On Maximal Functions With Curvature

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 H"{o}lder exponents, [ \left| \sup_{r

0} \ \frac{1}{r} \int_{0}^r \prod_{i=1}^n |f_i(x-\gamma_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$p \geq p_{\vec{\gamma}}$,where$1 \geq p_{\vec{\gamma}} > 1/n$ for certain curves. For instance, $p_{\vec{\gamma }}=1\mathrm{/}{n}^{+}$ for the case of fractional monomials, $$\vec{\gamma }(t)=(t^{{\alpha }_1},\dots ,t^{{\alpha }_n}),{\alpha }_1<\cdots <{\alpha }_n\mathrm{.}$$ 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 $\vec{\gamma }$, $$\underset{r\to 0}{\lim }\frac{1}{r}{\int }_0^{r}F(x_1-u_1(x)\cdot {\gamma }_1(t),\dots ,x_n-u_n(x)\cdot {\gamma }_n(t))dt=F(x_1,\dots ,x_n),a\mathrm{.}e\mathrm{.}$$ 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, $$\{x,x-{\gamma }_1(t),\dots ,x-{\gamma }_n(t)\}\subset A,$$ for some $t\ge c_{\vec{\gamma }}>0$ strictly bounded away from zero, depending on $\vec{\gamma }$.