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Journal articles on the topic 'Singular integral operator'

Author: Grafiati

Published: 4 June 2021

Last updated: 25 July 2025

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1

Lanzhe, Liu. "Continuity for some multilinear operators of integral operators on Triebel-Lizorkin spaces." International Journal of Mathematics and Mathematical Sciences 2004, no. 38 (2004): 2039–47. http://dx.doi.org/10.1155/s0161171204303121.

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The continuityfor some multilinear operators related to certain fractional singular integral operators on Triebel-Lizorkin spaces is obtained. The operators include Calderon-Zygmund singular integral operator and fractional integral operator.

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2

Potseiko, Pavel Gennadjevich, and Evgeny Alekseevich Rovba. "Approximations of one singular integral on an interval by Fourier-Chebyshev rational integral operators." Sbornik: Mathematics 215, no. 7 (2024): 953–92. http://dx.doi.org/10.4213/sm10030e.

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We study approximations on the interval $[-1,1]$ of singular integrals of the form $$ \widehat{f}(x)=\int_{-1}^{1}\frac{f(t)}{t-x}\sqrt{1-t^2} dt, \qquad x \in [-1,1], $$ by two rational integral operators related to each other in a certain sense. The first is the Fourier-Chebyshev integral operator associated with the Chebyshev-Markov system of rational functions. The second operator is its image under the transformation by the singular integral under consideration. Approximative properties of the corresponding polynomial analogues of both operators are studied in the case where the density o

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3

Hasanov, J. J., I. Ekincioglu, and C. Keskin. "A characterization for $B$-singular integral operator and its commutators on generalized weighted $B$-Morrey spaces." Carpathian Mathematical Publications 15, no. 1 (2023): 196–211. http://dx.doi.org/10.15330/cmp.15.1.196-211.

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We study the maximal operator $M_{\gamma}$ and the singular integral operator $A_{\gamma}$, associated with the generalized shift operator. The generalized shift operators are associated with the Laplace-Bessel differential operator. Our analysis is based on two weighted inequalities for the maximal operator, singular integral operators, and their commutators, related to the Laplace-Bessel differential operator in generalized weighted $B$-Morrey spaces.

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4

Liu, Lanzhe. "Estimates of multilinear singular integral operators and mean oscillation." Publications de l'Institut Math?matique (Belgrade) 95, no. 109 (2014): 201–14. http://dx.doi.org/10.2298/pim1409201l.

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We prove the boundedness properties for some multilinear operators related to certain integral operators from Lebesgue spaces to Orlicz spaces. The operators include Calder?n-Zygmund singular integral operator, Littlewood-Paley operator and Marcinkiewicz operator.

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5

Chen, Jiecheng, and Guoen Hu. "Compact Commutators of Rough Singular Integral Operators." Canadian Mathematical Bulletin 58, no. 1 (2015): 19–29. http://dx.doi.org/10.4153/cmb-2014-042-1.

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AbstractLet b ∊ BMO(ℝn) and TΩ be the singular integral operator with kernel Ω(x)/|x|n, where Ω is homogeneous of degree zero, integrable, and has mean value zero on the unit sphere Sn-1. In this paper, using Fourier transform estimates and approximation to the operator TΩ by integral operators with smooth kernels, it is proved that if b ∊ CMO(ℝn) and satisfies certain minimal size condition, then the commutator generated by b and TΩ is a compact operator on Lp(ℝn) for appropriate index p. The associated maximal operator is also considered.

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6

Hasanov, Javanshir J. "Φ-Admissible Sublinear Singular Operators and Generalized Orlicz-Morrey Spaces". Journal of Function Spaces 2014 (2014): 1–7. http://dx.doi.org/10.1155/2014/505237.

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We study the boundedness ofΦ-admissible sublinear singular operators on Orlicz-Morrey spacesMΦ,φℝn. These conditions are satisfied by most of the operators in harmonic analysis, such as the Hardy-Littlewood maximal operator and Calderón-Zygmund singular integral operator.

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7

Hu, Guoen, Yan Meng, and Dachun Yang. "Estimates for maximal singular integral operators in non-homogeneous spaces." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 136, no. 2 (2006): 351–64. http://dx.doi.org/10.1017/s0308210500004601.

