English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Tools annual

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Tools monthly

Global: Zendy Plus monthly

Presents the access of premium content as premium feature

Premium Content

Global: Zendy Plus annual

Global: Zendy Free Plan

We exhibit a class of "relatively curved" $\vec{\gamma}(t) :=(\gamma_1(t),\dots,\gamma_n(t))$, so that the pertaining multi-linear maximalfunction 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 $1 \geqp_{\vec{\gamma}} > 1/n$ for certain curves.  For instance, $p_{\vec{\gamma}} = 1/n^+$ for the case of fractionalmonomials, \[ \vec{\gamma}(t) = (t^{\alpha_1},\dots,t^{\alpha_n}), \; \; \;\alpha_1 < \dots < \alpha_n.\] Two sample applications of our method are asfollows:  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$\vec{\gamma}$, \[ \lim_{r \to 0} \ \frac{1}{r} \int_0^r F\big(x_1 - u_1(x)\cdot \gamma_1(t),\dots,x_n - u_n(x) \cdot \gamma_n(t) \big) \ dt =F(x_1,\dots,x_n), \; \; \; a.e. \] for every $F \in L^p(\mathbb{R}^n), \ p >1$.  Every appropriately normalized set $A \subset [0,1]$ of sufficiently largeHausdorff dimension contains the progression, \[ \{ x, x-\gamma_1(t),\dots,x -\gamma_n(t) \} \subset A, \] for some $t \geq c_{\vec{\gamma}} > 0$ strictlybounded away from zero, depending on $\vec{\gamma}$.

On Maximal Functions With Curvature