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Let M denote the maximal function along the polynomial curvep(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 withthe 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 ofoperators which is a mixture of stable semi-groups.

Logarithmic dimension bounds for the maximal function along a polynomial curve