On Maximal Functions With Curvature

Abstract

We exhibit a class of "relatively curved" $\vec{\gamma }(t):=({\gamma }_1(t),\dots ,{\gamma }_n(t))$, so that the pertaining multi-linear maximal function satisfies the sharp range of Hölder exponents, \[ \left\| \sup_{r > 0} \ \frac{1}{r} \int_{0}^r \prod_{i=1}^n |f_i(x-γ_i(t))| \ dt \right\|_{L^p(\mathbb{R})} \leq C \cdot \prod_{i=1}^n \| f_j \|_{L^{p_j}(\mathbb{R})} \] whenever $\frac{1}{p}={\sum }_{j=1}^n\frac{1}{p_j}$, where $p_j>1$ and $p\ge p_{\vec{\gamma }}$, where $1\ge p_{\vec{\gamma }}>1/n$ for certain curves. For instance, $p_{\vec{\gamma }}=1/n^{+}$ for the case of fractional monomials, \[ \vecγ(t) = (t^{α_1},\dots,t^{α_n}), \; \; \; α_1 < \dots < α_n.\] Two sample applications of our method are as follows: For any measurable $u_1,\dots ,u_n:{\mathbb{R}}^n\to \mathbb{R}$, with $u_i$ independent of the $i$th coordinate vector, and any relatively curved $\vec{\gamma }$, \[ \lim_{r \to 0} \ \frac{1}{r} \int_0^r F\big(x_1 - u_1(x) \cdot γ_1(t),\dots,x_n - u_n(x) \cdot γ_n(t) \big) \ dt = F(x_1,\dots,x_n), \; \; \; a.e. \] for every $F\in L^p({\mathbb{R}}^n),p>1$. Every appropriately normalized set $A\subset [0,1]$ of sufficiently large Hausdorff dimension contains the progression, \[ \{ x, x-γ_1(t),\dots,x - γ_n(t) \} \subset A, \] for some $t\ge c_{\vec{\gamma }}>0$ strictly bounded away from zero, depending on $\vec{\gamma }$.

Error with sum sets, to be corrected at a later date