Estimates for generalized fractional integrals associated with operators on Morrey--Campanato spaces

Let $\mathcal{L}$ be the infinitesimal generator of an analytic semigroup $\big\{e^{-t\mathcal L}\big\}_{t>0}$ satisfying the Gaussian upper bounds. For given $0<\alpha<n$, let $\mathcal L^{-\alpha/2}$ be the generalized fractional integral associated with $\mathcal{L}$, which is defined as \begin{equation*} \mathcal L^{-\alpha/2}(f)(x):=\frac{1}{\Gamma(\alpha/2)}\int_0^{+\infty} e^{-t\mathcal L}(f)(x)t^{\alpha/2-1}dt, \end{equation*} where $\Gamma(\cdot)$ is the usual gamma function. For a locally integrable function $b(x)$ defined on $\mathbb R^n$, the related commutator operator $\big[b,\mathcal L^{-\alpha/2}\big]$ generated by $b$ and $\mathcal{L}^{-\alpha/2}$ is defined by \begin{equation*} \big[b,\mathcal L^{-\alpha/2}\big](f)(x):=b(x)\cdot\mathcal{L}^{-\alpha/2}(f)(x)-\mathcal{L}^{-\alpha/2}(bf)(x). \end{equation*} A new class of Morrey--Campanato spaces associated with $\mathcal{L}$ is introduced in this paper. The authors establish some new estimates for the commutators $\big[b,\mathcal L^{-\alpha/2}\big]$ on Morrey--Campanato spaces. The corresponding results for higher-order commutators$\big[b,\mathcal L^{-\alpha/2}\big]^m$($m\in \mathbb{N}$) are also discussed.