Double Exponential Integrability of Convolution Operators in Generalized Lorentz–Zygmund Spaces

This paper provides estimates for an appropriate norm of the convolution of a function in a Lorentz space with one in a generalized Lorentz-Zygmund space. As a corollary, it is shown that the Riesz potential of a function in an appropriate generalized Lorentz-Zygmund space satisfies a 'double exponential' integrability condition. The results extend those of Brézis-Wainger on the convolution of functions in Lorentz spaces which lead to exponential integrability.