The relationship between Campanato spaces associated to operators and Morrey spaces and applications

Let $L$ be the infinitesimal generator of an analytic semigroup $\{e^{-tL}\}_{t\ge0}$ on $L^2({\mathbb R}^n)$ with suitable upper bounds on its heat kernels. Given $1\leq p<\infty$ and $\lambda \in (0, n)$, a function $f$ (with appropriate bound on its size $|f|$) belongs to Campanato space ${\mathscr{L}_L^{p,\lambda}({\mathbb{R}^n})}$ associated to an operator $L$, provided \begin{eqnarray*} \sup\limits_{x\in {\mathbb{R}^n}, \ r>0} r^{-\lambda}\int_{ B (x,r)} |f(y)-e^{-r^mL}f(y)|^p \ dy\leq C <\infty \end{eqnarray*} for a fixed positive constant $m$. These spaces ${\mathscr{L}_L^{p,\lambda}({\mathbb{R}^n})}$ associated to $L$ were introduced and studied in \cite{DXY}. In this article, we will show that for every $1\leq p<\infty$ and $0<\lambda0\}. $ As an application, we will study the problem of the characterization of Poisson integrals of Schr\"odinger operators with traces in Morrey spaces ${L}^{2,\lambda}(\mathbb{R}^n)$.