Some norm inequalities for commutators generated by the Riesz potentials on homogeneous variable exponent Herz-Morrey-Hardy spaces
Ferit Gurbuz
TL;DR
This work studies norm inequalities for commutators generated by the Riesz potential $I^{\beta}$ on homogeneous variable-exponent Herz-Morrey-Hardy spaces. By combining atomic decompositions with careful dyadic annulus estimates, the authors establish a boundedness result: for $0<q_1\le q_2<\infty$, $0\le\lambda<\infty$, $0<\beta<n$, and $1/p_2(\cdot)=1/p_1(\cdot)-\beta/n$, if $b\in BMO$, then $\|[b,I^{\beta}]f\|_{M\dot{K}_{p_2(\cdot ),\lambda }^{\alpha(\cdot ),q_2}} \lesssim \|b\|_{BMO}\|f\|_{HM\dot{K}_{p_1(\cdot ),\lambda }^{\alpha(\cdot ),q_1}}$ for all $f$ in $HM\dot{K}_{p_1(\cdot ),\lambda }^{\alpha(\cdot ),q_1}$. The result extends recent work on the boundedness of $I^{\beta}$ to the commutator case in this nonstandard function space setting and relies on the atomic structure and BMO control to handle the operator across scales.
Abstract
In harmonic analysis, the studies of inequalities of classical operators (= singular, maximal, Riesz potentials etc.) in various function spaces have a very important place. The maturation of many topics in the field of harmonic analysis, as a result of various needs and developments to respond to the problems of the time, has also led to the emergence of many studies and works on these topics. In [3], under some conditions, the boundedness of Riesz potential on homogeneous variable exponent Herz-Morrey-Hardy spaces has been given. Inspired by the work of [3], in this work, by the atomic decompositions, we obtain the boundedness of commutators generated by the Riesz potentials on homogeneous variable exponent Herz-Morrey-Hardy spaces.