Fourier multipliers on weighted $L^p$ spaces

Abstract

The paper provides a complement to the classical results on Fourier multipliers on L p spaces.In particular, we prove that if q ∈ (1, 2) and a function m : R → C is of bounded q-variation uniformly on the dyadic intervals in R, i.e., m ∈ V q (D), then m is a Fourier multiplier on L p (R, w dx) for every p ≥ q and every weight w satisfying Muckenhoupt's A p/q -condition.We also obtain a higherdimensional counterpart of this result as well as of a result by E. Berkson and T.A. Gillespie including the case of the V q (D) spaces with q > 2. New weighted estimates for modified Littlewood-Paley functions are also provided. 1 Introduction and statement of results 807 2 Proofs of Theorems B(i) and A 811 3 Proof of Theorem B(ii) 817 4 Higher-dimensional analogue of Theorem A 824

Locations

• Mathematical Research Letters - View - PDF
• arXiv (Cornell University) - View - PDF