$L^p$-improving properties of averages on polynomial curves and related integral estimates

In the combinatorial method proving of $L^p$-improving estimates for averages along curves pioneered by Christ \cite{christ1998}, it is desirable to estimate the average modulus (with respect to some uniform measure on a set) of a polynomial-like function from below using only the value of the function or its derivatives at some prescribed point. In this paper, it is shown that there is always a relatively large set of points (independent of the particular function to be integrated) for which such estimates are possible.

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Published 1 January 2009