Estimates for generalized fractional integrals associated with operators on Morrey--Campanato spaces
Cong Chen, Hua Wang
TL;DR
This work studies generalized fractional integrals $\mathcal{L}^{-\alpha/2}$ and their commutators $[b,\mathcal{L}^{-\alpha/2}]$ for operators $\mathcal{L}$ generating analytic semigroups with Gaussian bounds. It introduces operator-adapted Morrey–Campanato spaces $\mathcal{C}^{p,\gamma}_{\mathcal{L}}$ by using $e^{-t_{\mathcal{B}}\mathcal{L}}f$ as the averaging mechanism, and develops kernel estimates for the associated difference operator $(I-e^{-t\mathcal{L}})\mathcal{L}^{-\alpha/2}$ and a sharp maximal function $M^{\#}_{\mathcal{L}}$. The main contributions prove that $[b,\mathcal{L}^{-\alpha/2}]$ maps $\mathcal{M}^{p_2,\beta_2}$ to $\mathcal{C}^{q,\gamma}_{\mathcal{L}}$ when $0<\alpha<n$, $1/p_2< n/\alpha$, with $1/q=1/p_1+1/p_2-\alpha/n$ and $\gamma=\beta_1+\beta_2+\alpha$, and extend to higher-order commutators and related settings. These results generalize classical fractional integral and commutator bounds to a broad class of operators under Gaussian control, with potential applications in harmonic analysis and PDEs where the underlying semigroup replaces the Laplacian.
Abstract
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 $0<α<n$, let $\mathcal L^{-α/2}$ 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 $Γ(\cdot)$ is the usual gamma function. For a locally integrable function $b(x)$ defined on $\mathbb R^n$, the related commutator operator $\big[b,\mathcal L^{-α/2}\big]$ generated by $b$ and $\mathcal{L}^{-α/2}$ is defined by \begin{equation*} \big[b,\mathcal L^{-α/2}\big](f)(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 $\big[b,\mathcal L^{-α/2}\big]$ on Morrey--Campanato spaces. The corresponding results for higher-order commutators$\big[b,\mathcal L^{-α/2}\big]^m$($m\in \mathbb{N}$) are also discussed.