Related papers: Hardy inequalities in normal form

In this article we establish new improvements of the optimal Hardy inequality in the half space. We first add all possible linear combinations of Hardy type terms thus revealing the structure of this type of inequalities and obtaining best…

In this note we give several characterisations of weights for two-weight Hardy inequalities to hold on general metric measure spaces possessing polar decompositions. Since there may be no differentiable structure on such spaces, the…

A simple proof is provided to show that any bounded normal operator on a real Hilbert space is orthogonally equivalent to its transpose(adjoint). A structure theorem for invertible skew-symmetric operators, which is analogous to the finite…

We investigate the growth of the polynomial and multilinear Hardy--Littlewood inequalities. Analytical and numerical approaches are performed and, in particular, among other results, we show that a simple application of the best known…

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 prove certain generalization of Hardy's inequality where the "boundary defining function" is replaced by a polynomial defining a singular algebraic variety. An application is given on the existence of a small time heat trace expansion…

In the present paper we shall study a variational problem relating the weighted Hardy inequalities with sharp missing terms. As weights we treat non-doubling functions of the distance to the boundary of bounded domain.

Some q-analysis variants of Hardy type inequalities of the form \int_0^b (x^{\alpha-1} \int_0^x t^{-\alpha} f(t) d_qt)^p d_qx \leq C \int_0^b f^p(t) d_qt with sharp constant C are proved and discussed. A similar result with the…

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…

In this article, we look for the weight functions (say $g$) that admits the following generalized Hardy-Rellich type inequality: $ \int_{\Omega} g(x) u^2 dx \leq C \int_{\Omega} |\Delta u|^2 dx, \forall u \in \mathcal{D}^{2,2}_0(\Omega), $…

We characterize a weighted norm inequality which corresponds to the embedding of a class of absolutely continuous functions into the fractional order Sobolev space. The auxiliary result of the paper is of independent interest. It comprises…

For absolutely convergent series we state explicitly a one-sided summation estimate that can be viewed as the discrete analogue of the change of variable formula on the half line. This estimate is implicit in Pascal Lef\`evre's recent…

We investigate necessary and sufficient conditions on the weights for the Hardy-Rellich inequalities to hold, and propose a new way to use the notion of Bessel pair to establish the optimal Hardy-Rellich type inequalities. Our results…

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…

The aim of this paper is to find a broad family of means defined on a subinterval of $I \subset [0,+\infty)$ such that $$ \sum_{n=1}^\infty \mathscr{M}(a_1,\dots,a_n) <+\infty \quad\text{ for all }\quad a \in \ell_1(I).$$ Equivalently, the…

In this article, we determine conditions on the parameters of a generalized convolution operator such that it belongs to the Hardy space and to the space of bounded analytic functions. Results obtained are new and their usefulness is…

Let $1\leq p <\infty$ and $0 < q,r < \infty$. We characterize validity of the inequality for the composition of the Hardy operator, \begin{equation*} \bigg(\int_a^b \bigg(\int_a^x \bigg(\int_a^t f(s)ds \bigg)^q u(t) dt \bigg)^{\frac{r}{q}}…

This paper is devoted to Hardy type inequalities with remainders for compactly supported smooth functions on open sets in the Euclidean space. We establish new inequalities with weight functions depending on the distance function to the…

Associated to the class of restricted-weak type weights for the Hardy operator, we find a new class of Lorentz spaces for which the normability property holds. This result is analogous to the characterization given by Sawyer for the…

We introduce a natural generalization of a well studied integration operator acting on the family of Hardy spaces in the unit disc. We study the boundedness and compactness properties of the operator and finally we use these results to give…