Lp Function Decomposition on C°° Totally Real Submanifolds of C
نویسندگان
چکیده
For 1 < p < oo we show that LP functions defined on a C°° totally real submanifold of C" can be locally decomposed into the sum of boundary values of holomorphic functions in wedges such that the boundary values are in LP . The general case of a C°° totally real submanifold is reduced to the flat case of R" in C" by an almost analytic change of variables. LP results in the flat case are then obtained using Fourier multipliers. In transporting these results back to the manifold we lose analyticity, so it is necessary to solve a d problem in an appropriate domain. This gives holomorphy in the wedges but produces a C°° error on the edge. This C°° function is then holomorphically decomposed using the FBI transform with a careful analysis to check that the functions are C°° up to the edge and do not destroy the LP behavior. 0. Introduction A distribution defined on a totally real C°° submanifold of C" can be decomposed locally into the sum of boundary values of functions that are holomorphic in wedges. These holomorphic functions have polynomial growth near the edge and so the boundary values are understood in the sense of distributions. This result is found in the paper of Baouendi, Chang, and Treves [1]. In this paper we intend to show that if a function defined on a totally real C°° submanifold of C" is in an LP class for 1 < p < oo, then it can be decomposed so that the holomorphic functions have LP boundary values (Theorem 2). We first show by careful estimates using the FBI transform that a C°° function on the submanifold can be decomposed as in [ 1 ] so that the holomorphic functions are C°° up to the edge. The problem of LP decomposition is then reduced to the problem of C°° decomposition. This paper is mainly concerned with n > 1, however the method can be applied to the case n = 1, which corresponds to a curve in C. 1. Flat case The straight case is R" as a totally real submanifold of C" for which the LP decomposition is already known. For the convenience of the reader we sketch a proof of this fact and include some definitions that will be used later. Received by the editors January 28, 1991 and, in revised form, May 24, 1991. 1991 Mathematics Subject Classification. Primary 32F25; Secondary 32D15. ©1993 American Mathematical Society 0002-9939/93 $1.00+ $.25 per page
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