Shearlet systems have so far been only considered as a means to analyze $L^2$-functions defined on $\R^2$, which exhibit curvilinear singularities. However, in applications such as image processing or numerical solvers of partial differential equations the function to be analyzed or efficiently encoded is typically defined on a non-rectangular shaped bounded domain. Motivated by these applications, in this paper, we first introduce a novel model for cartoon-like images defined on a bounded domain. We then prove that compactly supported shearlet frames satisfying some weak decay and smoothness conditions, when orthogonally projected onto the bounded domain, do provide (almost) optimally sparse approximations of elements belonging to this model class.
In this paper, we study a newly developed shearlet system on bounded domains which yields frames for $H^s(Ω)$ for some $s\in \mathbb{N}$, $Ω\subset \mathbb{R}^2$. We will derive approximation rates with respect to $H^s(Ω)$ norms for functions whose derivatives admit smooth jumps along curves and demonstrate superior rates to those provided by pure wavelet systems. These improved approximation rates demonstrate the potential of the novel shearlet system for the discretization of partial differential equations. Therefore, we implement an adaptive shearlet-based algorithm for the solution of an e…
The shearlets are a special case of the wavelets with composite dilation that, among other things, have a basis-like structure and multi resolution analysis properties. These relatively new representation systems have encountered wide range of applications, generally surpassing the performance of their ancestors due to their directional sensitivity. However, little is known about their relation with spaces other than $L^2$. Here, we find a characterization of a kind of anisotropic inhomogeneous Triebel-Lizorkin spaces (to be defined) with the so called "shearlets on the cone" coefficients. We …
We provide a construction of multiscale systems on a bounded domain $Ω\subset \mathbb{R}^2$ coined boundary shearlet systems, which satisfy several properties advantageous for applications to imaging science and the numerical analysis of partial differential equations. More precisely, we construct boundary shearlet systems that form frames for the Sobolev spaces $H^s(Ω),s\in \mathbb{N} \cup \{0\},$ with controllable frame bounds and admit optimally sparse approximations for functions, which are smooth apart from a curve-like discontinuity. We show that the constructed systems allow incorporati…
It is known that the continuous wavelet transform of a function $f$ decays very rapidly near the points where $f$ is smooth, while it decays slowly near the irregular points. This property allows one to precisely identify the singular support of $f$. However, the continuous wavelet transform is unable to provide additional information about the geometry of the singular points. In this paper, we introduce a new transform for functions and distributions on $\R^2$, called the Continuous Shearlet Transform. This is defined by $\mathcal{S}\mathcal{H}_f(a,s,t) = \ip{f}{ψ_{ast}}$, where the analyzing…
Necessary and sufficient conditions for positive Toeplitz operators on the Bergman space of a minimal bounded homogeneous domain to be bounded or compact are described in terms of the Berezin transform, the averaging function and the Carleson property.
We describe the Bergman kernel of any bounded homogeneous domain in a minimal realization relating to the Bergman kernels of the Siegel disks. Taking advantage of this expression, we obtain substantial estimates of the Bergman kernel of the homogeneous domain.
Using an integral formula on a homogeneous Siegel domain, we show a necessary and sufficient condition for composition operators on the weighted Bergman space of a minimal bounded homogeneous domain to be compact. To describe the compactness of composition operators, we see a boundary behavior of the Bergman kernel.
We study the bounded and the compact weighted composition operators from the Bloch space into the weighted Banach spaces of holomorphic functions on bounded homogeneous domains, with particular attention to the unit polydisk. For bounded homogeneous domains, we characterize the bounded weighted composition operators and determine the operator norm. In addition, we provide sufficient conditions for compactness. For the unit polydisk, we completely characterize the compact weighted composition operators, as well as provide computable estimates on the operator norm.
Let $D$ be a bounded homogeneous domain in $\mathbb{C}^n$. In this paper, we study the bounded and the compact weighted composition operators mapping the Hardy space $H^\infty(D)$ into the Bloch space of $D$. We characterize the bounded weighted composition operators, provide operator norm estimates, and give sufficient conditions for compactness. We prove that these conditions are necessary in the case of the unit ball and the polydisk. We then show that if $D$ is a bounded symmetric domain, the bounded multiplication operators from $H^\infty(D)$ to the Bloch space of $D$ are the operators wh…
In this paper, we study the multiplication operators on the Bloch space of a bounded homogeneous domain in $\mathbb{C}^n$. Specifically, we characterize the bounded and the compact multiplication operators, establish estimates on the operator norm, and determine the spectrum. We prove that the only bounded multiplication operators on the Bloch space of the polydisk are those whose symbol is constant. Furthermore, we prove that for a large class of bounded symmetric domains, the isometric multiplication operators are those whose symbol is a constant of modulus one.
