Logarithmic dimension bounds for the maximal function along a polynomial curve

TL;DR

This paper proves that the L^2-norm of the maximal function along polynomial curves in R^d grows at most logarithmically with the dimension d, using a novel semi-group construction.

Contribution

It introduces a new semi-group approach to establish logarithmic bounds for the maximal function along polynomial curves in high dimensions.

Findings

L^2-norm of the maximal function grows at most logarithmically with dimension d

Constructs a parabolic semi-group of operators as a mixture of stable semi-groups

Provides explicit bounds with an absolute constant c

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.