On operator-valued Fourier multipliers

We give a simpler proof of a result on operator-valued Fourier multipliers on Lp([0,2π]d;X) using an induction argument based on a known result when d=1.     

Operator-valued Fourier Haar multipliers

Criteria are given to ensure the boundedness of Fourier Haar multiplier operators from Lp([0,1],X) to Lq([0,1],Y) where the Fourier Haar multiplier sequences come not from R, as in the classical setting, but rather from the space of bounded linear operators from a Banach space X into a Banach space Y.     

Fourier multipliers on compact groups

In a recent paperIan Inglis gives some sufficient conditions for a function on a totally disconnected compact Abelian group to be anL ^p Fourier multiplier. His proof depends on an interpolation theorem ofE. M. Stein. In this note we prove a generalization ofInglis' theorem. Our result is deduced from a factorization theorem, the proof of which is elementary, and a standard multiplier theorem.This work was done while the second-named author held a visiting appointment at the University of Washington. He wishes to thank ProfessorsE. Hewitt andR. R. Phelps for making the visit possible.     

Logarithmic inequalities for Fourier multipliers

In the paper we study LlogL estimates for Fourier multipliers resulting from modulation of the jumps of Lévy processes. We exhibit a class of functions $m:\mathbb R ^d \rightarrow \mathbb C $ , for which the corresponding multipliers $T_m$ satisfy the following estimate: for $K>1$ , any locally integrable function $f$ on $\mathbb R ^d$ and any Borel subset $A$ of $\mathbb R ^d$ , $$\begin{aligned} \int _{A}|T_m f(x)|\,\text{ d}x\le K\int _{\mathbb{R }d}\Psi (|f(x)|)\,\text{ d}x+\frac{|A|}{2(K-1)}, \end{aligned}$$ where $\Psi (t)=(t+1)\log (t+1)-t$ . We also present related lower bounds which arise from considering appropriate examples for the Beurling-Ahlfors operator.     

Wavelet decompositions of Fourier multipliers

We show that in terms of its weak topology, the space of Fourier multipliers for , , can be decomposed by band-limited wavelets belonging to the Schwartz class.

     

Fourier multipliers of Lipschitz spaces

Let G denote an infinite, compact, metrizable, 0-dimensional, Abelian group. The following are characterized: (i) the multipliers from one Lipschitz space Lip(α, p; G) to another Lipschitz space Lip(β, q; G) for 0 < α < β < ∞ and 1 ? p, q ? ∞; and (ii) the multipliers from Lip(α, p; G) to Lip(β, q; G) for 0 < β ? α < ∞ and 1 < q ? 2 ? p < ∞. Two special cases of (i), namely the case q = ∞ and the case p = 1, were obtained by the authors in an earlier publication (1981). A. Zygmund (J. Math. Mech.8 (1959), 889–895) and T. Mizuhara (Tôhoku Math. J.24 (1972), 263–268) have characterized the multipliers of certain Lipschitz spaces defined on the circle group.     

Unimodular Fourier multipliers for modulation spaces

We investigate the boundedness of unimodular Fourier multipliers on modulation spaces. Surprisingly, the multipliers with general symbol eiα|ξ|, where α∈[0,2], are bounded on all modulation spaces, but, in general, fail to be bounded on the usual Lp-spaces. As a consequence, the phase-space concentration of the solutions to the free Schrödinger and wave equations are preserved. As a byproduct, we also obtain boundedness results on modulation spaces for singular multipliers |ξ|^−δsin(α|ξ|) for 0?δ?α.     

Homogenous Fourier multipliers of Marcinkiewicz type

The second author was supported in part by a grant from the National Science Foundation (USA)     

Fourier multipliers of classical modulation spaces

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 with p(1,∞) and q[1,∞].     

Weighted estimates for conic Fourier multipliers

We prove a weighted inequality which controls conic Fourier multiplier operators in terms of lacunary directional maximal operators. By bounding the maximal operators, this enables us to conclude that the multiplier operators are bounded on \(L^p(\mathbb {R}^3)\) with \(1 .     

Fourier multipliers on the real Hardy spaces

We provide a variant of Hytönen’s embedding theorem, which allows us to extend and unify several sufficient conditions for a function to be a Fourier multiplier on the real Hardy spaces.     

Commutators for Fourier multipliers on Besov Spaces

If T is any bounded linear operator on Besov spaces B_p^σ_j,q_j(Rⁿ)(j=0,1, and 0<σ₁<σ<σ₀), it is proved that the commutator [T,T_μ]=TT_μ−T_μT is bounded on B_p^σ,q(Rⁿ), if T_μ is a Fourier multiplier such that μ is any (possibly unbounded) symbol with uniformly bounded variation on dyadic coronas.