Novel insights into the precise boundedness and Lipschitz continuity of Hardy-type operators within central Morrey spaces

Muhammad Asim; Ghada AlNemer; Muhammad Asim; Ghada AlNemer

doi:10.3934/math.20251046

AIMS Mathematics

2025, Volume 10, Issue 10: 23547-23563. doi: 10.3934/math.20251046

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Novel insights into the precise boundedness and Lipschitz continuity of Hardy-type operators within central Morrey spaces

Muhammad Asim ^{1
,
,},
Ghada AlNemer ²

1.
Department of NUSASH, National University of Technology (NUTECH), Islamabad 44000, Pakistan
2.
Department of Mathematical Sciences, College of Science, Princess Nourah bint Abdulrahman University, Riyadh 11671, Saudi Arabia

Received: 04 July 2025 Revised: 19 September 2025 Accepted: 25 September 2025 Published: 16 October 2025
MSC : 42B35, 26D10, 47B38, 47G10

In this study, precise evaluations of the optimal constants associated with weighted weak-type Hardy inequalities are established. The analysis extends to the computation of the weak-type operator norm of the fractional Hardy operator within the framework of weighted central Morrey spaces. Furthermore, it is demonstrated that the commutators of the Hardy operator exhibit boundedness on these weighted central Morrey spaces when the symbol functions reside in the weighted Lipschitz space.
- Hardy-type operators,
- sharp constants,
- Lipschitz space,
- weights
Citation: Muhammad Asim, Ghada AlNemer. Novel insights into the precise boundedness and Lipschitz continuity of Hardy-type operators within central Morrey spaces[J]. AIMS Mathematics, 2025, 10(10): 23547-23563. doi: 10.3934/math.20251046

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Abstract

In this study, precise evaluations of the optimal constants associated with weighted weak-type Hardy inequalities are established. The analysis extends to the computation of the weak-type operator norm of the fractional Hardy operator within the framework of weighted central Morrey spaces. Furthermore, it is demonstrated that the commutators of the Hardy operator exhibit boundedness on these weighted central Morrey spaces when the symbol functions reside in the weighted Lipschitz space.

References

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