A lower estimate for weak-type Fourier multipliers

Abstract

Asmar et al. [Note on norm convergence in the space of weak type multipliers. J Operator Theory. 1998;39(1):139–149] proved that the space of weak-type Fourier multipliers acting from into is continuously embedded into . We obtain a sharper result in the setting of abstract Lorentz spaces with built upon a Banach function space X on . We consider a source space and a target space in the class of admissible spaces . Let denote the space of Fourier multipliers acting from to . We show that if the space X satisfies the weak doubling property, then the space is continuously embedded into for every . This implies that is a quasi-Banach space for all choices of source and target spaces .