On the Hörmander–Mihlin theorem for mixed-norm Lebesgue spaces

Abstract

For some questions in the theory of partial differential equations, it is useful to consider the mixed-norm Lebesgue spaces introduced by Benedek and Panzone (1962). We show that Hörmander's results (1960) obtained on translation invariant operators on Lebesgue spaces extend to the mixed-norm case. We also revisit the Hörmander–Mihlin multiplier theorem in this setting, which was first obtained by Lizorkin (1970), where we particularly consider the form of the constants used in the estimates. This improvement allows us to prove the boundedness of classical pseudodifferential operators on mixed-norm Lebesgue spaces, as well as generalising the H-distributions, a recently introduced (2011, by Mitrović and the first author) extension of H-measures, to these spaces.