New Young Inequalities and Applications

Abstract

We establish upper bounds for the convolution operator acting between interpolation spaces. This gives new Young inequalities in the context of Lorentz–Karamata spaces, grand Lebesgue spaces and small Lebesgue spaces besides many other known results. Furthermore, we use this abstract Young inequality to prove a bilinear interpolation theorem for limit interpolation methods. Finally, we show applications to the study of bilinear multipliers.