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

Research paper by Liang Song, Lixin Yan

Indexed on: 17 Jun '15Published on: 17 Jun '15Published in: Mathematics - Analysis of PDEs



Abstract

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 $0<\lambda<n$, Campanato space $\mathscr{L}_L^{p,\lambda}(\mathbb{R}^n)$ (modulo the kernel spaces) coincides with Morrey space ${L}^{p,\lambda}({\mathbb{R}^n})$, i.e. $$\mathscr{L}_L^{p,\lambda}(\mathbb{R}^n)/\mathcal{K}_{L,p}= {L}^{p,\lambda}(\mathbb{R}^n), $$ where $ \mathcal{K}_{L,p}=\{f\in \mathcal{M}_p:\ e^{-tL}f(x)=f(x) \ {\rm for \ almost \ all} \ x\in\mathbb{R}^n \ {\rm and \ all} \ t>0\}. $ As an application, we will study the problem of the characterization of Poisson integrals of Schr\"odinger operators with traces in Morrey spaces ${L}^{2,\lambda}(\mathbb{R}^n)$.