Suppose that (M,d,m) is an unbounded metric measure space, which possesses two geometric properties, called "isoperimetric property" and "approximate midpoint property", and that the measure m is locally doubling. The isoperimetric property implies that the volume of balls grows at least exponentially with the radius. Hence the measure m is not globally doubling. In this paper we define an atomic Hardy space H1(m), where atoms are supported only on "small balls", and a corresponding space BMO(m) of functions of bounded mean oscillation, where the control is only on the oscillation over small balls. We prove that BMO(m) is the dual of H1(m) and that an inequality of John-Nirenberg type on small balls holds for functions in BMO(m). Furthermore, we show that the Lp(m) spaces are intermediate spaces between H1(m) and BMO(m), and we develop a theory of singular integral operators acting on function spaces on M. Finally, we show that our theory is strong enough to give H1(m)-L1(m) and L1(m)-BMO(m) estimates for various interesting operators on Riemannian manifolds and symmetric spaces which are unbounded on L1(m) and on L\infty(m).
We consider an infinite homogeneous tree $\mathcal V$ endowed with the usual metric $d$ defined on graphs and a weighted measure $μ$. The metric measure space $(\mathcal V,d,μ)$ is nondoubling and of exponential growth, hence the classical theory of Hardy and $BMO$ spaces does not apply in this setting. We introduce a space $BMO(μ)$ on $(\mathcal V,d,μ)$ and investigate some of its properties. We prove in particular that $BMO(μ)$ can be identified with the dual of a Hardy space $H^1(μ)$ introduced in a previous work and we investigate the sharp maximal function related with $BMO(μ)$.
We introduce a decreasing one-parameter family $\mathfrak{X}^γ(M)$, $γ>0$, of Banach subspaces of the Hardy-Goldberg space $\mathfrak{h}^1(M)$ on certain nondoubling Riemannian manifolds with bounded geometry and we investigate their properties. In particular, we prove that $\mathfrak{X}^{1/2}(M)$ agrees with the space of all functions in $\mathfrak{h}^1(M)$ whose Riesz transform is in $L^1(M)$, and we obtain the surprising result that this space does not admit an atomic decomposition.
H^1 and BMO for certain locally doubling metric measure spaces of finite measure
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。
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…
Let $\bfT$ is a certain tensor product of simple dyadic shifts defined below. We prove here that for dyadic bi-parameter commutator the following equivalence holds $ \|\bfT b-b \bfT \| \asymp \|b\|_{bmo^d}$. This result is well-known for many types of bi-parameter commutators, see \cite{FS}, \cite{DLWY} and \cite{DPSK} for more details.
A BMO-type representation formula for the total variation of SBV functions in Heisenberg groups
ETH Zürich（苏黎世联邦理工学院）；Dipartimento di Ingegneria Civile, Architettura, Territorio, Ambiente e di Matematica (DICATAM), via Branze, 38, 25123 Brescia, Italy（布雷西亚大学土木、建筑、领土、环境与数学工程系）；Dipartimento di Matematica “T. Levi-Civita”, Università di Padova, via Trieste 63, 35121 Padova, Italy（帕多瓦大学T. Levi-Civita数学系）
We construct Markov semi-groups $\mathcal{T}$ and associated BMO-spaces on a finite von Neumann algebra $(\mathcal{M}, τ)$ and obtain results for perturbations of commutators and non-commutative Lipschitz estimates. In particular, we prove that for any $A \in \mathcal{M}$ self-adjoint and $f: \mathbb{R} \rightarrow \mathbb{R}$ Lipschitz there is a Markov semi-group $\mathcal{T}$ such that for $x \in \mathcal{M}$, \[ \Vert [f(A), x] \Vert_{{\rm BMO}(\mathcal{M}, \mathcal{T})} \leq c_{abs} \Vert f' \Vert_\infty \Vert [A, x] \Vert_\infty. \] We obtain an analogue of this result for more general v…
The boundedness of certain sublinear operators with rough kernel generated by Calderón-Zygmund operators and their commutators on generalized weighted Morrey spaces
Norm inflation implies certain discontinuous dependence of the solution on the initial value. The well-posedness of the mild solution means the existence and uniqueness of the fixed points of the corresponding integral equation. For ${\rm BMO}^{-1}$, Auscher-Dubois-Tchamitchian proved that Koch-Tataru's solution is stable. In this paper, we construct a non-Gauss flow function to show that, for classic Navier-Stokes equations, wellposedness and norm inflation may have no conflict and stability may have meaning different to $L^{\infty}(({\rm BMO}^{-1})^{n})$.
