Some new estimates for generalized fractional integrals associated with operators on Morrey spaces

Let $\mathcal{L}$ be the infinitesimal generator of an analytic semigroup $\big\{e^{-t\mathcal L}:t>0\big\}$ on $L^2(\mathbb R^n)$ with Gaussian upper bounds, and suppose that $\mathcal{L}$ has a bounded holomorphic functional calculus on $L^2(\mathbb R^n)$. For given $0<\alpha<n$, let $\mathcal L^{-\alpha/2}$ be the generalized fractional integral associated with $\mathcal{L}$, which is given by \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. In the limiting Sobolev case $\lambda=n-\alpha p$ and $1\leq p<n/{\alpha}$, the author proves that the operator $\mathcal{L}^{-\alpha/2}$ is bounded from the Morrey space $M^{p,\lambda}(\mathbb R^n)$ into $\mathrm{BMO}_{\mathcal{L}}(\mathbb R^n)$, and is bounded from the vanishing Morrey space $VM^{p,\lambda}(\mathbb R^n)$ into $\mathrm{VMO}_{\mathcal{L}}(\mathbb R^n)$, where $\mathrm{BMO}_{\mathcal{L}}(\mathbb R^n)$ and $\mathrm{VMO}_{\mathcal{L}}(\mathbb R^n)$ are the spaces of bounded mean oscillation and vanishing mean oscillation associated with the operator $\mathcal{L}$, respectively. As a consequence, the author obtains that the operator $\mathcal{L}^{-\alpha/2}$ is bounded from $L^{p,\infty}(\mathbb R^n)$ into $\mathrm{BMO}_{\mathcal{L}}(\mathbb R^n)$ when $p=n/{\alpha}$ and $0<\alpha<n$. The proofs are based on pointwise kernel estimates of the operators $\mathcal L^{-\alpha/2}$ and $(I-e^{-t\mathcal L})\mathcal{L}^{-\alpha/2}$ for $0<\alpha<n$.