On the Boundedness of Generalized Fractional Integral Operators in Morrey Spaces and Camapanato Spaces associated with the Dunkl Operator on the Real line
Sumit Parashar, Saswata Adhikari
TL;DR
The paper investigates the boundedness of generalized Dunkl-type fractional integral operators $T_{ ho}^ ext{α}$ and their modified versions $ ilde T_{ ho}^ ext{α}$ on Campanato-type spaces associated with the Dunkl operator on $\mathbb{R}$. Building on prior work that established results for Morrey and BMO-type spaces, it extends boundedness results to generalized Dunkl-type Campanato spaces, deriving endpoint cases $p=1$ and general $1<p< ty$ results under suitable integral and doubling conditions on the kernel $ ho$ and the control functions $oldsymbol{ ext φ}$ and $oldsymbol{ ext ψ}$. The proofs hinge on the relationship between Dunkl translation and Bessel–Kingman hypergroup translation, maximal operator bounds in the hypergroup setting, and intricate decompositions of the kernels to obtain localized Campanato estimates. The findings enrich the harmonic analysis framework for Dunkl operators by establishing robust boundedness criteria for generalized fractional integrals in Campanato-type spaces, with potential applications to Dunkl-type PDEs and related analysis on hypergroup structures.
Abstract
It is known that the Dunkl-type fractional integral operator $I_β$ $(0 < β< 2α+ 2 =d_α)$ is bounded from $L^p(\R,dμ_α)$ to $L^q (\R, dμ_α)$ when $1 < p < \frac{d_α}β$ and $\frac{1}{p} - \frac{1}{q} = \fracβ{d_α}$. In \cite{spsa} , the authors introduced the generalized Dunkl-type fractional integral operator $T_ρ^α$ and it's modified version $\tilde{T}_ρ^α$ and extended the above boundedness results to the generalized Dunkl-type Morrey spaces and Dunkl-type $BMO_φ$ spaces. In this paper we investigate the boundedness of generalized Dunkl-type fractional integral operators and it's modified version mainly on the Dunkl-type Campanato space.