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Under the assumption that the Radon measure μ on Rd satisfies only some growth condition, the authors prove that, for the maximal singular integral operator associated with a singular integral whose kernel only satisfies a standard size condition and the Hörmander condition, its boundedness in Lebesgue spaces Lp(μ) for any p ∈ (1, ∞) is equivalent to its boundedness from L1(μ) into weak L1(μ). As an application, the authors verify that if the truncated singular integral operators are bounded on L2(μ) uniformly, then the associated maximal singular integral operator is also bounded on Lp(μ) for

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8

Оdinabekov, Jasur M. "On the Noethericity conditions and the index of some two–dimensional singular integral operators." Russian Universities Reports. Mathematics, no. 138 (2022): 164–74. http://dx.doi.org/10.20310/2686-9667-2022-27-138-164-174.

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The main problems in the theory of singular integral operators are the problems of boundedness, invertibility, Noethericity, and calculation of the index. The general theory of multidimensional singular integral operators over the entire space E_n was constructed by S.G. Mikhlin. It is known that in the two-dimensional case, if the symbol of an operator does not vanish, then the Fredholm theory holds. For operators over a bounded domain, the boundary of this domain significantly affects the solvability of the corresponding operator equations. In this paper, we consider two-dimensional singular

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9

Al-Qassem, H. M. "Weighted norm inequalities for a class of rough singular integrals." International Journal of Mathematics and Mathematical Sciences 2005, no. 5 (2005): 657–69. http://dx.doi.org/10.1155/ijmms.2005.657.

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Weighted norm inequalities are proved for a rough homogeneous singular integral operator and its corresponding maximal truncated singular operator. Our results are essential improvements as well as extensions of some known results on the weighted boundedness of singular integrals.

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10

Boimatov, K. Kh, and G. Dzhangibekov. "On a singular integral operator." Russian Mathematical Surveys 43, no. 3 (1988): 199–200. http://dx.doi.org/10.1070/rm1988v043n03abeh001746.

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11

Dashan, Fan, Lu Shanzhen, and Pan Yibiao. "A discrete singular integral operator." Acta Mathematica Sinica 14, no. 2 (1998): 235–44. http://dx.doi.org/10.1007/bf02560210.

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12

JI, UN CIG, and KALYAN B. SINHA. "INTEGRAL REPRESENTATION OF QUANTUM MARTINGALES." Infinite Dimensional Analysis, Quantum Probability and Related Topics 08, no. 01 (2005): 55–72. http://dx.doi.org/10.1142/s0219025705001858.

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A stochastic integral representation in terms of generalized integral kernel operator is proved for a wide class of quantum martingales which includes regular martingales and the martingales determined by the second quantization of integral operators containing singular kernels.

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13

Ekincioglu, Ismail, Vagif S. Guliyev, and Esra Kaya. "Bn-maximal operator and Bn-singular integral operators on variable exponent Lebesgue spaces." Mathematica Slovaca 70, no. 4 (2020): 893–902. http://dx.doi.org/10.1515/ms-2017-0401.

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AbstractIn this paper, we prove the boundedness of the Bn maximal operator and Bn singular integral operators associated with the Laplace-Bessel differential operator ΔBn on variable exponent Lebesgue spaces.

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14

Hawawsheh, Laith, and Mohammad Abudayah. "A boundedness result for Marcinkiewicz integral operator." Open Mathematics 18, no. 1 (2020): 829–36. http://dx.doi.org/10.1515/math-2020-0046.

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Abstract We extend a boundedness result for Marcinkiewicz integral operator. We find a new space of radial functions for which this class of singular integral operators remains {L}^{p} -bounded when its kernel satisfies only the sole integrability condition.

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15

Gu, Caixing, In Sung Hwang, Dong-O. Kang, and Woo Young Lee. "Normal singular Cauchy integral operators with operator-valued symbols." Journal of Mathematical Analysis and Applications 447, no. 1 (2017): 289–308. http://dx.doi.org/10.1016/j.jmaa.2016.10.003.

