Characterization of multilinear multipliers in terms of Sobolev space regularity

Abstract

We provide necessary and sufficient conditions for multilinear multiplier operators with symbols in L r -based product-type Sobolev spaces uniformly over all annuli to be bounded from products of Hardy spaces to a Lebesgue space. We consider the case 1 < r ≤ 2 and we characterize boundedness in terms of inequalities relating the Lebesgue indices (or Hardy indices), the dimension, and the regularity and integrability indices of the Sobolev space. The case r > 2 cannot be handled by known techniques and remains open. Our result not only extends but also establishes the sharpness of previous results of Miyachi, Nguyen, Tomita, and the first author [13,14,15,23], who only considered the case r = 2.