An improvement of the Marcinkiewicz multiplier theorem

Abstract

We provide an improvement of the Marcinkiewicz multiplier theorem on ℝⁿ by relaxing the integrability hypothesis on the multiplier. In particular, when r> 1 and min(s₁, …, s_n) > 1/r, we replace the product-type Sobolev norm ${\left\| {{{(I - \partial _1^2)}^{{s_1}/2}} \cdots {{(I - \partial _n^2)}^{{s_n}/2}}[\Phi \,g]} \right\|_{{L^r}}}$ by the larger Sobolev norm ${\left\| {{{(I - \partial _1^2)}^{{s_1}/2}} \cdots {{(I - \partial _n^2)}^{{s_n}/2}}[\Phi \,g]} \right\|_{{L^{q,1}}}}$ built upon the Lorentz space with first indices q = 1/min(s₁,…,s_n) and 1. Here s_i > 0 are distinct numbers and Φ is a function with compact support away from the origin.