Bilinear Multipliers on Banach Function Spaces

Oscar Blasco

doi:10.1155/2019/7639380

Abstract

Let X1,X2,X3 be Banach spaces of measurable functions in L0(R) and let m(ξ,η) be a locally integrable function in R2. We say that m∈BM(X1,X2,X3)(R) if Bm(f,g)(x)=∫R∫Rf^(ξ)g^(η)m(ξ,η)e2πi<ξ+η,x>dξdη, defined for f and g with compactly supported Fourier transform, extends to a bounded bilinear operator from X1×X2 to X3. In this paper we investigate some properties of the class BM(X1,X2,X3)(R) for general spaces which are invariant under translation, modulation, and dilation, analyzing also the particular case of r.i. Banach function spaces. We shall give some examples in this class and some procedures to generate new bilinear multipliers. We shall focus on the case m(ξ,η)=M(ξ-η) and find conditions for these classes to contain nonzero multipliers in terms of the Boyd indices for the spaces.

Highlights

Throughout the paper L0(Rn) stands for the space of complex svCua0pl(uRpeodnr)tmfaonerdatshvuearnasbipslahecinefsugnoacfttciinoofnninstiintdyue,ofriuenssepdfuecnotcinvtieoRlyn,n,SwCi(tRhc(nRc)onfmo) rpatanhcdet Schwartz class on Rn, and P(Rn) for the set of functions in S(Rn) such that supp fis compact
We present the definition of a bilinear multiplier we shall be dealing with
The bilinear versions of several classical linear operators appearing in Harmonic Analysis, such as the Hilbert transform or the fractional integral, are the motivation for the class of bilinear multipliers that we shall analyze in the paper

Summary

Introduction

Throughout the paper L0(Rn) stands for the space of complex svCua0pl(uRpeodnr)tmfaonerdatshvuearnasbipslahecinefsugnoacfttciinoofnninstiintdyue,ofriuenssepdfuecnotcinvtieoRlyn,n,SwCi(tRhc(nRc)onfmo) rpatanhcdet Schwartz class on Rn, and P(Rn) for the set of functions in S(Rn) such that supp fis compact. The bilinear versions of several classical linear operators appearing in Harmonic Analysis, such as the Hilbert transform or the fractional integral, are the motivation for the class of bilinear multipliers that we shall analyze in the paper. The boundedness results on Lp-spaces for the bilinear H and Iα shown took long that M(ξ) time to be = sign(ξ) achieved. In particular it was ∈ M(Lp1 ,Lp3 ,Lp3 )(Rn) for 1

Bilinear Multipliers

Bilinear Multipliers on Rearrangement Invariant Banach Function Spaces