Geometry and analysis of Dirichlet forms

Let $ \mathscr E $ be a regular, strongly local Dirichlet form on $L^2(X, m)$ and $d$ the associated intrinsic distance. Assume that the topology induced by $d$ coincides with the original topology on $ X$, and that $X$ is compact, satisfies a doubling property and supports a weak $(1, 2)$-Poincar\'e inequality. We first discuss the (non-)coincidence of the intrinsic length structure and the gradient structure. Under the further assumption that the Ricci curvature of $X$ is bounded from below in the sense of Lott-Sturm-Villani, the following are shown to be equivalent: (i) the heat flow of $\mathscr E$ gives the unique gradient flow of $\mathscr U_\infty$, (ii) $\mathscr E$ satisfies the Ne…

Radial Maximal Function Characterizations of Hardy Spaces on RD-Spaces and Their Applications

Let ${\mathcal X}$ be an RD-space with $\mu({\mathcal X})=\infty$, which means that ${\mathcal X}$ is a space of homogeneous type in the sense of Coifman and Weiss and its measure has the reverse doubling property. In this paper, we characterize the atomic Hardy spaces $H^p_{\rm at}(\{\mathcal X})$ of Coifman and Weiss for $p\in(n/(n+1),1]$ via the radial maximal function, where $n$ is the "dimension" of ${\mathcal X}$, and the range of index $p$ is the best possible. This completely answers the question proposed by Ronald R. Coifman and Guido Weiss in 1977 in this setting, and improves on a deep result of Uchiyama in 1980 on an Ahlfors 1-regular space and a recent result of Loukas Grafakos…

Pointwise characterizations of Besov and Triebel–Lizorkin spaces and quasiconformal mappings

Abstract In this paper, the authors characterize, in terms of pointwise inequalities, the classical Besov spaces B ˙ p , q s and Triebel–Lizorkin spaces F ˙ p , q s for all s ∈ ( 0 , 1 ) and p , q ∈ ( n / ( n + s ) , ∞ ] , both in R n and in the metric measure spaces enjoying the doubling and reverse doubling properties. Applying this characterization, the authors prove that quasiconformal mappings preserve F ˙ n / s , q s on R n for all s ∈ ( 0 , 1 ) and q ∈ ( n / ( n + s ) , ∞ ] . A metric measure space version of the above morphism property is also established.

L∞-variational problem associated to dirichlet forms

A quasiconformal composition problem for the Q-spaces

Given a quasiconformal mapping $f:\mathbb R^n\to\mathbb R^n$ with $n\ge2$, we show that (un-)boundedness of the composition operator ${\bf C}_f$ on the spaces $Q_{\alpha}(\mathbb R^n)$ depends on the index $\alpha$ and the degeneracy set of the Jacobian $J_f$. We establish sharp results in terms of the index $\alpha$ and the local/global self-similar Minkowski dimension of the degeneracy set of $J_f$. This gives a solution to [Problem 8.4, 3] and also reveals a completely new phenomenon, which is totally different from the known results for Sobolev, BMO, Triebel-Lizorkin and Besov spaces. Consequently, Tukia-V\"ais\"al\"a's quasiconformal extension $f:\mathbb R^n\to\mathbb R^n$ of an arbitr…

Hajłasz–Sobolev imbedding and extension

Abstract The author establishes some geometric criteria for a Hajlasz–Sobolev M ˙ ball s , p -extension (resp. M ˙ ball s , p -imbedding) domain of R n with n ⩾ 2 , s ∈ ( 0 , 1 ] and p ∈ [ n / s , ∞ ] (resp. p ∈ ( n / s , ∞ ] ). In particular, the author proves that a bounded finitely connected planar domain Ω is a weak α -cigar domain with α ∈ ( 0 , 1 ) if and only if F ˙ p , ∞ s ( R 2 ) | Ω = M ˙ ball s , p ( Ω ) for some/all s ∈ [ α , 1 ) and p = ( 2 − α ) / ( s − α ) , where F ˙ p , ∞ s ( R 2 ) | Ω denotes the restriction of the Triebel–Lizorkin space F ˙ p , ∞ s ( R 2 ) on Ω .

