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

Research paper by **Liang Song, Lixin Yan**

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

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join Sparrho today to stay on top of science

Discover, organise and share research that matters to you

Join for free

#### 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 ${L}^{2,\lambda}(\mathbb{R}^n)$.