Open Access𝐿log𝐿 results for the maximal operator in variable 𝐿^{𝑝} spaces
Author(s) -
David Cruz-Uribe,
Alberto Fiorenza
Publication year - 2008
Publication title -
transactions of the american mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.798
H-Index - 100
eISSN - 1088-6850
pISSN - 0002-9947
DOI - 10.1090/s0002-9947-08-04608-4
Subject(s) - algorithm , annotation , type (biology) , artificial intelligence , computer science , mathematics , biology , ecology
We generalize the classical L log L L\log L inequalities of Wiener and Stein for the Hardy-Littlewood maximal operator to variable L p L^p spaces where the exponent function p ( ⋅ ) p(\cdot ) approaches 1 1 in value. We prove a modular inequality with no assumptions on the exponent function, and a strong norm inequality if we assume the exponent function is log-Hölder continuous. As an application of our approach we give another proof of a related endpoint result due to Hästö.