On Maximal Functions With Curvature

We exhibit a class of "relatively curved" $\vecγ(t) := (γ_1(t),\dots,γ_n(t))$, 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 $p \geq p_{\vecγ}$, where $1 \geq p_{\vecγ} > 1/n$ for certain curves. For instance, $p_{\vecγ} = 1/n^+$ 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 $\vecγ$, $\lim_{r \to 0} \ \frac{1}{r} \int_0^r F\big(x_1 - u_1(x) \cdot γ_1(t),\dots,x_n - u_n(x) \cdot γ_n(t) \big) \ dt = F(x_1,\dots,x_n), \; \; \; a.e.$ 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 $t \geq c_{\vecγ} > 0$ strictly bounded away from zero, depending on $\vecγ$.

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