Weighted Hardy Type Inequalities Research Articles

Weighted Hardy type inequalities with Robin boundary conditions

In this paper, we establish a general weighted Hardy type inequality for the $ p- $Laplace operator with Robin boundary condition. We provide various concrete examples to illustrate our results for different weights. Furthermore, we present some Heisenberg-Pauli-Weyl type inequalities with boundary terms on balls centred at the origin with radius $ R $ in $ \mathbb{R} ^n $.

On discrete weighted Hardy type inequalities and properties of weighted discrete Muckenhoupt classes

In this paper, first we prove some new refinements of discrete weighted inequalities with negative powers on finite intervals. Next, by employing these inequalities, we prove that the self-improving property (backward propagation property) of the weighted discrete Muckenhoupt classes holds. The main results give exact values of the limit exponents as well as the new constants of the new classes. As an application, we establish the self-improving property (forward propagation property) of the discrete Gehring class.

A class of weighted Hardy type inequalities in RN\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mathbb {R}^N$$\\end{document}

In the paper we prove the weighted Hardy type inequality 1∫RNVφ2μ(x)dx≤∫RN|∇φ|2μ(x)dx+K∫RNφ2μ(x)dx,\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} \\int _{{{\\mathbb {R}}}^N}V\\varphi ^2 \\mu (x)dx\\le \\int _{\\mathbb {R}^N}|\ abla \\varphi |^2\\mu (x)dx +K\\int _{\\mathbb {R}^N}\\varphi ^2\\mu (x)dx, \\end{aligned}$$\\end{document}for functions φ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\varphi $$\\end{document} in a weighted Sobolev space Hμ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$H^1_\\mu $$\\end{document}, for a wider class of potentials V than inverse square potentials and for weight functions μ\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu $$\\end{document} of a quite general type. The case μ=1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\mu =1$$\\end{document} is included. To get the result we introduce a generalized vector field method. The estimates apply to evolution problems with Kolmogorov operators Lu=Δu+∇μμ·∇u\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\begin{aligned} Lu=\\varDelta u+\\frac{\ abla \\mu }{\\mu }\\cdot \ abla u \\end{aligned}$$\\end{document}perturbed by singular potentials.

A weighted Hardy type inequality and its applications

In this paper we prove a new Hardy type inequality and as a consequence we establish embedding results for a certain Sobolev space E1,p(R+n) defined on the upper half-space. Precisely, for 1<p<n we obtain an embedding from E1,p(R+n) into weighted Lebesgue spaces. In the border-line case p=n, we derive some Trudinger-Moser type inequalities, and in the case p>n we obtain a Morrey's type inequality.

Improved Hardy inequalities and weighted Hardy type inequalities with spherical derivatives

The first purpose of this paper is to set up several enhancements of the Hardy type inequalities on certain subspaces of the Sobolev spaces. The second aim is to explore the role of the spherical derivative in such improvements and to prove some weighted versions of the Hardy inequality with spherical derivative. As applications of our results, we establish several Hardy’s inequalities and the Hardy inequalities with spherical derivatives on half-spaces.

Weighted Hardy Type Inequalities and Parametric Lamb Equation

This paper is devoted to weighted Hardy type inequalities. Using the Bessel functions, we prove one-dimensional inequalities and give some remarks on extensions of the one-dimensional inequalities to $$n$$ -dimensional convex domains with finite inner radius. Constants in those inequalities depend on the roots of parametric Lamb equation for the Bessel function and turn out to be sharp in some particular cases. We establish the inequalities in $$L^{p}$$ spaces, with $$p\in[1,\infty)$$ . Also new $$L_{2}$$ - inequality with sharp constants is obtained.

