}\left(\int_{0}^{t}f\right)^{q}w(t)\,dt\right)^{\frac{1}{q}} \leq C \left(\int_{0}^{\infty}f^{p}v\right)^{\frac{1}{p}}. \end{equation*} The innovation is in the elementary nature of the proof and its versatility.
We consider the second best constant in the Hardy-Sobolev inequality on a Riemannian manifold. More precisely, we are interested with the existence of extremal functions for this inequality. This problem was tackled by Djadli-Druet [5] for Sobolev inequalities. Here, we establish the corresponding result for the singular case. In addition, we perform a blow-up analysis of solutions Hardy-Sobolev equations of minimizing type. This yields informations on the value of the second best constant in the related Riemannian functional inequality.
We prove Hardy inequalities for the conformally invariant fractional powers of the sublaplacian on the Heisenberg group $\mathbb{H}^n$. We prove two versions of such inequalities depending on whether the weights involved are non-homogeneous or homogeneous. In the first case, the constant arising in the Hardy inequality turns out to be optimal. In order to get our results, we will use ground state representations. The key ingredients to obtain the latter are some explicit integral representations for the fractional powers of the sublaplacian and a generalized result by M. Cowling and U. Haageru…
We calculate the regional fractional Laplacian on some power function on an interval. As an application, we prove Hardy inequality with an extra term for the fractional Laplacian on the interval with the optimal constant. As a result, we obtain the fractional Hardy inequality with best constant and an extra lower-order term for general domains, following the method developed by M. Loss and C. Sloane [arXiv:0907.3054v1 [math.AP]]
We prove the self-improvement of a pointwise $p$-Hardy inequality. The proof relies on maximal function techniques and a characterization of the inequality by curves.
We consider the best constant in the Rellich-Hardy inequality (with a radial power weight) for curl-free vector fields on $\mathbb{R}^N$, originally found by Tertikas-Zographopoulos \cite{Tertikas-Z} for unconstrained fields. This inequality is considered as an intermediate between Hardy-Leray and Rellich-Leray inequalities. Under the curl-free condition, we compute the new explicit best constant in the inequality and prove the non-attainability of the constant. This paper is a sequel to \cite{CF_MAAN,CF_Re}.
In this article, we prove a weighted Hardy inequality for $1<p<+\infty$ and dimension $d\ge1$. If $p>d$, then we can deduce from our weighted Hardy inequality a Poincaré inequality. The proof of the weighted Hardy inequality is based on the method of vector fields firstly introduced by Mitidieri \cite{MR1769903}. By the same method, we show for $1<p<+\infty$ and $d\ge1$ that weighted Caffarelli-Kohn-Nirenberg inequalities hold true. As an application of our weighted Hardy inequality, we prove a non-existence result for a p-Kolmogorov parabolic equation.
Two-sided bounds for the efficiency of the torsion function are obtained in terms of the square of the distance to the boundary function under the hypothesis that the Dirichlet Laplacian satisfies a strong Hardy inequality. Localisation properties of the torsion function are obtained under that hypothesis. An example is analysed in detail.
In the euclidean space, Sobolev and Hardy-Littlewood-Sobolev inequalities can be related by duality. In this paper, we investigate how to relate these inequalities using the flow of a fast diffusion equation in dimension $d\ge3$. The main consequence is an improvement of Sobolev's inequality when $d\ge5$, which involves the various terms of the dual Hardy-Littlewood-Sobolev inequality. In dimension $d=2$, Onofri's inequality plays the role of Sobolev's inequality and can also be related to its dual inequality, the logarithmic Hardy-Littlewood-Sobolev inequality, by a super-fast diffusion equat…
We prove all the maximizers of the sharp Hardy-Littlewood-Sobolev inequality are smooth. More generally, we show all the nonnegative critical functions are smooth, radial with respect to some points and strictly decreasing in the radial direction. In particular, this resolves all the cases left open by previous works of Chen, Li and Ou on the corresponding integral systems.
In this paper we deal with nonnegative distributional supersolutions for a class of linear elliptic equations involving inverse-square potentials and logarithmic weights. We prove sharp nonexistence results.
In this paper, we prove that the distance function of an open connected set in $\mathbb R^{n+1}$ with a $C^{2}$ boundary is superharmonic in the distribution sense if and only if the boundary is {\em weakly mean convex}. We then prove that Hardy inequalities with a sharp constant hold on {weakly mean convex} $C^{2}$ domains. Moreover, we show that the {weakly mean convexity} condition cannot be weakened. We also prove various improved Hardy inequalities on mean convex domains along the line of Brezis-Marcus \cite{BM}.
Let $Ω$ be a smooth bounded domain in $\mahbb R^N$ with $N\ge 3$ and let $Σ_k$ be a closed smooth submanifold of $δΩ$ of dimension $1\le k\le N-2$. In this paper we study the weighted Hardy inequality with weight function singular on $Σ_k$. In particular we provide sufficient and necessary conditions for existence of minimizers.
In this work, we have proved a version of the Hardy-Littlewood-Sobolev inequality for variable exponents. After we use the variational method to establish the existence of solution for a class of Choquard equations involving the $p(x)$-Laplacian operator.
We develop a geometric framework for Hardy's inequality on a bounded domain when the functions do vanish only on a closed portion of the boundary.
In this paper we obtain an inequality on the unit disc $B$ in the plane, which improves the classical Moser-Trudinger inequality and the classical Hardy inequality at the same time. Namely, there exists a constant $C_0>0$ such that \[ \int_B e^{\frac {4πu^2}{H(u)}} dx \le C_0 < \infty, \quad \forall\; u\in C^\infty_0(B),\] where $$H(u) := \int_B |\n u|^2 dx - \int_B \frac {u^2}{(1-|x|^2)^2} dx.$$ This inequality is a two dimensional analog of the Hardy-Sobolev-Maz'ya inequality in higher dimensions, which was recently intensively studied. We also prove that the supremum is achieved in a suitab…
There are at least two directions concerning the extension of classical sharp Hardy-Littlewood-Sobolev inequality: (1) Extending the sharp inequality on general manifolds; (2) Extending it for the negative exponent $λ=n-α$ (that is for the case of $α>n$). In this paper we confirm the possibility for the extension along the first direction by establishing the sharp Hardy-Littlewood-Sobolev inequality on the upper half space (which is conformally equivalent to a ball). The existences of extremal functions are obtained; And for certain range of the exponent, we classify all extremal functions via…
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