Boundedness Of Singular Integral Operators Research Articles

Riesz transform associated with the fractional Fourier transform and applications in image edge detection

The fractional Hilbert transform was introduced by Zayed [30, Zayed, 1998] and has been widely used in signal processing. In view of its connection with the fractional Fourier transform, Chen, the first, second and fourth authors of this paper in [6, Chen et al., 2021] studied the fractional Hilbert transform and other fractional multiplier operators on the real line. The present paper is concerned with a natural extension of the fractional Hilbert transform to higher dimensions: this extension is the fractional Riesz transform and is given by multiplication which a suitable chirp function on the fractional Fourier transform side. In addition to a thorough study of the fractional Riesz transform, in this work we also investigate the boundedness of singular integral operators with chirp functions on rotation invariant spaces, chirp Hardy spaces and their relation to chirp BMO spaces, as well as applications of the theory of fractional multipliers in partial differential equations. Through numerical simulation, we provide physical and geometric interpretations of high-dimensional fractional multipliers. Finally, we present an application of the fractional Riesz transforms in edge detection which verifies a hypothesis insinuated in [26, Xu et al., 2016]. In fact our numerical implementation confirms that amplitude, phase, and direction information can be simultaneously extracted by controlling the order of the fractional Riesz transform.

Weighted local Hardy spaces with variable exponents

AbstractThis paper defines local weighted Hardy spaces with variable exponents. Local Hardy spaces permit atomic decomposition, which is one of the main themes in this paper. A consequence is that the atomic decomposition is obtained for the functions in the Lebesgue spaces with exponentially decaying exponent. As applications, we obtain the boundedness of singular integral operators, the Littlewood–Paley characterization, and wavelet decomposition.

Singular integrals on $$C_{w^*}^{1,\alpha }$$ regular curves in Banach duals

The modern study of singular integral operators on curves in the plane began in the 1970s. Since then, there has been a vast array of work done on the boundedness of singular integral operators defined on lower dimensional sets in Euclidean spaces. In recent years, mathematicians have attempted to push these results into a more general metric setting particularly in the case of singular integral operators defined on curves and graphs in Carnot groups. Suppose $$X = Y^*$$ for a separable Banach space Y. Any separable metric space can be isometrically embedded in such a Banach space via the Kuratowski embedding. The $$w^*$$ -derivative $$\gamma '$$ of a curve $$\gamma :[a,b] \rightarrow X$$ at $$t \in [a,b]$$ satisfies $$\tfrac{\mathrm{d}}{\mathrm{d}s} \langle \gamma (s),y \rangle |_{s=t} = \langle \gamma '(t),y\rangle$$ for any $$y \in Y$$ . Suppose $$\Gamma = \gamma ([a,b])$$ is a curve in X whose $$w^*$$ -derivative is Hölder continuous and bounded away from 0. We prove that any convolution type singular integral operator associated with a 1-dimensional Calderón–Zygmund kernel which is uniformly $$L^2$$ -bounded on lines is $$L^p$$ -bounded along $$\Gamma$$ . We also prove a version of David’s “good lambda” theorem for upper regular measures on doubling metric spaces.

Rough singular integrals associated to polynomial curves

In this paper, the authors establish the boundedness of singular integral operators associated to polynomial curves as well as the related maximal operators with rough kernels Omega in H^{1}({mathrm{S}}^{n-1}) and hin Delta _{gamma }(mathbb{R}_{+}) for some gamma >1 on the Triebel–Lizorkin spaces. It should be pointed out that the bounds are independent of the coefficients of the polynomials in the definition of the operators. The main results of this paper not only improve and generalize essentially some known results but also complement some recent boundedness results.

Carleson measures and the boundedness of singular integral operators on Q-type spaces related to weights

Carleson measures and the boundedness of singular integral operators on Q-type spaces related to weights

Cones, rectifiability, and singular integral operators

Let $\\mu$ be a Radon measure on $\\mathbb{R}^d$. We define and study conical energies $\\mathcal{E}\_{\\mu,p}(x,V,\\alpha)$, which quantify the portion of $\\mu$ lying in the cone with vertex $x\\in\\mathbb{R}^d$, direction $V\\in G(d,d-n)$, and aperture $\\alpha\\in (0,1)$. We use these energies to characterize rectifiability and the big pieces of Lipschitz graphs property. Furthermore, if we assume that $\\mu$ has polynomial growth, we give a sufficient condition for $L^2(\\mu)$-boundedness of singular integral operators with smooth odd kernels of convolution type.

