$$\mathbb {Z}^n$$ 上的扩展索波列夫尺度

IF 0.9 3区 数学 Q2 MATHEMATICS
Ognjen Milatovic
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引用次数: 0

摘要

与 Mikhailets 和 Murach 对 \(\mathbb {R}^n\)上的 "扩展索波列夫尺度 "的定义类似,在晶格 \(\mathbb {Z}^n\)的背景下,我们定义了 "扩展索波列夫尺度"\(H^{\varphi }(\mathbb {Z}^n)\),其中 \(\varphi \)是一个在无穷远处为 RO 变化的函数。使用尺度 \(H^{\varphi }(\mathbb {Z}^n)\),我们就一对离散的索波列夫空间 \([H^{(s_0)}(\mathbb {Z}^n), H^{(s_1)}(\mathbb {Z}^n)]\),用 \(s_0<s_1\) 描述了所有作为插值空间的希尔伯特函数空间。我们利用这一插值结果得到了尺度 \(H^{\varphi }(\mathbb {Z}^n)\)背景下(离散)伪微分算子(PDOs)的映射性质和弗雷德霍尔性质。此外,从椭圆型的一阶正inite(离散)PDO A 开始,我们定义了 "扩展离散 A 尺度"(H^{\varphi }_{A}(\mathbb {Z}^n)),并证明它与尺度(H^{\varphi }(\mathbb {Z}^n))重合,直到规范等价。此外,我们还建立了尺度 \(H^{\varphi }(\mathbb {R}^n)\) 的其他几个性质的 \(\mathbb {Z}^n\)-analogues of several other properties of the scale \(H^{\varphi }(\mathbb {R}^n)\).
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Extended Sobolev scale on $$\mathbb {Z}^n$$

In analogy with the definition of “extended Sobolev scale" on \(\mathbb {R}^n\) by Mikhailets and Murach, working in the setting of the lattice \(\mathbb {Z}^n\), we define the “extended Sobolev scale" \(H^{\varphi }(\mathbb {Z}^n)\), where \(\varphi \) is a function which is RO-varying at infinity. Using the scale \(H^{\varphi }(\mathbb {Z}^n)\), we describe all Hilbert function-spaces that serve as interpolation spaces with respect to a pair of discrete Sobolev spaces \([H^{(s_0)}(\mathbb {Z}^n), H^{(s_1)}(\mathbb {Z}^n)]\), with \(s_0<s_1\). We use this interpolation result to obtain the mapping property and the Fredholmness property of (discrete) pseudo-differential operators (PDOs) in the context of the scale \(H^{\varphi }(\mathbb {Z}^n)\). Furthermore, starting from a first-order positive-definite (discrete) PDO A of elliptic type, we define the “extended discrete A-scale" \(H^{\varphi }_{A}(\mathbb {Z}^n)\) and show that it coincides, up to norm equivalence, with the scale \(H^{\varphi }(\mathbb {Z}^n)\). Additionally, we establish the \(\mathbb {Z}^n\)-analogues of several other properties of the scale \(H^{\varphi }(\mathbb {R}^n)\).

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来源期刊
CiteScore
2.20
自引率
9.10%
发文量
59
期刊介绍: The Journal of Pseudo-Differential Operators and Applications is a forum for high quality papers in the mathematics, applications and numerical analysis of pseudo-differential operators. Pseudo-differential operators are understood in a very broad sense embracing but not limited to harmonic analysis, functional analysis, operator theory and algebras, partial differential equations, geometry, mathematical physics and novel applications in engineering, geophysics and medical sciences.
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