New Orlicz variants of Hardy type inequalities with power, power-logarithmic, and power-exponential weights

Abstract

We obtain Hardy type inequalitiesand their Orlicz-norm counterpartswith an N-function M, power, power-logarithmic and power-exponential weights ω ρ, holding on suitable dilation invariant supersets of C ∞ 0 (R + ). Maximal sets of admissible functions are described. This paper is based on authors' earlier abstract results and applies them to particular classes of weights. MSC:26D10, 46E35

Caccioppoli--type estimates and Hardy--type inequalities derived from degenerated $p$--harmonic problems

“…Namely, they are achieved in the classical Hardy inequality (Section 5.1 in [40]); the Hardy-Poincaré inequality obtained in [41] due to [40], confirming some constants from [23] and [10] and establishing the optimal constants in further cases; the Poincaré inequality concluded from [40], confirmed to hold with best constant in Remark 7.6 in [19]. Moreover, the inequality in Theorem 5.5 in [40] can also be retrieved by the methods from [26] with the same constant, while some inequalities from Proposition 5.2 in [27] are comparable with Theorem 5.8 in [40]. In Theorem 4.3, we provide some extensions of Hardy-Poincaré inequalities from [41], which are proven in [29] by applying the results obtained in this paper.…”

Existence and uniqueness of the solution of linear degenerate PDEs in weighted Sobolev space with nonhomogeneous boundary condition

“…Remark It follows from that the assumptions (H2) and (H3) imply the following Poincaré inequality. (H1')For every where is given constant independent on v . Constructions of Poincaré inequality can be found for examples in the literature …”

Variable exponent Hardy-type inequalities in $\mathbb{R}^n$

“…Let us mention such branches as functional analysis, harmonic analysis, probability theory, and PDEs. Weighted versions of Hardy-type inequalities are also investigated on their own in the classical way [17,22,27], as well as in the various generalised frameworks [4,5,14,29].…”