Lipschitz continuity of Cheeger-harmonic functions in metric measure spaces

We use the heat equation to establish the Lipschitz continuity of Cheeger-harmonic functions in certain metric spaces. The metric spaces under consideration are those that are endowed with a doubling measure supporting a (1,2)-Poincaré inequality and in addition supporting a corresponding Sobolev-Poincaré-type inequality for the modification of the measure obtained via the heat kernel. Examples are given to illustrate the necessity of our assumptions on these spaces. We also provide an example to show that in the general setting the best possible regularity for the Cheeger-harmonic functions is Lipschitz continuity.     

Non-symmetric perturbations of symmetric Dirichlet forms

We provide a path-space integral representation of the semigroup associated with the quadratic form obtained by lower order perturbation of a symmetric local Dirichlet form. The representation is a combination of Feynman-Kac and Girsanov formulas, and extends previously known results in the framework of symmetric diffusion processes through the use of the Hardy class of smooth measures, which contains the Kato class of smooth measures.     

Harmonic functions of subordinate killed Brownian motion

In this paper we study harmonic functions of subordinate killed Brownian motion in a domain D. We first prove that, when the killed Brownian semigroup in D is intrinsic ultracontractive, all nonnegative harmonic functions of the subordinate killed Brownian motion in D are continuous and then we establish a Harnack inequality for these harmonic functions. We then show that, when D is a bounded Lipschitz domain, both the Martin boundary and the minimal Martin boundary of the subordinate killed Brownian motion in D coincide with the Euclidean boundary ∂D. We also show that, when D is a bounded Lipschitz domain, a boundary Harnack principle holds for positive harmonic functions of the subordinate killed Brownian motion in D.     

离体心脏的31P NMR谱

³¹P NMR谱自70年代后期开始被用于离体心脏中含磷化合物代谢的动态研究.^3lP NMR谱不仅可以鉴定心肌组织中浓度较高的含磷化合物,即ATP、磷酸肌酸和无机磷酸盐,而且可提供有关这些化合物化学环境(包括pH和Mg²⁺)和细胞内分布的信息.利用这一非损伤性的定量分析方法,可连续观测心肌的代谢并可同时测定离体心脏的机械功能.利用饱和转移技术还能对完整心脏中肌酸激酶和腺苷酸激酶反应的速率加一测定.本文综述介绍了近年来进行离体灌流心脏^3lP NMR谱测定的实验方法以及这一技术所提供的各种信息.     

A Problem of Totik on Fast Decreasing Polynomials

We solve a problem posed by V. Totik on the existence of fast-decreasing polynomials p _n of degree with p _n (0)=1 and for . For the largest c for which such polynomials exist was known. We give the solution for β > 2 . April 18, 1996.     

Quasi sure quadratic variation of local times of smooth semimartingales

In this paper, we prove that the process of the quadratic variation of local times of smooth semimartingales can be constructed as the quasi sure limit of the form ∑_Δn(L_t^a_i+1n−L_t^a_in)², where Δ_n=(a_iⁿ,a_i+1ⁿ) is a sequence of subdivisions of [a,b], a_iⁿ=i(b−a)/2ⁿ+a, i=0,1,…,2ⁿ.     

Wiener’s criterion for the unique solvability of the Dirichlet problem in arbitrary open sets with non-compact boundaries

This paper establishes a necessary and sufficient condition for the existence of a unique bounded solution to the classical Dirichlet problem in arbitrary open subset of R^N${\mathbb{R}}^N$ (N≥3$N\ge 3$) with a non-compact boundary. The criterion is the exact analogue of Wiener’s test for the boundary regularity of harmonic functions and characterizes the “thinness” of a complementary set at infinity. The Kelvin transformation counterpart of the result reveals that the classical Wiener criterion for the boundary point is a necessary and sufficient condition for the unique solvability of the Dirichlet problem in a bounded open set within the class of harmonic functions having a “fundamental solution” kind of singularity at the fixed boundary point. Another important outcome is that the classical Wiener’s test at the boundary point presents a necessary and sufficient condition for the “fundamental solution” kinds of singularities of the solution to the Dirichlet problem to be removable.     

Quasiminimizers in one dimension: integrability of the derivative,inverse function and obstacle problems

It is shown that a K-quasiminimizer u for the one-dimensional p-Dirichlet integral is a K′-quasiminimizer for the q-Dirichlet integral, 1  ≤  q  <  p ₁(p, K), where p ₁(p, K) > p; the exact value for p ₁(p, K) is obtained. The inverse function of a non-constant u is also K′′-quasiminimizer for the s-Dirichlet integral and the range of the exponent s is specified. Connections between quasiminimizers, superminimizers and solutions to obstacle problems are studied.     

Existence and multiplicity of solutions for semilinear elliptic equations with Hardy terms and Hardy–Sobolev critical exponents

Some existence and multiplicity results are obtained for solutions of semilinear elliptic equations with Hardy terms, Hardy–Sobolev critical exponents and superlinear nonlinearity by the variational methods and some analysis techniques.     

Univalence criteria for lifts of harmonic mappings to minimal surfaces

A general criterion in terms of the Schwarzian derivative is given for global univalence of the Weierstrass-Enneper lift of a planar harmonic mapping. Results on distortion and boundary regularity are also deduced. Examples are given to show that the criterion is sharp. The analysis depends on a generalized Schwarzian defined for conformai metrics and on a Schwarzian introduced by Ahlfors for curves. Convexity plays a central role.