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This article develops a novel approach to the representation of singular integral operators of Calderon-Zygmund type in terms of continuous model operators, in both the classical and the bi-parametric setting. The representation is realized as a finite sum of averages of wavelet projections of either cancellative or noncancellative type, which are themselves Calderon-Zygmund operators. Both properties are out of reach for the established dyadic-probabilistic technique. Unlike their dyadic counterparts, our representation reflects the additional kernel smoothness of the operator being analyzed. Our representation formulas lead naturally to a new family of $T(1)$ theorems on weighted Sobolev spaces whose smoothness index is naturally related to kernel smoothness. In the one parameter case, we obtain the Sobolev space analogue of the $A_2$ theorem; that is, sharp dependence of the Sobolev norm of $T$ on the weight characteristic is obtained in the full range of exponents. In the bi-parametric setting, where local average sparse domination is not generally available, we obtain quantitative $A_p$ estimates which are best known, and sharp in the range $max{p,p}geq 3$ for the fully cancellative case.
Let $Omega$ be homogeneous of degree zero and have mean value zero on the unit sphere ${S}^{n-1}$, $T_{Omega}$ be the convolution singular integral operator with kernel $frac{Omega(x)}{|x|^n}$. For $bin{rm BMO}(mathbb{R}^n)$, let $T_{Omega,,b}$ be th
Let $Omega$ be homogeneous of degree zero and have mean value zero on the unit sphere ${S}^{d-1}$, $T_{Omega}$ be the homogeneous singular integral operator with kernel $frac{Omega(x)}{|x|^d}$ and $T_{Omega,,b}$ be the commutator of $T_{Omega}$ with
We represent a bilinear Calderon-Zygmund operator at a given smoothness level as a finite sum of cancellative, complexity zero operators, involving smooth wavelet forms, and continuous paraproduct forms. This representation results in a sparse $T(1)$
Let $Pin Bbb Q_p[x,y]$, $sin Bbb C$ with sufficiently large real part, and consider the integral operator $ (A_{P,s}f)(y):=frac{1}{1-p^{-1}}int_{Bbb Z_p}|P(x,y)|^sf(x) |dx| $ on $L^2(Bbb Z_p)$. We show that if $P$ is homogeneous then for each charact
We establish $L^2$ boundedness of all nice parabolic singular integrals on Good Parabolic Graphs, aka {em regular} Lip(1,1/2) graphs. The novelty here is that we include non-homogeneous kernels, which are relevant to the theory of parabolic uniform r