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Journal articles on the topic 'Morrey spaces,integral operators,partial differential equations'
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1
Ragusa, Maria Alessandra, and Veli Shakhmurov. "Embedding of vector-valued Morrey spaces and separable differential operators." Bulletin of Mathematical Sciences 09, no. 02 (August 2019): 1950013. http://dx.doi.org/10.1142/s1664360719500139.
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The paper is the first part of a program devoted to the study of the behavior of operator-valued multipliers in Morrey spaces. Embedding theorems and uniform separability properties involving [Formula: see text]-valued Morrey spaces are proved. As a consequence, maximal regularity for solutions of infinite systems of anisotropic elliptic partial differential equations are established.
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Wang, Jiajia, Zunwei Fu, Shaoguang Shi, and Ling Mi. "Operator Inequalities of Morrey Spaces Associated with Karamata Regular Variation." Journal of Function Spaces 2017 (2017): 1–7. http://dx.doi.org/10.1155/2017/4618197.
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Karamata regular variation is a basic tool in stochastic process and the boundary blow-up problems for partial differential equations (PDEs). Morrey space is closely related to study of the regularity of solutions to elliptic PDEs. The aim of this paper is trying to bring together these two areas and this paper is intended as an attempt at motivating some further research on these areas. A version of Morrey space associated with Karamata regular variation is introduced. As application, some estimates of operators, especially one-sided operators, on these spaces are considered.
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Hasanov, Javanshir J., Rabil Ayazoglu, and Simten Bayrakci. "B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials on B-Morrey spaces." Open Mathematics 18, no. 1 (July 10, 2020): 715–30. http://dx.doi.org/10.1515/math-2020-0033.
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Abstract In this article, we consider the Laplace-Bessel differential operator {\Delta }_{{B}_{k,n}}=\mathop{\sum }\limits_{i=1}^{k}\left(\frac{{\partial }^{2}}{\partial {x}_{i}^{2}}+\frac{{\gamma }_{i}}{{x}_{i}}\frac{\partial }{\partial {x}_{i}}\right)+\mathop{\sum }\limits_{i=k+1}^{n}\frac{{\partial }^{2}}{\partial {x}_{i}^{2}},{\gamma }_{1}\gt 0,\ldots ,{\gamma }_{k}\gt 0. Furthermore, we define B-maximal commutators, commutators of B-singular integral operators and B-Riesz potentials associated with the Laplace-Bessel differential operator. Moreover, we also obtain the boundedness of the B-maximal commutator {M}_{b,\gamma } and the commutator {[}b,{A}_{\gamma }] of the B-singular integral operator and Hardy-Littlewood-Sobolev-type theorem for the commutator {[}b,{I}_{\alpha ,\gamma }] of the B-Riesz potential on B-Morrey spaces {L}_{p,\lambda ,\gamma } , when b\in {\text{BMO}}_{\gamma } .
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Fedorov, V. E., A. D. Godova, and B. T. Kien. "Integro-differential equations with bounded operators in Banach spaces." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 106, no. 2 (June 30, 2022): 93–107. http://dx.doi.org/10.31489/2022m2/93-107.
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The paper investigates integro-differential equations in Banach spaces with operators, which are a composition of convolution and differentiation operators. Depending on the order of action of these two operators, we talk about integro-differential operators of the Riemann—Liouville type, when the convolution operator acts first, and integro-differential operators of the Gerasimov type otherwise. Special cases of the operators under consideration are the fractional derivatives of Riemann—Liouville and Gerasimov, respectively. The classes of integro-differential operators under study also include those in which the convolution has an integral kernel without singularities. The conditions of the unique solvability of the Cauchy type problem for a linear integro-differential equation of the Riemann—Liouville type and the Cauchy problem for a linear integrodifferential equation of the Gerasimov type with a bounded operator at the unknown function are obtained. These results are used in the study of similar equations with a degenerate operator at an integro-differential operator under the condition of relative boundedness of the pair of operators from the equation. Abstract results are applied to the study of initial boundary value problems for partial differential equations with an integro-differential operator, the convolution in which is given by the Mittag-Leffler function multiplied by a power function.
