Let $Ω$ be a domain in $\Ri^d$ with boundary $Γ$${\!,}$ $d_Γ$ the Euclidean distance to the boundary and $H=-\divv(C\,\nabla)$ an elliptic operator with $C=(\,c_{kl}\,)>0$ where $c_{kl}=c_{lk}$ are real, bounded, Lipschitz functions. We assume that $C\sim c\,d_Γ^{\,δ}$ as $d_Γ\to0$ in the sense of asymptotic analysis where $c$ is a strictly positive, bounded, Lipschitz function and $δ\geq0$. We also assume that there is an $r>0$ and a $ b_{δ,r}>0$ such that the weighted Hardy inequality \[ \int_{Γ_{\!\!r}} d_Γ^{\,δ}\,|\nabla ψ|^2\geq b_{δ,r}^{\,2}\int_{Γ_{\!\!r}} d_Γ^{\,δ-2}\,| ψ|^2 \] is valid for all $ψ\in C_c^\infty(Γ_{\!\!r})$ where $Γ_{\!\!r}=\{x\inΩ: d_Γ(x)<r\}$. We then prove that the condition $(2-δ)/2<b_δ$ is sufficient for the essential self-adjointness of $H$ on $C_c^\infty(Ω)$ with $b_δ$ the supremum over $r$ of all possible $b_{δ,r}$ in the Hardy inequality. This result extends all known results for domains with smooth boundaries and also gives information on self-adjointness for a large family of domains with rough, e.g.\ fractal, boundaries.
We consider the problem of essential self-adjointness of the drift-diffusion operator $H=-\frac{1}ρ\nabla\cdot ρ\mathbb D\nabla +V$ on domains $Ω\subset \mathbb{R}^d$ with $\mathcal{C}^2$-boundary $\partial Ω$ and for large classes of coefficients $ρ,\; \mathbb{D}$ and $V$. We give criteria showing how the behavior as $x \rightarrow \partial Ω$ of these coefficients balances to ensure essential self-adjointness of $H$. On the way we prove a weak anisotropic Hardy inequality which is of independent interest.
In this paper we study various forms of the Hardy inequality for Dunkl operators, including the classical inequality, $L^p$ inequalities, an improved Hardy inequality, as well as the Rellich inequality and a special case of the Caffarelli-Kohn-Nirenberg inequality. As a consequence, one-dimensional many-particle Hardy inequalities for generalised root systems are proved, which in the particular case of root systems $A_{N-1}$ improve some well-known results.
We give a direct proof of fractional Hardy inequality by means of Littlewood-Paley decomposition and properties of singular homogeneous kernels of degree -$d$. A refinement when $q>2$ is proved.
Using elementary techniques, we prove sharp anisotropic Hardy-Littlewood inequalities for positive multilinear forms. In particular, we recover an inequality proved by F. Bayart in 2018.
The Hardy-Littlewood-Pólya inequality of majorization is extended to the framework of ordered Banach spaces. Several applications illustrating our main results are also included.
In this paper, we study the asymptotic behaviour of the sharp constant in discrete Hardy and Rellich inequality on the lattice $\mathbb{Z}^d$ as $d \rightarrow \infty$. In the process, we proved some Hardy-type inequalities for the operators $Δ^m$ and $\nabla(Δ^m)$ for non-negative integers $m$ on a $d$ dimensional torus. It turns out that the sharp constant in discrete Hardy and Rellich inequality grows as $d$ and $d^2$ respectively as $ d \rightarrow \infty$.
In this note we present a version of Hardy's inequality on a measure space $(X,μ)$ endowed with a measurable function $N\colon X\to \mathbb R$ which replaces the absolute value on $\mathbb R$ or $\mathbb R^n$, and, more generally, the distance function from a given point when $X$ is a metric space.
In this paper, using blow-up analysis, we prove a singular Hardy-Morser-Trudinger inequality, and find its extremal functions. Our results extend those of Wang-Ye (Adv. Math. 2012), Yang-Zhu ( Ann. Glob. Anal. Geom. 2016), Csató- Roy (Calc. Var. 2015), and Yang-Zhu (J. Funct. Anal. 2017).
Hardy's type inequality for the over critical exponent associated with the Dunkl transform
\begin{abstract} For the Hardy space $H^p(\mathbb{R}^{d})$, $ 0<p\leq 1,$ we shall improve a Hardy's type inequality associated with Dunkl transform respect to the measures $dμ_{k}$ homogeneous of degree $γ,$ on the strip $(2γ+d)(2-p)\leqσ<2γ+d+p(N+1),$ where $N =[(2γ+d)(1/p-1)]$ is the greatest integer not exceeding $(2γ+d)(1/ p -1).$ \end{abstract}
In this paper we focus our attention on an embedding result for a weighted Sobolev space that involves as weight the distance function from the boundary taken with respect to a general smooth gauge function $F$. Starting from this type of inequalities we prove some refined Hardy-type inequalities.
