Galia Dafni

Some new tent spaces and duality theorems for fractional Carleson measures and Qα(Rn)

Several duality questions for fractional Carleson measures and the spaces Qα(Rn) are resolved using a new type of tent spaces. These tent spaces are defined in terms of Choquet integrals with respect to Hausdorff capacity. A predual for Qα(Rn) is then defined as a space of distributions containing the Hardy space H1, and an atomic decomposition is proved.

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

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

A Div-Curl Lemma in BMO on a Domain

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).

Analysis and geometry of metric measure spaces lecture notes of the 50th Séminaire de Mathématiques Supérieures (SMS), Montréal, 2011

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.

A div-curl decomposition for the local Hardy space

A decomposition theorem for the local Hardy space of Goldberg, in terms of nonhomogeneous div-curl quantities, is proved via a dual result for the space bmo.

The space JN(p): Nontriviality and duality

Abstract We study a function space J N p based on a condition introduced by John and Nirenberg as a variant of BMO. It is known that L p ⊂ J N p ⊊ L p , ∞ , but otherwise the structure of J N p is largely a mystery. Our first main result is the construction of a function that belongs to J N p but not L p , showing that the two spaces are not the same. Nevertheless, we prove that for monotone functions, the classes J N p and L p do coincide. Our second main result describes J N p as the dual of a new Hardy kind of space H K p ′ .

A Note on Nonhomogenous Weighted Div-Curl Lemmas

We prove some nonhomogeneous versions of the div-curl lemma in the context of weighted spaces. Namely, assume the vector fields \(\mathbf{V},\mathbf{W}\!\!: \mathbb{R}^{n}\rightarrow \mathbb{R}^{n}\), along with their distributional divergence and curl, respectively, lie in L μ p and L ν q , \(\frac{1} {p} + \frac{1} {q} = 1\), where μ and ν are in certain Muckenhoupt weight classes. Then the resulting scalar product V ⋅ W is in the weighted local Hardy space \(h_{\omega }^{1}(\mathbb{R}^{n})\), for \(\omega =\mu ^{\frac{1} {p} }\nu ^{\frac{1} {q} }\) in \(A_{1+ \frac{1} {n} }\).