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Register a new user$ell^p(mathbb{Z}^d)$-Improving Properties and Sparse Bounds for Discrete Spherical Maximal Averages
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Authors
Robert Kesler
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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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We prove an expanded range of $ell ^{p}(mathbb{Z}^d)$-improving properties and sparse bounds for discrete spherical maximal means in every dimension $dgeq 6$. Essential elements of the proofs are bounds for high exponent averages of Ramanujan and restricted Kloosterman sums.
Let $ lambda ^2 in mathbb N $, and in dimensions $ dgeq 5$, let $ A_{lambda } f (x)$ denote the average of $ f ;:; mathbb Z ^{d} to mathbb R $ over the lattice points on the sphere of radius $lambda$ centered at $x$. We prove $ ell ^{p}$ improving properties of $ A_{lambda }$. begin{equation*} lVert A_{lambda }rVert_{ell ^{p} to ell ^{p}} leq C_{d,p, omega (lambda ^2 )} lambda ^{d ( 1-frac{2}p)}, qquad tfrac{d-1}{d+1} < p leq frac{d} {d-2}. end{equation*} It holds in dimension $ d =4$ for odd $ lambda ^2 $. The dependence is in terms of $ omega (lambda ^2 )$, the number of distinct prime factors of $ lambda ^2 $. These inequalities are discre
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.
Let $fin ell^2(mathbb Z)$. Define the average of $ f$ over the square integers by $ A_N f(x):=frac{1}{N}sum_{k=1}^N f(x+k^2) $. We show that $ A_N$ satisfies a local scale-free $ ell ^{p}$-improving estimate, for $ 3/2 < p leq 2$: begin{equation*} N ^{-2/p} lVert A_N f rVert _{ p} lesssim N ^{-2/p} lVert frVert _{ell ^{p}}, end{equation*} provided $ f$ is supported in some interval of length $ N ^2 $, and $ p =frac{p} {p-1}$ is the conjugate index. The inequality above fails for $ 1< p < 3/2$. The maximal function $ A f = sup _{Ngeq 1} |A_Nf| $ satisfies a similar sparse bound. Novel weighted and vector valued inequalities for $ A$ follow. A critical step in the proof requires the control of a logarithmic average over $ q$ of a function $G(q,x)$ counting the number of square roots of $x$ mod $q$. One requires an estimate uniform in $x$.
Consider averages along the prime integers $ mathbb P $ given by begin{equation*} mathcal{A}_N f (x) = N ^{-1} sum_{ p in mathbb P ;:; pleq N} (log p) f (x-p). end{equation*} These averages satisfy a uniform scale-free $ ell ^{p}$-improving estimate. For all $ 1< p < 2$, there is a constant $ C_p$ so that for all integer $ N$ and functions $ f$ supported on $ [0,N]$, there holds begin{equation*} N ^{-1/p }lVert mathcal{A}_N frVert_{ell^{p}} leq C_p N ^{- 1/p} lVert frVert_{ell^p}. end{equation*} The maximal function $ mathcal{A}^{ast} f =sup_{N} lvert mathcal{A}_N f rvert$ satisfies $ (p,p)$ sparse bounds for all $ 1< p < 2$. The latter are the natural variants of the scale-free bounds. As a corollary, $ mathcal{A}^{ast} $ is bounded on $ ell ^{p} (w)$, for all weights $ w$ in the Muckenhoupt $A_p$ class. No prior weighted inequalities for $ mathcal{A}^{ast} $ were known.
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