Related papers: Littlewood-Paley theory and the T(1) theorem with …

Let $m,n\ge 1$ and $g_{\lambda_1,\lambda_2}^*$ be the bi-parameter Littlewood-Paley $g_{\lambda}^{*}$-function defined by $$ g_{\lambda_1,\lambda_2}^*(f)(x)= \bigg(\iint_{\R^{m+1}_{+}} \big(\frac{t_2}{t_2 + |x_2 - y_2|}\big)^{m \lambda_2}…

The article proves an assertion analogous to the Littlewood-Paley theorem for the orthoprojectors onto mutually orthogonal subspaces of piecewise polynomial functions on the cube $ I^d. $ This assertion provides an upper estimate for the…

Local $Tb$ theorems with $L^p$ type testing conditions, which are not scale invariant, have been studied widely in the case of the Lebesgue measure. Until very recently, local $Tb$ theorems in the non-homogeneous case had only been proved…

Let $L = \Delta + V$ be a Schr\"odinger operator with a non-negative potential $V$ on a complete Riemannian manifold $M$. We prove that the vertical Littlewood-Paley-Stein functional associated with $L$ is bounded on $L^p(M)$ {\it if and…

In this article we use Littlewood-Paley-Stein theory to prove two versions of Dunkl multiplier theorem when the multiplier $ m $ satisfies a modified H\"ormander condition. When $ m $ is radial we give a simple proof of a known result. For…

We consider certain Littlewood-Paley operators and prove characterization of some function spaces in terms of those operators. When treating weighted Lebesgue spaces, a generalization to weighted spaces will be made for H\"ormander's…

We develop a wide general theory of bilinear bi-parameter singular integrals $T$. First, we prove a dyadic representation theorem starting from $T1$ assumptions and apply it to show many estimates, including $L^p \times L^q \to L^r$…

We establish a new $T1$ theorem for the compactness of bi-parameter Calder\'on-Zygmund singular integral operators. Namely, we show that if a bi-parameter CZO $T$ satisfies the product weak compactness property, the mixed weak…

For a fixed singular Borel probability measure $\mu$ on $\mathbb{T}$, we give several characterizations of when an entire function is the Fourier transform of some $f \in L^2(\mu)$. The first characterization is given in terms of criteria…

We introduce a new sparse $T1$ theorem that estimates the dual pair associated with a Calderon-Zygmund operator by a sub-bilinear form supported on a sparse family of cubes. The main result in the paper improves previous sparse $T1$…

In this paper we prove new inequalities describing the relationship between the "size" of a function on a compact homogeneous manifold and the "size" of its Fourier coefficients. These inequalities can be viewed as noncommutative versions…

We study the properties of reflectionless measures for an $s$-dimensional Calder\'on-Zygmund operator $T$ acting in $\mathbb{R}^d$, where $s\in (0,d)$. Roughly speaking, these are the measures $\mu$ for which $T(\mu)$ is constant on the…

Let $n_1,n_2\ge 1, \lambda_1>1$ and $\lambda_2>1$. For any $x=(x_1,x_2) \in \mathbb {R}^n\times\mathbb{R}^m$, let $g$ and $g_{\vec{\lambda}}^*$ be the bi-parameter Littlewood-Paley square functions defined by \begin{align*} g(f)(x)=…

We prove a Littlewood-type theorem on random analytic functions for not necessarily independent Gaussian processes. We show that if we randomize a function in the Hardy space $H^2(\dd)$ by a Gaussian process whose covariance matrix $K$…

The famous $T1$ theorem for classical Calder\'on-Zygmund operators is a characterisation for their boundedness in $L^{2}$. In the bi-parameter case, on the other hand, the current $T1$ theorem is merely a collection of sufficient…

In this paper we develop a general theoretical tool for the establishment of the boundedness of notoriously difficult operators (such as potentials) on certain specific types of rearrangement-invariant function spaces from analogous…

In this paper we solve a long standing problem about the bilinear $T1$ theorem to characterize the (weighted) compactness of bilinear Calder\'{o}n-Zygmund operators. Let $T$ be a bilinear operator associated with a standard bilinear…

We give a new formulation of the $T1$ theorem for compactness of Calder\'on-Zygmund singular integral operators. In particular, we prove that a Calder\'on-Zygmund operator $T$ is compact on $L^2(\mathbb{R}^n)$ if and only if $T1,T^*1\in…

In this paper, we first prove that the Littlewood-Paley $g$-function, related to the convolution corresponding to the composition of pseudo-differential operator and evolution system associated with pseudo-differential operators, is a…

It is well-known that Littlewood-Paley operators formed with respect to lacunary sets of finite order are bounded on $L^p (\mathbb{R})$ for all $1<p<\infty$. In this note it is shown that $$ \| S_{\mathcal{I}_{E_2}} \|_{L^p (\mathbb{R})…