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

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