Related papers: Dyadic representation theorem using smooth wavelet…

Let $G$ be a locally compact second countable groupoid with a Haar system. In this article, we introduce the induced representation of $G$ from a continuous unitary representation of a closed wide subgroupoid $H$ with a Haarsystem provided…

This note is intended to reformulate the Dixmier-Malliavin theorem about smooth group representations in the language of bornological vector spaces, instead of topological vector spaces. This language turns out to allow a more general…

Given a measure $\mu$ of polynomial growth, we refine a deep result by David and Mattila to construct an atomic martingale filtration of $\mathrm{supp}(\mu)$ which provides the right framework for a dyadic form of nondoubling harmonic…

We study several fundamental operators in harmonic analysis related to Jacobi expansions, including Riesz transforms, imaginary powers of the Jacobi operator, the Jacobi-Poisson semigroup maximal operator and Littlewood-Paley-Stein square…

We develop a technique of proving standard estimates in the setting of Laguerre function expansions of convolution type, which works for all admissible type multi-indices $\alpha$ in this context. This generalizes a simpler method existing…

Any Calderon-Zygmund operator T is pointwise dominated by a convergent sum of positive dyadic operators. We give an elementary self-contained proof of this fact, which is simpler than the probabilistic arguments used for all previous…

A famous theorem of Dixmier-Malliavin asserts that every smooth, compactly-supported function on a Lie group can be expressed as a finite sum in which each term is the convolution, with respect to Haar measure, of two such functions. We…

We construct a family of n+1 dyadic filtrations in R^n, so that every Euclidean ball B is contained in some cube Q of our family satisfying diam(Q) \le c_n diam(B) for some dimensional constant c_n. Our dyadic covering is optimal on the…

With the aim of completing the previous study by A. Or{\l}owski and the author concerning intertwining maps between induced representations and conjugation representation, termed here weighted class operators, we compute the latter…

Representation theorems relate seemingly complex objects to concrete, more tractable ones. In this paper, we take advantage of the abstraction power of category theory and provide a general representation theorem for a wide class of…

This exposition presents a self-contained proof of the $A_2$ theorem, the quantitatively sharp norm inequality for singular integral operators in the weighted space $L^2(w)$. The strategy of the proof is a streamlined version of the…

We obtain the off-diagonal Muckenhoupt-Wheeden conjecture for Calder\'on-Zygmund operators. Namely, given $1<p<q<\infty$ and a pair of weights $(u,v)$, if the Hardy-Littlewood maximal function satisfies the following two weight…

Smoothed Wigner transforms have been used in signal processing, as a regularized version of the Wigner transform, and have been proposed as an alternative to it in the homogenization and / or semiclassical limits of wave equations. We…

We prove a boundedness criterion for a class of dyadic multilinear forms acting on two-dimensional functions. Their structure is more general than the one of classical multilinear Calder\'{o}n-Zygmund operators as several functions can now…

The aim of this paper is to give a wavelet series representation of Linear Multifractional Stable Motion (LMSM in brief), which is more explicit than that introduced in (Ayache & Hamonier 2012). Instead of using Daubechies wavelet, which is…

In recent years, it has been well understood that a Calder\'on-Zygmund operator $T$ is pointwise controlled by a finite number of dyadic operators of a very simple structure (called the sparse operators). We obtain a similar pointwise…

In this work we extend Lacey's domination theorem to prove the pointwise control of bilinear Calder\'on--Zygmund operators with Dini--continuous kernel by sparse operators. The precise bounds are carefully tracked following the spirit in a…

We present a framework based on modified dyadic shifts to prove multiple results of modern singular integral theory under mild kernel regularity. Using new optimized representation theorems we first revisit a result of Figiel concerning the…

In this paper, we introduce a class of singular integral operators which generalize Calder\'on-Zygmund operators to the more general case, where the set of singular points of the kernel need not to be the diagonal, but instead, it can be a…

The construction of generalized continuous wavelet transforms on locally compact abelian groups $A$ from quasi-regular representations of a semidirect product group $G = A \rtimes H$ acting on ${\rm L}^2(A)$ requires the existence of a…