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Lorentz-Shimogaki and Boyd theorems for weighted Lorentz spaces

Elona Agora, Jorge Antezana, María J. Carro, Javier Soria

TL;DR

This poster presents a probabilistic procedure for estimating the intensity of the response of the immune system to certain types of injuries sustained during the natural disasters.

Abstract

We prove the Lorentz-Shimogaki and Boyd theorems for the spaces $Λ^p_u(w)$. As a consequence, we give the complete characterization of the strong boundedness of $H$ on these spaces in terms of some geometric conditions on the weights $u$ and $w$, whenever $p>1$. For these values of $p$, we also give the complete solution of the weak-type boundedness of the Hardy-Littlewood operator on $Λ^p_u(w)$.

Lorentz-Shimogaki and Boyd theorems for weighted Lorentz spaces

TL;DR

This poster presents a probabilistic procedure for estimating the intensity of the response of the immune system to certain types of injuries sustained during the natural disasters.

Abstract

We prove the Lorentz-Shimogaki and Boyd theorems for the spaces . As a consequence, we give the complete characterization of the strong boundedness of on these spaces in terms of some geometric conditions on the weights and , whenever . For these values of , we also give the complete solution of the weak-type boundedness of the Hardy-Littlewood operator on .
Paper Structure (5 sections, 21 theorems, 119 equations, 2 figures)

This paper contains 5 sections, 21 theorems, 119 equations, 2 figures.

Table of Contents

  1. Introduction and motivation
  2. The case $u=1$: $\Lambda^p(w)$
  3. The Lorentz-Shimogaki theorem for $\Lambda^p_u(w)$
  4. The Boyd theorem for $\Lambda^p_u(w)$
  5. Weak and strong boundedness of Hardy-Littlewood maximal operator in weighted Lorentz spaces

Key Result

Theorem 1.1

crs:crs If $0<p<\infty$, is bounded if and only if there exists $q\in(0,p)$ such that, for every finite family of disjoint intervals $(I_j)_{j=1}^J$, and every family of measurable sets $(S_j)_{j=1}^{J}$, with $S_j\subset I_j$, for every $j$, we have that

Figures (2)

  • Figure 1: $f_{S,I}$ when $m=1$.
  • Figure 2: $f_{S,I}$ when $m=2$.

Theorems & Definitions (41)

  • Theorem 1.1
  • Remark 1.2
  • Remark 1.3
  • Proposition 2.1
  • Theorem 2.2
  • Corollary 2.3
  • Lemma 2.4
  • proof
  • proof : Proof of Corollary \ref{['ari']}
  • Theorem 2.5
  • ...and 31 more