Pseudodifferential Operators on Variable Lebesgue Spaces
Alexei Yu. Karlovich, Ilya M. Spitkovsky

TL;DR
This paper establishes boundedness and Fredholmness of pseudodifferential operators on variable Lebesgue spaces with broad classes of exponents, extending previous results to less regular exponent functions.
Contribution
It generalizes known boundedness and Fredholm criteria for pseudodifferential operators to variable Lebesgue spaces with minimal regularity assumptions on the exponent functions.
Findings
Boundedness of perators with symbols in Hörmander classes on variable Lebesgue spaces.
Fredholmness of perators under certain conditions on the symbol and variable exponents.
Extension of previous results to broader classes of variable exponents.
Abstract
Let be the class of bounded away from one and infinity functions such that the Hardy-Littlewood maximal operator is bounded on the variable Lebesgue space . We show that if belongs to the H\"ormander class with , , then the pseudodifferential operator is bounded on the variable Lebesgue space provided that . Let be the class of variable exponents represented as where , , and . We prove that if slowly oscillates at infinity in the first variable, then the condition \[ \lim_{R\to\infty}\inf_{|x|+|\xi|\ge R}|a(x,\xi)|>0…
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