Let $1\leq k\leq n$. We prove that $k$-dimensional intrinsic Lipschitz graphs in the Heisenberg group $\mathbb{H}^n$ satisfy a geometric lemma $\mathrm{GLem}(\beta_{2,\mathcal{V}_k},p)$ for horizontal $\beta$-numbers with an exponent $p=p(k)$. Previously, this result was known only in the case $k=1$; our proof recovers the sharp exponent $p=4$ in this setting. For $k>1$, we adapt an integral geometric approach originally developed by Orponen for Euclidean and parabolic Lipschitz functions. In addition, for $k=n$, we show how to deduce a geometric lemma with $p=4$ directly from an isotropic Dorronsoro theorem in $\mathbb{R}^{2n}$ using a Poincar\'e inequality. Building on the new geometric lemmas, we establish a necessary condition for $k$-regular sets in $\mathbb{H}^n$ to admit corona decompositions by intrinsic Lipschitz graphs. The condition is known to be sufficient by earlier work of the last two authors together with Pinamonti. It involves additional flatness coefficients besides $\beta_{2,\mathcal{V}_k}$. Along the way, we therefore extend the known stability results for geometric lemmas under the ``big pieces''functor to a larger class of coefficients.
Let $X$ be an $n$-dimensional generalized Grassmannian not isomorphic to $\mathbb{P}^n$. We prove that $k(X)\le n-1$, where $k(X)$ denotes the maximal integer such that every uniform bundle on $X$ of rank at most $k(X)$ is homogeneous. In particular, for smooth quadrics $\mathbb{Q}^n$, we have $k(\mathbb{Q}^n)=n-1$ for...
For every $1\leq p<\infty$ and even integer $m\geq4$, we determine the optimal order of the $p$-moment torus inequality for $L_1$: it is $m^p n^{(1-p/2)_+}+n$, with comparison constants independent of $p,m,n$ after taking $p$-th roots. Consequently, for every $2\leq q<\infty$ and $1\leq p\leq q$, the corresponding metr...
Let $X=G/K$ be a symmetric space of noncompact type, of dimension $n$ and rank $r$, and let $Y=\Gamma\backslash X$. Sarnak's local bound for an $L^2$-normalized spherical joint eigenfunction with regular tempered parameter of size $T$ is $\|\phi\|_\infty\ll T^{(n-r)/2}$. We prove $o(T^{(n-r)/2})$ locally uniformly on e...
We study the central Hardy--Littlewood maximal operator $M_c$ in $\mathbb{R}^n$ for all $n\geq1$, with particular emphasis on the question of its injectivity. More precisely, we consider whether there exist two nonnegative functions $f,g\in L^1(\mathbb{R}^n)$ such that $\|f-g\|_1>0$ and $M_cf=M_cg$. Along the way, we o...
Let $S^k_d(n)$ denote the maximum number of regular $(k-1)$-simplices spanned by $n$ points in $\mathbb{R}^d$. For all fixed $r\geq k\geq3$, the exact value of $S^k_{2r}(n)$ was recently determined by Dumitrescu and the authors for all sufficiently large $n$ when $k=3$, and conditionally on an optimization problem when...
We study the $(n-2)$-plane Kakeya problem in $\mathbb{R}^n$ ($n\geq4$) subject to convex Wolff axioms. In other words, we study finite families of $\delta$-neighborhoods of $(n-2)$-plane segments ($\delta$-flakes) in $\mathbb{R}^n$, with the property that not too many of them can be contained inside a common convex set...
Chun-Kit Lai, Yi-Xuan Pang, Rahul Sethi et al.· 0 citations
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