Exact Norms and Morrey Estimates for Logarithmic Hardy Operators and Their Commutators on the Half-Line
- 1Departamento de Matemática y Estadística, Universidad Nacional Pedro Henríquez Ureña, Santo Domingo 1423, Dominican Republic
- 2Instituto de Matemática, Facultad de Ciencias, Universidad Autónoma de Santo Domingo, Santo Domingo 1356, Dominican Republic
Abstract
We study logarithmic Hardy operators and their commutators on the half-line Morrey spaces defined using all bounded intervals. For 1≤p<∞, 0<λ<1, and γ>−1, the positive dilation principle gives the exact norm Γ(γ+1)[p/(p+λ−1)]γ+1, attained by a power function. We obtain exact norms for Hardy iterates and a reverse inequality for nonnegative nonincreasing functions. For the fractional operator, we prove a mapping theorem between distinct Morrey spaces, establish the necessary scaling relation, and give explicit upper bounds throughout a logarithmic range that is optimal for finiteness at every point. We distinguish this pointwise threshold from the almost-everywhere mapping endpoint. For the one-sided point-value symbol class, we prove embeddings into ordinary and central mean-oscillation spaces and identify its relation to the symmetric point-value class through even extension. This symbol condition yields explicit logarithmic and fractional commutator estimates. The symbol b(x)=logx raises the logarithmic order by one and gives an exact commutator norm with which the general upper bound can be compared.
Related
- Estimates of Weighted Hardy Operators and Commutators on Total Morrey Spaces
- Commutators of a class of Ψ-bounded oscillation operators on abstract weighted Orlicz-Morrey spaces endowed with ball-basis and applications
- Endpoint estimates for commutators of singular integral operators associated with admissible functions
- Agmon estimates for the difference of exact and approximate Dirichlet eigenfunctions for difference operators
- Integral Operators on Fractional Cesàro--Morrey Spaces over ℚp
BibTeX
@article{moraferreras2026exact,
title = {{Exact Norms and Morrey Estimates for Logarithmic Hardy Operators and Their Commutators on the Half-Line}},
author = {Francis Mora-Ferreras and Manuel Montero and Victor Ramírez},
journal = {Mathematics},
year = {2026},
month = oct,
doi = {10.3390/math14203824},
url = {https://www.mdpi.com/2227-7390/14/20/3824},
note = {CC BY 4.0},
}
RIS
TY - JOUR
TI - Exact Norms and Morrey Estimates for Logarithmic Hardy Operators and Their Commutators on the Half-Line
AU - Francis Mora-Ferreras
AU - Manuel Montero
AU - Victor Ramírez
PY - 2026
DA - 2026/10/21
JO - Mathematics
DO - 10.3390/math14203824
UR - https://www.mdpi.com/2227-7390/14/20/3824
AB - We study logarithmic Hardy operators and their commutators on the half-line Morrey spaces defined using all bounded intervals. For 1≤p<∞, 0<λ<1, and γ>−1, the positive dilation principle gives the exact norm Γ(γ+1)[p/(p+λ−1)]γ+1, attained by a power function. We obtain exact norms for Hardy iterates and a reverse inequality for nonnegative nonincreasing functions. For the fractional operator, we prove a mapping theorem between distinct Morrey spaces, establish the necessary scaling relation, and give explicit upper bounds throughout a logarithmic range that is optimal for finiteness at every point. We distinguish this pointwise threshold from the almost-everywhere mapping endpoint. For the one-sided point-value symbol class, we prove embeddings into ordinary and central mean-oscillation spaces and identify its relation to the symmetric point-value class through even extension. This symbol condition yields explicit logarithmic and fractional commutator estimates. The symbol b(x)=logx raises the logarithmic order by one and gives an exact commutator norm with which the general upper bound can be compared.
SN - 2227-7390
ER -