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$$L_{\beta ,\gamma } =- {\text {div}}D^{d+1+\gamma -n} \nabla$$${L}_{\beta ,\gamma }=-\text{div}{D}^{d+1+\gamma -n}\nabla$associated to a domain$$\Omega \subset {\mathbb {R}}^n$$$\Omega \subset {R}^{n}$with a uniformly rectifiable boundary$$\Gamma$$$\Gamma$of dimension$$d < n-1$$$d, the now usual distance to the boundary$$D = D_\beta$$$D={D}_{\beta }$given by$$D_\beta (X)^{-\beta } = \int _{\Gamma } |X-y|^{-d-\beta } d\sigma (y)$$${D}_{\beta }{\left(X\right)}^{-\beta }={\int }_{\Gamma }{|X-y|}^{-d-\beta }d\sigma \left(y\right)$for$$X \in \Omega$$$X\in \Omega$, where$$\beta >0$$$\beta >0$and$$\gamma \in (-1,1)$$$\gamma \in \left(-1,1\right)$. In this paper we show that the Green functionGfor$$L_{\beta ,\gamma }$$${L}_{\beta ,\gamma }$, with pole at infinity, is well approximated by multiples of$$D^{1-\gamma }$$${D}^{1-\gamma }$, in the sense that the function$$\big | D\nabla \big (\ln \big ( \frac{G}{D^{1-\gamma }} \big )\big )\big |^2$$$|D\nabla \left(ln\left(\frac{G}{{D}^{1-\gamma }}\right)\right){|}^{2}$satisfies a Carleson measure estimate on$$\Omega$$$\Omega$. We underline that the strong and the weak results are different in nature and, of course, at the level more »

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10369381
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Mathematische Annalen
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0025-5831
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Let us fix a primepand a homogeneous system ofmlinear equations$$a_{j,1}x_1+\dots +a_{j,k}x_k=0$$${a}_{j,1}{x}_{1}+\cdots +{a}_{j,k}{x}_{k}=0$for$$j=1,\dots ,m$$$j=1,\cdots ,m$with coefficients$$a_{j,i}\in \mathbb {F}_p$$${a}_{j,i}\in {F}_{p}$. Suppose that$$k\ge 3m$$$k\ge 3m$, that$$a_{j,1}+\dots +a_{j,k}=0$$${a}_{j,1}+\cdots +{a}_{j,k}=0$for$$j=1,\dots ,m$$$j=1,\cdots ,m$and that every$$m\times m$$$m×m$minor of the$$m\times k$$$m×k$matrix$$(a_{j,i})_{j,i}$$${\left({a}_{j,i}\right)}_{j,i}$is non-singular. Then we prove that for any (large)n, any subset$$A\subseteq \mathbb {F}_p^n$$$A\subseteq {F}_{p}^{n}$of size$$|A|> C\cdot \Gamma ^n$$$|A|>C·{\Gamma }^{n}$contains a solution$$(x_1,\dots ,x_k)\in A^k$$$\left({x}_{1},\cdots ,{x}_{k}\right)\in {A}^{k}$to the given system of equations such that the vectors$$x_1,\dots ,x_k\in A$$${x}_{1},\cdots ,{x}_{k}\in A$are all distinct. Here,Cand$$\Gamma$$$\Gamma$are constants only depending onp,mandksuch that$$\Gamma $\Gamma . The crucial point here is the condition for the vectors$$x_1,\dots ,x_k$$${x}_{1},\cdots ,{x}_{k}$in the solution$$(x_1,\dots ,x_k)\in A^k$$$\left({x}_{1},\cdots ,{x}_{k}\right)\in {A}^{k}$to be distinct. If we relax this condition and only demand that$$x_1,\dots ,x_k$$${x}_{1},\cdots ,{x}_{k}$are not all equal, then the statement would follow easily from Tao’s slice rank polynomial method. However, handling the distinctness condition is much harder, and requires a new approach. While all previous combinatorial applications of the slice rank polynomial method have relied on the slice rank of diagonal tensors, we use a slice rank argument for a non-diagonal tensor in combination with combinatorial and probabilistic arguments. 2. Abstract A study of possible superconducting phases of graphene has been constructed in detail. A realistic tight binding model, fit to ab initio calculations, accounts for the Li-decoration of graphene with broken lattice symmetry, and includessanddsymmetry Bloch character that influences the gap symmetries that can arise. The resulting seven hybridized Li-C orbitals that support nine possible bond pairing amplitudes. The gap equation is solved for all possible gap symmetries. One band is weakly dispersive near the Fermi energy along Γ → Mwhere its Bloch wave function has linear combination of$${d}_{{x}^{2}-{y}^{2}}$$${d}_{{x}^{2}-{y}^{2}}$anddxycharacter, and is responsible for$${d}_{{x}^{2}-{y}^{2}}$$${d}_{{x}^{2}-{y}^{2}}$anddxypairing with lowest pairing energy in our model. These symmetries almost preserve properties from a two band model of pristine graphene. Another part of this band, alongK → Γ, is nearly degenerate with uppersband that favors extendedswave pairing which is not found in two band model. Upon electron doping to a critical chemical potentialμ1 = 0.22 eVthe pairing potential decreases, then increases until a second critical valueμ2 = 1.3 eV at which a phase transition to a distorteds-wave occurs. The distortion ofd- or s-wave phases are a consequence of decoration which is not appear in two band pristine model. In the pristine graphene these phases convert to usuald-wave or extendeds-wave pairing. 