18,179 research outputs found
Weighted norm inequalities for the bilinear maximal operator on variable Lebesgue spaces
We extend the theory of weighted norm inequalities on variable Lebesgue
spaces to the case of bilinear operators. We introduce a bilinear version of
the variable \A_\pp condition, and show that it is necessary and sufficient
for the bilinear maximal operator to satisfy a weighted norm inequality. Our
work generalizes the linear results of the first author, Fiorenza and
Neugebauer \cite{dcu-f-nPreprint2010} in the variable Lebesgue spaces and the
bilinear results of Lerner {\em et al.} \cite{MR2483720} in the classical
Lebesgue spaces. As an application we prove weighted norm inequalities for
bilinear singular integral operators in the variable Lebesgue spaces.Comment: Revised based on anonymous referee's reports. A number of typos and
small errors corrected. One conjecture added to introductio
Embeddings between grand, small and variable Lebesgue spaces
We give conditions on the exponent function that imply the
existence of embeddings between grand, small and variable Lebesgue spaces. We
construct examples to show that our results are close to optimal. Our work
extends recent results by the second author, Rakotoson and Sbordone.Comment: Final version to appear in Math. Note
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