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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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We prove a sharp, global-in-time Strichartz estimate for the Schrodinger equation on the cylinder $mathbb{R}timesmathbb{T}$.
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{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.
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