Some New Sobolev-Type Theorems for the Rough Riesz Potential Operator on Grand Variable Herz Spaces

Ghada AlNemer, Ghada Ali Basendwah, Babar Sultan () and Ioan-Lucian Popa ()
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Ghada AlNemer: Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, Riyadh 11671, Saudi Arabia
Ghada Ali Basendwah: Department of Mathematics, King Abdulaziz University, Jeddah 21589, Saudi Arabia
Babar Sultan: Department of Mathematics, Quaid-I-Azam University, Islamabad 45320, Pakistan
Ioan-Lucian Popa: Department of Computing, Mathematics and Electronics, “1 Decembrie 1918” University of Alba Iulia, 510009 Alba Iulia, Romania

Mathematics, 2025, vol. 13, issue 11, 1-19

Abstract: 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.

Keywords: Lebesgue spaces; weighted estimates; Riesz potential; grand variable Herz spaces (search for similar items in EconPapers)
JEL-codes: C (search for similar items in EconPapers)
Date: 2025
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