In this paper, a new two-dimensional Hardy type inequality is given in terms of pseudo-analysis dealing with set-valued functions. The first one is given for a pseudo-integral of set-valued function where pseudo-addition and pseudo-multiplication are constructed by a monotone continuous function $g:[0, \infty ]\to[0, \infty]$. Another is given by the semiring $([0, 1], \max, \odot)$ where pseudo-multiplication is generated by an increasing continuous function.
We prove a sharp integral inequality that generalizes the well known Hardy type integral inequality for negative exponents. We also give sharp applications in two directions for Muckenhoupt weights on R. This work refines the results that appear in [9].
In this paper we describe the Euler semigroup $\{e^{-t\mathbb{E}^{*}\mathbb{E}}\}_{t>0}$ on homogeneous Lie groups, which allows us to obtain various types of the Hardy-Sobolev and Gagliardo-Nirenberg type inequalities for the Euler operator $\mathbb{E}$. Moreover, the sharp remainder terms of the Sobolev type inequality, maximal Hardy inequality and $|\cdot|$-radial weighted Hardy-Sobolev type inequality are established.
In this paper we study an extension problem for the Laplace-Beltrami operator on Riemannian symmetric spaces of noncompact type and use the solution to prove Hardy-type inequalities for fractional powers of the Laplace-Beltrami operator. Next, we study the mapping properties of the extension operator. In the last part we prove Poincaré-Sobolev inequalities on these spaces.
Equivalent and attained version of Hardy's inequality in $\mathbb{R}^n$
研究了全空间$\mathbb{R}^n$中的Hardy不等式与Sobolev-Lorentz空间中的嵌入不等式之间的联系.特别地，我们完善了[A. Alvino，Sulla diseguaglianza di Sobolev in spazi di Lorentz，（1977）] and [G. Talenti，An inequality between $u^*$ and $|{\rm{grad}} u^*|$，（1992）]通过建立同样在$p < q<\infty$范围内的Sobolev-Lorentz拟线性方程$\|\nabla\,\cdot\,\|_{p,q}$的最优嵌入不等式，自Alvino的工作以来，该范围基本上保持开放。最佳嵌入常数的可达性也进行了研究，以及极限情况下，当$q=\infty$。在这里，我们惊奇地发现Hardy不等式等价于相应的Sobolev-Marcinkiewicz嵌入不等式.此外，后者原来是达到所谓的“鬼”极值函数[Brezis-Vázquez，爆破解决方案的一些非线性椭圆问题，（1977）]，在鲜明对比的哈代不等式，这是从来没有达到。从这个意义上说，我们的功能方法似乎比经典的索博列夫设置更自然，回答了Brezis和Vázquez提出的问题。
In this note besides two abstract versions of the Vitali Covering Lemma an abstract Hardy-Littlewood Maximal Inequality, generalizing weak type (1,1) maximal function inequality, associated to any outer measure and a family of subsets on a set is introduced. The inequality is (effectively) satisfied if and only if a special numerical constant called Hardy-Littelwood maximal constant is finite. Two general sufficient conditions for the finiteness of this constant are given and upper bounds for this constant associated to the family of (centered) balls in homogeneous spaces, family of dyadic cub…
In this paper, we estimate an operator norm of dilation operators on block spaces ($\mathfrak{B}_{r,α}(\mathbb{Q}_p)$) over $p$-adic field. With this estimate, we establish the boundedness of $p$-adic Hardy-Hilbert type integral operator on $\mathfrak{B}_{r,α}(\mathbb{Q}_p)$. Moreover as application to our result, we obtain the $p$-adic Hilbert inequality, $p$-adic Hardy inequality and $p$-adic Hardy-Littlewood-Pólya inequality on $\mathfrak{B}_{r,α}(\mathbb{Q}_p)$.
This note contains two simple observations. First, by the weak factorization of product $H^1$ (Ferguson--Lacey, Lacey--Terwilleger), we obtain a multi-parameter analogue of Hardy's inequality. Second, as a dual statement, the Fourier transform of an essentially bounded function belongs to a certain product $\mathrm{BMO}$-Sobolev space.
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.
In this article, we give a short proof of Hardy's inequality for Hermite expansions of functions in the classical Hardy spaces $H^p({\mathbb R^n})$, by using an atomic decomposition of the Hardy spaces associated with the Hermite operators. When the space dimension is $1$, we obtain a new estimate of Hardy's inequality for Hermite expansions in $H^p({\mathbb R})$ for the range $0<p<1.$
The Hardy-Littlewood-Polya inequality of majorization is extended for the ω-m-star-convex functions to the framework of ordered Banach spaces. Several open problems which seem of interest for further extensions of the Hardy-Littlewood-Polya inequality are also included.
A method of proving Hardy's type inequality for orthogonal expansions is presented in a rather general setting. Then sharp multi-dimensional Hardy's inequality associated with the Laguerre functions of convolution type is proved for type index $\al\in[-1/2,\infty)^d$. The case of the standard Laguerre functions is also investigated. Moreover, the sharp analogues of Hardy's type inequality involving $L^1$ norms in place of $H^1$ norms are obtained in both settings.
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.
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 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 $q \geq \max(p_1,p_2)$.
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 …
