ﻻ يوجد ملخص باللغة العربية
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 $x$.
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 pr
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.
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 t
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{e