Hardy-Littlewood maximal, generalized Bessel-Riesz and generalized fractional integral operators in generalized Morrey and $BMO_φ$ spaces associated with Dunkl operator on the real line
Sumit Parashar, Saswata Adhikari
TL;DR
This work extends harmonic analysis in the Dunkl setting to generalized Morrey and $BMO_\phi$ spaces on $\mathbb{R}$ by establishing boundedness results for a family of Dunkl-type operators. Using Dunkl translations, generalized convolutions, and kernel estimates, it proves $L^{p,\phi}$-to-$L^{q,\psi}$ boundedness for the Dunkl-type Hardy-Littlewood maximal operator $M^\alpha$, the Bessel-Riesz operators $I_{\beta,\gamma}^\alpha$, and their generalized versions $I_{\tilde{\rho},\gamma}^\alpha$ as well as the generalized fractional integral $T_\rho^\alpha$, plus $BMO_\phi$ to $BMO_\psi$ boundedness for a modified operator $\tilde{T}_\rho^\alpha$. The results hinge on kernel norms in $L^t$ and $L^{s,t}$ spaces, doubling and growth conditions on the Morrey-type control functions $\phi$, $\psi$, and $\tilde{\rho}$, and properties of the Dunkl transform and translation. Together, these findings generalize classical Riesz and Hardy-Littlewood theory to the Dunkl framework with weight $d\mu_\alpha$ and provide a robust toolset for analysis on Dunkl-type Morrey and $BMO_\phi$ spaces. The paper further demonstrates how a modified fractional operator preserves mean oscillation control between $BMO_\phi$ and $BMO_\psi$, enhancing the applicability to Dunkl-type PDEs and related harmonic analysis problems.
Abstract
The analysis of Morrey spaces, generalized Morrey spaces and $BMO_φ$ spaces related to the Dunkl operators on $\mathbb{R}$ are covered in this paper. We prove the boundedness of the Hardy-Littlewood maximal operators, Bessel-Riesz operators, generalized Bessel-Riesz operators, and generalized fractional integral operators associated with Dunkl operators on $\mathbb{R}$ in the generalized Dunkl-type Morrey spaces. Further, we derive the boundedness of the modified version of the generalized fractional integral operators associated with the Dunkl operators on $\mathbb{R}$ in Dunkl-type $BMO_φ$ spaces.