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Register a new user$ell^p$-improving inequalities for Discrete Spherical Averages
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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
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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 $delta>0.$
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 $x$.
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.
For a polynomial $P$ mapping the integers into the integers, define an averaging operator $A_{N} f(x):=frac{1}{N}sum_{k=1}^N f(x+P(k))$ acting on functions on the integers. We prove sufficient conditions for the $ell^{p}$-improving inequality begin{equation*} |A_N f|_{ell^q(mathbb{Z})} lesssim_{P,p,q} N^{-d(frac{1}{p}-frac{1}{q})} |f|_{ell^p(mathbb{Z})}, qquad N inmathbb{N}, end{equation*} where $1leq p leq q leq infty$. For a range of quadratic polynomials, the inequalities established are sharp, up to the boundary of the allowed pairs of $(p,q)$. For degree three and higher, the inequalities are close to being sharp. In the quadratic case, we appeal to discrete fractional integrals as studied by Stein and Wainger. In the higher degree case, we appeal to the Vinogradov Mean Value Theorem, recently established by Bourgain, Demeter, and Guth.
Let $ Lambda $ denote von Mangoldts function, and consider the averages begin{align*} A_N f (x) &=frac{1}{N}sum_{1leq n leq N}f(x-n)Lambda(n) . end{align*} We prove sharp $ ell ^{p}$-improving for these averages, and sparse bounds for the maximal function. The simplest inequality is that for sets $ F, Gsubset [0,N]$ there holds begin{equation*} N ^{-1} langle A_N mathbf 1_{F} , mathbf 1_{G} rangle ll frac{lvert Frvert cdot lvert Grvert} { N ^2 } Bigl( operatorname {Log} frac{lvert Frvert cdot lvert Grvert} { N ^2 } Bigr) ^{t}, end{equation*} where $ t=2$, or assuming the Generalized Riemann Hypothesis, $ t=1$. The corresponding sparse bound is proved for the maximal function $ sup_N A_N mathbf 1_{F}$. The inequalities for $ t=1$ are sharp. The proof depends upon the Circle Method, and an interpolation argument of Bourgain.
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