Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Fractional Orlicz-Hardy Inequalities

T. V. Anoop, Prosenjit Roy, Subhajit Roy

Abstract

We establish the weighted fractional Orlicz-Hardy inequalities for various Orlicz functions. Further, we identify the critical cases for each Orlicz function and prove the weighted fractional Orlicz-Hardy inequalities with logarithmic correction. Moreover, we discuss the analogous results in the local case. In the process, for any Orlicz function $Φ$ and for any $Λ>1$, the following inequality is established $$ Φ(a+b)\leq λΦ(a)+\frac{C( Φ, Λ )}{(λ-1)^{p_Φ^+-1}}Φ(b),\;\;\;\forall\,a,b\in [0,\infty),\,\forall\,λ\in (1,Λ], $$ where $p_Φ^+:=\sup\big\{t\varphi(t)/Φ(t):t>0\big\},$ $\varphi$ is the right derivatives of $Φ$ and $C( Φ, Λ )$ is a positive constant that depends only on $Φ$ and $Λ.$

On Fractional Orlicz-Hardy Inequalities

Abstract

We establish the weighted fractional Orlicz-Hardy inequalities for various Orlicz functions. Further, we identify the critical cases for each Orlicz function and prove the weighted fractional Orlicz-Hardy inequalities with logarithmic correction. Moreover, we discuss the analogous results in the local case. In the process, for any Orlicz function and for any , the following inequality is established where is the right derivatives of and is a positive constant that depends only on and
Paper Structure (7 sections, 13 theorems, 150 equations, 1 figure)

This paper contains 7 sections, 13 theorems, 150 equations, 1 figure.

Table of Contents

  1. Introduction
  2. preliminaries
  3. Properties of an Orlicz function
  4. Some function spaces:
  5. Weighted fractional Orlicz-Hardy inequalities
  6. Weighted Orlicz-Hardy inequalities in local case
  7. Concluding remarks and examples:

Key Result

Theorem 1.2

Let $N\geq 1,\,s\in (0,1)$, $\alpha_1, \, \alpha_2\in \mathbb{R}$, and let $\gamma:=s-\alpha_1-\alpha_2.$ For an Orlicz function $\Phi$, if $\gamma<N/p^\oplus_\Phi$, then Hardy holds and if $\gamma> N/p^\ominus_\Phi$, then Hardy fails. Furthermore, if $p^\ominus_\Phi$ is attained, then Hardy fails a

Figures (1)

  • Figure :

Theorems & Definitions (31)

  • Definition 1.1: Orlicz function
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Remark 1.5
  • Remark 1.6
  • Remark 1.7
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • ...and 21 more