与耦合分数傅立叶变换相关的 $$L^p$$ -Sobolev 空间和耦合势算子

IF 0.9 3区 数学 Q2 MATHEMATICS
Shraban Das, Kanailal Mahato, Sourav Das
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引用次数: 0

摘要

本文致力于研究耦合势算子 \(J_{s}^{\alpha , \beta }\) 和涉及耦合分数傅里叶变换(CFrFT)的相应 \(L^p\)-Sobolev 空间。引入了 Schwartz 型空间 \(\mathcal {S}_{\alpha , \beta }\) 。此外,还定义了伪微分算子,并导出了另一个积分表示。此外,还证明了与 CFrFT 相关联的伪微分算子是二维分数傅里叶变换的更广义化。得到了与 CFrFT 相关的伪微分算子的 \(L^p\) 规范不等式。耦合势算子 \(J_{s}^{\alpha , \beta }\) 被定义为与精确符号相关的伪微分算子。算子(J_{s}^{\alpha , \beta }\ )被扩展到分布空间。算子 \(J_{s}^{\alpha , \beta }\) 的 \(L^p\)-Sobolev 有界性结果得到了证明。引入了空间 \(H^{m,\alpha ,\beta }_{p}\) 和 \(\mathcal {H}^{m,\alpha ,\beta }_{p}/),作为应用,证明了某类微分方程的解属于这些空间。
本文章由计算机程序翻译,如有差异,请以英文原文为准。

$$L^p$$ -Sobolev spaces and coupled potential operators associated with coupled fractional Fourier transform

$$L^p$$ -Sobolev spaces and coupled potential operators associated with coupled fractional Fourier transform

This paper is devoted in investigations concerning the study of the coupled potential operator \(J_{s}^{\alpha , \beta }\) and corresponding \(L^p\)-Sobolev spaces involving coupled fractional Fourier transform (CFrFT). The Schwartz type space \(\mathcal {S}_{\alpha ,\beta }\) is introduced. Moreover, pseudo-differential operator is defined and derived one more integral representation. Further, it is shown that pseudo-differential operator associated with CFrFT is more generalization as of two dimensional fractional Fourier transform. The \(L^p\) norm inequality for the pseudo-differential operator associated with CFrFT is obtained. The coupled potential operator \(J_{s}^{\alpha , \beta }\) is defined as a pseudo-differential operator related with a precise symbol. The operator \(J_{s}^{\alpha , \beta }\) is extended to a space of distributions. An \(L^p\)-Sobolev boundedness result for the operator \(J_{s}^{\alpha , \beta }\) is shown. The spaces \(H^{m,\alpha ,\beta }_{p}\) and \(\mathcal {H}^{m,\alpha ,\beta }_{p}\) introduced and as an application, it is shown that the solutions of certain class of differential equations belong to these spaces.

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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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