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16

Yuan, Hongfen, Guohong Shi, and Xiushen Hu. "Boundary Value Problems for the Perturbed Dirac Equation." Axioms 13, no. 4 (2024): 238. http://dx.doi.org/10.3390/axioms13040238.

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The perturbed Dirac operators yield a factorization for the well-known Helmholtz equation. In this paper, using the fundamental solution for the perturbed Dirac operator, we define Cauchy-type integral operators (singular integral operators with a Cauchy kernel). With the help of these operators, we investigate generalized Riemann and Dirichlet problems for the perturbed Dirac equation which is a higher-dimensional generalization of a Vekua-type equation. Furthermore, applying the generalized Cauchy-type integral operator F˜λ, we construct the Mann iterative sequence and prove that the iterati

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17

Anastassiou, George A. "Multivariate Perturbed Hyperbolic Tangent-Activated Singular Integral Approximation." Mathematics 12, no. 17 (2024): 2700. http://dx.doi.org/10.3390/math12172700.

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Here we study the quantitative multivariate approximation of perturbed hyperbolic tangent-activated singular integral operators to the unit operator. The engaged neural network activation function is both parametrized and deformed, and the related kernel is a density function on RN. We exhibit uniform and Lp, p≥1 approximations via Jackson-type inequalities involving the first Lp modulus of smoothness, 1≤p≤∞. The differentiability of our multivariate functions is covered extensively in our approximations. We continue by detailing the global smoothness preservation results of our operators. We

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18

Polosin, A. A. "On One Degenerating Singular Integral Operator." Differential Equations 57, no. 10 (2021): 1413–17. http://dx.doi.org/10.1134/s0012266121100165.

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19

Guoen, Hu. "On a multilinear singular integral operator." Approximation Theory and its Applications 11, no. 4 (1995): 90–107. http://dx.doi.org/10.1007/bf02836833.

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20

Qiyu, Sun. "Two problems about singular integral operator." Approximation Theory and its Applications 7, no. 2 (1991): 83–98. http://dx.doi.org/10.1007/bf02845193.

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21

Alexander G, Ramm. "New definition of singular integral operator." Annals of Mathematics and Physics 6, no. 2 (2023): 097–99. http://dx.doi.org/10.17352/amp.000087.

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Let D be a connected bounded domain in ℝ2, S be its boundary which is closed, connected and smooth or S = (-∞,∞). Let , f∈L1(S), z = x+iy. The singular integral operator , t∈S, is defined in a new way. This definition simplifies the proof of the existence of Φ(t). Necessary and sufficient conditions are given for f∈L1(S) to be boundary value of an analytic in D function. The Sokhotsky-Plemelj formulas are derived for f∈L1(S). Our new definition allows one to treat singular boundary values of analytic functions.

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22

Chen, Jiecheng, Dashan Fan, and Yiming Ying. "Certain Operators with Rough Singular Kernels." Canadian Journal of Mathematics 55, no. 3 (2003): 504–32. http://dx.doi.org/10.4153/cjm-2003-021-4.

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AbstractWe study the singular integral operatordefined on all test functions f, where b is a bounded function, α ≥ 0, Ω (yʻ) is an integrable function on the unit sphere Sn-1 satisfying certain cancellation conditions. We prove that, for 1 < p < ∞, TΩ,α extends to a bounded operator from the Sobolev space to the Lebesgue space Lp with Ω being a distribution in the Hardy space Hq(Sn-1) where . The result extends some known results on the singular integral operators. As applications, we obtain the boundedness for TΩ,α on the Hardy spaces, as well as the boundedness for the truncated maxima

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23

Meskhi, Alexander. "On a measure of non-compactness for singular integrals." Journal of Function Spaces and Applications 1, no. 1 (2003): 35–43. http://dx.doi.org/10.1155/2003/927590.

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It is proved that there exists no weight pair(v, w)for which a singular integral operator is compact from the weighted Lebesgue spaceLwp(Rn)toLvp(Rn). Moreover, a measure of non-compatness for this operator is estimated from below. Analogous problems for Cauchy singular integrals defined on Jordan smooth curves are studied.