This paper is concerned with an explicit value of the embedding constant from $W^{1,q}(Ω)$ to $L^{p}(Ω)$ for a bounded domain $Ω\subset\mathbb{R}^N~(N\in\mathbb{N})$, where $1\leq q\leq p\leq \infty$. To obtain this value, we previously proposed a formula for estimating the embedding constant on bounded and unbounded Lipschitz domains by estimating the norm of Stein's extension operator, in the article (K. Tanaka, K. Sekine, M. Mizuguchi, and S. Oishi, Estimation of Sobolev-type embedding constant on domains with minimally smooth boundary using extension operator, Journal of Inequalities and A…
We give estimates for the essential norms of a positive Toeplitz operator on the Bergman space of a minimal bounded homogeneous domain in terms of the Berezin transform or the averaging function of the symbol. Using these estimates, we also give necessary and sufficient conditions for the positive Toeplitz operators to be compact.
We generalize the Hartogs triangle to a class of bounded Hartogs domains, and we prove that the corresponding Bergman projections are bounded on $L^p$ if and only if $p$ is in the range $(\frac{2n}{n+1},\frac{2n}{n-1})$.
We show that Sobolev space $W^1_1(Ω)$ of any planar one-connected domain $Ω$ has the Bounded Approximation property. The result holds independently from the properties of the boundary of $Ω$. The prove is based on a new decomposition of a planar domain.
We provide a large family of atoms for Bergman spaces on irreducible bounded symmetric domains. This vastly generalizes results by Coifman and Rochberg from 1980. The atomic decompositions are derived using the holomorphic discrete series representations for the domain, and the approach is inspired by recent advances in wavelet and coorbit theory. This approach also settles the relation between atomic decompositions for the bounded and unbounded realizations of the domain.
Elliptic estimates in Hardy classes are proved on domains with minimally smooth boundary. The methodology is different from the original methods of Chang/Krantz/Stein.
This is our third work on Bergman-type operator over bounded domains. In the previous two articles, we systematically study the boundedness, compactness and Schatten membership of Bergman-type on the Hilbert unit ball. In the present paper, we investigate singular integral operators induced by the Bergman kernel and Szegö kernel on the irreducible bounded symmetric domain in its standard Harish-Chandra realization. We completely characterize when Bergman-type operators and Szegö-type operators belong to Schatten class operator ideals by several analytic numerical invariants of the bounded symm…
In this paper, we study the weighted compositon operators on weighted Bergman spaces of bounded symmetric domains. The necessary and sufficient conditions for a weighted composition operator $W_{ϕ,ψ}$ to be bounded and compact are studied by using the Carleson measure techniques. In the last section, we study the Schatten p-class weighted composition operators.
In this paper, we present the current results in the study of weighted composition operators on the Bloch space of bounded homogeneous domains in $\mathbb{C}^n$ with particular emphasis on the issues of boundedness and compactness. We also discuss the bounded and the compact weighted composition operators from the Bloch space to the Hardy space $H^\infty$.
We consider the Stokes resolvent problem in a two-dimensional bounded Lipschitz domain $Ω$ subject to homogeneous Dirichlet boundary conditions. We prove $\mathrm{L}^p$-resolvent estimates for $p$ satisfying the condition $\lvert 1 / p - 1 / 2 \rvert < 1 / 4 + \varepsilon$ for some $\varepsilon > 0$. We further show that the Stokes operator admits the property of maximal regularity and that its $\mathrm{H}^{\infty}$-calculus is bounded. This is then used to characterize domains of fractional powers of the Stokes operator. Finally, we give an application to the regularity theory of weak solutio…
We show that there are many sets in the boundary of a bounded symmetric domain that determine the values and norm of holomorphic functions on the domain having continuous extensions to the boundary. We provide an analogue of the Bergmann-Shilov boundary for finite rank JB*-triples.
We study the existence and the uniqueness of a solution $\fy$ to the linear Fokker-Planck equation $-Δ\fy + ÷(\fy \F) = f$ in a bounded domain of $\R^d$ when $\F$ is a "confinement" vector field acting for instance like the inverse of the distance to the boundary. An illustration of the obtained results is given within the framework of fluid mechanics and polymer flows.
We define and study a noncommutative Fourier transform on every homogeneous complex bounded domain. We then give an application in noncommutative differential geometry by defining noncommutative Baumslag-Solitar tori.
