Positive kernel operators in $$L^{p(x)}$$ spaces

Abstract

A characterization of a weight $$v$$ governing the boundedness/compactness of the weighted kernel operator $$K_v$$ in variable exponent Lebesgue spaces $$L^{p(\cdot )}$$ is established under the log-Holder continuity condition on exponents of spaces. The kernel operator involves, for example, weighted variable parameter fractional integral operators. The distance between $$K_v$$ and the class of compact integral operators acting from $$L^{p(\cdot )}$$ to $$L^{q(\cdot )}$$ (measure of non-compactness) is also estimated from above and below.