Pseudo-Differential Operators on $${\mathbb {Z}}^n$$ Z n with Applications to Discrete Fractional Integral Operators

Cardona, Duván

doi:10.1007/s41980-018-00195-y

Cited by 9 publications

(1 citation statement)

References 27 publications

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“…Ruzhansky and Turunen [57], and Cardona, Messiouene and Senoussaoui [10] studied some of mapping properties for the periodic Fourier integral operators. Pseudo-differential operators on Z n (discrete pseudo-differential operators) were introduced by Molahajloo in [47], and some of their properties were developed in the last few years, see [8,16,50,51,52,53,55]. However, Botchway, Kibiti, and Ruzhansky in their recent fundamental work [5] investigated the discrete pseudo-differential calculus and its applications to difference equations.…”

Section: Introductionmentioning

confidence: 99%

$$L^p$$-Boundedness and $$L^p$$-Nuclearity of Multilinear Pseudo-differential Operators on $${\mathbb {Z}}^n$$ and the Torus $${\mathbb {T}}^n$$

Cardona

Kumar

2019

J Fourier Anal Appl

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In this article, we begin a systematic study of the boundedness and the nuclearity properties of multilinear periodic pseudo-differential operators and multilinear discrete pseudo-differential operators on L p -spaces. First, we prove analogues of known multilinear Fourier multipliers theorems (proved by Coifman and Meyer, Grafakos, Tomita, Torres, Kenig, Stein, Fujita, Tao, etc.) in the context of periodic and discrete multilinear pseudo-differential operators. For this, we use the periodic analysis of pseudo-differential operators developed by Ruzhansky and Turunen. Later, we investigate the s-nuclearity, 0 < s ≤ 1, of periodic and discrete pseudo-differential operators. To accomplish this, we classify those s-nuclear multilinear integral operators on arbitrary Lebesgue spaces defined on σ-finite measures spaces. We also study similar properties for periodic Fourier integral operators. Finally, we present some applications of our study to deduce the periodic Kato-Ponce inequality and to examine the s-nuclearity of multilinear Bessel potentials as well as the s-nuclearity of periodic Fourier integral operators admitting suitable types of singularities.MSC 2010. Primary 58J40.; Secondary 47B10, 47G30, 35S30. ContentsDate: Received: xxxxxx; Revised: yyyyyy; Accepted: zzzzzz. 2010 Mathematics Subject Classification. 58J40.; Secondary 47B10, 47G30, 35S30 .

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Section: Introductionmentioning

confidence: 99%

$$L^p$$-Boundedness and $$L^p$$-Nuclearity of Multilinear Pseudo-differential Operators on $${\mathbb {Z}}^n$$ and the Torus $${\mathbb {T}}^n$$

Cardona

Kumar

2019

J Fourier Anal Appl

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The nuclear trace of periodic vector‐valued pseudo‐differential operators with applications to index theory

Cardona

Kumar

2021

Mathematische Nachrichten

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In this paper we investigate the nuclear trace of vector‐valued Fourier multipliers on the torus and its applications to the index theory of periodic pseudo‐differential operators. First we characterise the nuclearity of pseudo‐differential operators acting on Bochner integrable functions. In this regards, we consider the periodic and the discrete cases. We go on to address the problem of finding sharp sufficient conditions for the nuclearity of vector‐valued Fourier multipliers on the torus. We end our investigation with two index formulae. First, we express the index of a vector‐valued Fourier multiplier in terms of its operator‐valued symbol and then we use this formula for expressing the index of certain elliptic operators belonging to periodic Hörmander classes.

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Weighted periodic and discrete pseudo-differential Operators

Dasgupta,

Mohan,

Mondal

2024

Monatsh Math

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Pseudo-Differential Operators on $${\mathbb {Z}}^n$$ Z n with Applications to Discrete Fractional Integral Operators

Cited by 9 publications

References 27 publications

$$L^p$$-Boundedness and $$L^p$$-Nuclearity of Multilinear Pseudo-differential Operators on $${\mathbb {Z}}^n$$ and the Torus $${\mathbb {T}}^n$$

$$L^p$$-Boundedness and $$L^p$$-Nuclearity of Multilinear Pseudo-differential Operators on $${\mathbb {Z}}^n$$ and the Torus $${\mathbb {T}}^n$$

The nuclear trace of periodic vector‐valued pseudo‐differential operators with applications to index theory

Weighted periodic and discrete pseudo-differential Operators

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