Class Of Singular Integral Operators Research Articles

Singular integral operators on product domains along twisted surfaces

We introduce a class of singular integral operators on product domains along twisted surfaces. We prove that the operators are bounded on Lp provided that the kernels satisfy weak conditions.

Boundedness of singular integral operators on weak Herz type spaces with variable exponent

In this paper, the authors define the weak Herz spaces and the weak Herz-type Hardy spaces with variable exponent. As applications, the authors establish the boundedness for a class of singular integral operators including some critical cases.

Symbolic Computation Applied to the Study of the Kernel of Special Classes of Paired Singular Integral Operators

Operator theory has many applications in several main scientific research areas (structural mechanics, aeronautics, quantum mechanics, ecology, probability theory, electrical engineering, among others) and the importance of its study is globally acknowledged. On the study of the operator’s kernel some progress has been achieved for some specific classes of singular integral operators whose properties allow the use of particular strategies. However, the existing algorithms allow, in general, to study the dimension of the kernel of some classes of singular integral operators but are not designed to be implemented on a computer. The main goal of this paper is to show how the symbolic and numeric capabilities of a computer algebra system can be used to study the kernel of special classes of paired singular integral operators with essentially bounded coefficients defined on the unit circle. It is described how some factorization algorithms can be used to compute the dimension of the kernel of special classes of singular integral operators. The analytical algorithms [ADimKerPaired-Scalar], [AKerPaired-Scalar], and [ADimKerPaired-Matrix] are presented. The design of these new algorithms was focused on the possibility of implementing on a computer all the extensive symbolic and numeric calculations present in the algorithms. For the essentially bounded hermitian coefficients case, there exist some relations with Hankel operators. The paper contains some interesting and nontrivial examples obtained with the use of a computer algebra system.

Multi‐parameter estimates via operator‐valued shifts

AbstractWe prove the mixed‐norm ‐boundedness of a general class of singular integral operators having a multi‐parameter singularity and acting on vector‐valued (UMD Banach lattice‐valued) functions. Moreover, families of such operators with uniform assumptions are shown to be not only uniformly bounded but ‐bounded, a genuinely stronger property that is often needed in applications. Previous results of this nature only dealt with convolution‐type or slightly more general paraproduct‐free singular integrals. In contrast, our analysis specifically targets the array of different partial paraproducts that arise in the multi‐parameter setting by interpreting them as paraproduct‐valued one‐parameter operators. This new point‐of‐view provides a conceptual simplification over the existing representation results for multi‐parameter operators, which is a key to the proof of the boundedness of these operators.

Quantitative Weighted Bounds for a Class of Singular Integral Operators

In this article, the authors consider the weighted bounds for the singular integral operator defined by $${T_A}f(x) = {\rm{p}}.{\rm{v}}.\int_{\mathbb{R}^{n}} {{{{\rm{\Omega }}(x - y)} \over {{\rm{|}}x - y{{\rm{|}}^{n + 1}}}}\left( {A(x) - A(y) - \nabla A(y)} \right)f(y){\rm{d}}y} ,$$ where Ω is homogeneous of degree zero and has vanishing moment of order one, and A is a function on ℝn such that ▽A ∈ BMO(ℝn). By sparse domination, the authors obtain some quantitative weighted bounds for Ta when Ω satisfies regularity condition of Lr-Dini type for some r ∈ (1, ∞).

The Noetherian Conditions and the Index of Some Class of Singular Integral Operators over a Bounded Simply Connected Domain

Necessary and sufficient conditions to be Noetherian are obtained for two-dimensional singular integral operators on Lebesgue spaces with a weight coefficient. A formula for calculation of their index is given.

The BMO-Dirichlet problem for elliptic systems in the upper half-space and quantitative characterizations of VMO

We prove that for any homogeneous, second order, constant complex coefficient elliptic system $L$, the Dirichlet problem in $\mathbb{R}^{n}_{+}$ with boundary data in BMO is well-posed in the class of functions $u$ with $d\mu_u(x',t):=|\nabla u(x',t)|^2\,t\,dx'dt$ being a Carleson measure. We establish a Fatou type theorem guaranteeing the existence of the pointwise nontangential boundary trace for smooth null-solutions $u$ of such systems satisfying the said Carleson measure condition. These imply that BMO can be characterized as the collection of nontangential pointwise traces of smooth null-solutions $u$ to the elliptic system $L$ with the property that $\mu_u$ is a Carleson measure. We establish a regularity result for the BMO-Dirichlet problem in the upper-half space: the nontangential pointwise trace of any given smooth null-solutions of $L$ satisfying the above Carleson measure condition belongs to Sarason's space VMO if and only if $\mu_u$ satsifies a vanishing Carleson measure condition. Moreover, we obtain the well-posedness of the Dirichlet problems when the boundary data are prescribed in Morrey-Campanato and solutions are required to satisfy a vanishing Carleson measure condition of fractional order. As a consequence, we characterize the space VMO as the closure in BMO of classes of smooth functions contained in BMO within which uniform continuity may be suitably quantified (such as the class of smooth functions satisfying a H\"older or Lipschitz condition), improving Sarason's classical result describing VMO as the closure in BMO of the space of uniformly continuous functions with bounded mean oscillations. Finally, we show that any Calder\'on-Zygmund operator $T$ satisfying $T(1)=0$ extends as a bounded linear operator in $\mathrm{VMO}$, and characterize the membership to $\mathrm{VMO}$ via the action of various classes of singular integral operators.

