Lipschitz spaces and the fractional integral in the context of the variable Lebesgue space L^{p(·)}

MAURICIO RAMSEYER

Sevilla

Workshop; Harmonic Analysis, Metric Spaces and Applications to P.D.E.; 2011

Instituto de Matemática de la Universidad de Sevilla

In recent years, function spaces with variable exponent and related differential equations have attracted the attention of many researchers.In this context, we study the fractional integral operator I_{\alpha} and its relation to certain variable Lipschitz spaces.\\For a measurable function p: R^n --> (1,\infty) (called the exponent), we consider L^{p(·)} with Luxemburg norm and introduce the associated space \mathfrak{L}_{\alpha,p(·)} where 0 < \alpha < n.\\For a certain range of values of the exponent, I_{\alpha} is not defined.In such cases, we prove that this operator can be extended to a bounded linear operator \tilde{I}_{\alpha} from L^{p(·)} into \mathfrak{L}_{\alpha,p(·)}.