Old and New Morry Spaces via Heat Kernel Bounds

Given $p\in [1,\infty)$ and $\lambda\in (0,n)$, we study Morrey space ${\rm L}^{p,\lambda}({\mathbb R}^n)$ of all locally integrable complex-valued functions $f$ on ${\mathbb R}^n$ such that for every open Euclidean ball $B\subset{\mathbb R}^n$ with radius $r_B$ there are numbers $C=C(f)$ (depending on $f$) and $c=c(f,B)$ (relying upon $f$ and $B$) satisfying $$r_B^{-\lambda}\int_{B}|f(x)-c|^pdx\le C$$ and derive old and new, two essentially different cases arising from either choosing $c=f_B=|B|^{-1}\int_{B}f(y)dy$ or replacing $c$ by $P_{t_B}(x)=\int_{t_B}p_{t_B}(x,y)f(y)dy$ -- where $t_B$ is scaled to $r_B$ and $p_t(\cdot,\cdot)$ is the kernel of the infinitesimal generator $L$ of an analytic semigroup $\{e^{-tL}\}_{t\ge0}$ on ${\rm L}^2({\mathbb R}^n)$. Consequently, we are led to simultaneously characterize the old and new Morrey spaces, but also to show that for a suitable operator $L$, the new Morrey space is equivalent to the old one.