Singular Integrals Associated with Zygmund Dilations

The main purpose of this paper is to study multi-parameter singular integral operators which commute with Zygmund dilations. Motivated by some explicit examples of singular integral operators studied in Ricci and Stein (Ann Inst Fourier (Grenoble) 42:637–670, 1992), Fefferman and Pipher (Am J Math 11:337–369, 1997), and Nagel and Wainger (Am J Math 99:761–785, 1977), we introduce a class of singular integral operators on {mathbb {R}}^3 associated with Zygmund dilations by providing suitable version of regularity conditions and cancellation conditions on convolution kernels, and then show the boundedness for this class of operators on L^p, 1<p<infty .

Weighted Inequalities for Schrödinger Type Singular Integrals

Related to the Schrodinger operator $$L=-\Delta +V$$ , the behaviour on $$L^p$$ of several first and second order Riesz transforms was studied by Shen (Ann Inst Fourier (Grenoble) 45(2):513–546, 1995). Under his assumptions on V, a critical radius function $$\rho :X\rightarrow {\mathbb {R}}^+$$ can be associated, with the property that its variation is controlled by powers. Given such a function, we introduce a class of singular integral operators whose kernels have some extra decay related to $$\rho $$ . We analyse their behaviour on weighted $$L^p$$ and BMO-type spaces. Here, the weights as well as the regularity spaces depend only on the critical radius function. When our results are set back into the Schrodinger context, we obtain weighted inequalities for all the Riesz transforms initially appearing in Shen (1995). Concerning the action of Schrodinger singular integrals on regularity spaces, we extend some previous work of Ma et al.

Singular integral operators on tent spaces: a Calderón–Zygmund theory and weak-type endpoint estimates

We propose a Calderon–Zygmund-type extrapolation theory for sublinear operators acting on the so-called tent spaces introduced by Coifman et al. (J Funct Anal 62(2):304–335, 1985). As an application, we prove endpoint weak-type estimates for the article [referred to as Auscher et al. (J Evol Equ 12(4):741–765, 2012)]. The main ingredient in establishing this extrapolation theory is the use of some Calderon–Zygmund-type decompositions in tent spaces. In applying this abstract theory to the class of singular integral operators on tent spaces as considered in [3], we shall use certain Hardy–Littlewood embeddings for tent space functions. These embeddings are also interesting in themselves. Applications to maximal regularity operators on tent spaces are also discussed.

Multi-parameter singular integral operators and representation theorem

We formulate a class of singular integral operators in arbitrarily many parameters using mixed type characterizing conditions. The main result we prove for this class of operators is a multi-parameter representation theorem stating that a generic operator in our class can be represented as an average of sums of dyadic shifts, which implies a new multi-parameter T 1 theorem as a byproduct. This extends the representation principles of Hytönen’s and Martikainen’s to the multi-parameter setting. Furthermore, equivalence between ours and Journé’s class of multi-parameter operators is established, whose proof requires the multiparameter T 1 theorem.

Exploring the Spectra of Some Classes of Singular Integral Operators with Symbolic Computation

Spectral theory has many applications in several main scientific research areas (structural mechanics, aeronautics, quantum mechanics, ecology, probability theory, electrical engineering, among others) and the importance of its study is globally acknowledged. In recent years, several software applications were made available to the general public with extensive capabilities of symbolic computation. These applications, known as computer algebra systems (CAS), allow to delegate to a computer all, or a significant part, of the symbolic calculations present in many mathematical algorithms. In our work we use the CAS Mathematica to implement for the first time on a computer analytical algorithms developed by us and others within the Operator Theory. The main goal of this paper is to show how the symbolic computation capabilities of Mathematica allow us to explore the spectra of several classes of singular integral operators. For the one-dimensional case, nontrivial rational examples, computed with the automated process called [ASpecPaired-Scalar], are presented. For the matrix case, nontrivial essentially bounded and rational examples, computed with the analytical algorithms [AFact], [SInt], and [ASpecPaired-Matrix], are presented. In both cases, it is possible to check, for each considered paired singular integral operator, if a complex number (chosen arbitrarily) belongs to its spectrum.

WEIGHTED HARDY SPACES ON SPACE OF HOMOGENEOUS TYPE WITH APPLICATIONS

In this paper, we develop a theory of weighted Hardy spaces $H^p_\omega$ on spaces of homogeneous type and prove that certain class of singular integral operators are bounded from $H^p_\omega$ to itself and from $H^p_\omega$ to $L^p_\omega$. As an application, we give weighted endpoint estimates for Nagel-Stein's NIS operators studided in [26].