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Nguyen, Thieu Huy, Vu Thi Ngoc Ha, and Trinh Xuan Yen. "Admissible integral manifolds for partial neutral functional-differential equations." Ukrains’kyi Matematychnyi Zhurnal 74, no. 10 (November 27, 2022): 1364–87. http://dx.doi.org/10.37863/umzh.v74i10.6257.
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UDC 517.9 We prove the existence and attraction property for admissible invariant unstable and center-unstable manifolds of admissible classes of solutions to the partial neutral functional-differential equation in Banach space X of the form & ∂ ∂ t F u t = A ( t ) F u t + f ( t , u t ) , t ≥ s , t , s ∈ ℝ , & u s = ϕ ∈ 𝒞 : = C ( [ - r ,0 ] , X ) under the conditions that the family of linear partial differential operators ( A ( t ) ) t ∈ ℝ generates the evolution family ( U ( t , s ) ) t ≥ s with an exponential dichotomy on the whole line ℝ ; the difference operator F : 𝒞 → X is bounded and linear, and the nonlinear delay operator f satisfies the φ -Lipschitz condition, i.e., ‖ f ( t , ϕ ) - f ( t , ψ ) ‖ ≤ φ ( t ) ‖ ϕ - ψ ‖ 𝒞 for ϕ , ψ ∈ 𝒞 , where φ ( ⋅ ) belongs to an admissible function space defined on ℝ . We also prove that an unstable manifold of the admissible class attracts all other solutions with exponential rates. Our main method is based on the Lyapunov – Perron equation combined with the admissibility of function spaces. We apply our results to the finite-delayed heat equation for a material with memory.
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CHKADUA, O., S. E. MIKHAILOV, and D. NATROSHVILI. "ANALYSIS OF DIRECT SEGREGATED BOUNDARY-DOMAIN INTEGRAL EQUATIONS FOR VARIABLE-COEFFICIENT MIXED BVPs IN EXTERIOR DOMAINS." Analysis and Applications 11, no. 04 (June 18, 2013): 1350006. http://dx.doi.org/10.1142/s0219530513500061.
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Direct segregated systems of boundary-domain integral equations are formulated for the mixed (Dirichlet–Neumann) boundary value problems for a scalar second-order divergent elliptic partial differential equation with a variable coefficient in an exterior three-dimensional domain. The boundary-domain integral equation system equivalence to the original boundary value problems and the Fredholm properties and invertibility of the corresponding boundary-domain integral operators are analyzed in weighted Sobolev spaces suitable for infinite domains. This analysis is based on the corresponding properties of the BVPs in weighted Sobolev spaces that are proved as well.
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Providas, Efthimios, and Ioannis Nestorios Parasidis. "A Procedure for Factoring and Solving Nonlocal Boundary Value Problems for a Type of Linear Integro-Differential Equations." Algorithms 14, no. 12 (November 28, 2021): 346. http://dx.doi.org/10.3390/a14120346.
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The aim of this article is to present a procedure for the factorization and exact solution of boundary value problems for a class of n-th order linear Fredholm integro-differential equations with multipoint and integral boundary conditions. We use the theory of the extensions of linear operators in Banach spaces and establish conditions for the decomposition of the integro-differential operator into two lower-order integro-differential operators. We also create solvability criteria and derive the unique solution in closed form. Two example problems for an ordinary and a partial intergro-differential equation respectively are solved.
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Chen, Jiao, Wei Ding, and Guozhen Lu. "Boundedness of multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces." Forum Mathematicum 32, no. 4 (July 1, 2020): 919–36. http://dx.doi.org/10.1515/forum-2019-0319.
