engleski

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

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

Hörmander-Mihlin theorem ; mixed-norm Lebesgue space

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