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24

Gunawan, Hendra, Denny Ivanal Hakim, Yoshihiro Sawano, and Idha Sihwaningrum. "Weak Type Inequalities for Some Integral Operators on Generalized Nonhomogeneous Morrey Spaces." Journal of Function Spaces and Applications 2013 (2013): 1–12. http://dx.doi.org/10.1155/2013/809704.

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We prove weak type inequalities for some integral operators, especially generalized fractional integral operators, on generalized Morrey spaces of nonhomogeneous type. The inequality for generalized fractional integral operators is proved by using two different techniques: one uses the Chebyshev inequality and some inequalities involving the modified Hardy-Littlewood maximal operator and the other uses a Hedberg type inequality and weak type inequalities for the modified Hardy-Littlewood maximal operator. Our results generalize the weak type inequalities for fractional integral operators on ge

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25

Fan, Dashan. "A Singular Integral on L2(Rn)." Canadian Mathematical Bulletin 37, no. 2 (1994): 197–201. http://dx.doi.org/10.4153/cmb-1994-029-0.

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AbstractWe consider a convolution singular integral operator associated to a kernel K(x) = b(x)Ω(x)|x|-n, and prove that if b ∊ L∞(ℝn) is a radial function and Ω ∊ H(Σn-1) with mean zero condition (1), then is a bounded linear operator in the space L2(ℝn).

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26

Akbulut, Ali, Vagif Guliyev, and Rza Mustafayev. "On the boundedness of the maximal operator and singular integral operators in generalized Morrey spaces." Mathematica Bohemica 137, no. 1 (2012): 27–43. http://dx.doi.org/10.21136/mb.2012.142786.

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27

Anastassiou, George A. "Uniform approximation by smooth Picard multivariate singular integral operators revisited." Acta Universitatis Sapientiae, Mathematica 16, no. 1 (2025): 42–58. https://doi.org/10.47745/ausm-2024-0003.

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In this article we reexamine the uniform approximation properties of smooth Picard multivariate singular integral operators over R N, N ≥ 1. We establish their convergence to the unit operator with rates. The estimates are pointwise and uniform. The established inequalities involve the multivariate first modulus of continuity. Our approach is based on a new multivariate trigonometric Taylor formula. At first we present in detail the general theory of uniform approximation by general smooth multivariate singular integral operators, which then is applied to the Picard operators case.

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28

MOUROU, MOHAMED A., and KHALIFA TRIMÈCHE. "TRANSMUTATION OPERATORS AND PALEY–WIENER THEOREM ASSOCIATED WITH A SINGULAR DIFFERENTIAL-DIFFERENCE OPERATOR ON THE REAL LINE." Analysis and Applications 01, no. 01 (2003): 43–70. http://dx.doi.org/10.1142/s0219530503000090.

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We consider a singular differential-difference operator Λ on the real line which includes, as particular case, the Dunkl operator associated with the reflection group Z2 on R. We exhibit a Laplace integral representation for the eigenfunctions of the operator Λ. From this representation, we construct a pair of integral transforms which turn out to be transmutation operators of Λ into the first derivative operator d/dx. We exploit these transmutation operators to develop a new commutative harmonic analysis on the real line corresponding to the operator Λ. In particular, we establish a Paley–Wie

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29

FANG, CHENGLONG. "CHARACTERIZATIONS OF COMMUTATORS OF SINGULAR INTEGRAL OPERATORS ON MORREY TRIEBEL-LIZORKIN." Mathematical Reports 25(75), no. 3 (2023): 425–39. http://dx.doi.org/10.59277/mrar.2023.25.75.3.425.

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In this paper, we obtain the characterizations of Morrey Triebel-Lizorkin spaces by two families of operators. Applying the characterizations of Morrey TriebelLizorkin spaces, it is proved that b is a Lipschitz function if and only if the commutator [b, T] is bounded from Morrey spaces to Morrey Triebel-Lizorkin spaces, where T is singular integral operator or Riesz potential operator.

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30

Mohammad, Najem A., and Mohammad Shami Hasso. "On Solution of Singular Integral Equations by Operator Method." International Frontier Science Letters 14 (March 2019): 41–48. http://dx.doi.org/10.18052/www.scipress.com/ifsl.14.41.

