### A class of Fourier multipliers on H¹(ℝ²)

An integral criterion for being an ${H}^{1}\left({\mathbb{R}}^{2}\right)$ Fourier multiplier is proved. It is applied in particular to suitable regular functions which depend on the product of variables.

Skip to main content (access key 's'),
Skip to navigation (access key 'n'),
Accessibility information (access key '0')

An integral criterion for being an ${H}^{1}\left({\mathbb{R}}^{2}\right)$ Fourier multiplier is proved. It is applied in particular to suitable regular functions which depend on the product of variables.

Hörmander’s famous Fourier multiplier theorem ensures the ${L}_{p}$-boundedness of $F(-{\Delta}_{\mathbb{R}}D)$ whenever $F\in \mathscr{H}\left(s\right)$ for some $s\>\frac{D}{2}$, where we denote by $\mathscr{H}\left(s\right)$ the set of functions satisfying the Hörmander condition for $s$ derivatives. Spectral multiplier theorems are extensions of this result to more general operators $A\ge 0$ and yield the ${L}_{p}$-boundedness of $F\left(A\right)$ provided $F\in \mathscr{H}\left(s\right)$ for some $s$ sufficiently large. The harmonic oscillator $A=-{\Delta}_{\mathbb{R}}+{x}^{2}$ shows that in general $s\>\frac{D}{2}$ is not sufficient even if $A$ has a heat kernel satisfying gaussian estimates. In this paper,...

It is proved that if $m:{\mathbb{R}}^{d}\to \u2102$ satisfies a suitable integral condition of Marcinkiewicz type then m is a Fourier multiplier on the ${H}^{1}$ space on the product domain ${\mathbb{R}}^{{d}_{1}}\times ...\times {\mathbb{R}}^{{d}_{k}}$. This implies an estimate of the norm $N(m,{L}^{p}\left({\mathbb{R}}^{d}\right)$ of the multiplier transformation of m on ${L}^{p}\left({\mathbb{R}}^{d}\right)$ as p→1. Precisely we get $N(m,{L}^{p}\left({\mathbb{R}}^{d}\right))\lesssim {(p-1)}^{-k}$. This bound is the best possible in general.

We define a new type of multiplier operators on ${L}^{p}{(}^{N})$, where ${}^{N}$ is the N-dimensional torus, and use tangent sequences from probability theory to prove that the operator norms of these multipliers are independent of the dimension N. Our construction is motivated by the conjugate function operator on ${L}^{p}{(}^{N})$, to which the theorem applies as a particular example.

Riesz function technique is used to prove a multiplier theorem for the Hankel transform, analogous to the classical Hörmander-Mihlin multiplier theorem (Hörmander (1960)).

We strengthen the Carleson-Hunt theorem by proving ${L}^{p}$ estimates for the $r$-variation of the partial sum operators for Fourier series and integrals, for $r>\mathrm{\U0001d696\U0001d68a\U0001d6a1}\{{p}^{\text{'}},2\}$. Four appendices are concerned with transference, a variation norm Menshov-Paley-Zygmund theorem, and applications to nonlinear Fourier transforms and ergodic theory.

We prove a weighted vector-valued weak type (1,1) inequality for the Bochner-Riesz means of the critical order. In fact, we prove a slightly more general result.

Various new sufficient conditions for representation of a function of several variables as an absolutely convergent Fourier integral are obtained. The results are given in terms of ${L}_{p}$ integrability of the function and its partial derivatives, each with a different p. These p are subject to certain relations known earlier only for some particular cases. Sharpness and applications of the results obtained are also discussed.

In the two-parameter setting, we say a function belongs to the mean little BMO if its mean over any interval and with respect to any of the two variables has uniformly bounded mean oscillation. This space has been recently introduced by S. Pott and the present author in relation to the multiplier algebra of the product BMO of Chang-Fefferman. We prove that the Cotlar-Sadosky space $bmo{(}^{N})$ of functions of bounded mean oscillation is a strict subspace of the mean little BMO.

Let K be a Calderón-Zygmund kernel and P a real polynomial defined on ℝⁿ with P(0) = 0. We prove that convolution with Kexp(i/P) is continuous on L²(ℝⁿ) with bounds depending only on K, n and the degree of P, but not on the coefficients of P.

We study the problem of ${L}^{p}$-boundedness ($1\<p\<\infty $) of operators of the form $m({L}_{1},\cdots ,{L}_{n})$ for a commuting system of self-adjoint left-invariant differential operators ${L}_{1},\cdots ,{L}_{n}$ on a Lie group $G$ of polynomial growth, which generate an algebra containing a weighted subcoercive operator. In particular, when $G$ is a homogeneous group and ${L}_{1},\cdots ,{L}_{n}$ are homogeneous, we prove analogues of the Mihlin-Hörmander and Marcinkiewicz multiplier theorems.

We define homogeneous classes of x-dependent anisotropic symbols ${\u1e60}_{\gamma ,\delta}^{m}\left(A\right)$ in the framework determined by an expansive dilation A, thus extending the existing theory for diagonal dilations. We revisit anisotropic analogues of Hörmander-Mikhlin multipliers introduced by Rivière [Ark. Mat. 9 (1971)] and provide direct proofs of their boundedness on Lebesgue and Hardy spaces by making use of the well-established Calderón-Zygmund theory on spaces of homogeneous type. We then show that x-dependent symbols in...