Problems in harmonic analysis related to curvature
Elias M. Stein, Stephen Wainger · Bulletin of the American Mathematical Society · 1978
THEOREM A.Ifn>3 and f is locally in L p ,p > n/{n -1), Hm f f(x -ey) do(y) = f{x) a.e.e->0 J This is false ifp 2. It is to be understood that part of the assertion of Theorem A is that the limit above is well defined for almost every JC.Incidentally, the analogue of Theorem A with the sphere replaced by the boundary of a cube is false.This should be the first hint to the reader that curvature will play a decisive role in these matters.For our second question we let y(t) be a continuous curve in R n with y(0) = 0. We ask, doesThe answer to ( 2) is yes if y(t) is a half-line, and this is, in fact, an easy consequence of the one-dimensional theory.We shall (in Part III) give examples of C 00 curves y(t) such that the answer to (2) is negative even if we restrict ourselves to the class of bounded functions.Thus to obtain positive answers to question (2) it is necessary to restrict attention to a subclass of curves.Again curvature is crucial.We obtain positive results if y has an appropriate amount of curvature.Let us say that a C 00 curve y(t) in R n is well-curved if y(0) = 0 and a segment of the curve containing the origin lies in the subspace of R n spanned by \.Then Jim I f o h f(x-y(t))dt=f(x) a.e.provided y(t) is well-curved.To see the full significance of Theorems A and B, we shall, in §2, review relevant classical aspects of singular integral theory, differentiation theory, and especially the relation between them.This will suggest the control of appropriate maximal functions is the basic underlying analytical problem, and that there should be singular integral operators related to the curves y(t).These operators are defined by 3C/(*) =ƒ_/(*-Y(0) f • {% is well defined as a principal value integral if ƒ is smooth.)