Boundedness of pseudodifferential operators on Banach function spaces

Karlovich, Alexei Yu. "Boundedness of pseudodifferential operators on Banach function spaces." Operator Theory, Operator Algebras and Applications. Operator Theory: Advances and Applications, 242. Eds. Maria Amélia Bastos, Amarino Lebre, Stefan Samko, and Ilya M. Spitkovsky. Basel: Birkhäuser/Springer, 2014. 185-195.


We show that if the Hardy-Littlewood maximal operator is bounded on a separable Banach function space \(X(\mathbb{R}^n)\) and on its associate space \(X'(\mathbb{R}^n)\), then a pseudodifferential operator \(\operatorname{Op}(a)\) is bounded on \(X(\mathbb{R}^n)\) whenever the symbol \(a\) belongs to the Hörmander class \(S_{\rho,\delta}^{n(\rho-1)}\) with \(0<\rho\le 1\), \(0\le\delta<1\) or to the the Miyachi class \(S_{\rho,\delta}^{n(\rho-1)}(\varkappa,n)\) with \(0\le\delta\le\rho\le 1\), \(0\le\delta<1\), and \(\varkappa>0\). This result is applied to the case of variable Lebesgue spaces \(L^{p(\cdot)}(\mathbb{R}^n)\).

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