Multilinear Fourier multipliers on variable Lebesgue spaces

Abstract

In this paper, we study the properties of a bilinear multiplier space. We give a necessary condition for a continuous bounded function to be a bilinear multiplier on variable exponent Lebesgue spaces, and we prove the localization theorem of multipliers on variable exponent Lebesgue spaces. Moreover, we present a Mihlin-Hörmander type theorem for multilinear Fourier multipliers on weighted variable Lebesgue spaces and give some applications. MSC: 42B15; 42B20; 42B25

Kato-Ponce inequalities on weighted and variable Lebesgue spaces

“…A slightly different version of Corollary 4.2 for the smaller class of symbols σ satisfying sup k∈Z σ k H s (R 2n ) < ∞ with n < s ≤ 2n was proved in [36]. The proof again used linear extrapolation and required the additional hypothesis that r(•) r ∈ B.…”