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Register a new userSharp Davies-Gaffney-Grigoryan Lemma on Graphs
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In this note, we prove the sharp Davies-Gaffney-Grigoryan lemma for minimal heat kernels on graphs.
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We prove a variant of the Davies-Gaffney-Grigoryan Lemma for the continuous time heat kernel on graphs. We use it together with the Li-Yau inequality to obtain strong heat kernel estimates for graphs satisfying the exponential curvature dimension inequality.
We prove the generalized Margulis lemma with a uniform index bound on an Alexandrov $n$-space $X$ with curvature bounded below, i.e., small loops at $pin X$ generate a subgroup of the fundamental group of unit ball $B_1(p)$ that contains a nilpotent subgroup of index $le w(n)$, where $w(n)$ is a constant depending only on the dimension $n$. The proof is based on the main ideas of V.~Kapovitch, A.~Petrunin, and W.~Tuschmann, and the following results: (1) We prove that any regular almost Lipschitz submersion constructed by Yamaguchi on a collapsed Alexandrov space with curvature bounded below is a Hurewicz fibration. We also prove that such fibration is uniquely determined up to a homotopy equivalence. (2) We give a detailed proof on the gradient push, improving the universal pushing time bound given by V.~Kapovitch, A.~Petrunin, and W.~Tuschmann, and justifying in a specific way that the gradient push between regular points can always keep away from extremal subsets.
Let $L$ be a one-to-one operator of type $omega$ in $L^2(mathbb{R}^n)$, with $omegain[0,,pi/2)$, which has a bounded holomorphic functional calculus and satisfies the Davies-Gaffney estimates. Let $p(cdot): mathbb{R}^nto(0,,1]$ be a variable exponent function satisfying the globally log-H{o}lder continuous condition. In this article, the authors introduce the variable Hardy space $H^{p(cdot)}_L(mathbb{R}^n)$ associated with $L$. By means of variable tent spaces, the authors establish the molecular characterization of $H^{p(cdot)}_L(mathbb{R}^n)$. Then the authors show that the dual space of $H^{p(cdot)}_L(mathbb{R}^n)$ is the BMO-type space ${rm BMO}_{p(cdot),,L^ast}(mathbb{R}^n)$, where $L^ast$ denotes the adjoint operator of $L$. In particular, when $L$ is the second order divergence form elliptic operator with complex bounded measurable coefficients, the authors obtain the non-tangential maximal function characterization of $H^{p(cdot)}_L(mathbb{R}^n)$ and show that the fractional integral $L^{-alpha}$ for $alphain(0,,frac12]$ is bounded from $H_L^{p(cdot)}(mathbb{R}^n)$ to $H_L^{q(cdot)}(mathbb{R}^n)$ with $frac1{p(cdot)}-frac1{q(cdot)}=frac{2alpha}{n}$ and the Riesz transform $ abla L^{-1/2}$ is bounded from $H^{p(cdot)}_L(mathbb{R}^n)$ to the variable Hardy space $H^{p(cdot)}(mathbb{R}^n)$.
We show that the deficiency indices of the minimal Gaffney Laplacian on an infinite locally finite metric graph are equal to the number of finite volume graph ends. Moreover, we provide criteria, formulated in terms of finite volume graph ends, for the Gaffney Laplacian to be closed.
In this paper, we prove a sharp Meis Lemma with assuming the bases of the underlying general dyadic grids are different. As a byproduct, we specify all the possible cases of adjacent general dyadic systems with different bases. The proofs have connections with certain number-theoretic properties.
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