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Register a new userSparse bounds for discrete singular Radon transforms
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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.
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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.
Consider the discrete cubic Hilbert transform defined on finitely supported functions $f$ on $mathbb{Z}$ by begin{eqnarray*} H_3f(n) = sum_{m ot = 0} frac{f(n- m^3)}{m}. end{eqnarray*} We prove that there exists $r <2$ and universal constant $C$ such that for all finitely supported $f,g$ on $mathbb{Z}$ there exists an $(r,r)$-sparse form ${Lambda}_{r,r}$ for which begin{eqnarray*} left| langle H_3f, g rangle right| leq C {Lambda}_{r,r} (f,g). end{eqnarray*} This is the first result of this type concerning discrete harmonic analytic operators. It immediately implies some weighted inequalities, which are also new in this setting.
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.
We prove sparse bounds for the spherical maximal operator of Magyar, Stein and Wainger. The bounds are conjecturally sharp, and contain an endpoint estimate. The new method of proof is inspired by ones by Bourgain and Ionescu, is very efficient, and has not been used in the proof of sparse bounds before. The Hardy-Littlewood Circle method is used to decompose the multiplier into major and minor arc components. The efficiency arises as one only needs a single estimate on each element of the decomposition.
We exhibit a range of $ell ^{p}(mathbb{Z}^d)$-improving properties for the discrete spherical maximal average in every dimension $dgeq 5$. The strategy used to show these improving properties is then adapted to establish sparse bounds, which extend the discrete maximal theorem of Magyar, Stein, and Wainger to weighted spaces. In particular, the sparse bounds imply that the discrete spherical maximal average is a bounded map from $ell^2(w)$ into $ell^2(w)$ provided $w^{frac{d}{d-4}+delta}$ belongs to the Muckenhoupt class $A_2$ for some $delta>0.$
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