$BMO$, the space of functions of bounded mean oscillation, was first introduced by F. John and L. Nirenberg in 1961. It became a focus of attention when C. Fefferman proved that $BMO$ is the dual of the (real) Hardy space $H^1$ in 1971. In the past 30 years, this space was studied extensively by many mathematicians. With the help of $BMO$, many phenomena can be characterized clearly. In this review we discuss the connections between $BMO$ functions, the sharp function operator, Carleson measures, atomic decompositions and commutator operators in $\bf{R}^n$. We strive to cover some of the main developments in the theory, including $BMO$ in a bounded Lipschitz domain in $\bf{R}^n$ and in the product space $\bf{R}\times \bf{R}$.

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Functions of bounded mean oscillation

Hardy spaces and divergence operators on strongly Lipschitz domains of Rn

We adapt Forman's discretization of Ricci curvature to the case of undirected networks, both weighted and unweighted, and investigate the measure in a variety of model and real-world networks. We find that most nodes and edges in model and real networks have a negative curvature. Furthermore, the distribution of Forman curvature of nodes and edges is narrow in random and small-world networks, while the distribution is broad in scale-free and real-world networks. In most networks, Forman curvature is found to display significant negative correlation with degree and centrality measures. However, Forman curvature is uncorrelated with clustering coefficient in most networks. Importantly, we find that both model and real networks are vulnerable to targeted deletion of nodes with highly negative Forman curvature. Our results suggest that Forman curvature can be employed to gain novel insights on the organization of complex networks.

Forman curvature for complex networks

Le produit d'une fonction a oscillation moyenne bornee avec une fonction de l'espace de Hardy H 1 n'est pas integrable en general. Nous montrons toutefois qu'on peut lui donner un sens en tant que distribution temperee, ceci grâce a la dualite H 1 , BMO. Cette distribution peut de plus s'ecrire comme la somme d'une fonction integrable et d'une distribution appartenant a un espace de Hardy-Orlicz adapte. Lorsqu'on considere un tel produit pour les fonctions holomorphes du disque unite, cet enonce possede une reciproque: toute fonction holomorphe de l'espace de Hardy-Orlicz considere peut s'ecrire comme un tel produit.

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On the Product of Functions in BMO and H$^\text{1}$

We establish pointwise characterizations of functions in the Hardy-Sobolev spaces $ H^{1,p}$ within the range $p\in (n/(n+1),1]$. In particular, a locally integrable function $u$ belongs to $ H^{1,p}(\real^n)$ if and only if $u\in L^p(\real^n)$ and it satisfies the Haj\l{}asz type condition $$ |u(x)-u(y)|\leq |x-y|(h(x)+h(y)),\quad x,y\in \real^n\setminus E, $$ where $E$ is a set of measure zero and $h\in L^p(\real^n)$. We also investigate Hardy-Sobolev spaces on subdomains and extend Hardy inequalities to the case $p\leq 1.$

http://intlpress.com/site/pub/files/_fulltext/journals/mrl/2008/0015/0004/MRL-2008-0015-0004-a011.pdf

Pointwise characterizations of Hardy-Sobolev functions

We study two different local Hp spaces, 0 < p ≤ 1, on a smooth domain in Rn, by means of maximal functions and atomic decomposition. We prove the regularity in these spaces, as well as in the corresponding dual BMO spaces, of the Dirichlet and Neumann problems for the Laplacian. 0. Introduction Let Ω be a bounded domain in R, with smooth boundary. The L regularity of elliptic boundary value problems on Ω, for 1 < p < ∞, is a classical result in the theory of partial differential equations (see e.g. [ADN]). In the situation of the whole space without boundary, i.e. where Ω is replaced by R, the results for L, 1 < p < ∞, extend to the Hardy spaces H when 0 < p ≤ 1 and to BMO. Thus it is a natural question to ask whether the L regularity of elliptic boundary value problems on a domain Ω has an H and BMO analogue, and what are the H and BMO spaces for which it holds. This question was previously studied in [CKS], where partial results were obtained and were framed in terms of a pair of spaces, hr(Ω) and h p z(Ω). These spaces, variants of those defined in [M] and [JSW], are, roughly speaking, the “largest” and “smallest” h spaces that can be associated to a domain Ω. Our purpose here is to substantially extend the previous results by determining those h spaces on Ω which are particularly applicable to boundary value problems. These spaces allow one to prove sharp results (preservation of the appropriate h spaces) for all values of p, 0 < p ≤ 1, as well as the preservation of corresponding spaces of BMO functions. 0.1. Motivation and statement of results. There are two approaches to defining the appropriate Hardy spaces on Ω. Recall that the spaces H(R), for p < 1, are spaces of distributions. Thus one approach is to look at the problem from the point of view of distributions on Ω. If we denote by D(Ω) the space of smooth functions with compact support in Ω, and by D′(Ω) its dual, we can consider the space of distributions in D′(Ω) which are the restriction to Ω of distributions in H(R) (or in h(R), the local Hardy spaces defined in [G].) These spaces were studied in [M] (for arbitrary open sets) and in [CKS] (for Lipschitz domains), where they were denoted hr(Ω) (the r stands for “restriction”.) While one is able to prove regularity results for the Dirichlet problem for these spaces when p is near 1 (see [CKS]), these spaces are no longer appropriate when p Received by the editors September 5, 1996 and, in revised form, March 20, 1997. 1991 Mathematics Subject Classification. Primary 35J25, 42B25; Secondary 46E15, 42B30. c ©1999 American Mathematical Society

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Hardy spaces, BMO, and boundary value problems for the Laplacian on a smooth domain in ^{}

The space JNp: Nontriviality and duality

This book contains lecture notes from most of the courses presented at the 50th anniversary edition of the Seminaire de Mathematiques Superieure in Montreal. This 2011 summer school was devoted to the analysis and geometry of metric measure spaces, and featured much interplay between this subject and the emergent topic of optimal transportation. In recent decades, metric measure spaces have emerged as a fruitful source of mathematical questions in their own right, and as indispensable tools for addressing classical problems in geometry, topology, dynamical systems, and partial differential equations. The summer school was designed to lead young scientists to the research frontier concerning the analysis and geometry of metric measure spaces, by exposing them to a series of minicourses featuring leading researchers who highlighted both the state-of-the-art and some of the exciting challenges which remain. This volume attempts to capture the excitement of the summer school itself, presenting the reader with glimpses into this active area of research and its connections with other branches of contemporary mathematics.

Analysis and Geometry of Metric Measure Spaces

We study, in the setting of a doubling metric measure space, the local bmo space and Hardy space h1 defined by Goldberg. We state a John–Nirenberg type inequality for the local bmo space and give two proofs, via a good-lambda inequality and via duality. We also prove the boundedness of the Hardy–Littlewood maximal function from bmo to bmo. Finally, we give characterizations of bmo and h1 using alternative mean-oscillation and moment conditions.

Some characterizations of local bmo and h 1 on metric measure spaces

Let Ω ⊂ R n be a Lipschitz domain. There are two BMO spaces, BMO r (Ω) and BMO z (Ω), which can be defined on Ω. The first part of this paper is a survey of some results for functions in these two spaces. The second part contains a div-curl-type lemma for BMO r (Ω) and BMO z (Ω).

Galia Dafni

Papers

Hardy spaces, BMO, and boundary value problems for the Laplacian on a smooth domain in ^{}

The space JNp: Nontriviality and duality

Analysis and Geometry of Metric Measure Spaces

Some characterizations of local bmo and h 1 on metric measure spaces

A Div-Curl Lemma in BMO on a Domain