Showing papers on "Singular integral published in 2023"
2 citations
TL;DR: In this article , a weak-type (1, 1)-constraint criterion for vector-valued singular integral operators with rough kernels was established, where the authors considered a complex interpolation space between a Hilbert space H and a UMD space X.
Abstract: In this paper, we establish a weak‐type (1,1) boundedness criterion for vector‐valued singular integral operators with rough kernels. As applications, we obtain weak‐type (1,1) bounds for the convolution singular integral operator taking value in the Banach space Y with a rough kernel, the maximal operator taking vector value in Y with a rough kernel and several square functions with rough kernels. Here, Y=[H,X]θ$Y=[H,X]_\theta$ is a complex interpolation space between a Hilbert space H and a UMD space X.
1 citations
1 citations
TL;DR: The class of variable anisotropic singular integral operators associated to a continuous multi-level ellipsoid cover was introduced by Dahmen et al. as mentioned in this paper , which is an extension of the classical isotropic singular integral operator on general expansive matrices.
Abstract: We introduce the class of variable anisotropic singular integral operators associated to a continuous multi-level ellipsoid cover $$\Theta $$ of $$\mathbb {R}^n$$ introduced by Dahmen et al. (Constr Approx 31:149–194, 2010). This is an extension of the classical isotropic singular integral operators on $${\mathbb R}^n$$ of arbitrary smoothness and their anisotropic analogues for general expansive matrices introduced by the first author Bownik (Mem Am Math Soc 164(781):1–122, 2003). We establish the boundedness of variable anisotropic singular integral operators T on the Hardy spaces with pointwise variable anisotropy $$H^p(\Theta )$$ , which were developed by Dekel et al. (J Fourier Anal Appl 17:1066–1107, 2011). In contrast with the general theory of Hardy spaces on spaces of homogenous type, our results work in the full range $$0
1 citations
1 citations
TL;DR: In this article , the authors studied singular integral operators with kernels that are more singular than standard Calderón-Zygmund kernels, but less singular than bi-parameter products of Calderón and Zygmund.
Abstract: Abstract We study singular integral operators with kernels that are more singular than standard Calderón–Zygmund kernels, but less singular than bi-parameter product Calderón–Zygmund kernels. These kernels arise as restrictions to two dimensions of certain three-dimensional kernels adapted to so-called Zygmund dilations, which is part of our motivation for studying these objects. We make the case that such kernels can, in many ways, be seen as part of the extended realm of standard kernels by proving that they satisfy both a T 1 theorem and commutator estimates in a form reminiscent of the corresponding results for standard Calderón–Zygmund kernels. However, we show that one-parameter weighted estimates, in general, fail.
1 citations
TL;DR: In this paper , the boundedness of generalized generalized generalized Marcinkiewicz integrals on product spaces is established for weak conditions on singular kernels. Butler et al. used Yano's extrapolation technique to obtain boundedness for generalized generalized generalization of generalized generalizations of Marcinkiewić integrals.
Abstract: In this article, suitable estimates for a class of rough generalized Marcinkiewicz integrals on product spaces are established. By these estimates, together with employing Yano’s extrapolation technique, we obtain the boundedness of the aforementioned integral operators under weak conditions on singular kernels. A number of known previous results on Marcinkiewicz as well as generalized Marcinkiewicz operators over a symmetric space are essentially improved or extended.
1 citations
TL;DR: In this paper , the authors present analytical expressions for the computation of singular integrals obtained in the discretisation of boundary integral equations for the Laplace and creeping flow problems with triangular or quadrilateral boundary elements with linear interpolation of the potential and constant interpolations of the flux.
Abstract: In this paper we present analytical expressions for the computation of singular integrals obtained in the discretisation of boundary integral equations for the Laplace and creeping flow (Stokes) problems with triangular or quadrilateral boundary elements with linear interpolation of the potential and constant interpolation of the flux. We compare the singular integrals computed with the presented analytical expressions with the same integrals computed with numerical quadrature and find that a considerably larger computational effort has to be made for numerical quadrature to achieve high accuracy than with the analytical expression. Furthermore, we show that the accuracy in solving a Laplace test case and a creeping flow test case using analytical expressions for singular integrals is better than the accuracy achieved with numerical quadrature. The analytical expressions are listed in the appendix of the paper and their implementation in computer code is available online.
TL;DR: In this article , the generalized singular integral operator with rough kernel and the approximation problem for the generalized surface quasi-geostrophic equation were studied and uniform Lp−Lq estimates with respect to a parameter β were obtained.
