An official website of the United States government
Abstract In this paper we prove a higher dimensional analogue of Carleson’s$$\varepsilon ^{2}$$ conjecture. Given two arbitrary disjoint Borel sets$$\Omega ^{+},\Omega ^{-}\subset \mathbb{R}^{n+1}$$ , and$$x\in \mathbb{R}^{n+1}$$ ,$$r>0$$ , we denote$$ \varepsilon _{n}(x,r) := \frac{1}{r^{n}}\, \inf _{H^{+}} \mathcal{H}^{n} \left ( ((\partial B(x,r)\cap H^{+}) \setminus \Omega ^{+}) \cup (( \partial B(x,r)\cap H^{-}) \setminus \Omega ^{-})\right ), $$ where the infimum is taken over all open affine half-spaces$$H^{+}$$ such that$$x \in \partial H^{+}$$ and we define$$H^{-}= \mathbb{R}^{n+1} \setminus \overline{H^{+}}$$ . Our first main result asserts that the set of points$$x\in \mathbb{R}^{n+1}$$ where$$ \int _{0}^{1} \varepsilon _{n}(x,r)^{2} \, \frac{dr}{r}< \infty $$ is$$n$$ -rectifiable. For our second main result we assume that$$\Omega ^{+}$$ ,$$\Omega ^{-}$$ are open and that$$\Omega ^{+}\cup \Omega ^{-}$$ satisfies the capacity density condition. For each$$x \in \partial \Omega ^{+} \cup \partial \Omega ^{-}$$ and$$r>0$$ , we denote by$$\alpha ^{\pm }(x,r)$$ the characteristic constant of the (spherical) open sets$$\Omega ^{\pm }\cap \partial B(x,r)$$ . We show that, up to a set of$$\mathcal{H}^{n}$$ measure zero,$$x$$ is a tangent point for both$$\partial \Omega ^{+}$$ and$$\partial \Omega ^{-}$$ if and only if$$ \int _{0}^{1} \min (1,\alpha ^{+}(x,r) + \alpha ^{-}(x,r) -2) \frac{dr}{r} < \infty . $$ The first result is new even in the plane and the second one improves and extends to higher dimensions the$$\varepsilon ^{2}$$ conjecture of Carleson.
more »
« less
Green function estimates on complements of low-dimensional uniformly rectifiable sets
Abstract It has been recently established in David and Mayboroda (Approximation of green functions and domains with uniformly rectifiable boundaries of all dimensions.arXiv:2010.09793) that on uniformly rectifiable sets the Green function is almost affine in the weak sense, and moreover, in some scenarios such Green function estimates are equivalent to the uniform rectifiability of a set. The present paper tackles a strong analogue of these results, starting with the “flagship degenerate operators on sets with lower dimensional boundaries. We consider the elliptic operators$$L_{\beta ,\gamma } =- {\text {div}}D^{d+1+\gamma -n} \nabla $$ associated to a domain$$\Omega \subset {\mathbb {R}}^n$$ with a uniformly rectifiable boundary$$\Gamma $$ of dimension$$d < n-1$$ , the now usual distance to the boundary$$D = D_\beta $$ given by$$D_\beta (X)^{-\beta } = \int _{\Gamma } |X-y|^{-d-\beta } d\sigma (y)$$ for$$X \in \Omega $$ , where$$\beta >0$$ and$$\gamma \in (-1,1)$$ . In this paper we show that the Green functionGfor$$L_{\beta ,\gamma }$$ , with pole at infinity, is well approximated by multiples of$$D^{1-\gamma }$$ , in the sense that the function$$\big | D\nabla \big (\ln \big ( \frac{G}{D^{1-\gamma }} \big )\big )\big |^2$$ satisfies a Carleson measure estimate on$$\Omega $$ . We underline that the strong and the weak results are different in nature and, of course, at the level of the proofs: the latter extensively used compactness arguments, while the present paper relies on some intricate integration by parts and the properties of the “magical distance function from David et al. (Duke Math J, to appear).