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AbstractAfter the celebrated work of L. Hörmander on the one-parameter pseudo-differential operators, the applications of pseudo-differential operators have played an important role in partial differential equations, geometric analysis, harmonic analysis, theory of several complex variables and other branches of modern analysis. For instance, they are used to construct parametrices and establish the regularity of solutions to PDEs such as the {\overline{\partial}} problem. The study of Fourier multipliers, pseudo-differential operators and Fourier integral operators has stimulated further such applications. It is well known that the one-parameter pseudo-differential operators are {L^{p}({\mathbb{R}^{n}})} bounded for {1<p<\infty}, but only bounded on local Hardy spaces {h^{p}({\mathbb{R}^{n}})} introduced by Goldberg in [D. Goldberg, A local version of real Hardy spaces, Duke Math. J. 46 1979, 1, 27–42] for {0<p\leq 1}. Though much work has been done on the {L^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} boundedness for {1<p<\infty} and Hardy {H^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} boundedness for {0<p\leq 1} for multi-parameter Fourier multipliers and singular integral operators, not much has been done yet for the boundedness of multi-parameter pseudo-differential operators in the range of {0<p\leq 1}. The main purpose of this paper is to establish the boundedness of multi-parameter pseudo-differential operators on multi-parameter local Hardy spaces {h^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} for {0<p\leq 1} recently introduced by Ding, Lu and Zhu in [W. Ding, G. Lu and Y. Zhu, Multi-parameter local Hardy spaces, Nonlinear Anal. 184 2019, 352–380].
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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 integral operators with continuous coefficients over a bounded domain. Such operators are used in many problems in the theory of partial differential equations. In this regard, it is of interest to establish criteria for the considered operators to be Noetherian in the form of explicit conditions on their coefficients. The paper establishes effective necessary and sufficient conditions for two-dimensional singular integral operators to be Noetherian in Lebesgue spaces L_p (D) (considered over the field of real numbers), 1<p<∞, and formulas for calculating indices are given. The method developed by R.V. Duduchava [Duduchava R. On multidimensional singular integral operators. I: The half-space case; II: The case of compact manifolds // J. Operator Theory, 1984, v. 11, 41–76 (I); 199–214 (II)]. In this case, the study of the Noetherian properties of operators is reduced to the factorization of the corresponding matrix-functions and finding their partial indices.
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Sitnik, Sergey M., Vladimir E. Fedorov, Nikolay V. Filin, and Viktor A. Polunin. "On the Solvability of Equations with a Distributed Fractional Derivative Given by the Stieltjes Integral." Mathematics 10, no. 16 (August 18, 2022): 2979. http://dx.doi.org/10.3390/math10162979.
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Linear equations in Banach spaces with a distributed fractional derivative given by the Stieltjes integral and with a closed operator A in the right-hand side are considered. Unlike the previously studied classes of equations with distributed derivatives, such kinds of equations may contain a continuous and a discrete part of the integral, i.e., a standard integral of the fractional derivative with respect to its order and a linear combination of fractional derivatives with different orders. Resolving families of operators for such equations are introduced into consideration, and their properties are studied. In terms of the resolvent of the operator A, necessary and sufficient conditions are obtained for the existence of analytic resolving families of the equation under consideration. A perturbation theorem for such a class of operators is proved, and the Cauchy problem for the inhomogeneous equation with a distributed fractional derivative is studied. Abstract results are applied for the research of the unique solvability of initial boundary value problems for partial differential equations with a distributed derivative with respect to time.
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Jendoubi, Ch. "On the theory of integral manifolds for some delayed partial differential equations with nondense domain}." Ukrains’kyi Matematychnyi Zhurnal 72, no. 6 (June 17, 2020): 776–89. http://dx.doi.org/10.37863/umzh.v72i6.6020.