01 Feb 2023
TL;DR: In this article , the authors consider a non-radial kernel with properties similar to those from the classical theory of singular integrals and show that the Dunkl convolution operator is a bounded operator on the space of homogeneous types.
Abstract: On $\mathbb R^N$ equipped with a root system $R$, multiplicity function $k \geq 0$, and the associated measure $dw(\mathbf{x})=\prod_{\alpha \in R}|\langle \mathbf{x},\alpha\rangle|^{k(\alpha)}\,d\mathbf{x}$, we consider a (non-radial) kernel ${K}(\mathbf{x})$ which has properties similar to those from the classical theory of singular integrals and the Dunkl convolution operator $\mathbf{T}f=f*K$ associated with ${K}$. Assuming that $b$ belongs to the ${\rm BMO}$ space on the space of homogeneous type $X=(\mathbb{R}^N,\|\cdot\|,dw)$, we prove that the commutator $[b,\mathbf{T}]f(\mathbf{x})=b(\mathbf{x})\mathbf{T}f(\mathbf{x})-\mathbf{T}(bf)(\mathbf{x})$ is a bounded operator on $L^p(dw)$ for all $1
TL;DR: In this article , a numerical method for solving weakly singular Volterra integral equations of the first kind is presented. But this method is restricted to the case where the integral equation is first converted into algebraic form using the two-dimensional Laplace transform and then derived the series expansion for large values.
TL;DR: In this article , a Green-type solution for an anisotropic strip with an unloaded boundary was constructed by using the Schwartz algorithm and the Fourier integral transform (FIFT) and the Lekhnitskii potentials with isolated poles.
Abstract: By the method of successive approximations (Schwartz algorithm) and the Fourier integral transform, we construct a Green-type solution for an anisotropic strip with unloaded boundary. This solution is expressed in terms of the Lekhnitskii potentials with isolated poles. On the basis of the constructed solutions, we deduce singular integral equations for anisotropic plates with holes such that the boundary conditions imposed on the sides of the strip are identically satisfied. These equations are numerically solved by the method of mechanical quadratures. We also analyze the stress concentration in the vicinity of holes of different shapes contained in composite plates.
26 Jan 2023
TL;DR: In this article , the authors study the boundedness of singular integrals with mixed homogeneities on the Hardy space and show that the same can be achieved for non-standard convolutional singular integral integrals.
Abstract: This paper is motivated by Phong and Stein's paper on non-standard singular integrals with mixed homogeneities. Our purpose is to study these new non-standard convolution singular integrals and establish the boundedness of these singular integrals on the Hardy spaces.
TL;DR: In this article , the necessity of BMO for boundedness of commutators of singular integral operators on weighted Lebesgue spaces is investigated, and the results relax the restriction on the weight class to general multiple weights.
Abstract: In this paper, the necessity of BMO for boundedness of commutators of multilinear singular integral operators on weighted Lebesgue spaces is investigated. The results relax the restriction on the weight class to general multiple weights, which can be rega
01 Jan 2023
TL;DR: In this article , the singular stochastic Volterra integral equations with Mittag-Leffler kernels were studied and the θ-Maruyama method was proposed for solving the equations.
Abstract: This paper focusses on the singular stochastic Volterra integral equations with Mittag–Leffler kernels. Some qualitative properties of the solution are given under local Lipschitz condition, which include uniqueness and existence, boundedness of pth moments, Hölder continuity and continuous dependence on the initial value. The θ-Maruyama method is proposed for solving the equations. The strong convergence results of this method are obtained under global Lipschitz condition and local Lipschitz condition, respectively. Some numerical examples are given to verify the theoretical results.
TL;DR: In this article , it was shown that singular integral operators of convolution type are bounded on a rearrangement-invariant Banach function space if and only if their Boyd indices are nontrivial.
Abstract: We prove that nondegenerate singular integral operators of convolution type are bounded on a rearrangement-invariant Banach function space $X(\mathbb{R}^d)$ if and only if its Boyd indices are nontrivial, extending the result by David Boyd (1966) for the Hilbert transform.
05 Apr 2023
TL;DR: In this paper , the authors proposed an iterative decomposition method for finding the fractional integrals of fractional differential and integro-differential equations as the inverse operators of the respective differential operator.
Abstract: In the paper, on Iterative Decomposition method is proposed to approximate analytical solution of Singular Differential, Integral and Integro-differential equations the method involves deriving a formula for finding the fractional integrals of fractional differential and integro-differential equations as the inverse operators of the respective differential operator. A recursive formula, devoid of linearization or discretization was derived, which uses the reduces at previous points to obtained the next iteration. The solution was then presented as an infinite series of easily comparable terms which converge, even for every few terms. The convergence of the method is established.
