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

Abstract: 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 $0<α<n$, let $\mathcal L^{-α/2}$ be the generalized fractional integral associated with $\mathcal{L}$, which is given by \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. In the limiting Sobolev case $λ=n-αp$ and $1\leq p<n/α$, the author proves that the operator $\mathcal{L}^{-α/2}$ is bounded from the Morrey space $M^{p,λ}(\mathbb R^n)$ into $\mathrm{BMO}{\mathcal{L}}(\mathbb R^n)$, and is bounded from the vanishing Morrey space $VM^{p,λ}(\mathbb R^n)$ 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 $\mathcal{L}^{-α/2}$ is bounded from $L^{p,\infty}(\mathbb R^n)$ into $\mathrm{BMO}_{\mathcal{L}}(\mathbb R^n)$ when $p=n/α$ and $0<α<n$. The proofs are based on pointwise kernel estimates of the operators $\mathcal L^{-α/2}$ and $(I-e^{-t\mathcal L})\mathcal{L}^{-α/2}$ for $0<α<n$.