Related papers: $L^p$-$L^q$ Multipliers on commutative hypergroups

We consider degenerate differential operators $A = \displaystyle{\sum_{k,j=1}^d \partial_k (a_{kj} \partial_j)}$ on $L^2(\mathbb{R}^d)$ with real symmetric bounded measurable coefficients. Given a function $\chi \in…

The unit sphere $\mathbb{S}$ in $\mathbb{C}^n$ is equipped with the tangential Cauchy-Riemann complex and the associated Laplacian $\Box_b$. We prove a H\"ormander spectral multiplier theorem for $\Box_b$ with critical index $n-1/2$, that…

For $1<p<\infty$ and $0<s<1$, let $\mathcal{Q}^p_ s (\mathbb{T})$ be the space of those functions $f$ which belong to $ L^p(\mathbb{T})$ and satisfy \[ \sup_{I\subset…

We investigate the boundedness of unimodular Fourier multipliers on modulation spaces. Surprisingly, the multipliers with general symbol $e^{i|\xi|^\alpha}$, where $\alpha\in[0, 2]$, are bounded on all modulation spaces, but, in general,…

This dissertation studies the Fourier restriction, which is to find the range of the constants p, q such that the L^q norm on a chosen subset of the Fourier domain is bounded above by the L^p norm in a spacial domain, up to some constant…

We prove a two-sided transference theorem between $L^{p}$ spherical multipliers on the compact symmetric space $U/K$ and $L^{p}$ multipliers on the vector space $i\mathfrak{p},$ where the Lie algebra of $U$ has Cartan decomposition…

This paper is devoted to the study of pointwise convergence of Fourier series for group von Neumann algebras and quantum groups. It is well-known that a number of approximation properties of groups can be interpreted as summation methods…

We obtain a sharp $L^2\times L^2 \to L^1$ boundedness criterion for a class of bilinear operators associated with a multiplier given by a signed sum of dyadic dilations of a given function, in terms of the $L^q$ integrability of this…

Let $L$ be a second order divergence form elliptic operator with complex bounded measurable coefficients. The operators arising in connection with $L$, such as the heat semigroup and Riesz transform, are not, in general, of…

On a general Lie group $G$ endowed with a sub-Riemannian structure and of local dimension $d$, we characterize the pointwise multipliers of Triebel--Lizorkin spaces $F^{p,q}_{\alpha}$ for $p,q\in (1,\infty)$ and $\alpha>d/p$, and those of…

We give a class of Fourier multipliers with non-symmetric symbols and explicit norm bounds on $L^p$ spaces by using the stochastic calculus of L\'evy processes and Burkholder-Wang estimates for differentially subordinate martingales.

We give various conditions for Hermite pseudo-multipliers to be bounded on $L^2(\mathbb{R}^n)$. As a by-product we also give results on $L^p(\mathbb{R}^n)$, as well as new results for pseudo-multipliers for the Gaussian measure setting. One…

We show that the cone multiplier satisfies local $L^p$-$L^q$ bounds only in the trivial range $1\leq q\leq 2\leq p\leq\infty$. To do so, we suitably adapt to this setting the proof of Fefferman for the ball multiplier. As a consequence we…

The classical Stein-Tomas restriction theorem is equivalent to the statement that the spectral measure $dE(\lambda)$ of the square root of the Laplacian on $\RR^n$ is bounded from $L^p(\RR^n)$ to $L^{p'}(\RR^n)$ for $1 \leq p \leq…

We prove uniform $L^p$ bounds for multilinear operators which are given by multipliers whose symbols are singular on a one dimensional subspace. The novelty is that these bounds are uniform in the choice of the subspace.

We provide asymptotically sharp bounds for the $L_p$ norms of the Fourier multipliers with the symbols $e^{i\lambda \varphi(\xi/|\xi|)}$, where $\lambda\in \mathbb{R}$ is a large parameter.

This article explores weighted $(L^p, L^q)$ inequalities for the Fourier transform in rank one Riemannian symmetric spaces of noncompact type. We establish both necessary and sufficient conditions for these inequalities to hold. To prove…

We prove a Mikhlin--H\"ormander multiplier theorem for the partial harmonic oscillator $H_{\textup{par}}=-\pa_\rho^2-\Delta_x+|x|^2$ for $(\rho, x)\in\R\times\R^d$ by using the Littlewood--Paley $g$ and $g^\ast$ functions and the associated…

The $L^p$ boundedness theory of convolution operators is \linebreak based on an initial $L^2\to L^2$ estimate derived from the Fourier transform. The corresponding theory of multilinear operators lacks such a simple initial estimate in view…

In this paper we obtain some noncommutative multiplier theorems and maximal inequalities on semigroups. As applications, we obtain the corresponding individual ergodic theorems. Our main results extend some classical results of Stein and…