Weighted Sobolev spaces and pseudodifferential operators with smooth symbols

Abstract

Let ${u^\# }$ be the Fefferman-Stein sharp function of $u$, and for $1 < r < \infty$, let ${M_r}u$ be an appropriate version of the Hardy-Littlewood maximal function of $u$. If $A$ is a (not necessarily homogeneous) pseudodifferential operator of order $0$, then there is a constant $c > 0$ such that the pointwise estimate ${(Au)^\# }(x) \leqslant c{M_r}u(x)$ holds for all $x \in {R^n}$ and all Schwartz functions $u$. This estimate implies the boundedness of $0$-order pseudodifferential operators on weighted ${L^p}$ spaces whenever the weight function belongs to Muckenhoupt’s class ${A_p}$. Having established this, we construct weighted Sobolev spaces of fractional order in ${R^n}$ and on a compact manifold, prove a version of Sobolev’s theorem, and exhibit coercive weighted estimates for elliptic pseudodifferential operators.