Pub Date : 2023-12-08DOI: 10.1007/s11868-023-00575-x
Abimbola Abolarinwa, Michael Ruzhansky
This paper is primarily devoted to a class of interpolation inequalities of Hardy and Rellich types on the Heisenberg group (mathbb {H}^n). Consequently, several weighted Hardy type, Heisenberg–Pauli–Weyl uncertainty principle and Hardy–Rellich type inqualities are established on (mathbb {H}^n). Moreover, new weighted Sobolev type embeddings are derived. Finally, an integral inequality for vector fields in a domain of the Heisenberg group is obtained, leading to several specific weighted Hardy type inequalities by making careful choices of vector fields.
{"title":"Generalised Hardy type and Rellich type inequalities on the Heisenberg group","authors":"Abimbola Abolarinwa, Michael Ruzhansky","doi":"10.1007/s11868-023-00575-x","DOIUrl":"https://doi.org/10.1007/s11868-023-00575-x","url":null,"abstract":"<p>This paper is primarily devoted to a class of interpolation inequalities of Hardy and Rellich types on the Heisenberg group <span>(mathbb {H}^n)</span>. Consequently, several weighted Hardy type, Heisenberg–Pauli–Weyl uncertainty principle and Hardy–Rellich type inqualities are established on <span>(mathbb {H}^n)</span>. Moreover, new weighted Sobolev type embeddings are derived. Finally, an integral inequality for vector fields in a domain of the Heisenberg group is obtained, leading to several specific weighted Hardy type inequalities by making careful choices of vector fields.</p>","PeriodicalId":48793,"journal":{"name":"Journal of Pseudo-Differential Operators and Applications","volume":"14 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138556449","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-12-08DOI: 10.1007/s11868-023-00574-y
Khrystyna Mamalyha, Mykhailo Osypchuk
The aim of this paper is a behavior investigation of pseudo-gradients with respect to the spatial variable of a single-layer potential if the spatial point tends to some point in the carrier surface of the potential. The potentials connect to the generator of an isotropic (alpha )-stable stochastic process with the power (alpha in (1,2]). This generator is the fractional Laplacian of the order (alpha ). Pseudo-gradient or fractional gradient is the pseudo-differential operator of the gradient type. Its order (beta ) is a positive number less than (alpha ). The jump theorem is known in the case of (beta =alpha -1). We present here the corresponding results in the cases of (beta <alpha -1) and (beta >alpha -1). In the first case there are no jumps, and in the second case there are no finite limits of the fractional gradients of the single-layer potential, when the spatial argument tends to some point placed on the potential carrier.
{"title":"On single-layer potentials, pseudo-gradients and a jump theorem for an isotropic $$alpha $$ -stable stochastic process","authors":"Khrystyna Mamalyha, Mykhailo Osypchuk","doi":"10.1007/s11868-023-00574-y","DOIUrl":"https://doi.org/10.1007/s11868-023-00574-y","url":null,"abstract":"<p>The aim of this paper is a behavior investigation of pseudo-gradients with respect to the spatial variable of a single-layer potential if the spatial point tends to some point in the carrier surface of the potential. The potentials connect to the generator of an isotropic <span>(alpha )</span>-stable stochastic process with the power <span>(alpha in (1,2])</span>. This generator is the fractional Laplacian of the order <span>(alpha )</span>. Pseudo-gradient or fractional gradient is the pseudo-differential operator of the gradient type. Its order <span>(beta )</span> is a positive number less than <span>(alpha )</span>. The jump theorem is known in the case of <span>(beta =alpha -1)</span>. We present here the corresponding results in the cases of <span>(beta <alpha -1)</span> and <span>(beta >alpha -1)</span>. In the first case there are no jumps, and in the second case there are no finite limits of the fractional gradients of the single-layer potential, when the spatial argument tends to some point placed on the potential carrier.\u0000</p>","PeriodicalId":48793,"journal":{"name":"Journal of Pseudo-Differential Operators and Applications","volume":"28 1","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-12-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138556452","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-27DOI: 10.1007/s11868-023-00570-2
Sudip Deb, Anupam Das
This article contains modified version of fixed point theorems with the help of freshly made contraction operators and the solvability of a fractional hybrid differential equation involving Caputo fractional derivative by using modified Darbo’s fixed point theorem in the space of continuous tempered functions. The fundamental tool used in the solvability of the fractional hybrid differential equation is the measure of noncompactness approach. Our results indicate the importance of fixed point theorem for finding solutions of the fractional hybrid differential equation and improve some known results, which have wider applications as well. Also, there are two suitable examples are given to illustrate our results.
