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Some weighted norm inequalities for the Fourier transform of functions with vanishing moments

1987; American Mathematical Society; Volume: 300; Issue: 2 Linguagem: Inglês

10.1090/s0002-9947-1987-0876464-4

1088-6850

Cora Sadosky, Richard L. Wheeden,

Differential Equations and Boundary Problems

Weighted LP norm inequalities are derived between a function and its Fourier transform in case the function has vanishing moments up to some order.For weights of the form IxIY, the results concern values of y which are outside the range which is normally considered, 1. Introduction.Weighted norm inequalities for the Fourier transform with power weights have natural constraints on the exponents, as indicated in Pitt's theorem [6], which asserts for example that (I)if 1 < P < 00 and max{O, p -2} ~ Y < P -1.The result fails for y outside this range.We will show, however, that (1) holds for y > p -1, y"* kp -1 for k = 1,2, ... , provided that enough moments of f vanish.For example, an immediate consequence of Theorem 1 below is that (1) is valid for p -1 < y < 2 P -1 for all f having mean value zero (d.[2], where analytic functions in the unit circle are considered).The case y = p -1 is excluded, even with this restriction on f, as shown by the counterexample in §5.We work with functions in .9"0.0' the class of Schwartz functions whose Fourier transforms have compact support not containing the origin.Note that all the moments of a function in .9"0,0vanish: foo f{x)xidx = 0, j = 0,1,2, ... , fE.9"o,o.-00 .9"0,0 is dense in all the weighted spaces that we will consider, and the Fourier transform operator has a natural extension to functions (not necessarily locally integrable) in these spaces: see §4.In what follows, if 1 < P < 00, A P stands for the class of nonnegative, locally integrable functions w on RI such that C~I i w{x) dx )C~I i w{xrl/(p-l) dx rl ~ A < 00 for all intervals I c RI.