Analysis Seminar: Necessary Conditions for Bounded Operators on the Variable Lebesgue Spaces

Abstract: The variable Lebesgue spaces are a generalization of the classical Lebesgue spaces where the exponent p is replaced by a variable exponent function p(x). Due to their application to PDEs with non-standard growth conditions, which model phenomena such as elasticity and electrorheological fluids, they have garnered a great deal of attention in the past few decades. A natural question to ask is for which exponent functions p(x) are classical operators, such as maximal functions and singular integrals, bounded between them. A great deal of work was done in the 2000's proving sufficient conditions on p(x) for such operators to be bounded, but in general these conditions fail to be necessary. The question of necessary conditions was left open aside for the Hardy-Littlewood maximal operator and the Riesz transforms. In this talk we will discuss recent results we have obtained which establish necessary conditions for fractional maximal operators and fractional singular integral operators to be bounded on the variable Lebesgue spaces which both sharpen and generalize known results.

Host: Alan Chang