Logarithmic dimension bounds for the maximal function along a polynomial curve
Classical Analysis and ODEs
2013-10-14 v2
Abstract
Let M denote the maximal function along the polynomial curve p(t)=(t,t^2,...,t^d) in R^d: M(f)=sup_{r>0} (1/2r) \int_{|t|<r} |f(x-p(t))| dt. We show that the L^2-norm of this operator grows at most logarithmically with the parameter d: ||M||_2 < c log d ||f||_2, where c>0 is an absolute constant. The proof depends on the explicit construction of a "parabolic" semi-group of operators which is a mixture of stable semi-groups.
Cite
@article{arxiv.0810.4508,
title = {Logarithmic dimension bounds for the maximal function along a polynomial curve},
author = {Ioannis Parissis},
journal= {arXiv preprint arXiv:0810.4508},
year = {2013}
}
Comments
15 pages, final version, small typos and notational inconsistencies corrected, to appear in J. Geom. Anal