Weighted inequalities and vector-valued Calderón-Zygmund operators on non-homogeneous spaces
The paper deals with weighted inequalities of the form \[ \int_X|Tf|^pu d\mu \leq \int_X|f|^pv d\mu\tag \(*\) \] where \(X\) is a non-homogeneous space (i.e., a separable metric space endowed with a measure \(\mu\) such that \(\mu (B(x,r))\leq Cr^n)\) and \(T\) is a Calderón-Zygmund operator. Specifically, it is studied the problem of finding conditions on \(v\) (resp. on \(u\)) which imply the existence of a function \(u\) (resp. \(v\)) such that \((\ast)\) holds. Sufficient conditions are given for arbitrary Calderón-Zygmund operators. It is seen that these conditions are also necessary for the Cauchy integral operator. In order to obtain the above mentioned sufficient conditions, the authors prove vector-valued versions of the weak and strong type inequalities for Calderón -Zygmund operators on non-homogeneous spaces obtained by \textit{F. Nazarov, S. Treil} and \textit{A. Volberg} [``Weak type estimates and Cotlar inequalities for Calderón-Zygmund operators on non-homogeneous spaces'', Int. Math. Res. Not. 1998, No. 9, 463-487 (1998; Zbl 0918.42009)]. Then, they use the well-known close connection between vector-valued inequalities and weighted inequalities to finish the proof.
scientific article; zbMATH DE number 1563899
scientific article