{"title":"与耦合分数傅立叶变换相关的 $$L^p$$ -Sobolev 空间和耦合势算子","authors":"Shraban Das, Kanailal Mahato, Sourav Das","doi":"10.1007/s11868-024-00642-x","DOIUrl":null,"url":null,"abstract":"<p>This paper is devoted in investigations concerning the study of the coupled potential operator <span>\\(J_{s}^{\\alpha , \\beta }\\)</span> and corresponding <span>\\(L^p\\)</span>-Sobolev spaces involving coupled fractional Fourier transform (CFrFT). The Schwartz type space <span>\\(\\mathcal {S}_{\\alpha ,\\beta }\\)</span> 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 <span>\\(L^p\\)</span> norm inequality for the pseudo-differential operator associated with CFrFT is obtained. The coupled potential operator <span>\\(J_{s}^{\\alpha , \\beta }\\)</span> is defined as a pseudo-differential operator related with a precise symbol. The operator <span>\\(J_{s}^{\\alpha , \\beta }\\)</span> is extended to a space of distributions. An <span>\\(L^p\\)</span>-Sobolev boundedness result for the operator <span>\\(J_{s}^{\\alpha , \\beta }\\)</span> is shown. The spaces <span>\\(H^{m,\\alpha ,\\beta }_{p}\\)</span> and <span>\\(\\mathcal {H}^{m,\\alpha ,\\beta }_{p}\\)</span> introduced and as an application, it is shown that the solutions of certain class of differential equations belong to these spaces.</p>","PeriodicalId":48793,"journal":{"name":"Journal of Pseudo-Differential Operators and Applications","volume":"8 1","pages":""},"PeriodicalIF":0.9000,"publicationDate":"2024-09-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"$$L^p$$ -Sobolev spaces and coupled potential operators associated with coupled fractional Fourier transform\",\"authors\":\"Shraban Das, Kanailal Mahato, Sourav Das\",\"doi\":\"10.1007/s11868-024-00642-x\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>This paper is devoted in investigations concerning the study of the coupled potential operator <span>\\\\(J_{s}^{\\\\alpha , \\\\beta }\\\\)</span> and corresponding <span>\\\\(L^p\\\\)</span>-Sobolev spaces involving coupled fractional Fourier transform (CFrFT). The Schwartz type space <span>\\\\(\\\\mathcal {S}_{\\\\alpha ,\\\\beta }\\\\)</span> 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 <span>\\\\(L^p\\\\)</span> norm inequality for the pseudo-differential operator associated with CFrFT is obtained. The coupled potential operator <span>\\\\(J_{s}^{\\\\alpha , \\\\beta }\\\\)</span> is defined as a pseudo-differential operator related with a precise symbol. The operator <span>\\\\(J_{s}^{\\\\alpha , \\\\beta }\\\\)</span> is extended to a space of distributions. An <span>\\\\(L^p\\\\)</span>-Sobolev boundedness result for the operator <span>\\\\(J_{s}^{\\\\alpha , \\\\beta }\\\\)</span> is shown. The spaces <span>\\\\(H^{m,\\\\alpha ,\\\\beta }_{p}\\\\)</span> and <span>\\\\(\\\\mathcal {H}^{m,\\\\alpha ,\\\\beta }_{p}\\\\)</span> introduced and as an application, it is shown that the solutions of certain class of differential equations belong to these spaces.</p>\",\"PeriodicalId\":48793,\"journal\":{\"name\":\"Journal of Pseudo-Differential Operators and Applications\",\"volume\":\"8 1\",\"pages\":\"\"},\"PeriodicalIF\":0.9000,\"publicationDate\":\"2024-09-12\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Pseudo-Differential Operators and Applications\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://doi.org/10.1007/s11868-024-00642-x\",\"RegionNum\":3,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Pseudo-Differential Operators and Applications","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s11868-024-00642-x","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
$$L^p$$ -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.
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.