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Consider averages along the prime integers P given by ANf(x)=N-1 p:p≤N(logp)f(x-p). A N f(x) = {N-1*∑ ∈ P:p ≤ N} (p)f(x-p).} These averages satisfy a uniform scale-free p-improving estimate. For all 1 < p < 2, there is a constant Cp so that for all integer N and functions f supported on [0, N], there holds N-1/p′-ANf-p′≤CpN-1/p-f-p. {N-1/p'ANf |ℓ p'≤ C_p N-1p| f |ℓ p. The maximal function A∗f = supN|AN 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 Ap(w), for all weights w in the Muckenhoupt Ap class. No prior weighted inequalities for A∗ were known.

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Concrete Operators

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