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  1. ; ; ; (, Forum of Mathematics, Sigma)
    Abstract Let$$P_1, \ldots , P_m \in \mathbb {K}[\mathrm {y}]$$be polynomials with distinct degrees, no constant terms and coefficients in a general local field$$\mathbb {K}$$. We give a quantitative count of the number of polynomial progressions$$x, x+P_1(y), \ldots , x + P_m(y)$$lying in a set$$S\subseteq \mathbb {K}$$of positive density. The proof relies on a general$$L^{\infty }$$inverse theorem which is of independent interest. This inverse theorem implies a Sobolev improving estimate for multilinear polynomial averaging operators which in turn implies our quantitative estimate for polynomial progressions. This general Sobolev inequality has the potential to be applied in a number of problems in real, complex andp-adic analysis. 
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  2. ; ; (, Annals of Mathematics)
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  3. ; ; ; (, Concrete Operators)
    Abstract Consider averages along the prime integers ℙ given by {\mathcal{A}_N}f(x) = {N^{ - 1}}\sum\limits_{p \in \mathbb{P}:p \le N} {(\log p)f(x - p).} These averages satisfy a uniform scale-free ℓ 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 {N^{ - 1/p'}}{\left\| {{\mathcal{A}_N}f} \right\|_{\ell p'}} \le {C_p}{N^{ - 1/p}}{\left\| f \right\|_{\ell p}}. The maximal function 𝒜 * f = sup N |𝒜 N f | satisfies ( p , p ) sparse bounds for all 1 < p < 2. The latter are the natural variants of the scale-free bounds. As a corollary, 𝒜 * is bounded on ℓ p ( w ), for all weights w in the Muckenhoupt 𝒜 p class. No prior weighted inequalities for 𝒜 * were known. 
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