Logarithmic dimension bounds for the maximal function along a polynomial curve

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Parissis, Ioannis
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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.
Comment: 15 pages, final version, small typos and notational inconsistencies corrected, to appear in J. Geom. Anal
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Mathematics - Classical Analysis and ODEs, 42B20, 42B25, 42B35, 43A15
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