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# Mathematical Research Letters

## Volume 16 (2009)

### Number 6

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

Pages: 971 – 989

DOI: http://dx.doi.org/10.4310/MRL.2009.v16.n6.a5

#### Author

#### Abstract

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.