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In this paper, we study the exact solution of singular integral equations using two methods, including Adomian Decomposition Method and Elzaki Transform Method. We propose an analytical method for solving singular integral equations and system of singular integral equations, and have some goals in our paper related to suggested technique for solving singular integral equations. The primary goal is for giving analytical solutions of such equations with simple steps, another goal is to compare the suggested method with other methods used in this study.

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31

Casper, W. Riley, F. Alberto Grünbaum, Milen Yakimov, and Ignacio Zurrián. "Reflective prolate-spheroidal operators and the KP/KdV equations." Proceedings of the National Academy of Sciences 116, no. 37 (2019): 18310–15. http://dx.doi.org/10.1073/pnas.1906098116.

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Commuting integral and differential operators connect the topics of signal processing, random matrix theory, and integrable systems. Previously, the construction of such pairs was based on direct calculation and concerned concrete special cases, leaving behind important families such as the operators associated to the rational solutions of the Korteweg–de Vries (KdV) equation. We prove a general theorem that the integral operator associated to every wave function in the infinite-dimensional adelic GrassmannianGradof Wilson always reflects a differential operator (in the sense ofDefinition 1bel

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32

Sitnik, Sergei M., and Shakhobiddin T. Karimov. "Solution of the Goursat Problem for a Fourth-Order Hyperbolic Equation with Singular Coefficients by the Method of Transmutation Operators." Mathematics 11, no. 4 (2023): 951. http://dx.doi.org/10.3390/math11040951.

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In this paper, the method of transmutation operators is used to construct an exact solution of the Goursat problem for a fourth-order hyperbolic equation with a singular Bessel operator. We emphasise that in many other papers and monographs the fractional Erdélyi-Kober operators are used as integral operators, but our approach used them as transmutation operators with additional new properties and important applications. Specifically, it extends its properties and applications to singular differential equations, especially with Bessel-type operators. Using this operator, the problem under cons

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33

Gao, Feng, and Xiao-Jun Yang. "Fractional Maxwell fluid with fractional derivative without singular kernel." Thermal Science 20, suppl. 3 (2016): 871–77. http://dx.doi.org/10.2298/tsci16s3871g.

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In this paper we propose a new model for the fractional Maxwell fluid within fractional Caputo-Fabrizio derivative operator. We present the fractional Maxwell fluid in the differential form for the first time. The analytical results for the proposed model with the fractional Losada-Nieto integral operator are given to illustrate the efficiency of the fractional order operators to the line viscoelasticity.

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34

Rahman, Gauhar, Muhammad Samraiz, Cetin Yildiz, Thabet Abdeljawad, and Manar A. Alqudah. "New Generalized Results for Modified Atangana-Baleanu Fractional Derivatives and Integral Operators." European Journal of Pure and Applied Mathematics 18, no. 1 (2025): 5697. https://doi.org/10.29020/nybg.ejpam.v18i1.5697.

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In this current study, first we establish the modified power Atangana-Baleanu fractional derivative operators (MPC) in both the Caputo and Riemann-Liouville (MPRL) senses. Using the convolution approach and Laplace transformation, the so-called modified power fractional Caputo and R-L derivative operators with non-singular kernels are introduced. We establish theboundedness of the modified Caputo fractional derivative operator in this study. The fractional differential equations are solved with the generalised Laplace transform (GLT). In addition, the corresponding form of the fractional integ

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35

Dong, Jianfeng, Jizheng Huang, and Heping Liu. "Boundedness of Singular Integrals on Hardy Type Spaces Associated with Schrödinger Operators." Journal of Function Spaces 2015 (2015): 1–11. http://dx.doi.org/10.1155/2015/409215.

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LetL=-Δ+Vbe a Schrödinger operator onRn,n≥3, whereV≢0is a nonnegative potential belonging to the reverse Hölder classBn/2. The Hardy type spacesHLp, n/(n+δ) <p≤1,for someδ>0, are defined in terms of the maximal function with respect to the semigroup{e-tL}t>0. In this paper, we investigate the bounded properties of some singular integral operators related toL, such asLiγand∇L-1/2, on spacesHLp. We give the molecular characterization ofHLp, which is used to establish theHLp-boundedness of singular integrals.