3. Abstract The softmax policy gradient (PG) method, which performs gradient ascent under softmax policy parameterization, is arguably one of the de facto implementations of policy optimization in modern reinforcement learning. For$$\gamma $$$\gamma$-discounted infinite-horizon tabular Markov decision processes (MDPs), remarkable progress has recently been achieved towards establishing global convergence of softmax PG methods in finding a near-optimal policy. However, prior results fall short of delineating clear dependencies of convergence rates on salient parameters such as the cardinality of the state space$${\mathcal {S}}$$$S$and the effective horizon$$\frac{1}{1-\gamma }$$$\frac{1}{1-\gamma }$, both of which could be excessively large. In this paper, we deliver a pessimistic message regarding the iteration complexity of softmax PG methods, despite assuming access to exact gradient computation. Specifically, we demonstrate that the softmax PG method with stepsize$$\eta $$$\eta$can take$$\begin{aligned} \frac{1}{\eta } |{\mathcal {S}}|^{2^{\Omega \big (\frac{1}{1-\gamma }\big )}} ~\text {iterations} \end{aligned}$$$\begin{array}{c}\frac{1}{\eta }{|S|}^{{2}^{\Omega \left(\frac{1}{1-\gamma }\right)}}\phantom{\rule{0ex}{0ex}}\text{iterations}\end{array}$to converge, even in the presence of a benign policy initialization and an initial state distribution amenable to exploration (so that the distribution mismatch coefficient is not exceedingly large). This is accomplished by characterizing the algorithmic dynamics over a carefully-constructed MDP containing only three actions. Our exponential lower bound hints at the necessity of carefully adjusting update rules or enforcing proper regularization inmore » 4. Abstract Let$$X\rightarrow {{\mathbb {P}}}^1$$$X\to {P}^{1}$be an elliptically fiberedK3 surface, admitting a sequence$$\omega _{i}$$${\omega }_{i}$of Ricci-flat metrics collapsing the fibers. LetVbe a holomorphicSU(n) bundle overX, stable with respect to$$\omega _i$$${\omega }_{i}$. Given the corresponding sequence$$\Xi _i$$${\Xi }_{i}$of Hermitian–Yang–Mills connections onV, we prove that, ifEis a generic fiber, the restricted sequence$$\Xi _i|_{E}$$${\Xi }_{i}{|}_{E}$converges to a flat connection$$A_0$$${A}_{0}$. Furthermore, if the restriction$$V|_E$$${V|}_{E}$is of the form$$\oplus _{j=1}^n{\mathcal {O}}_E(q_j-0)$$${\oplus }_{j=1}^{n}{O}_{E}\left({q}_{j}-0\right)$forndistinct points$$q_j\in E$$${q}_{j}\in E$, then these points uniquely determine$$A_0$$${A}_{0}$. 5. Abstract We study the structure of the Liouville quantum gravity (LQG) surfaces that are cut out as one explores a conformal loop-ensemble$$\hbox {CLE}_{\kappa '}$$${\text{CLE}}_{{\kappa }^{\prime }}$for$$\kappa '$$${\kappa }^{\prime }$in (4, 8) that is drawn on an independent$$\gamma $$$\gamma$-LQG surface for$$\gamma ^2=16/\kappa '$$${\gamma }^{2}=16/{\kappa }^{\prime }$. The results are similar in flavor to the ones from our companion paper dealing with$$\hbox {CLE}_{\kappa }$$${\text{CLE}}_{\kappa }$for$$\kappa $$$\kappa$in (8/3, 4), where the loops of the CLE are disjoint and simple. In particular, we encode the combined structure of the LQG surface and the$$\hbox {CLE}_{\kappa '}$$${\text{CLE}}_{{\kappa }^{\prime }}$in terms of stable growth-fragmentation trees or their variants, which also appear in the asymptotic study of peeling processes on decorated planar maps. This has consequences for questions that do a priori not involve LQG surfaces: In our paper entitled “CLE Percolations” described the law of interfaces obtained when coloring the loops of a$$\hbox {CLE}_{\kappa '}$$${\text{CLE}}_{{\kappa }^{\prime }}$independently into two colors with respective probabilitiespand$$1-p$$$1-p$. This description was complete up to one missing parameter$$\rho $$$\rho$. The results of the present paper about CLE on LQG allow us to determine its value in terms ofpand$$\kappa '$$${\kappa }^{\prime }$. It shows in particular that$$\hbox {CLE}_{\kappa '}$$${\text{CLE}}_{{\kappa }^{\prime }}$and$$\hbox {CLE}_{16/\kappa '}$$${\text{CLE}}_{16/{\kappa }^{\prime }}$are related via a continuum analog of the Edwards-Sokal coupling between$$\hbox {FK}_q${\text{FK}}_{q}$percolation and theq-state Potts model (which makes sense evenmore »