ﻻ يوجد ملخص باللغة العربية
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.
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*}
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
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
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
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.