Positivity Conditions for Operators with Difference Kernels in Reflexive Spaces
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POSITIVITY CONDITIONS FOR OPERATORS WITH DIFFERENCE KERNELS IN REFLEXIVE SPACES S. N. Askhabov
UDC 517.983
Abstract. Using methods of the theory of discrete and intFourier transforms, we obtain necessary and sufficient conditions for of the positivity of linear discrete, integral, and integro-differential operators with difference kernels in the spaces p and Lp for 1 < p < ∞ and present examples illustrating the results obtained. Keywords and phrases: positive operator, convolution operator, generalized operator of potential type, singular operator, integro-differential operator. AMS Subject Classification: 47G10, 47G20
CONTENTS 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2. Discrete, Integral, and Integro-Differential Convolution Operators . 3. Singular Integral and Integro-Differential Operators in Lp () . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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Introduction
The notion of a positive operator is closely related to the notion of a positive-definite (in the Bochner sense) function, which plays a key role in harmonic analysis and other branches of modern mathematics (see [7, p. 176]). The positivity property of discrete, integral, and integro-differential operators has many applications in the study of corresponding linear and nonlinear equations in various Banach spaces (see, e.g., the monographs [2, 3, 9, 10, 19, 20] and the references therein). In this paper, we study linear discrete, integral, and integro-differential operators with difference kernels in real spaces p and Lp (the cases of complex and weighted spaces are specially indicated). We find necessary and sufficient conditions for the positivity of such operators and give examples that illustrate the results obtained. We recall some definitions and introduce the necessary notation. Let X be a real Banach space and X ∗ be the dual space. We denote by y, x the value of a linear continuous functional y ∈ X ∗ on an element x ∈ X. If X is a Hilbert space, then y, x coincides with the scalar product (y, x). A linear operator A acting from X to X ∗ is said to be positive in the space X if for all x ∈ X, the inequality Ax, x ≥ 0 is fulfilled. As usual, by R, Z, and N we denote the sets of all real, integer, and natural numbers, respectively, and by p = p/(p − 1) the number conjugate to p.
Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 149, Proceedings of the International Conference “Actual Problems of Applied Mathematics and Physics,” Kabardino-Balkaria, Nalchik, May 17–21, 2017, 2018. c 2020 Springer Science+Business Media, LLC 1072–3374/20/2505–0717
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Discrete, Integral, and Integro-Differential Convolution Operators
2.1. Discrete convolution operators in p . We denote by p , p ≥ 1, the Banach space consisting of all real number sequences ϕ = {ϕk }∞ k=−∞ such that ∞
|ϕ
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