more »
« less
- Award ID(s):
- 1839077
- PAR ID:
- 10369381
- Publisher / Repository:
- Springer Science + Business Media
- Date Published:
- Journal Name:
- Mathematische Annalen
- Volume:
- 385
- Issue:
- 3-4
- ISSN:
- 0025-5831
- Page Range / eLocation ID:
- p. 1797-1821
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
More Like this
-
-
Abstract We introduce a distributional Jacobian determinant \det DV_{\beta}(Dv)in dimension two for the nonlinear complex gradient V_{\beta}(Dv)=\lvert Dv\rvert^{\beta}(v_{x_{1}},-v_{x_{2}})for any \beta>-1, whenever v\in W^{1\smash{,}2}_{\mathrm{loc}}and \beta\lvert Dv\rvert^{1+\beta}\in W^{1\smash{,}2}_{\mathrm{loc}}.This is new when \beta\neq 0.Given any planar ∞-harmonic function 𝑢, we show that such distributional Jacobian determinant \det DV_{\beta}(Du)is a nonnegative Radon measure with some quantitative local lower and upper bounds.We also give the following two applications. Applying this result with \beta=0, we develop an approach to build up a Liouville theorem, which improves that of Savin.Precisely, if 𝑢 is an ∞-harmonic function in the whole \mathbb{R}^{2}with \liminf_{R\to\infty}\inf_{c\in\mathbb{R}}\frac{1}{R}\barint_{B(0,R)}\lvert u(x)-c\rvert\,dx<\infty,then u=b+a\cdot xfor some b\in\mathbb{R}and a\in\mathbb{R}^{2}.Denoting by u_{p}the 𝑝-harmonic function having the same nonconstant boundary condition as 𝑢, we show that \det DV_{\beta}(Du_{p})\to\det DV_{\beta}(Du)as p\to\inftyin the weak-⋆ sense in the space of Radon measure.Recall that V_{\beta}(Du_{p})is always quasiregular mappings, but V_{\beta}(Du)is not in general.more » « less
-
Abstract LetXbe a compact normal complex space of dimensionnandLbe a holomorphic line bundle onX. Suppose that$$\Sigma =(\Sigma _1,\ldots ,\Sigma _\ell )$$ is an$$\ell $$ -tuple of distinct irreducible proper analytic subsets ofX,$$\tau =(\tau _1,\ldots ,\tau _\ell )$$ is an$$\ell $$ -tuple of positive real numbers, and let$$H^0_0(X,L^p)$$ be the space of holomorphic sections of$$L^p:=L^{\otimes p}$$ that vanish to order at least$$\tau _jp$$ along$$\Sigma _j$$ ,$$1\le j\le \ell $$ . If$$Y\subset X$$ is an irreducible analytic subset of dimensionm, we consider the space$$H^0_0 (X|Y, L^p)$$ of holomorphic sections of$$L^p|_Y$$ that extend to global holomorphic sections in$$H^0_0(X,L^p)$$ . Assuming that the triplet$$(L,\Sigma ,\tau )$$ is big in the sense that$$\dim H^0_0(X,L^p)\sim p^n$$ , we give a general condition onYto ensure that$$\dim H^0_0(X|Y,L^p)\sim p^m$$ . WhenLis endowed with a continuous Hermitian metric, we show that the Fubini-Study currents of the spaces$$H^0_0(X|Y,L^p)$$ converge to a certain equilibrium current onY. We apply this to the study of the equidistribution of zeros inYof random holomorphic sections in$$H^0_0(X|Y,L^p)$$ as$$p\rightarrow \infty $$ .more » « less
-
Abstract We investigate the low moments$$\mathbb {E}[|A_N|^{2q}],\, 0 of secular coefficients$$A_N$$ of the critical non-Gaussian holomorphic multiplicative chaos, i.e. coefficients of$$z^N$$ in the power series expansion of$$\exp (\sum _{k=1}^\infty X_kz^k/\sqrt{k})$$ , where$$\{X_k\}_{k\geqslant 1}$$ are i.i.d. rotationally invariant unit variance complex random variables. Inspired by Harper’s remarkable result on random multiplicative functions, Soundararajan and Zaman recently showed that if each$$X_k$$ is standard complex Gaussian,$$A_N$$ features better-than-square-root cancellation:$$\mathbb {E}[|A_N|^2]=1$$ and$$\mathbb {E}[|A_N|^{2q}]\asymp (\log N)^{-q/2}$$ for fixed$$q\in (0,1)$$ as$$N\rightarrow \infty $$ . We show that this asymptotics holds universally if$$\mathbb {E}[e^{\gamma |X_k|}]<\infty $$ for some$$\gamma >2q$$ . As a consequence, we establish the universality for the tightness of the normalized secular coefficients$$A_N(\log (1+N))^{1/4}$$ , generalizing a result of Najnudel, Paquette, and Simm. Another corollary is the almost sure regularity of some critical non-Gaussian holomorphic chaos in appropriate Sobolev spaces. Moreover, we characterize the asymptotics of$$\mathbb {E}[|A_N|^{2q}]$$ for$$|X_k|$$ following a stretched exponential distribution with an arbitrary scale parameter, which exhibits a completely different behavior and underlying mechanism from the Gaussian universality regime. As a result, we unveil a double-layer phase transition around the critical case of exponential tails. Our proofs combine Harper’s robust approach with a careful analysis of the (possibly random) leading terms in the monomial decomposition of$$A_N$$ .more » « less
-
Abstract Let$$\mathbb {F}_q^d$$ be thed-dimensional vector space over the finite field withqelements. For a subset$$E\subseteq \mathbb {F}_q^d$$ and a fixed nonzero$$t\in \mathbb {F}_q$$ , let$$\mathcal {H}_t(E)=\{h_y: y\in E\}$$ , where$$h_y:E\rightarrow \{0,1\}$$ is the indicator function of the set$$\{x\in E: x\cdot y=t\}$$ . Two of the authors, with Maxwell Sun, showed in the case$$d=3$$ that if$$|E|\ge Cq^{\frac{11}{4}}$$ andqis sufficiently large, then the VC-dimension of$$\mathcal {H}_t(E)$$ is 3. In this paper, we generalize the result to arbitrary dimension by showing that the VC-dimension of$$\mathcal {H}_t(E)$$ isdwhenever$$E\subseteq \mathbb {F}_q^d$$ with$$|E|\ge C_d q^{d-\frac{1}{d-1}}$$ .more » « less
