Morrey space

Abstract:

For $1 \leq p < \infty$, $\Omega$ an open and bounded subset of ${R^n}$, and a nonincreasing and nonnegative function $\varphi$ defined in $(0,{\rho _0}]$, ${\rho _0} = \operatorname {diam} \Omega$, we introduce the space $\mathcal {M}_{\varphi ,0}^p(\Omega )$ of locally integrable functions satisfying \[ {\inf _{c \in C}}\left \{ {\int \limits _{B({x_0},\rho ) \cap \Omega } {|f(x) - c{|^p}dx} } \right \} \leq A|B({x_0},\rho )|{\varphi ^p}(\rho )\] for every ${x_0} \in \Omega$ and $0 < \rho \leq {\rho _0}$, where $|B({x_0},\rho )|$ denotes the volume of the ball centered in ${x_0}$ and radius $\rho$. The constant $A > 0$ does not depend on $({x_0},\rho )$. (i) We list some results on the structure, regularity, and density properties of the space so defined. (ii) $\mathcal {M}_{\varphi ,0}^p$ is represented as the dual of an atomic space.