On the isoperimetric constant, covariance inequalities and $L_{p}$-Poincaré inequalities in dimension one

First, we derive in dimension one a new covariance inequality of $L_{1}-L_{\infty}$ type that characterizes the isoperimetric constant as the best constant achieving the inequality. Second, we generalize our result to $L_{p}-L_{q}$ bounds for the covariance. Consequently, we recover Cheeger’s inequality without using the co-area formula. We also prove a generalized weighted Hardy type inequality that is needed to derive our covariance inequalities and that is of independent interest. Finally, we explore some consequences of our covariance inequalities for $L_{p}$-Poincaré inequalities and moment bounds. In particular, we obtain optimal constants in general $L_{p}$-Poincaré inequalities for measures with finite isoperimetric constant, thus generalizing in dimension one Cheeger’s inequality, which is a $L_{p}$-Poincaré inequality for $p=2$, to any real $p\geq1$.

On Hardy-type inequalities for weighted means

The aim of this paper is to establish weighted Hardy type inequality in a broad family of means. In other words, for a fixed vector of weights $(\lambda_n)_{n=1}^\infty$ and a weighted mean $\mathscr{M}$, we search for the smallest number $C$ such that $$\sum_{n=1}^{\infty} \lambda_n \mathscr{M} \big((x_1,\dots,x_n),(\lambda_1,\dots,\lambda_n)\big) \le C \sum_{n=1}^{\infty} \lambda_nx_n \text{ for all admissible }x.$$ The main results provide a definite answer in the case when $\mathcal{M}$ is monotone and satisfies the weighted counterpart of the Kedlaya inequality. In particular, if $\mathcal{M}$ is symmetric, Jensen-concave, and the sequence $\big(\tfrac{\lambda_n}{\lambda_1+\cdots+\lambda_n}\big)$ is nonincreasing. In addition, it is proved that if $\mathcal{M}$ is a symmetric and monotone mean, then the biggest possible weighted Hardy constant is achieved if $\lambda$ is the constant vector.

General weighted Hardy type inequalities related to Baouendi-Grushin operators

ABSTRACTIn this paper, we derive a sufficient condition on a pair of nonnegative weight functions and w in so that the general weighted Hardy type inequality with a remainder termholds for all . Here and is the sub-elliptic gradient. It is worth emphasizing here that our unifying method may be readily used to recover most of the previously known sharp weighted Hardy and Heisenberg–Pauli–Weyl type inequalities as well as to construct other new inequalities with an explicit constant. Furthermore, we also obtain new results on two-weight Hardy type inequalities with remainder terms on smooth bounded domains in via a non-linear partial differential inequality.

A unified approach to weighted Hardy type inequalities on Carnot groups

We find a simple sufficient criterion on a pair of nonnegative weight functions \begin{document}$V(x)$\end{document} and \begin{document}$W(x) $\end{document} on a Carnot group \begin{document}$\mathbb{G},$\end{document} so that the general weighted \begin{document}$L^{p}$\end{document} Hardy type inequality \begin{document}$\begin{equation*}\int_{\mathbb{G}}V\left( x\right) \left\vert \nabla _{\mathbb{G}}\phi \left(x\right) \right\vert ^{p}dx\geq \int_{\mathbb{G}}W\left( x\right) \left\vert\phi \left( x\right) \right\vert ^{p}dx\end{equation*}$ \end{document} is valid for any \begin{document}$φ ∈ C_{0}^{∞ }(\mathbb{G})$\end{document} and \begin{document}$p>1.$\end{document} It is worth noting here that our unifying method may be readily used both to recover most of the previously known weighted Hardy and Heisenberg-Pauli-Weyl type inequalities as well as to construct other new inequalities with an explicit best constant on \begin{document}$\mathbb{G}.$\end{document} We also present some new results on two-weight \begin{document}$L^{p}$\end{document} Hardy type inequalities with remainder terms on a bounded domain \begin{document}$Ω$\end{document} in \begin{document}$\mathbb{G}$\end{document} via a differential inequality.

Weighted Hardy type inequalities for supremum operators on the cones of monotone functions

The complete characterization of the weighted $L^{p}-L^{r}$ inequalities of supremum operators on the cones of monotone functions for all $0< p,r\leq \infty$ is given.

