Singular integrals along submanifolds of finite type.

Abstract

Theorem A. Let T8 be given as above. Suppose that (i) 8 is of finite type at 0, and (ii) ∈C1(Sn−1). Then, for 1 0 such that ‖T8f ‖Lp(Rd ) ≤ Cp‖f ‖Lp(Rd ) (1.4) for every f ∈Lp(Rd). It is well known that T8 may fail to be bounded on L for any p if condition (i) is removed (the precise definition of a finite type mapping will be reviewed in the next section). The purpose of this paper is to establish the L boundedness of T8 when condition (ii) is replaced by the following weaker condition: (ii′ ) ∈Lq(Sn−1) for some q > 1. This yields the following theorem.