Boundedness for a Class of Singular Integral Operators on Both Classical and Product Hardy Spaces

We found that the classical Calderón-Zygmund singular integral operators are bounded on both the classical Hardy spaces and the product Hardy spaces. The purpose of this paper is to extend this result to a more general class. More precisely, we introduce a class of singular integral operators including the classical Calderón-Zygmund singular integral operators and show that they are bounded on both the classical Hardy spaces and the product Hardy spaces.

Calderón‐Zygmund operators with non‐diagonal singularity

In this paper, we introduce a class of singular integral operators which generalize Calderón‐Zygmund operators to the more general case, where the set of singular points of the kernel need not to be the diagonal, but instead, it can be a general hyper curve. We show that such operators have similar properties as ordinary Calderón‐Zygmund operators. In particular, we prove that they are of weak‐type (1, 1) and strong type for .

Atomic decompositions of weighted Hardy-Morrey spaces

We obtain the Fefferman-Stein vector-valued maximal inequalities on Morrey spaces generated by weighted Lebesgue spaces. Using these inequalities, we introduce and define the weighted Hardy-Morrey spaces by using the Littlewood-Paley functions. We also establish the non-smooth atomic decompositions for the weighted Hardy-Morrey spaces and, as an application of the decompositions, we obtain the boundedness of a class of singular integral operators on the weighted Hardy-Morrey spaces.

Regularizations of general singular integral operators

In the theory of singular integral operators significant effort is often required to rigorously define such an operator. This is due to the fact that the kernels of such operators are not locally integrable on the diagonal s=t , so the integral formally defining the operator T or its bilinear form \langle Tf, g \rangle is not well defined (the integrand in not in L^1 ) even for “nice” f and g . However, since the kernel only has singularities on the “diagonal” s=t , the bilinear form \langle Tf, g \rangle is well defined, say, for bounded compactly supported functions with separated supports. One of the standard ways to interpret the boundedness of a singular integral operators is to consider the regularized kernel [ K_\varepsilon(s, t) = K(s, t), m((s-t)/\varepsilon), ] where the cut-off function m is 0 in a neighborhood of the origin, so the integral operators T_\varepsilon with kernel K_\varepsilon are well defined (at least on a dense set). Then instead of asking about the boundedness of the operator T , which is not well defined, one can ask about uniform boundedness (in \varepsilon ) of the regularized operators T_\varepsilon . For the standard regularizations one usually considers truncated operators T_\varepsilon with m(s) =\mathbf{1}_{[1, \infty)} (|s|) , although smooth cut-off functions were also considered in the literature. The main result of the paper is that for a wide class of singular integral operators (including the classical Calderón–Zygmund operators in nonhomogeneous two weight settings), the so called restricted L^p boundedness, i.e., the uniform estimate [ |\langle Tf, g \rangle| \le C ,|f| p ,|g| {p'} ] for bounded compactly supported f and g with separated supports implies the uniform L^p -boundedness of regularized operators T_\varepsilon for any reasonable choice of smooth cut-off function m . For example, any m \in C^\infty(\mathbb{R}^N) }, m\equiv 0 in a neighborhood of 0 , and such that 1-m is compactly supported would work. If the kernel K satisfies some additional assumptions (which are satisfied for classical singular integral operators like the Hilbert transform, Cauchy transform, Ahlfors–Beurling transform, and generalized Riesz transforms), then the restricted L^p boundedness also implies the uniform L^p boundedness of the classical truncated operators T_\varepsilon ( m(s) =\mathbf{1}_{[1, \infty)} (|s|) ).

Homogeneous Triebel-Lizorkin Spaces on Stratified Lie Groups

Homogeneous Triebel-Lizorkin spaces with full range of parameters are introduced on stratified Lie groups in terms of Littlewood-Paley-type decomposition. It is shown that the scale of these spaces is independent of the choice of Littlewood-Paley-type decomposition and the sub-Laplacian used for the construction of the decomposition. Some basic properties of these spaces are given. As the main result of this paper, boundedness of a class of singular integral operators on these function spaces is obtained.

ROUGH SINGULAR INTEGRALS ASSOCIATED TO SUBMANIFOLDS

We investigate the $L^p$ boundedness for a class of singular integral operators associated to submanifolds, including surfaces of revolution, under the $L(\log L)(S^{n-1})$ or Block space condition on the kernel functions. Our results improve known results.

ROUGH SINGULAR INTEGRALS ASSOCIATED TO SUBMANIFOLDS

We investigate the $L^p$ boundedness for a class of singular integral operators associated to submanifolds, including surfaces of revolution, under the $L(\log L)({{S}^{n-1}})$ or Block space condition on the kernel functions. Our results improve known results.