Logarithmic Dimension Bounds for the Maximal Function Along a Polynomial Curve

Abstract

Let ℳ denote the maximal function along the polynomial curve (γ 1 t,…,γ d t d ): $$\mathcal{M}(f)(x)=\sup_{r>0}\frac{1}{2r}\int_{|t|\leq r}|f(x_1-\gamma_1t,\ldots,x_d-\gamma_dt^d)|\,dt.$$ We show that the L 2 norm of this operator grows at most logarithmically with the parameter d: $$\Vert \mathcal{M}f\Vert _{L^2(\mathbb{R}^d)}\leq c\log d\Vert f\Vert _{L^2(\mathbb{R}^d)},$$ where c>0 is an absolute constant. The proof depends on the explicit construction of a “parabolic” semigroup of operators which is a mixture of stable semigroups.