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 $\{e^{-t\mathcal{L}}:t>0\}$ 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 $0<\hat{I}\pm <n$, let ${\mathcal{L}}^{-\hat{I}\pm /2}$ 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 $\hat{I}“(\cdot )$ is the usual gamma function. In the limiting Sobolev case $\hat{I}»=n-\hat{I}\pm p$ and $1\le p<n/\hat{I}\pm$, the author proves that the operator ${\mathcal{L}}^{-\hat{I}\pm /2}$ is bounded from the Morrey space $M^{p,\hat{I}»}({\mathbb{R}}^n)$ into ${\mathrm{BMO}}_{\mathcal{L}}({\mathbb{R}}^n)$, and is bounded from the vanishing Morrey space $V{M}^{p,\hat{I}»}({\mathbb{R}}^n)$ 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 ${\mathcal{L}}^{-\hat{I}\pm /2}$ is bounded from $L^{p,\infty }({\mathbb{R}}^n)$ into ${\mathrm{BMO}}_{\mathcal{L}}({\mathbb{R}}^n)$ when $p=n/\hat{I}\pm$ and $0<\hat{I}\pm <n$. The proofs are based on pointwise kernel estimates of the operators ${\mathcal{L}}^{-\hat{I}\pm /2}$ and $(I-e^{-t\mathcal{L}}){\mathcal{L}}^{-\hat{I}\pm /2}$ for $0<\hat{I}\pm <n$.

19 pages