New results on restriction of Fourier multipliers (original) (raw)

Fourier multipliers of classical modulation spaces

Applied and Computational Harmonic Analysis, 2006

Based on the observation that translation invariant operators on modulation spaces are convolution operators we use techniques concerning pointwise multipliers for generalized Wiener amalgam spaces in order to give a complete characterization of the Fourier multipliers of modulation spaces. We deduce various applications, among them certain convolution relations between modulation spaces, as well as a short proof for a generalization of the main result of a recent paper by Bènyi et al., see [À. Bènyi, L. Grafakos, K. Gröchenig, K.A. Okoudjou, A class of Fourier multipliers for modulation spaces, Appl. Comput. Harmon. Anal. 19 (1) (2005) 131-139]. Finally, we show that any function with ([d/2] + 1)-times bounded derivatives is a Fourier multiplier for all modulation spaces M p,q (R d) with p ∈ (1, ∞) and q ∈ [1, ∞].

Weak-Type Boundedness of the Fourier Transform on Rearrangement Invariant Function Spaces

Proceedings of the Edinburgh Mathematical Society

We study several questions about the weak-type boundedness of the Fourier transform ℱ on rearrangement invariant spaces. In particular, we characterize the action of ℱ as a bounded operator from the minimal Lorentz space Λ(X) into the Marcinkiewicz maximal space M(X), both associated with a rearrangement invariant space X. Finally, we also prove some results establishing that the weak-type boundedness of ℱ, in certain weighted Lorentz spaces, is equivalent to the corresponding strong-type estimates.

A class of Fourier multipliers for modulation spaces

Applied and Computational Harmonic Analysis, 2005

We prove the boundedness of a general class of Fourier multipliers, in particular of the Hilbert transform, on modulation spaces. In general, however, the Fourier multipliers in this class fail to be bounded on L p spaces. The main tools are Gabor frames and methods from time-frequency analysis.

Functions which operate on algebras of Fourier multipliers

Tohoku Mathematical Journal, 1995

We study functions which operate on a Banach space of bounded functions defined on a discrete space. As a consequence we characterize functions which operate on the algebra of the translation invariant operators from L P {G) to L 2 {G) for 1 <p<2 and for a compact abelian group G.

Modulation Spaces, Multipliers Associated with the Special Affine Fourier Transform

Complex Analysis and Operator Theory

We study some fundamental properties of the special affine Fourier transform (SAFT) in connection with the Fourier analysis and time-frequency analysis. We introduce the modulation space M r,s A in connection with SAFT and prove that if a bounded linear operator between new modulation spaces commutes with A-translation, then it is a A-convolution operator. We also establish Hörmander multiplier theorem and Littlewood-Paley theorem associated with the SAFT.

Boundedness of classical operators on rearrangement-invariant spaces

2019

We study the behaviour on rearrangement-invariant spaces of such classical operators of interest in harmonic analysis as the Hardy-Littlewood maximal operator (including the fractional version), the Hilbert and Stieltjes transforms, and the Riesz potential. The focus is on sharpness questions, and we present characterisations of the optimal domain (or range) partner spaces when the range (domain) is fixed. When a rearrangement-invariant partner space exists at all, a complete characterisation of the situation is given. We illustrate the results with a variety of examples of sharp particular results involving customary function spaces.

Free Fourier Multipliers associated with the firstSegment

arXiv: Operator Algebras, 2019

We study Fourier multipliers on free group mathbbFinfty\mathbb{F}_\inftymathbbFinfty associated with the first segment of the reduced words, and prove that they are completely bounded on the noncommutative LpL^pLp spaces Lp(hatmathbbFinfty)L^p(\hat{\mathbb{F}}_\infty)Lp(hatmathbbFinfty) iff their restriction on Lp(hatmathbbF_1)=Lp(mathbbT)L^p(\hat{\mathbb{F}}_1)=L^p(\mathbb{T})Lp(hatmathbbF_1)=Lp(mathbbT) are completely bounded. As a consequence, every classical Mikhlin multiplier extends to a LpL^pLp Fourier multiplier on free groups for all $1