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Discrete H\" older and reversed Hardy-type inequalities in Lorentz sequence spaces

Sorina Barza, Anca-Nicoleta Marcoci, Liviu-Gabriel Marcoci

Abstract

We establish new optimal reversed Hardy-type inequalities on the cone of decreasing sequences from $\ell^p$-spaces with power weights, as well as estimates between different norms in Lorentz spaces of sequences. Based on these inequalities, we derive a sharp Hölder-type inequality in Lorentz sequence spaces that complements the previously considered case of functions.

Discrete H\" older and reversed Hardy-type inequalities in Lorentz sequence spaces

Abstract

We establish new optimal reversed Hardy-type inequalities on the cone of decreasing sequences from -spaces with power weights, as well as estimates between different norms in Lorentz spaces of sequences. Based on these inequalities, we derive a sharp Hölder-type inequality in Lorentz sequence spaces that complements the previously considered case of functions.

Paper Structure

This paper contains 5 sections, 15 theorems, 96 equations.

Table of Contents

  1. Introduction
  2. Definitions and notations
  3. Reversed Hardy inequalities with power weights for decreasing sequences
  4. Equivalence between the dual and maximal norm
  5. Hölder type inequalities

Key Result

Theorem 2.1

Let $\varphi=(\varphi_n)_{n}$ be a positive sequence and $\Phi_n=\sum_{i=1}^n \varphi_i$. Let $x=(x_n)_{n}$ be a positive sequence and suppose that Then, there exists a unique nonnegative sequence $x^\circ=(x^\circ_n)_n$ satisfying the following conditions: (a) $\left(\frac{x_n^\circ }{\varphi_n}\right)_n$ is decreasing; (b) $x \prec x^\circ$; (c) The set $\{n:\, x^\circ_n\neq x_n\}=\cup_{k=1}^\i

Theorems & Definitions (25)

  • Theorem 2.1: Theorem 3.3, BarzaMarcociPersson2012
  • Theorem 2.2: Theorem 4.2, BarzaMarcociPersson2012
  • Lemma 3.1: see R, Lemma 2
  • Lemma 3.2
  • proof
  • Lemma 3.3: R, Lemma 1
  • Lemma 3.4
  • proof
  • Theorem 3.5
  • proof
  • ...and 15 more