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UDC 517.9 Integral manifolds are very useful in studying dynamics of nonlinear evolution equations. In this paper, we consider the nondensely-defined partial differential equation ⅆ u ⅆ t = ( A + B ( t ) ) u ( t ) + f ( t , u t ) , t ∈ R , ( 1 ) where ( A , D ( A ) ) satisfies the Hille – Yosida condition, ( B ( t ) ) t ∈ R is a family of operators in ℒ ( D ( A ) ¯ , X ) satisfying some measurability and boundedness conditions, and the nonlinear forcing term f satisfies ‖ f ( t , ϕ ) - f ( t , ψ ) ‖ ≤ φ ( t ) ‖ ϕ - ψ ‖ 𝒞 , here, φ belongs to some admissible spaces and ϕ , ψ ∈ 𝒞 : = C ( [ - r ,0 ] , X ) . We first present an exponential convergence result between the stable manifold and every mild solution of (1). Then we prove the existence of center-unstable manifolds for such solutions.Our main methods are invoked by the extrapolation theory and the Lyapunov – Perron method based on the admissible functions properties.
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Riedle, Markus. "Stochastic integration with respect to cylindrical Lévy processes in Hilbert spaces: An L2 approach." Infinite Dimensional Analysis, Quantum Probability and Related Topics 17, no. 01 (March 2014): 1450008. http://dx.doi.org/10.1142/s0219025714500088.
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In this work stochastic integration with respect to cylindrical Lévy processes with weak second moments is introduced. It is well known that a deterministic Hilbert–Schmidt operator radonifies a cylindrical random variable, i.e. it maps a cylindrical random variable to a classical Hilbert space valued random variable. Our approach is based on a generalisation of this result to the radonification of the cylindrical increments of a cylindrical Lévy process by random Hilbert–Schmidt operators. This generalisation enables us to introduce a Hilbert space valued random variable as the stochastic integral of a predictable stochastic process with respect to a cylindrical Lévy process. We finish this work by deriving an Itô isometry and by considering shortly stochastic partial differential equations driven by cylindrical Lévy processes.
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Kumar, Dr Amresh, and Dr Ram Kishore Singh. "A Role of Hilbert Space in Sampled Data to Reduced Error Accumulation by Over Sampling Then the Computational and Storage Cost Increase Using Signal Processing On 2-Sphere Dimension”." International Journal of Scientific Research and Management 8, no. 05 (May 15, 2020): 386–96. http://dx.doi.org/10.18535/ijsrm/v8i05.ec02.
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Hilbert Space has wide usefulness in signal processing research. It is pitched at a graduate student level, but relies only on undergraduate background material. The needs and concerns of the researchers In engineering differ from those of the pure science. It is difficult to put the finger on what distinguishes the engineering approach that we have taken. In the end, if a potential use emerges from any result, however abstract, then an engineer would tend to attach greater value to that result. This may serve to distinguish the emphasis given by a mathematician who may be interested in the proof of a fundamental concept that links deeply with other areas of mathematics or is a part of a long-standing human intellectual endeavor not that engineering, in comparison, concerns less intellectual pursuits. The theory of Hilbert spaces was initiated by David Hilbert (1862-1943), in the early of twentieth century in the context of the study of "Integral equations". Integral equations are a natural complement to differential equations and arise, for example, in the study of existence and uniqueness of function which are solution of partial differential equations such as wave equation. Convolution and Fourier transform equation also belongs to this class. Geometric intuition plays an important role in many aspects of Hilbert space theory. Exact analogs of the Pythagorean theorem and parallelogram law hold in Hilbert space. At a deeper level, perpendicular projection onto a subspace that is the analog of "dropping the altitude" of a triangle plays a significant role in optimization problem and other aspects of the theory. An element of Hilbert space can be uniquely specified by its co-ordinates with respect to a set of coordinate axes that is an orthonormal basis, in analogy with Cartesian coordinates in the plane. When that set of axes is countably infinite, this means that the Hilbert space can also usefully be thought in terms of infinite sequences that are square summable. Linear operators on Hilbert space are ply transformations that stretch the space by different factors in mutually perpendicular directions in a sense that is made precise by the study of their spectral theory. In brief Hilbert spaces are the means by which the ordinary experience of Euclidean concepts can be extended meaningfully into idealized constructions of more complex abstract mathematics. However, in brief, the usual application demand for Hilbert spaces are integral and differential equations, generalized functions and partial differential equations, quantum mechanics, orthogonal polynomials and functions, optimization and approximation theory. In signal processing which is the main objective of the present thesis and engineering. Wavelets and optimization problem that has been dealt in the present thesis, optimal control, filtering and equalization, signal processing on 2- sphere, Shannon information theory, communication theory, linear and non-linear theory and many more is application domain of the Hilbert space.