CYP3 phylogenomics: evidence for positive selection of CYP3A4 and CYP3A7.

CYP3A metabolizes 50% of currently prescribed drugs and is frequently involved in clinically relevant drug interactions. The understanding of roles and regulations of the individual CYP3A genes in pharmacology and physiology is incomplete.Using genomic sequences from 16 species we investigated the evolution of CYP3 genomic loci over a period of 450 million years.CYP3A genes in amniota evolved from two ancestral CYP3A genes. Upon the emergence of eutherian mammals, one of them was lost, whereas, the other acquired a novel genomic environment owing to translocation. In primates, CYP3A underwent rapid evolutionary changes involving multiple gene duplications, deletions, pseudogenizations, and …

A Rademacher type theorem for Hamiltonians H(x, p) and an application to absolute minimizers

AbstractWe establish a Rademacher type theorem involving Hamiltonians H(x, p) under very weak conditions in both of Euclidean and Carnot-Carathéodory spaces. In particular, H(x, p) is assumed to be only measurable in the variable x, and to be quasiconvex and lower-semicontinuous in the variable p. Without the lower-semicontinuity in the variable p, we provide a counter example showing the failure of such a Rademacher type theorem. Moreover, by applying such a Rademacher type theorem we build up an existence result of absolute minimizers for the corresponding $$L^\infty $$ L ∞ -functional. These improve or extend several known results in the literature.

Geometry and analysis of Dirichlet forms (II)

Abstract Given a regular, strongly local Dirichlet form E , under assumption that the lower bound of the Ricci curvature of Bakry–Emery, the local doubling and local Poincare inequalities are satisfied, we obtain that: (i) the intrinsic differential and distance structures of E coincide; (ii) the Cheeger energy functional Ch d E is a quadratic norm. This shows that (ii) is necessary for the Riemannian Ricci curvature defined by Ambrosio–Gigli–Savare to be bounded from below. This together with some recent results of Ambrosio–Gigli–Savare yields that the heat flow gives a gradient flow of Boltzman–Shannon entropy under the above assumptions. We also obtain an improvement on Kuwada's duality …

The $Q_{ \alpha}$-restriction problem

Everywhere differentiability of viscosity solutions to a class of Aronsson's equations

For any open set $\Omega\subset\mathbb R^n$ and $n\ge 2$, we establish everywhere differentiability of viscosity solutions to the Aronsson equation $$ =0 \quad \rm in\ \ \Omega, $$ where $H$ is given by $$H(x,\,p)==\sum_{i,\,j=1}^na^{ij}(x)p_i p_j,\ x\in\Omega, \ p\in\mathbb R^n, $$ and $A=(a^{ij}(x))\in C^{1,1}(\bar\Omega,\mathbb R^{n\times n})$ is uniformly elliptic. This extends an earlier theorem by Evans and Smart \cite{es11a} on infinity harmonic functions.

A characterization of Hajłasz–Sobolev and Triebel–Lizorkin spaces via grand Littlewood–Paley functions

Abstract In this paper, we establish the equivalence between the Hajlasz–Sobolev spaces or classical Triebel–Lizorkin spaces and a class of grand Triebel–Lizorkin spaces on Euclidean spaces and also on metric spaces that are both doubling and reverse doubling. In particular, when p ∈ ( n / ( n + 1 ) , ∞ ) , we give a new characterization of the Hajlasz–Sobolev spaces M ˙ 1 , p ( R n ) via a grand Littlewood–Paley function.

Morrey–Sobolev Extension Domains

We show that every uniform domain of R n with n ≥ 2 is a Morrey-Sobolev W 1, p-extension domain for all p ∈ [1, n), and moreover, that this result is essentially best possible for each p ∈ [1, n) in the sense that, given a simply connected planar domain or a domain of R n with n ≥ 3 that is quasiconformal equivalent to a uniform domain, if it is a W 1, p-extension domain, then it must be uniform. peerReviewed