DUALITIES OF VARIABLE ANISOTROPIC HARDY SPACES AND BOUNDEDNESS OF SINGULAR INTEGRAL OPERATORS

DUALITIES OF VARIABLE ANISOTROPIC HARDY SPACES AND BOUNDEDNESS OF SINGULAR INTEGRAL OPERATORS

Boundedness of Singular Integral Operators on Local Hardy Spaces and Dual Spaces

The purpose of this paper is to provide necessary and sufficient conditions of the boundedness for singular integrals on the local Hardy space and its dual. Particularly the singular integrals considered in this paper include the pseudo-differential operators $T_{\sigma }f(x)=\int \limits \sigma (x \xi )e^{2\pi ix\xi }\hat {f}(\xi )d\xi $ with $\sigma \in S_{1 0}^{0}$ . As a consequence our results give another proof of the boundedness of the pseudo-differential operators on the local Hardy space (Goldberg Duke Math. J. 46(1) 27–42 1979).

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.

Boundedness of Singular Integral Operators with Operator-Valued Kernels and Maximal Regularity of Sectorial Operators in Variable Lebesgue Spaces

This paper is devoted to the maximal regularity of sectorial operators in Lebesgue spaces Lp⋅ with a variable exponent. By extending the boundedness of singular integral operators in variable Lebesgue spaces from scalar type to abstract-valued type, the maximal Lp⋅−regularity of sectorial operators is established. This paper also investigates the trace of the maximal regularity space E01,p⋅I, together with the imbedding property of E01,p⋅I into the range-varying function space C−I,X1−1/p⋅,p⋅. Finally, a type of semilinear evolution equations with domain-varying nonlinearities is taken into account.

The Boundedness on Mixed Hardy Spaces

The boundedness of operators on Hardy spaces is usually given by atomic decomposition. In this paper, we obtain the boundedness of singular integral operators in mixed Journé class on mixed Hardy spaces by a direct method.

A global Tb Theorem for compactness and boundedness of Calderón-Zygmund operators

We prove a Tb Theorem that characterizes all Calderón-Zygmund operators that extend compactly on Lp(Rn), 1<p<∞. The result, whose proof does not require the property of accretivity, can be used to prove compactness of the Double Layer Potential operator on a wide class of domains.The study also provides conditions for boundedness of singular integral operators by means of non-accretive testing functions.

Weak Morrey spaces with applications

AbstractThe main object of this investigation is to study weak Morrey spaces. Block spaces, which are preduals of weak Morrey spaces, are characterized. Besides, the Fatou property of block spaces is proved. Finally, as an application, we study the boundedness of singular integral operators in weak Morrey spaces.

Boundary value problems for modified Helmholtz equations and applications

We investigate factorizations of modified Helmholtz equations in Clifford algebra $Cl(V_{3,3})$ . Using the method of fundamental solutions for modified Helmholtz equations and Clifford calculus, we obtain some integral representation theorem in Clifford analysis. The boundedness of singular integral operators in Holder space is given. Moreover, we establish solvability conditions of Riemann type problems for modified Helmholtz equations in Clifford analysis. As applications, we solve a kind of singular integral equations. The explicit representation of the solution is also given.

相关于微分算子的(Musielak-)Orlicz-Hardy空间的实变理论及其应用

As is well-known, because of their important applications in several branches of mathematics, such as harmonic analysis and partial differential equations, the theory of Calderon-Zygmund singular integral operators and the real-variable theory of various function spaces, which could characterize the boundedness of those operators, turn into one of the main contents of modern harmonic analysis. However, the real-variable theory of classical function spaces has been no longer suitable for characterizing the boundedness of singular integral operators associated with some more general differential operators than Laplace operators. Thus, for different operators, it has become one of the very active research fields of harmonic analysis in recent years to develop the real-variable theory of function spaces, which are suitable to those operators and could characterize the boundedness of singular integral operators associated with those operators. In this article, we study the realvariable theory of (Musielak-)Orlicz-Hardy spaces associated with some differential operators, including secondorder divergence form elliptic operators and Schrodinger operators as special cases, on n -dimensional Euclidean space R n , strongly domains of R n or metric spaces with doubling measure, and its applications to the boundedness of operators.

