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Sparse domination of singular Radon transform

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 Added by Bingyang Hu
 Publication date 2019
  fields
and research's language is English
 Authors Bingyang Hu




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The purpose of this paper is to study the sparse bound of the operator of the form $f mapsto psi(x) int f(gamma_t(x))K(t)dt$, where $gamma_t(x)$ is a $C^infty$ function defined on a neighborhood of the origin in $(x, t) in mathbb R^n times mathbb R^k$, satisfying $gamma_0(x) equiv x$, $psi$ is a $C^infty$ cut-off function supported on a small neighborhood of $0 in mathbb R^n$ and $K$ is a Calderon-Zygmund kernel suppported on a small neighborhood of $0 in mathbb R^k$. Christ, Nagel, Stein and Wainger gave conditions on $gamma$ under which $T: L^p mapsto L^p (1<p<infty)$ is bounded. Under the these same conditions, we prove sparse bounds for $T$, which strengthens their result. As a corollary, we derive weighted norm estimates for such operators.



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We show that discrete singular Radon transforms along a certain class of polynomial mappings $P:mathbb{Z}^dto mathbb{Z}^n$ satisfy sparse bounds. For $n=d=1$ we can handle all polynomials. In higher dimensions, we pose restrictions on the admissible polynomial mappings stemming from a combination of interacting geometric, analytic and number-theoretic obstacles.
We present a general approach to sparse domination based on single-scale $L^p$-improving as a key property. The results are formulated in the setting of metric spaces of homogeneous type and avoid completely the use of dyadic-probabilistic techniques as well as of Christ-Hytonen-Kairema cubes. Among the applications of our general principle, we recover sparse domination of Dini-continuous Calderon-Zygmund kernels on spaces of homogeneous type, we prove a family of sparse bounds for maximal functions associated to convolutions with measures exhibiting Fourier decay, and we deduce sparse estimates for Radon transforms along polynomial submanifolds of $mathbb R^n$.
116 - Xiangxing Tao , Guooen Hu 2019
Let $Omega$ be homogeneous of degree zero, have mean value zero and integrable on the unit sphere, and $mu_{Omega}$ be the higher-dimensional Marcinkiewicz integral defined by $$mu_Omega(f)(x)= Big(int_0^inftyBig|int_{|x-y|leq t}frac{Omega(x-y)}{|x-y|^{n-1}}f(y)dyBig|^2frac{dt}{t^3}Big)^{1/2}. $$ In this paper, the authors establish a bilinear sparse domination for $mu_{Omega}$ under the assumption $Omegain L^{infty}(S^{n-1})$. As applications, some quantitative weighted bounds for $mu_{Omega}$ are obtained.
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251 - Robert Kesler , Dario Mena 2017
Consider the discrete quadratic phase Hilbert Transform acting on $ell^{2}$ finitely supported functions $$ H^{alpha} f(n) : = sum_{m eq 0} frac{e^{2 pi ialpha m^2} f(n - m)}{m}. $$ We prove that, uniformly in $alpha in mathbb{T}$, there is a sparse bound for the bilinear form $langle H^{alpha} f , g rangle$. The sparse bound implies several mapping properties such as weighted inequalities in an intersection of Muckenhoupt and reverse Holder classes.
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