{"title":"Modified version of fixed point theorems and their applications on a fractional hybrid differential equation in the space of continuous tempered functions","authors":"Sudip Deb, Anupam Das","doi":"10.1007/s11868-023-00570-2","DOIUrl":"https://doi.org/10.1007/s11868-023-00570-2","url":null,"abstract":"<p>This article contains modified version of fixed point theorems with the help of freshly made contraction operators and the solvability of a fractional hybrid differential equation involving Caputo fractional derivative by using modified Darbo’s fixed point theorem in the space of continuous tempered functions. The fundamental tool used in the solvability of the fractional hybrid differential equation is the measure of noncompactness approach. Our results indicate the importance of fixed point theorem for finding solutions of the fractional hybrid differential equation and improve some known results, which have wider applications as well. Also, there are two suitable examples are given to illustrate our results.</p>","PeriodicalId":48793,"journal":{"name":"Journal of Pseudo-Differential Operators and Applications","volume":"15 10","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138513025","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-09DOI: 10.1007/s11868-023-00569-9
Do Lan, Tran Van Tuan
{"title":"Long time behavior of solutions for time-fractional pseudo-parabolic equations involving time-varying delays and superlinear nonlinearities","authors":"Do Lan, Tran Van Tuan","doi":"10.1007/s11868-023-00569-9","DOIUrl":"https://doi.org/10.1007/s11868-023-00569-9","url":null,"abstract":"","PeriodicalId":48793,"journal":{"name":"Journal of Pseudo-Differential Operators and Applications","volume":" 21","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135241556","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-06DOI: 10.1007/s11868-023-00567-x
Zhiyong Wang, Pengtao Li, Yu Liu
{"title":"Weighted norm inequalities related to fractional Schrödinger operators","authors":"Zhiyong Wang, Pengtao Li, Yu Liu","doi":"10.1007/s11868-023-00567-x","DOIUrl":"https://doi.org/10.1007/s11868-023-00567-x","url":null,"abstract":"","PeriodicalId":48793,"journal":{"name":"Journal of Pseudo-Differential Operators and Applications","volume":"13 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135634619","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-11-01DOI: 10.1007/s11868-023-00566-y
Nabin Kumar Sahu, Shalini Chauhan, Ram N. Mohapatra
{"title":"Representations of frames via iterative actions of operators in tensor product spaces","authors":"Nabin Kumar Sahu, Shalini Chauhan, Ram N. Mohapatra","doi":"10.1007/s11868-023-00566-y","DOIUrl":"https://doi.org/10.1007/s11868-023-00566-y","url":null,"abstract":"","PeriodicalId":48793,"journal":{"name":"Journal of Pseudo-Differential Operators and Applications","volume":"15 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135326127","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-28DOI: 10.1007/s11868-023-00565-z
Minggang Fei, Jiali Zhou
In this paper, we prove the sharp Pitt’s inequality for the Weinstein transform. As an application, the Beckner’s logarithmic uncertainty principle for the Weinstein transform is established.
{"title":"Sharp Pitt’s inequality and Beckner’s logarithmic uncertainty principle for the Weinstein transform","authors":"Minggang Fei, Jiali Zhou","doi":"10.1007/s11868-023-00565-z","DOIUrl":"https://doi.org/10.1007/s11868-023-00565-z","url":null,"abstract":"<p>In this paper, we prove the sharp Pitt’s inequality for the Weinstein transform. As an application, the Beckner’s logarithmic uncertainty principle for the Weinstein transform is established.</p>","PeriodicalId":48793,"journal":{"name":"Journal of Pseudo-Differential Operators and Applications","volume":"16 3","pages":""},"PeriodicalIF":1.1,"publicationDate":"2023-10-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"138513024","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-24DOI: 10.1007/s11868-023-00564-0
B.-Wolfgang Schulze, Jörg Seiler
Abstract Pseudodifferential operators on the half-space associated with classical symbols of order zero without transmission property are shown to belong to the so-called edge algebra.
摘要给出了半空间上具有经典零阶无传输性质符号的伪微分算子属于所谓的边缘代数。
{"title":"Truncation quantization in the edge calculus","authors":"B.-Wolfgang Schulze, Jörg Seiler","doi":"10.1007/s11868-023-00564-0","DOIUrl":"https://doi.org/10.1007/s11868-023-00564-0","url":null,"abstract":"Abstract Pseudodifferential operators on the half-space associated with classical symbols of order zero without transmission property are shown to belong to the so-called edge algebra.","PeriodicalId":48793,"journal":{"name":"Journal of Pseudo-Differential Operators and Applications","volume":"14 3","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135273212","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-10-16DOI: 10.1007/s11868-023-00560-4
Fatemeh Esmaeelzadeh
{"title":"Localization operators related to Stockwell transforms on locally compact abelian groups","authors":"Fatemeh Esmaeelzadeh","doi":"10.1007/s11868-023-00560-4","DOIUrl":"https://doi.org/10.1007/s11868-023-00560-4","url":null,"abstract":"","PeriodicalId":48793,"journal":{"name":"Journal of Pseudo-Differential Operators and Applications","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-10-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"136078706","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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