Boundedness Of Fourier Multipliers Research Articles

– MULTIPLIERS ON COMMUTATIVE HYPERGROUPS

AbstractThe main purpose of this paper is to prove Hörmander’s $L^p$ – $L^q$ boundedness of Fourier multipliers on commutative hypergroups. We carry out this objective by establishing the Paley inequality and Hausdorff–Young–Paley inequality for commutative hypergroups. We show the $L^p$ – $L^q$ boundedness of the spectral multipliers for the generalised radial Laplacian by examining our results on Chébli–Trimèche hypergroups. As a consequence, we obtain embedding theorems and time asymptotics for the $L^p$ – $L^q$ norms of the heat kernel for generalised radial Laplacian.

Calderón–Zygmund operators, Bochner–Riesz means, and parametric Marcinkiewicz integrals on Hardy–Morrey spaces with variable exponents

We obtain the boundedness of Calderón–Zygmund operators on the entire Hardy–Morrey spaces with variable exponents. We obtain this result by refining the extrapolation theory. This refined extrapolation theory also gives the boundedness of Fourier multipliers, Bochner–Riesz means, and parametric Marcinkiewicz integrals on Hardy–Morrey spaces with variable exponents.

$$L^{p}$$–$$L^{q}$$ Boundedness of Fourier Multipliers on Fundamental Domains of Lattices in $$\mathbb {R}^d$$

In this paper we study the \(L^{p}\)–\(L^{q}\) boundedness of Fourier multipliers on the fundamental domain of a lattice in \(\mathbb {R}^{d}\) for \(1< p,q < \infty \) under the classical Hörmander condition. First, we introduce Fourier analysis on lattices and have a look at possible generalisations. We then prove the Hausdorff–Young inequality, Paley’s inequality and the Hausdorff–Young–Paley inequality in the context of lattices. This amounts to a quantitative version of the \(L^{p}\)–\(L^{q}\) boundedness of Fourier multipliers. We will show that this delivers also some \(L^{p}\)-estimates by using some standard Lebesgue space embeddings, which come from the finite measure of the fundamental domain. Moreover, the Paley inequality allows us to prove the Hardy–Littlewood inequality. As an application we treat some Sobolev embedding results, which also indicate the sharpness of our main inequalities.

Identification of anisotropic mixed-norm Hardy spaces and certain homogeneous Triebel–Lizorkin spaces

Let S(Rn) be the Schwartz class on Rn and S∞(Rn)≔ϕ∈S(Rn):∫Rnxαϕ(x)dx=0for any multi-indexα∈({0,1,…})n,and S′(Rn) and S∞′(Rn) be their dual spaces, respectively. Let a→≔(a1,…,an)∈[1,∞)n, p→≔(p1,…,pn)∈(0,1]n, and Ha→p→(Rn)⊂S′(Rn) be the anisotropic mixed-norm Hardy space, associated with an anisotropic quasi-homogeneous norm |⋅|a→, defined via the non-tangential grand maximal function. In this article, via the known atomic characterizations and Lusin area function characterizations of Ha→p→(Rn), the authors first establish a new atomic characterization of Ha→p→(Rn) in terms of S∞′(Rn). Applying this atomic characterization, the authors then obtain the Littlewood–Paley g-function characterization of Ha→p→(Rn) in terms of S∞′(Rn), which further induces the identification of Ha→p→(Rn) and certain anisotropic homogeneous mixed-norm Triebel–Lizorkin spaces. As applications, the authors obtain the dual theorem on some anisotropic homogeneous mixed-norm Triebel–Lizorkin spaces and the boundedness of Fourier multipliers as well as anisotropic pseudo-differential operators on Ha→p→(Rn). All these results are new even for isotropic mixed-norm Hardy spaces on Rn.

On the failure of the Hörmander multiplier theorem in a limiting case

We discuss the Hörmander multiplier theorem for $L^p$ boundedness of Fourier multipliers in which the multiplier belongs to a fractional Sobolev space with smoothness $s$. We show that this theorem does not hold in the limiting case $|1/p-1/2|=s/n$.

Equivalent Quasi-Norms of Besov–Triebel–Lizorkin-Type Spaces via Derivatives

In this paper, applying the Peetre maximal function characterizations and the boundedness of Fourier multipliers on Besov-type and Triebel–Lizorkin-type spaces, as well as Besov–Morrey and Triebel–Lizorkin–Morrey spaces, the authors present some equivalent quasi-norms of these spaces in terms of derivatives.

Remarks on the Unimodular Fourier Multipliers onα-Modulation Spaces

We study the boundedness properties of the Fourier multiplier operatoreiμ(D)onα-modulation spacesMp,qs,α (0≤α&lt;1)and Besov spacesBp,qs(Mp,qs,1). We improve the conditions for the boundedness of Fourier multipliers with compact supports and for the boundedness ofeiμ(D)onMp,qs,α. Ifμis a radial functionϕ(|ξ|)andϕsatisfies some size condition, we obtain the sufficient and necessary conditions for the boundedness ofeiϕ(|D|)betweenMp1,q1s1,αandMp2,q2s2,α.

Fourier Multipliers on Triebel-Lizorkin-Type Spaces

The authors study the mapping properties of Fourier multipliers, with symbols satisfying some generalized Hörmander's condition, on Triebel- Lizorkin-type spaces and Triebel-Lizorkin-Hausdorff spaces. To this end, the authors first establish a new characterization of these spaces via some generalized (weighted)gλ∗functions, which essentially improves the known result for Triebel-Lizorkin spaces even whenτ=0. Applying this new characterization, the authors then obtain the boundedness of Fourier multipliers on Triebel-Lizorkin-type spaces and Triebel-Lizorkin-Hausdorff spaces, which also give a new proof of the Sobolev embedding theorems for these spaces.

Singular operators and Fourier multipliers in weighted Lebesgue spaces with variable index

Mikhlin’s ideas and results related to the theory of spaces Lρp(·) with nonstandard growth are developed. These spaces are called Lebesgue spaces with variable index; they are used in mechanics, the theory of differential equations, and variational problems. The boundedness of Fourier multipliers and singular operators on the spaces Lρp(·) are considered. All theorems are derived from an extrapolation theorem due to Rubio de Francia. The considerations essentially use theorems on the boundedness of operators and maximal Hardy-Littlewood functions on Lebesgue spaces with constant index.