在前一篇文章中，作者对具有无限测度的无界度量测度空间$(M,ρ,m)$建立了一个H^1-BMO理论，该空间是局部加倍的，并且满足两个几何性质，称为“近似中点”性质和“等周”性质。本文对有限测度空间发展了一个类似的理论。我们证明了所有在无限测度情形下成立的结果在有限测度情形下都有对应的结果。最后，我们证明了该理论适用于一类无界完备有限测度黎曼流形和一类（R^n，ρ_n，m_n）形式的度量测度空间，其中dm_n =e^{-ε} dx，ρ_n为$ is the Riemannian metric corresponding to the length element ds^2=(1+ |gradϕ|)^2(dx_1^2+...+dx_n^2)$ .这推广了高斯空间的最后两个作者以前的工作。
英文摘要：
In a previous paper the authors developed a H^1-BMO theory for unbounded metric measure spaces $(M,ρ,m)$ of infinite measure that are locally doubling and satisfy two geometric properties, called "approximate midpoint" property and "isoperimetric" property. In this paper we develop a similar theory for spaces of finite measure. We prove that all the results that hold in the infinite measure case have their counterparts in the finite measure case. Finally, we show that the theory applies to a class of unbounded, complete Riemannian manifolds of finite measure and to a class of metric measure spaces of the form (R^n,ρ_ϕ, m_ϕ), where dm_ϕ=e^{-ϕ} dx and ρ_ϕ$ is the Riemannian metric corresponding to the length element ds^2=(1+ |gradϕ|)^2(dx_1^2+...+dx_n^2)$. This generalizes previous work of the last two authors for the Gauss space.
We prove that for a domain $Ω\subset \mathbb{R}^n$, being $(ε,δ)$ in the sense of Jones is equivalent to being an extension domain for bmo$(Ω)$, the nonhonomogeneous version of the space of function of bounded mean oscillation on $Ω$. In the process we demonstrate that these conditions are equivalent to local versions of two other conditions characterizing uniform domains, one involving the presence of length cigars between nearby points and the other a local version of the quasi-hyperbolic uniform condition. Our results show that the definition of bmo$(Ω)$ is closely connected to the geometry…
令$X$是一个度量空间，配备有一个度量$d$和一个满足加倍性质的非负Borel测度$μ$，并令$\{\mathcal{A}_t\}_{t>0}$，是单位元的广义逼近，例如$\{\mathcal{A}_t\}$是一个全纯半群$e^{-tL}$，具有高斯上界，由$L^2(X)$上的算子$L$生成。本文引入并研究了与\{ \mathcal{A}_t \} $. We show that for these spaces, the weighted John-Nirenberg inequality holds and we establish an interpolation theorem in scale of weighted $ L^p $ spaces. As applications, we prove the boundedness of two singular integrals with non-smooth kernels on the weighted BMO space BMO$ _ \mathcal{A}（X，w）$相关的加权BMO空间BMO $_\mathcal{A}(X,w)$.
We prove a radial maximal function characterisation of the local atomic Hardy space h^1(M) on a Riemannian manifold M with positive injectivity radius and Ricci curvature bounded from below. As a consequence, we show that an integrable function belongs to h^1(M) if and only if either its local heat maximal function or its local Poisson maximal function are integrable. A key ingredient is a decomposition of Hölder cut-offs in terms of an appropriate class of approximations of the identity, which we obtain on arbitrary Ahlfors-regular metric measure spaces and generalises a previous result of A.…
We give conditions for boundedness of Hausdorff operators on real Hardy spaces $H^1$ over homogeneous spaces of locally compact groups with local doubling property. The special case of the hyperbolic plane is considered.
