Related papers: Change of variable and discrete Hardy inequality
The celebrated Hardy inequality can be written in the form $$\int_0^\infty \mathcal{P}_p \big(f|_{[0,x]}\big)dx \le (1-p)^{-1/p} \int_0^\infty f(x)\:dx \qquad \text{ for }p\in(0,1)\text{ and }f \in L^1\text{ with }f\ge0,$$ where…
A simple normal form for Hardy operators is introduced that unifies and simplifies the theory of weighted Hardy inequalities. A straightforward transition to normal form is given that applies to the various Hardy operators and their duals,…
Using recent results concerning the homogenization and the Hardy property of weighted means, we establish sharp Hardy constants for concave and monotone weighted quasideviation means and for a few particular subclasses of this broad family.…
We give a partial negative answer to a question left open in a previous work by Brasco and the first and third-named authors concerning the sharp constant in the fractional Hardy inequality on convex sets. Our approach has a geometrical…
We study finite sections of weighted Hardy's inequality following the approach of De Bruijn. Similar to the unweighted case, we obtain an asymptotic expression for the optimal constant.
We obtain the sharp factor of the two-sides estimates of the optimal constant in generalized Hardy's inequality with two general Borel measures on $\mathbb{R}$, which generalizes and unifies the known continuous and discrete cases.
The main aim of this note is to prove sharp weighted integral Hardy inequality and conjugate integral Hardy inequality on homogeneous Lie groups with any quasi-norm for the range $1<p\leq q<\infty.$ We also calculate the precise value of…
An absolutely convergent double series representation for the density of the supremum of $\alpha$-stable Levy process is given in [3, Theorem 2] for almost all irrational $\alpha$. This result cannot be made stronger in the following sense:…
Although the Hardy inequality corresponding to one quadratic singularity, with optimal constant, does not admit any extremal function, it is well known that such a potential can be improved, in the sense that a positive term can be added to…
We consider a multidimensional version of an inequality due to Leray as a substitute for Hardy's inequality in the case $p=n\geq2.$ In this paper we provide an optimal Sobolev-type improvement of this substitute, analogous to the…
In this paper we study some improvements of the classical Hardy inequality. We add to the right hand side of the inequality a term which depends on some Lorentz norms of $u$ or of its gradient and we find the best values of the constants…
When studying the weighted Hardy-Rellich inequality in $L^2$ with the full gradient replaced by the radial derivative the best constant becomes trivially larger or equal than in the first situation. Our contribution is to determine the new…
We consider the Hardy constant associated with a domain in the $n$-dimensional Euclidean space and we study its variation upon perturbation of the domain. We prove a Fr\'{e}chet differentiability result and establish a Hadamard-type formula…
We prove a sharp integral inequality valid for non-negative functions defined on $[0,1]$, with given $L^1$ norm. This is in fact a generalization of the well known integral Hardy inequality. We prove it as a consequence of the respective…
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…
This paper surveys some of our recent progress on Hardy-type inequa\-lities which consist of a well-known topic in Harmonic Analysis. In the first section, we recall the original probabilistic motivation dealing with the stability speed in…
Hardy-Littlewood-Sobolev (HLS) Inequality fails in the "critical" case: \mu=n. However, for discrete HLS, we can derive a finite form of HLS inequality with logarithm correction for a critical case: \mu=n and p=q, by limiting the inequality…
We establish a new improvement of the classical $L^p$-Hardy inequality on the multidimensional Euclidean space in the supercritical case. Recently, in [14], there has been a new kind of development of the one dimensional Hardy inequality.…
Motivated by a discrete inequality problem proposed by Duanyang Zhang as Problem 6 of the 2022 Spring NSMO, we prove a median version of Hardy's inequality. For a nonnegative function $f\in L^p(0,\infty)$, $p>1$, let $A(t)$ be the average…
We show that the sharp constant in the classical $n$-dimensional Hardy-Leray inequality can be improved for axisymmetric divergence-free fields, and find its optimal value. The same result is obtained for $n=2$ without the axisymmetry…