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36

Keles, Seyda, and Mehriban N. Omarova. "Boundedness of vector-valued B-singular integral operators in Lebesgue spaces." Open Mathematics 15, no. 1 (2017): 987–1002. http://dx.doi.org/10.1515/math-2017-0081.

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Abstract We study the vector-valued B-singular integral operators associated with the Laplace-Bessel differential operator $$\triangle_{B}=\sum\limits_{k=1}^{n-1}\frac{\partial^{2}}{\partial x_{k}^{2}}+(\frac{\partial^{2}}{\partial x_{n}^{2}}+\frac{2v}{x_{n}}\frac{\partial}{\partial x_{n}}) , v>0.$$ We prove the boundedness of vector-valued B-singular integral operators A from $L_{p,v}(\mathbb{R}_{+}^{n}, H_{1}) \,{\rm to}\, L_{p,v}(\mathbb{R}_{+}^{n}, H_{2}),$ 1 < p < ∞, where H1 and H2 are separable Hilbert spaces.

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37

Al-Hasan, Abdelnaser J., and Dashan Fan. "Lp-Boundedness of a Singular Integral Operator." Canadian Mathematical Bulletin 41, no. 4 (1998): 404–12. http://dx.doi.org/10.4153/cmb-1998-054-5.

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AbstractLet b(t) be an L∞ function on R, Ω(y′) be an H1 function on the unit sphere satisfying the mean zero property (1) and Qm(t) be a real polynomial on R of degree m satisfying Qm(0) = 0. We prove that the singular integral operatoris bounded in Lp(Rn) for 1 < p < ∞, and the bound is independent of the coefficients of Qm(t).

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38

Fan, Dashan, and Yibiao Pan. "A singular integral operator with rough kernel." Proceedings of the American Mathematical Society 125, no. 12 (1997): 3695–703. http://dx.doi.org/10.1090/s0002-9939-97-04111-7.

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39

Ding, Shusen, and Bing Liu. "A singular integral of the composite operator." Applied Mathematics Letters 22, no. 8 (2009): 1271–75. http://dx.doi.org/10.1016/j.aml.2009.01.041.

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40

Alexander, G. Ramm. "New definition of a singular integral operator." Annals of Communications in Mathematics 6, no. 4 (2023): 220–24. https://doi.org/10.5281/zenodo.10445478.

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Let D be a connected bounded domain in R2 , S be its boundary which is closed, connected and smooth or S = (−∞, ∞). Let Φ(z) = 1 2πi R S f(s)ds s−z , f ∈ L1 (S), z = x+iy. The singular integral operator Af := 1 iπ R S f(s)ds s−t , t ∈ S, is defined in a new way. This definition simplifies the proof of the existence of Φ(t). Necessary and sufficient conditions are given for f ∈ L1 (S) to be boundary value of an analytic in D function. The Sokhotsky-Plemelj formulas are derived for f ∈ L1 (S)

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41

Chai, Yan, Yaoyao Han, and Kai Zhao. "Herz-Type Hardy Spaces Associated with Operators." Journal of Function Spaces 2018 (July 17, 2018): 1–10. http://dx.doi.org/10.1155/2018/1296837.

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Suppose L is a nonnegative, self-adjoint differential operator. In this paper, we introduce the Herz-type Hardy spaces associated with operator L. Then, similar to the atomic and molecular decompositions of classical Herz-type Hardy spaces and the Hardy space associated with operators, we prove the atomic and molecular decompositions of the Herz-type Hardy spaces associated with operator L. As applications, the boundedness of some singular integral operators on Herz-type Hardy spaces associated with operators is obtained.

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42

Keskin, Cansu, and Havva Nur Turkak. "Sharp $B$-Maximal Function Estimates and Boundedness for Some Integral Operators to the Inequalities." Journal of New Theory, no. 51 (June 30, 2025): 65–75. https://doi.org/10.53570/jnt.1696750.