Multilinear Singular Integrals and Commutators on Herz Space with Variable Exponent

The paper establishes some sufficient conditions for the boundedness of singular integral operators and their commutators from products of variable exponent Herz spaces to variable exponent Herz spaces.

An Odd Rearrangement ofL1(Rn)

We introduce an odd rearrangementf*defined byπ(f)(x)=f*(x)=sgn(x1)f*(νn|x|n),x∈Rn, wheref*is a decreasing rearrangement of the measurable functionf. With the help of this odd rearrangement, we show that for eachf∈L1(Rn), there exists ag∈H1(Rn)such thatdf=dg, wheredfis an distribution function off. Moreover, we study the boundedness of singular integral operators when they are restricted to odd rearrangement ofL1(Rn), and we give some results on Hilbert transform.

Boundedness criterion of Journéʼs class of singular integrals on multiparameter Hardy spaces

This article concerns nonconvolutional type operators (also known as Journéʼs type operators) associated with a multiparameter family of dilations given by (x1,x2,…,xm)→(δ1x1,δ2x2,…,δmxm) where x1∈Rn1,x2∈Rn2,…,xm∈Rnm and m⩾3. We are especially interested in the boundedness of such operators on the multiparameter Hardy spaces. This work is motivated by Pipherʼs result on the boundedness of these operators from the multiparameter Hp spaces to Lp spaces for 0<p⩽1, and Journéʼs counter-example which shows that the number of parameter plays a crucial role in boundedness of singular integral operators on multiparameter Hardy spaces. Journéʼs work shows that there is a sharp difference between the situations for two and three or more parameters. We establish in this paper the necessary and sufficient conditions under which the singular integral operators in Journéʼs class are bounded on multiparameter Hp spaces (0<p⩽1) with arbitrary number of parameters.

Singular integrals and weighted Triebel-Lizorkin and Besov spaces of arbitrary number of parameters

Though the theory of Triebel-Lizorkin and Besov spaces in one-parameter has been developed satisfactorily, not so much has been done for the multiparameter counterpart of such a theory. In this paper, we introduce the weighted Triebel-Lizorkin and Besov spaces with an arbitrary number of parameters and prove the boundedness of singular integral operators on these spaces using discrete Littlewood-Paley theory and Calderon’s identity. This is inspired by the work of discrete Littlewood-Paley analysis with two parameters of implicit dilations associated with the flag singular integrals recently developed by Han and Lu [12]. Our approach of derivation of the boundedness of singular integrals on these spaces is substantially different from those used in the literature where atomic decomposition on the one-parameter Triebel-Lizorkin and Besov spaces played a crucial role. The discrete Littlewood-Paley analysis allows us to avoid using the atomic decomposition or deep Journe’s covering lemma in multiparameter setting.

Multiparameter Hardy space theory on Carnot-Carathéodory spaces and product spaces of homogeneous type

This paper is inspired by the work of Nagel and Stein in which the L p L^p ( 1 &gt; p &gt; ∞ ) (1&gt;p&gt;\infty ) theory has been developed in the setting of the product Carnot-Carathéodory spaces M ~ = M 1 × ⋯ × M n \widetilde {M}=M_1\times \cdots \times M_n formed by vector fields satisfying Hörmander’s finite rank condition. The main purpose of this paper is to provide a unified approach to develop the multiparameter Hardy space theory on product spaces of homogeneous type. This theory includes the product Hardy space, its dual, the product B M O BMO space, the boundedness of singular integral operators and Calderón-Zygmund decomposition and interpolation of operators. As a consequence, we obtain the endpoint estimates for those singular integral operators considered by Nagel and Stein (2004). In fact, we will develop most of our theory in the framework of product spaces of homogeneous type which only satisfy the doubling condition and some regularity assumption on the metric. All of our results are established by introducing certain Banach spaces of test functions and distributions, developing discrete Calderón identity and discrete Littlewood-Paley-Stein theory. Our methods do not rely on the Journé-type covering lemma which was the main tool to prove the boundedness of singular integrals on the classical product Hardy spaces.