{"title":"Ultradistributions on $$ {\\mathbb {R}}_{+}^{n}$$ and solvability and hypoellipticity through series expansions of ultradistributions","authors":"Stevan Pilipović, Ɖorđe Vučković","doi":"10.1007/s11868-024-00636-9","DOIUrl":null,"url":null,"abstract":"<p>In the first part we analyze space <span>\\({\\mathcal {G}}^*({\\mathbb {R}}^{n}_+)\\)</span> and its dual through Laguerre expansions when these spaces correspond to a general sequence <span>\\(\\{M_p\\}_{p\\in {\\mathbb {N}}_0}\\)</span>, where * is a common notation for the Beurling and Roumieu cases of spaces. In the second part we are solving equation of the form <span>\\(Lu=f,\\; L=\\sum _{j=1}^ka_jA_j^{h_j}+cE^{d}_y+bP(x,D_x),\\)</span> where <i>f</i> belongs to the tensor product of ultradistribution spaces over compact manifolds without boundaries as well as ultradistribution spaces on <span>\\({\\mathbb {R}}^n_+\\)</span> and <span>\\({\\mathbb {R}}^m\\)</span>; <span>\\(A_j, j=1,...,k\\)</span>, <span>\\(E_y\\)</span> and <span>\\(P(x,D_x)\\)</span> are operators whose eigenfunctions form orthonormal basis of corresponding <span>\\(L^2-\\)</span>space. The sequence space representation of solutions enable us to study the solvability and the hypoellipticity in the specified spaces of ultradistributions.\n</p>","PeriodicalId":48793,"journal":{"name":"Journal of Pseudo-Differential Operators and Applications","volume":null,"pages":null},"PeriodicalIF":0.9000,"publicationDate":"2024-08-26","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/s11868-024-00636-9","RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In the first part we analyze space \({\mathcal {G}}^*({\mathbb {R}}^{n}_+)\) and its dual through Laguerre expansions when these spaces correspond to a general sequence \(\{M_p\}_{p\in {\mathbb {N}}_0}\), where * is a common notation for the Beurling and Roumieu cases of spaces. In the second part we are solving equation of the form \(Lu=f,\; L=\sum _{j=1}^ka_jA_j^{h_j}+cE^{d}_y+bP(x,D_x),\) where f belongs to the tensor product of ultradistribution spaces over compact manifolds without boundaries as well as ultradistribution spaces on \({\mathbb {R}}^n_+\) and \({\mathbb {R}}^m\); \(A_j, j=1,...,k\), \(E_y\) and \(P(x,D_x)\) are operators whose eigenfunctions form orthonormal basis of corresponding \(L^2-\)space. The sequence space representation of solutions enable us to study the solvability and the hypoellipticity in the specified spaces of ultradistributions.
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.