Vector Valued Inequalities for Strongly Singular Calderón-Zygmund Operators.

In this article we consider a theory of vector valued strongly singular operators. Our results include Lp, Hp and BMO continuity results. Moreover, as is well known, vector valued estimates are closely related to weighted norm inequalities. These results are developed in the first four sections of our paper. In section 5 we use our vector valued singular integrals to estimate the corresponding maximal operators. Finally in section 6 we discuss applications to weighted norm inequalities for pseudo-differential operators with symbols in the classes described above. Our results in this direction are related to some recent work by Chanillo and Torchinsky [3]. We refer to section 2 for a review of the necessary definitions and notation to be followed in the paper