Estimates for operators in mixed weighted $L\sp p$-spaces

Abstract

A weighted Marcinkiewicz interpolation theorem is proved. If T is simultaneously of weak type (p,, q,), i = 0, 1; 1 1. The result is applied to obtain weighted estimates for the Laplace and Fourier transform, as well as the Riesz potential. 1. In [8] we proved that if T is simultaneously of weak type (pi, qi), i = 0, 1; 1 po 1. As always, new interpolation results imply new results for a number of classical operators. Here we apply the results to the Laplace transform, complementing those obtained in [1, Theorem 2.4], the Fourier transform, extending results in [8, 10-12] and apply it to obtain a new inequality with applications in signal analysis [6]. Further we obtain a weighted extension of Sobolev's theorem. In [9, Theorem 1.1] it was shown that the only bounded translation invariant operators from LP to Lq, q 1. In the next section we introduce notation, give a preliminary result and prove the main result (Theorem 2.2). ?3 contains applications. Throughout, p' denotes the conjugate index of p and is related top by p + p' = pp' with p' = + 00 if p = 1, similarly for other letters. Further, constants are denoted by C and may be different at different appearances, but are always independent of the function in question. Z denotes the set of integers. 2. Let u and v be nonnegative weight functions defined on S2 c R, and T a sublinear operator defined on Lebesgue measurable functions on U2. We denote by LP= L P(9), 0 1 or 0 O . It is well known [2, 4] that P E [LiP, Lq], 1 O 0 where here and in the sequel XE(X) 1 if x E, XE\J\~0 i f x ~F is the characteristic function. The corresponding result for the dual operator also holds, only now the characteristic functions in the above integrals are interchanged. In case 0 0. If P E [LP, Lq], fix a positive increasing sequence {Xk}k, z and a given sequence of nonnegative numbers { ak} This content downloaded from 207.46.13.124 on Wed, 22 Jun 2016 05:32:16 UTC All use subject to http://about.jstor.org/terms OPERATORS IN MIXED WEIGHTED LP-SPACES 485 and define f(x) = FakXk(x) v(x) k