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Trofymenko, Olga, and Yuliia Perevierzieva. "Mean value theorems for polyharmonic functions." Proceedings of the Institute of Applied Mathematics and Mechanics NAS of Ukraine 35 (January 28, 2022): 173–78. http://dx.doi.org/10.37069/10.37069/1683-4720-2021-35-13.
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The problem of characterization of polyharmonic functions by mean value expressions is analyzed. A sufficient condition for the biharmonic function is formulated and proved in the paper. Mean value theorems and their operators have various applications in function theory (approximation of functions, description of functional spaces) and in the qualitative theory of linear differential equations with partial derivatives (boundary properties, elimination of features, differential properties of solutions, etc.). The main properties of polyharmonic functions, in particular, properties with mean value are investigated in the work. The results of L. Salcman and V. Volchkov, which generalize the classical mean value theorems, are analyzed. These results also reveal deep connections between complex analysis, the theory of linear differential equations with partial derivatives, harmonic analysis, integral geometry and the theory of special functions. The new directions of research in the theory of polyharmonic functions, differential equations, as well as in computational mathematics are established with the mean value theorems (algorithms "walk on spheres"\ , application of the Monte Carlo method). Particular attention is paid to the case of biharmonic functions. It is found that when performing the equality with the mean value on the circle from the set on the complex plane, the function of class $C^2$ on the specified set satisfies the Laplace equation of the second order, i.e. the function is polyharmonic of the second order. By using a smooth function with a compact support, the required function is obtained as a weak solution of the corresponding Laplace equation. In the future, this result can be used to establish a sufficient condition of biharmonicity of a twice continuously differentiated function on a plane that satisfies the above-mentioned relation only for a pair of positive values of $r_1$, $r_2$. Thus, it is possible to formulate a new theorem on two radii.
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Castro, Alejandro J., Anders Israelsson, and Wolfgang Staubach. "Regularity of Fourier integral operators with amplitudes in general Hörmander classes." Analysis and Mathematical Physics 11, no. 3 (June 5, 2021). http://dx.doi.org/10.1007/s13324-021-00552-x.
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AbstractWe prove the global $$L^p$$ L p -boundedness of Fourier integral operators that model the parametrices for hyperbolic partial differential equations, with amplitudes in classical Hörmander classes $$S^{m}_{\rho , \delta }(\mathbb {R}^n)$$ S ρ , δ m ( R n ) for parameters $$0\le \rho \le 1$$ 0 ≤ ρ ≤ 1 , $$0\le \delta <1$$ 0 ≤ δ < 1 . We also consider the regularity of operators with amplitudes in the exotic class $$S^{m}_{0, \delta }(\mathbb {R}^n)$$ S 0 , δ m ( R n ) , $$0\le \delta < 1$$ 0 ≤ δ < 1 and the forbidden class $$S^{m}_{\rho , 1}(\mathbb {R}^n)$$ S ρ , 1 m ( R n ) , $$0\le \rho \le 1.$$ 0 ≤ ρ ≤ 1 . Furthermore we show that despite the failure of the $$L^2$$ L 2 -boundedness of operators with amplitudes in the forbidden class $$S^{0}_{1, 1}(\mathbb {R}^n)$$ S 1 , 1 0 ( R n ) , the operators in question are bounded on Sobolev spaces $$H^s(\mathbb {R}^n)$$ H s ( R n ) with $$s>0.$$ s > 0 . This result extends those of Y. Meyer and E. M. Stein to the setting of Fourier integral operators.