Boundedness of nonregular bilinear pseudodifferential operators on variable Besov and Triebel--Lizorkin spaces

Abstract

In this article, we introduce two bilinear, non-regular symbol classes\linebreak $BS\mathcal B^m_δ(r,μ,v;N,λ)$ and $BS\mathcal F^m_δ(r,μ,v;N,λ)$, where $m\in\mathbb R$, $0\leqδ<1$, $0<μ,v\leq\infty$, $1\leqλ\leq2$, $r>n/μ$, and $N>2n/λ$, with $μ<\infty$ for the latter class. For $μ\geq1$, $(1-δ)r\geq n/μ$, and sufficiently large $N$, we establish boundedness of the associated bilinear pseudodifferential operators on Besov and Triebel--Lizorkin spaces with variable smoothness, integrability and summation exponents. To the best of the authors' knowledge, these operator estimates are new even on the classical Besov and Triebel--Lizorkin spaces with constant exponents. As an application, we establish the multiplication algebra property for both variable exponent spaces.