01 Jan 2023
TL;DR: In this paper , it was shown that the Cauchy problem for a number of abstract singular equations with fractional derivatives reduces to a simpler problem for non-singular equations.
Abstract: With the help of integral representations of the Poisson type, it is established that the Cauchy problem for a number of abstract singular equations with fractional derivatives reduces to a simpler problem for a non-singular equation.
12 May 2023
TL;DR: In this article , it was shown that if the homogeneous singular integral operator of degree zero and integrable on S^{d-1} has mean value zero, then for all ρ ∈ (1,\,\infty) the maximal operator enjoys a bilinear sparse domination with bound $Cr'\|\Omega\|''Omega''|_{L^{\infty}(S^{d -1})}, where ρ(t)=t\log\log ({\rm e}^2+t)
Abstract: Let $\Omega$ be homogeneous of degree zero, integrable on $S^{d-1}$ and have mean value zero, $T_{\Omega}$ be the homogeneous singular integral operator with kernel $\frac{\Omega(x)}{|x|^d}$ and $T_{\Omega}^*$ be the maximal operator associated to $T_{\Omega}$. In this paper, the authors prove that if $\Omega\in L^{\infty}(S^{d-1})$, then for all $r\in (1,\,\infty)$, $T_{\Omega}^*$ enjoys a $(L^\Phi,\,L^r)$ bilinear sparse domination with bound $Cr'\|\Omega\|_{L^{\infty}(S^{d-1})}$, where $\Phi(t)=t\log\log ({\rm e}^2+t)$.
01 Jan 2023
TL;DR: Yang and Lin this article studied the boundedness of multilinear strongly singular integral operators with generalized kernels and Lipschitz functions, and showed that the smoothness condition of kernel functions with strongly singular Calderón-Zygmund operators is weak.
Abstract: Abstract In [S. Yang and Y. Lin, Multilinear strongly singular integral operators with generalized kernels and applications, AIMS Math. 6 2021, 12, 13533–13551], the authors of the present paper further weaken the smoothness condition of kernel functions with multilinear strongly singular Calderón–Zygmund operators of [Y. Lin, Multilinear theory of strongly singular Calderón–Zygmund operators and applications, Nonlinear Anal. 192 2020, Article ID 111699]. They defined a new class of multilinear strongly singular integral operators, and studied its weighted L p {L^{p}} boundedness, variable exponent L p ( ⋅ ) {L^{p(\cdot)}} boundedness and endpoint estimates. In this paper, we naturally consider the boundedness of multilinear commutators and multilinear iterated commutators, which are generated by multilinear strongly singular integral operators with generalized kernels and Lipschitz functions. Our results include the corresponding results of multilinear strongly singular Calderón–Zygmund operators and classical multilinear Calderón–Zygmund operators, respectively.
TL;DR: In this paper , a residual-based numerical scheme for solving a system of Cauchy-type singular integral equations of index minus N using Chebyshev polynomials of the first and second kind was developed.
TL;DR: In this article , the authors presented a solution to the bending problem for an infinite isotropic plate with a circular rigid inclusion and arbitrarily oriented straight through cracks, where the faces of all cracks were in smooth contact along a region of constant width near the upper surface of the plate.
Abstract: Abstract The article presents a solution to the bending problem for an infinite isotropic plate with a circular rigid inclusion and arbitrarily oriented straight through cracks. It is assumed that under the action of an external load at infinity, the faces of all cracks are in smooth contact along a region of constant width near the upper surface of the plate. The solution is obtained using methods of the theory of functions of complex variable and complex potentials and is reduced to a system of singular integral equations, which is numerically solved by the mechanical quadrature method. On the basis of numerical analysis, the graphical dependences of the contact force, forces and moments intensity factors are constructed at various parameters of the problem.
TL;DR: In this paper , the problem of solving the transverse shear of a plate along the edges of the holes and weakened by a two-periodic system of rectilinear through cracks with plastic end zones, collinear to the abscissa and ordinate axes of unequal length, is considered.
Abstract: The problem of solving the problem of the transverse shear of a plate along the edges of the holes and weakened by a two-periodic system of rectilinear through cracks with plastic end zones, collinear to the abscissa and ordinate axes of unequal length, is considered. General representations of solutions are constructed that describe a class of problems with a doubly periodic distribution of moments outside circular holes and straight-line cracks with end zones of plastic deformations. Satisfying the boundary conditions, the solution of the problem of the plate shear theory is reduced to two infinite systems of algebraic equations and two singular integral equations. Then each singular integral equation is reduced to a finite system of linear algebraic equations.