Following Malý's definition of absolutely continuous functions of several variables, we consider $Q$-absolutely continuous mappings $f\colon X\to V$ between a doubling metric measure space $X$ and a Banach space $V$. The relation between these mappings and Sobolev mappings $f\in N^{1,p}(X;V)$ for $p\ge Q$ is investigated. In particular, a locally $Q$-absolutely continuous mapping on an Ahlfors $Q$-regular space is a continuous mapping in $N^{1,Q}_{\rm{loc}}(X;V)$, as well as differentiable almost everywhere in terms of Cheeger derivatives provided $V$ satisfies the Radon-Nikodym property. Conv…
Let $f:=(f_n)_{n\in \mathbb{Z}_+}$ and $g:=(g_n)_{n\in \mathbb{Z}_+}$ be two martingales related to the probability space $(Ω,\mathcal F,\mathbb P)$ equipped with the filtration $(\mathcal F_n)_{n\in \mathbb{Z}_+}.$ Assume that $f$ is in the martingale Hardy space $H_1$ and $g$ is in its dual space, namely the martingale $\rm BMO.$ Then the semi-martingale $f\cdot g:=(f_ng_n)_{n\in \mathbb{Z}_+}$ may be written as the sum $$f\cdot g=G(f, g)+L( f,g).$$ Here $L( f,g):=(L( f,g)_n)_{n\in\mathbb{Z}_+}$ with $L( f,g)_n:=\sum_{k=0}^n(f_k-f_{k-1})(g_k-g_{k-1)})$ for any $n\in\mathbb{Z}_+$, where $f_{-…
A regular symmetric operator on a Hilbert module is self-adjoint whenever there exists a suitable approximate identity. We say an operator is 'locally bounded' if the composition of the operator with each element in the approximate identity is bounded. We prove that the perturbation of a regular self-adjoint operator by a locally bounded symmetric operator is again regular and self-adjoint. We use this result to show that the Kasparov class represented by an unbounded Kasparov module is stable under locally bounded perturbations. As an application, we show that we obtain a converse to the 'dou…
In this note we show that locally $p$-admissible measures on $\mathbb{R}$ necessarily come from local Muckenhoupt $A_p$ weights. In the proof we employ the corresponding characterization of global $p$-admissible measures on $\mathbb{R}$ in terms of global $A_p$ weights due to Björn, Buckley and Keith, together with tools from analysis in metric spaces, more specifically preservation of the doubling condition and Poincaré inequalities under flattening, due to Durand-Cartagena and Li. As a consequence, the class of locally $p$-admissible weights on $\mathbb{R}$ is invariant under addition and sa…
Calderón-Zygmund theory has been traditionally developed on metric measure spaces satisfying additional regularity properties. In the lack of good metrics, we introduce a new approach for general measure spaces which admit a Markov semigroup satisfying purely algebraic assumptions. We shall construct an abstract form of "Markov metric" governing the Markov process and the naturally associated BMO spaces, which interpolate with the Lp-scale and admit endpoint inequalities for Calderón-Zygmund operators. Motivated by noncommutative harmonic analysis, this approach gives the first form of Calderó…
We study the so-called John-Nirenberg space that is a generalization of functions of bounded mean oscillation in the setting of metric measure spaces with a doubling measure. Our main results are local and global John-Nirenberg inequalities, which give weak type estimates for the oscillation of a function. We consider medians instead of integral averages throughout, and thus functions are not a priori assumed to be locally integrable. Our arguments are based on a Calderón-Zygmund decomposition and a good-$λ$ inequality for medians. A John-Nirenberg inequality up to the boundary is proven by us…
Given $p \in (1,\infty)$, let $(\operatorname{X},\operatorname{d},μ)$ be a metric measure space with uniformly locally doubling measure $μ$ supporting a weak local $(1,p)$-Poincaré inequality. For each $θ\in [0,p)$, we characterize the trace space of the Sobolev $W^{1}_{p}(\operatorname{X})$-space to lower codimension-$θ$ content regular closed subsets $S \subset \operatorname{X}$. In particular, if the space $(\operatorname{X},\operatorname{d},μ)$ is Ahlfors $Q$-regular for some $Q \geq 1$ and $p \in (Q,\infty)$, then we get an intrinsic description of the trace-space of the Sobolev $W^{1}_{p…
We study the well-posedness of Cauchy problems on the upper half space $\mathbb{R}^{n+1}_+$ associated to higher order systems $\partial_t u =(-1)^{m+1}\mbox{div}_m A\nabla ^m u$ with bounded measurable and uniformly elliptic coefficients. We address initial data lying in $L^p$ ($1<p<\infty$) and $BMO$ ($p=\infty$) spaces and work with weak solutions. Our main result is the identification of a new well-posedeness class, given for $p\in(1,\infty]$ by distributions satisfying $\nabla^m u \in T^{p,2}_m$, where $T^{p,2}_m$ is a parabolic version of the tent space of Coifman--Meyer--Stein. In the r…
In this paper we fix $1\le p<\infty$ and consider $(\Om,d,μ)$ be an unbounded, locally compact, non-complete metric measure space equipped with a doubling measure $μ$ supporting a $p$-Poincaré inequality such that $\Om$ is a uniform domain in its completion $\bar\Om$. We realize the trace of functions in the Dirichlet-Sobolev space $D^{1,p}(\Om)$ on the boundary $\partial\Om$ as functions in the homogeneous Besov space $HB^α_{p,p}(\partial\Om)$ for suitable $α$; here, $\partial\Om$ is equipped with a non-atomic Borel regular measure $ν$. We show that if $ν$ satisfies a $θ$-codimensional condit…
