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

In this article, we introduce two bilinear, non-regular symbol classes\linebreak $BS{\mathcal{B}}_{\delta }^m(r,\mu ,v;N,\lambda )$ and $BS{\mathcal{F}}_{\delta }^m(r,\mu ,v;N,\lambda )$, where $m\in \mathbb{R}$, $0\le \delta <1$, $0<\mu ,v\le \mathrm{\infty }$, $1\le \lambda \le 2$, $r>n\mathrm{/}\mu$, and $N>2n\mathrm{/}\lambda$, with $\mu <\mathrm{\infty }$ for the latter class. For $\mu \ge 1$, $(1-\delta )r\ge n\mathrm{/}\mu$, 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.