Weighted Hardy–Littlewood–Sobolev inequalities on the upper half space

In this paper, we establish a weighted Hardy–Littlewood–Sobolev (HLS) inequality on the upper half space using a weighted Hardy type inequality on the upper half space with boundary term, and discuss the existence of extremal functions based on symmetrization argument. As an application, we can show a weighted Sobolev–Hardy trace inequality with [Formula: see text]-biharmonic operator.

Weighted Hardy and Rellich type inequalities on Riemannian manifolds

In this paper we present new results on two-weight Hardy, Hardy–Poincare and Rellich type inequalities with remainder terms on a complete noncompact Riemannian Manifold M. The method we use is flexible enough to obtain more weighted Hardy type inequalities. Our results improve and include many previously known results as special cases.

Sharp constants in hardy type inequalities

We prove new weighted Hardy type inequalities with sharp constants and describe their applications to inequalities in multidimensional domains.

Weighted hardy type integral inequalities involving many functions

The object of this paper is to derive some new weighted Hardy type integral inequalities involving many functions and to obtain a classical weighted Hardy type inequality involving many functions.

The weighted Stieltjes inequality and applications

AbstractLet 1 &lt; p ≤ q &lt; ∞. Inspired by some results concerning characterization of weighted Hardy type inequalities, where the equivalence of four scales of integral conditions was proved, we use related ideas to find some new equivalent scales of integral conditions related to the Stieltjes transform. By applying our result to weighted inequalities for the Stieltjes transform we obtain four new scales of conditions for characterization of this inequality. We also derive a new characterization for the solvability of a Riccati type equation and show via our new results that this characterization can be done in infinite many ways via our four scales of equivalent conditions.

On equivalent conditions for the general weighted Hardy type inequality in space $L^{p(\cdot)}$

We study the Hardy type two-weighted inequality for the multidimensional Hardy operator in the norms of generalized Lebesgue spaces L^{p(\cdot)}(\mathbb{R}^{n}) . In this way we prove equivalent conditions for L^{p(\cdot)}\rightarrow L^{q(\cdot)} boundedness of Hardy operator in the case of exponents q(0)\geq p(0) , q(\infty)\geq p\left(\infty \right) . We also prove that the condition for such inequality to hold coincides with condition for validity of two weighted Hardy inequalities with constant exponents, if we require the exponents to be regular near zero and at infinity.

On a general weighted Hardy type inequality in the variable exponent Lebesgue spaces

We study the Hardy type, two-weight inequality for the multidimensional Hardy operator in the variable exponent Lebesgue space Lp(.)(ℝn). We prove equivalent conditions for Lp(.)→Lq(.) boundness of the Hardy operator in the case of so called “mixed” exponents: q(0)≥p(0), q(∞)<p(∞) or q(0)<p(0), q(∞)≥p(∞). We show that a necessary and sufficient condition for such an inequality to hold coincides with conditions for the validity of two weight Hardy inequalities with constant exponents, provided that the exponents are regular at zero and at infinity.

Bounds for Hardy type differences

Let A k be an integral operator defined by $$ A_k f\left( x \right) = \frac{1} {{K\left( x \right)}}\int_{\Omega _2 } {k\left( {x,y} \right)f\left( y \right)d\mu _2 \left( y \right),} $$ where k: Ω1 × Ω2 → ℝ is a general nonnegative kernel, (Ω1, Σ1, µ1), (Ω2, Σ2, µ2) are measure spaces with σ-finite measures and $$ K\left( x \right) = \int_{\Omega _2 } {k\left( {x,y} \right)d\mu _2 \left( y \right),} x \in \Omega _1 . $$ In this paper improvements and reverses of new weighted Hardy type inequalities with integral operators of such type are stated and proved. New Cauchy type mean is introduced and monotonicity property of this mean is proved.

On the weighted Hardy type inequality in a fixed domain for functions vanishing on the part of the boundary

We derive and discuss a new two-dimensional weighted Hardy-type inequality in a rectangle for the class of functions from the Sobolev space H 1 vanishing on small alternating pieces of the boundary. � p dt � p p −1 � pa