Estimates of Weighted Hardy Operators and Commutators on Total Morrey Spaces
- 1School of Mathematics and Statistics, Yili Normal University, Yining 835000, China
- 2Hubei Key Laboratory of Applied Mathematics, Hubei University, Wuhan 430062, China
Abstract
We determine the exact operator norms of the weighted Hardy operator Uψ and the weighted Cesàro operator Vψ on Total Morrey spaces Lp,λ,μ(Rn). We also study their commutators with symbols b∈BMO(Rn) satisfying the additional dilation condition sup0<t<1∥b−b(t·)∥L∞(Rn)log(2/t)<∞. Assuming in addition that the corresponding underlying weighted Hardy or Cesàro operator is bounded, we obtain necessary and sufficient integral conditions on the weight function ensuring the boundedness of the first-order and higher-order commutators for every symbol satisfying this dilation condition. The results reflect the two-scale structure of Total Morrey spaces: the weighted Hardy operator is governed by the large-scale parameter μ, whereas the weighted Cesàro operator is governed by the small-scale parameter λ. The exact operator norm formulas extend their classical Morrey-space counterparts, while the commutator criteria are obtained for BMO symbols satisfying the stated additional dilation condition.
Related
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BibTeX
@article{hu2026estimates,
title = {{Estimates of Weighted Hardy Operators and Commutators on Total Morrey Spaces}},
author = {Zhenghao Hu and Suixin He},
journal = {Axioms},
year = {2026},
month = oct,
doi = {10.3390/axioms15100785},
url = {https://www.mdpi.com/2075-1680/15/10/785},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Estimates of Weighted Hardy Operators and Commutators on Total Morrey Spaces
AU - Zhenghao Hu
AU - Suixin He
PY - 2026
DA - 2026/10/19
JO - Axioms
DO - 10.3390/axioms15100785
UR - https://www.mdpi.com/2075-1680/15/10/785
AB - We determine the exact operator norms of the weighted Hardy operator Uψ and the weighted Cesàro operator Vψ on Total Morrey spaces Lp,λ,μ(Rn). We also study their commutators with symbols b∈BMO(Rn) satisfying the additional dilation condition sup0<t<1∥b−b(t·)∥L∞(Rn)log(2/t)<∞. Assuming in addition that the corresponding underlying weighted Hardy or Cesàro operator is bounded, we obtain necessary and sufficient integral conditions on the weight function ensuring the boundedness of the first-order and higher-order commutators for every symbol satisfying this dilation condition. The results reflect the two-scale structure of Total Morrey spaces: the weighted Hardy operator is governed by the large-scale parameter μ, whereas the weighted Cesàro operator is governed by the small-scale parameter λ. The exact operator norm formulas extend their classical Morrey-space counterparts, while the commutator criteria are obtained for BMO symbols satisfying the stated additional dilation condition.
SN - 2075-1680
ER -