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

TL;DR

This paper establishes boundedness properties of generalized fractional integrals linked to operators on Morrey spaces, connecting them to BMO and VMO spaces under certain conditions.

Contribution

It provides new estimates for fractional integrals associated with operators on Morrey spaces, including boundedness results into BMO and VMO spaces.

Findings

Boundedness from Morrey to BMO spaces for certain parameters.

Boundedness from vanishing Morrey to VMO spaces.

Extension to boundedness from weak Lebesgue spaces to BMO.

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 $0<\alpha<n$, 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 $0<\alpha<n$.