Some New Sobolev-Type Theorems for the Rough Riesz Potential Operator on Grand Variable Herz Spaces
In this paper, our first objective is to define the idea of grand variable Herz spaces. Then, our main goal is to prove boundedness results for operators, including the rough Riesz potential operator of variable order and the fractional Hardy operators, on grand variable Herz spaces under some prope...
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Article |
| Language: | English |
| Published: |
MDPI AG
2025-06-01
|
| Series: | Mathematics |
| Subjects: | |
| Online Access: | https://www.mdpi.com/2227-7390/13/11/1873 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1850160816661200896 |
|---|---|
| author | Ghada AlNemer Ghada Ali Basendwah Babar Sultan Ioan-Lucian Popa |
| author_facet | Ghada AlNemer Ghada Ali Basendwah Babar Sultan Ioan-Lucian Popa |
| author_sort | Ghada AlNemer |
| collection | DOAJ |
| description | In this paper, our first objective is to define the idea of grand variable Herz spaces. Then, our main goal is to prove boundedness results for operators, including the rough Riesz potential operator of variable order and the fractional Hardy operators, on grand variable Herz spaces under some proper assumptions. To prove the boundedness results, we use Holder-type and Minkowski inequalities. In the proof of the main result, we use different techniques. We divide the summation into different terms and estimate each term under different conditions. Then, by combining the estimates, we prove that the rough Riesz potential operator of variable order and the fractional Hardy operators are bounded on grand variable Herz spaces. It is easy to show that the rough Riesz potential operator of variable order generalizes the Riesz potential operator and that the fractional Hardy operators are generalized versions of simple Hardy operators. So, our results extend some previous results to the more generalized setting of grand variable Herz spaces. |
| format | Article |
| id | doaj-art-726d4cebb6a044b781712409657b09c0 |
| institution | OA Journals |
| issn | 2227-7390 |
| language | English |
| publishDate | 2025-06-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj-art-726d4cebb6a044b781712409657b09c02025-08-20T02:23:04ZengMDPI AGMathematics2227-73902025-06-011311187310.3390/math13111873Some New Sobolev-Type Theorems for the Rough Riesz Potential Operator on Grand Variable Herz SpacesGhada AlNemer0Ghada Ali Basendwah1Babar Sultan2Ioan-Lucian Popa3Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, Riyadh 11671, Saudi ArabiaDepartment of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi ArabiaDepartment of Mathematics, Quaid-I-Azam University, Islamabad 45320, PakistanDepartment of Computing, Mathematics and Electronics, “1 Decembrie 1918” University of Alba Iulia, 510009 Alba Iulia, RomaniaIn this paper, our first objective is to define the idea of grand variable Herz spaces. Then, our main goal is to prove boundedness results for operators, including the rough Riesz potential operator of variable order and the fractional Hardy operators, on grand variable Herz spaces under some proper assumptions. To prove the boundedness results, we use Holder-type and Minkowski inequalities. In the proof of the main result, we use different techniques. We divide the summation into different terms and estimate each term under different conditions. Then, by combining the estimates, we prove that the rough Riesz potential operator of variable order and the fractional Hardy operators are bounded on grand variable Herz spaces. It is easy to show that the rough Riesz potential operator of variable order generalizes the Riesz potential operator and that the fractional Hardy operators are generalized versions of simple Hardy operators. So, our results extend some previous results to the more generalized setting of grand variable Herz spaces.https://www.mdpi.com/2227-7390/13/11/1873Lebesgue spacesweighted estimatesRiesz potentialgrand variable Herz spaces |
| spellingShingle | Ghada AlNemer Ghada Ali Basendwah Babar Sultan Ioan-Lucian Popa Some New Sobolev-Type Theorems for the Rough Riesz Potential Operator on Grand Variable Herz Spaces Mathematics Lebesgue spaces weighted estimates Riesz potential grand variable Herz spaces |
| title | Some New Sobolev-Type Theorems for the Rough Riesz Potential Operator on Grand Variable Herz Spaces |
| title_full | Some New Sobolev-Type Theorems for the Rough Riesz Potential Operator on Grand Variable Herz Spaces |
| title_fullStr | Some New Sobolev-Type Theorems for the Rough Riesz Potential Operator on Grand Variable Herz Spaces |
| title_full_unstemmed | Some New Sobolev-Type Theorems for the Rough Riesz Potential Operator on Grand Variable Herz Spaces |
| title_short | Some New Sobolev-Type Theorems for the Rough Riesz Potential Operator on Grand Variable Herz Spaces |
| title_sort | some new sobolev type theorems for the rough riesz potential operator on grand variable herz spaces |
| topic | Lebesgue spaces weighted estimates Riesz potential grand variable Herz spaces |
| url | https://www.mdpi.com/2227-7390/13/11/1873 |
| work_keys_str_mv | AT ghadaalnemer somenewsobolevtypetheoremsfortheroughrieszpotentialoperatorongrandvariableherzspaces AT ghadaalibasendwah somenewsobolevtypetheoremsfortheroughrieszpotentialoperatorongrandvariableherzspaces AT babarsultan somenewsobolevtypetheoremsfortheroughrieszpotentialoperatorongrandvariableherzspaces AT ioanlucianpopa somenewsobolevtypetheoremsfortheroughrieszpotentialoperatorongrandvariableherzspaces |