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Galia Dafni

Publications and source records attributed to Galia Dafni.

15 recordsLinked to original sources
arXiv2025

A Dyadic Approach to Weak Characterizations of Function Spaces

Weak-type quasi-norms are defined using the mean oscillation or the mean of a function on dyadic cubes, providing discrete analogues and variants of the corresponding quasi-norms on the upper half-space previously considered in the literature. Comparing the resulting function spaces to known function spaces such as $\dot{W}^{1,p}(\rn)$, $\JNp$, $\Lp$ and weak-$\Lp$ gives new embeddings and characterizations of these spaces. Examples are provided to prove the sharpness of the results.

math.FA↗
arXiv2023

$h^1$ boundedness of Localized Operators and Commutators with bmo and lmo

We first consider two types of localizations of singular integral operators of convolution type, and show, under mild decay and smoothness conditions on the auxiliary functions, that their boundedness on the local Hardy space $h^1(\mathbb{R}^n)$ is equivalent. We then study the boundedness on $h^1(\mathbb{R}^n)$ of the commutator $[b,T]$ of an inhomogeneous singular integral operator with $b$ in $bmo(\mathbb{R}^n)$, the nonhomogeneous space of functions of bounded mean oscillation. We define local analogues of the atomic space $H^1_b(\mathbb{R}^n)$ introduced by Pérez in the case of the homogeneous Hardy space and $BMO$, including a variation involving atoms with approximate cancellation conditions. For such an atom $a$, we prove integrability of the associated commutator maximal function and of $[b,T](a)$. For $b$ in $lmo(\mathbb{R}^n)$, this gives $h^1$ to $L^1$ boundedness of $[b,T]$. Finally, under additional approximate cancellation conditions on $T$, we show boundedness to $h^1$.

math.FA↗
arXiv2022

Necessary cancellation conditions for the boundedness of operators on local Hardy spaces

In this work we present necessary cancellation conditions for the continuity of linear operators in $h^p(\mathbb{R}^n)$, $0<p\leq 1$, that map atoms into pseudo-molecules. Our necessary condition, expressed in terms of the $T^{\ast}$ condition, is the same as the one recently proved sufficient in [3], thus providing a necessary and sufficient cancellation condition for the boundedness of inhomogeneous Calderón--Zygmund type operators

math.AP↗
arXiv2022

Local vanishing mean oscillation

We consider various notions of vanishing mean oscillation on a (possibly unbounded) domain $Ω\subset \mathbb{R}^n$, and prove an analogue of Sarason's theorem, giving sufficient conditions for the density of bounded Lipschitz functions in the nonhomogeneous space $\rm{vmo}(Ω)$. We also study $\rm{cmo}(Ω)$, the closure in $\rm{bmo}(Ω)$ of the continuous functions with compact support in $Ω$. Using these approximation results, we prove that there is a bounded extension from $\rm{vmo}(Ω)$ and $\rm{cmo}(Ω)$ to the corresponding spaces on $\mathbb{R}^n$, if and only if $Ω$ is a locally uniform domain.

math.AP↗
arXiv2022

Locally uniform domains and extension of bmo functions

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 of the domain.

math.FA↗
arXiv2022

Vanishing Mean Oscillation and Continuity of Rearrangements

We study the decreasing rearrangement of functions in VMO, and show that for rearrangeable functions, the mapping f -> f* preserves vanishing mean oscillation. Moreover, as a map on BMO, while bounded, it is not continuous, but continuity holds at points in VMO (under certain conditions). This also applies to the symmetric decreasing rearrangement. Many examples are included to illustrate the results.

math.FA↗
arXiv2021

Mean oscillation bounds on rearrangements

We use geometric arguments to prove explicit bounds on the mean oscillation for two important rearrangements on $\mathbb{R}^n$. For the decreasing rearrangement $f^*$ of a rearrangeable function $f$ of bounded mean oscillation (BMO) on cubes, we improve a classical inequality of Bennett--DeVore--Sharpley, $\|f^*\|_{BMO(\mathbb{R}_+)}\leq C_n \|f\|_{BMO(\mathbb{R}^n)}$, by showing the growth of $C_n$ in the dimension $n$ is not exponential but at most of the order of $\sqrt{n}$. This is achieved by comparing cubes to a family of rectangles for which one can prove a dimension-free Calderón--Zygmund decomposition. By comparing cubes to a family of polar rectangles, we provide a first proof that an analogous inequality holds for the symmetric decreasing rearrangement, $Sf$.