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In this paper, we first establish the relation between $B$-maximal and sharp $B$-maximal functions generated by the generalized translation operator connected with the Laplace-Bessel differential operator. We then prove some sharp $B$-maximal function estimates and present an application using these sharp estimates to study singular integral operators. We finally obtain the boundedness of the Littlewood-Paley $g$-function related to the Laplace-Bessel differential operator on generalized $B$-Morrey spaces.

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43

Lyalinov, Mikhail Anatol'evich. "On eigenfunctions of the essential spectrum of the model problem for the Schrödinger operator with singular potential." Sbornik: Mathematics 214, no. 10 (2023): 1415–41. http://dx.doi.org/10.4213/sm9861e.

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We are concerned with generalized eigenfunctions of the continuous (essential) spectrum for the Schrödinger operator with singular $\delta$-potential that has support on the sides of an angle in the plane. Operators of this kind appear in quantum-mechanical models for quantum state destruction of two point-interacting quantum particles of which one is reflected by a potential barrier. We propose an approach capable of constructing integral representations for eigenfunctions in terms of the solution of a functional-difference equation with spectral parameter. Solutions of this equation are stu

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44

Anastassiou, George A. "Degree of Lp Approximation Using Activated Singular Integrals." Symmetry 16, no. 8 (2024): 1022. http://dx.doi.org/10.3390/sym16081022.

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In this article we present the Lp, p≥1, approximation properties of activated singular integral operators over the real line. We establish their approximation to the unit operator with rates. The kernels here come from neural network activation functions and we employ the related density functions. The derived inequalities use the high order Lp modulus of smoothness.

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45

Blaya, Ricardo Abreu, Juan Bory Reyes, and Boris Kats. "Cauchy integral and singular integral operator over closed Jordan curves." Monatshefte für Mathematik 176, no. 1 (2014): 1–15. http://dx.doi.org/10.1007/s00605-014-0656-9.

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Aykol, Canay, and Esra Kaya. "B−maximal operators, B−singular integral operators and B−Riesz potentials in variable exponent Lorentz spaces." Filomat 37, no. 17 (2023): 5765–74. http://dx.doi.org/10.2298/fil2317765a.

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In this paper, we prove the boundedness of B?maximal operator, B?singular integral operator and B?Riesz potential in the variable exponent Lorentz space Lp(?),q(?),?(Rn k,+). As a consequence of the boundedness of B?Riesz potentials in variable exponent Lorentz spaces, we also obtain that B?fractional maximal operators are bounded in Lp(?),q(?),?(Rn k,+).

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Liu, Lanzhe. "Weighted boundedness for toeplitz type operators associated to singular integral operator with non-smooth kernel." Filomat 30, no. 9 (2016): 2489–502. http://dx.doi.org/10.2298/fil1609489l.

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In this paper, the weighted boundedness of the Toeplitz type operator associated to some singular integral operator with non-smooth kernel on Lebesgue spaces are obtained. To do this, some weighted sharp maximal function inequalities for the operator are proved.

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Kang, Hyeonbae, and Jin Keun Seo. "L2-boundedness of the cauchy transform on smooth non-Lipschitz curves." Nagoya Mathematical Journal 130 (June 1993): 123–47. http://dx.doi.org/10.1017/s0027763000004463.

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Wang, Wei, and Jingshi Xu. "Precompact Sets, Boundedness, and Compactness of Commutators for Singular Integrals in Variable Morrey Spaces." Journal of Function Spaces 2017 (2017): 1–9. http://dx.doi.org/10.1155/2017/3764142.

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We give sufficient conditions for subsets to be precompact sets in variable Morrey spaces. Then we obtain the boundedness of the commutator generated by a singular integral operator and a BMO function on the variable Morrey spaces. Finally, we discuss the compactness of the commutator generated by a singular integral operator and a BMO function on the variable Morrey spaces.

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Anastassiou, George A. "Quantitative Uniform Approximation by Activated Singular Operators." Mathematics 12, no. 14 (2024): 2152. http://dx.doi.org/10.3390/math12142152.

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In this article, we study the approximation properties of activated singular integral operators over the real line. We establish their convergence to the unit operator with rates. The kernels here derive from neural network activation functions and their corresponding density functions. The estimates are mostly sharp, and they are pointwise and uniform. The derived inequalities involve the higher order modulus of smoothness.

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