A lower estimate for weak-type Fourier multipliers

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 (Formula presented.) into (Formula presented.) is continuously embedded into (Formula presented.). We obtain a sharper result in the setting of abstract Lorentz spaces (Formula presented.) with (Formula presented.) built upon a Banach function space X on (Formula presented.). We consider a source space (Formula presented.) and a target space (Formula presented.) in the class of admissible spaces (Formula presented.). Let (Formula presented.) denote the space of Fourier multipliers acting from (Formula presented.) to (Formula presented.). We show that if the space X satisfies the weak doubling property, then the space (Formula presented.) is continuously embedded into (Formula presented.) for every (Formula presented.). This implies that (Formula presented.) is a quasi-Banach space for all choices of source and target spaces (Formula presented.).