math.FA↗
arXiv2020

BMO and the John-Nirenberg Inequality on Measure Spaces

We study the space BMO in the general setting of a measure space $\mathbb{X}$ with a fixed collection $\mathscr{G}$ of measurable sets of positive and finite measure, consisting of functions of bounded mean oscillation on sets in $\mathscr{G}$. The aim is to see how much of the familiar BMO machinery holds when metric notions have been replaced by measure-theoretic ones. In particular, three aspects of BMO are considered: its properties as a Banach space, its relation with Muckenhoupt weights, and the John-Nirenberg inequality. We give necessary and sufficient conditions on a decomposable measure space $\mathbb{X}$ for BMO to be a Banach space modulo constants. We also develop the notion of a Denjoy family $\mathscr{G}$, which guarantees that functions in BMO satisfy the John-Nirenberg inequality on the elements of $\mathscr{G}$.

math.FA↗
arXiv2020

Geometric maximal operators and BMO on product bases

We consider the problem of the boundedness of maximal operators on BMO on shapes in $\mathbb{R}^n$. We prove that for bases of shapes with an engulfing property, the corresponding maximal function is bounded from BMO to BLO, generalising a known result of Bennett for the basis of cubes. When the basis of shapes does not possess an engulfing property but exhibits a product structure with respect to lower-dimensional shapes coming from bases that do possess an engulfing property, we show that the corresponding maximal function is bounded from BMO to a space we define and call rectangular BLO.

math.FA↗
arXiv2019

BMO on shapes and sharp constants

We consider a very general definition of BMO on a domain in $\mathbb{R}^n$, where the mean oscillation is taken with respect to a basis of shapes, i.e. a collection of open sets covering the domain. We examine the basic properties and various inequalities that can be proved for such functions, with special emphasis on sharp constants. For the standard bases of shapes consisting of balls or cubes (classic BMO), or rectangles (strong BMO), we review known results, such as the boundedness of rearrangements and its consequences. Finally, we prove a product decomposition for BMO when the shapes exhibit some product structure, as in the case of strong BMO.

math.FA↗
arXiv2018

On the extension of VMO functions

We consider functions of vanishing mean oscillation on a bounded domain $Ω$ and prove a $\rm{VMO}$ analogue of the extension theorem of P. Jones for $\rm{BMO}(Ω)$. We show that if $Ω$ satisfies the same condition imposed by Jones (i.e.\ is a uniform domain), there is a linear extension map from $\rm{VMO}(Ω)$ to $\rm{VMO}(\mathbb{R}^n)$ which is bounded in the $\rm{BMO}$ norm. Moreover, if such an extension map exists from $\rm{VMO}(Ω)$ to $\rm{BMO}(\mathbb{R}^n)$, then the domain is uniform.

math.FA↗
arXiv2018

The space $JN_p$: nontriviality and duality

We study a function space $JN_p$ based on a condition introduced by John and Nirenberg as a variant of BMO. It is known that $L^p\subset JN_{p}\subsetneq L^{p,\infty}$, but otherwise the structure of $JN_p$ is largely a mystery. Our first main result is the construction of a function that belongs to $JN_p$ but not $L^p$, showing that the two spaces are not the same. Nevertheless, we prove that for monotone functions, the classes $JN_{p}$ and $L^p$ do coincide. Our second main result describes $JN_p$ as the dual of a new Hardy kind of space $HK_{p'}$.

math.FA↗
arXiv2010

An atomic decomposition of the Hajłasz Sobolev space $\Mone$ on manifolds

Several possible notions of Hardy-Sobolev spaces on a Riemannian manifold with a doubling measure are considered. Under the assumption of a Poincaré inequality, the space $\Mone$, defined by Hajłasz, is identified with a Hardy-Sobolev space defined in terms of atoms. Decomposition results are proved for both the homogeneous and the nonhomogeneous spaces.

math.DG↗