Weighted norm inequalities for pseudo-differential operators
If \(f\in L^ 1_{loc}({\mathbb{R}}^ n)\), put \[ f^{\#}f(x)=\sup_{x\in Q}\inf_{c\in {\mathbb{C}}}| Q|^{-1}\int_{Q}| f(y)-c| dy,\quad M_ p f(x)=\sup_{x\in Q}(| Q|^{-1}\int_{Q}| f(y)|^ p dy)^{1/p}, \] where Q ranges over all cubes with sides parallel to the coordinate axes, containing x. An inequality of the type \((Tf)^{\#}(x)\leq BM_ pf(x)\), valid for all \(f\in C_ 0^{\infty}({\mathbb{R}}^ n)\), where T is a linear operator of \(C_ 0^{\infty}({\mathbb{R}}^ n)\) to \(L^ 1_{loc}({\mathbb{R}}^ n)\), is called a Fefferman-Stein inequality. The author proves Fefferman-Stein inequalities for various classes of pseudo-differential operators by studying their kernels. Weighted \(L^ p\) estimates are deduced.
scientific article; zbMATH DE number 4028042