eBook ISBN:  9781470404819 
Product Code:  MEMO/187/877.E 
List Price:  $68.00 
MAA Member Price:  $61.20 
AMS Member Price:  $40.80 
eBook ISBN:  9781470404819 
Product Code:  MEMO/187/877.E 
List Price:  $68.00 
MAA Member Price:  $61.20 
AMS Member Price:  $40.80 

Book DetailsMemoirs of the American Mathematical SocietyVolume: 187; 2007; 128 ppMSC: Primary 42; Secondary 26; 46; 47
The main objective of this work is to bring together two well known and, a priori, unrelated theories dealing with weighted inequalities for the HardyLittlewood maximal operator \(M\). For this, the authors consider the boundedness of \(M\) in the weighted Lorentz space \(\Lambda^p_u(w)\). Two examples are historically relevant as a motivation: If \(w=1\), this corresponds to the study of the boundedness of \(M\) on \(L^p(u)\), which was characterized by B. Muckenhoupt in 1972, and the solution is given by the so called \(A_p\) weights. The second case is when we take \(u=1\). This is a more recent theory, and was completely solved by M.A. Ariño and B. Muckenhoupt in 1991. It turns out that the boundedness of \(M\) on \(\Lambda^p(w)\) can be seen to be equivalent to the boundedness of the Hardy operator \(A\) restricted to decreasing functions of \(L^p(w)\), since the nonincreasing rearrangement of \(Mf\) is pointwise equivalent to \(Af^*\). The class of weights satisfying this boundedness is known as \(B_p\).
Even though the \(A_p\) and \(B_p\) classes enjoy some similar features, they come from very different theories, and so are the techniques used on each case: Calderón–Zygmund decompositions and covering lemmas for \(A_p\), rearrangement invariant properties and positive integral operators for \(B_p\).
This work aims to give a unified version of these two theories. Contrary to what one could expect, the solution is not given in terms of the limiting cases above considered (i.e., \(u=1\) and \(w=1\)), but in a rather more complicated condition, which reflects the difficulty of estimating the distribution function of the HardyLittlewood maximal operator with respect to general measures.

Table of Contents

Chapters

Introduction

1. Boundedness of operators on characteristic functions and the Hardy operator

2. Lorentz spaces

3. The HardyLittlewood maximal operator in weighted Lorentz spaces


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The main objective of this work is to bring together two well known and, a priori, unrelated theories dealing with weighted inequalities for the HardyLittlewood maximal operator \(M\). For this, the authors consider the boundedness of \(M\) in the weighted Lorentz space \(\Lambda^p_u(w)\). Two examples are historically relevant as a motivation: If \(w=1\), this corresponds to the study of the boundedness of \(M\) on \(L^p(u)\), which was characterized by B. Muckenhoupt in 1972, and the solution is given by the so called \(A_p\) weights. The second case is when we take \(u=1\). This is a more recent theory, and was completely solved by M.A. Ariño and B. Muckenhoupt in 1991. It turns out that the boundedness of \(M\) on \(\Lambda^p(w)\) can be seen to be equivalent to the boundedness of the Hardy operator \(A\) restricted to decreasing functions of \(L^p(w)\), since the nonincreasing rearrangement of \(Mf\) is pointwise equivalent to \(Af^*\). The class of weights satisfying this boundedness is known as \(B_p\).
Even though the \(A_p\) and \(B_p\) classes enjoy some similar features, they come from very different theories, and so are the techniques used on each case: Calderón–Zygmund decompositions and covering lemmas for \(A_p\), rearrangement invariant properties and positive integral operators for \(B_p\).
This work aims to give a unified version of these two theories. Contrary to what one could expect, the solution is not given in terms of the limiting cases above considered (i.e., \(u=1\) and \(w=1\)), but in a rather more complicated condition, which reflects the difficulty of estimating the distribution function of the HardyLittlewood maximal operator with respect to general measures.

Chapters

Introduction

1. Boundedness of operators on characteristic functions and the Hardy operator

2. Lorentz spaces

3. The HardyLittlewood maximal operator in weighted Lorentz spaces