The distribution function in the Morrey space

Abstract:

For $1 \leqslant p \leqslant \infty$, we consider $p$-integrable functions on a finite cube ${Q_0}$ in ${{\mathbf {R}}^n}$, satisfying \[ {\left ( {\frac {1} {{|Q|}}\int _Q {|f(x) - {f_Q}{|^p}dx} } \right )^{1/p}} \leqslant C\varphi (|Q|)\] for every parallel subcube $Q$ of ${Q_0}$, where $|Q|$ denotes the volume of $Q$, ${f_Q}$ is the mean value of $f$ over $Q$ and $\varphi (t)$ is a nonnegative function defined in $(0,\infty )$, such that $\varphi (t)$ is nonincreasing near zero, $\varphi (t) \to \infty$ as $t \to 0$, and $t{\varphi ^p}(t)$ is nondecreasing near zero. The constant $C$ does not depend on $Q$. Let $g$ be a nonnegative $p$-integrable function $g:(0,1) \to {\mathbf {R}}$ such that $g$ is nonincreasing and $g(t) \to \infty$ as $t \to 0$. We prove here that there exist a cube ${Q_0}$ and a function $f$ satisfying condition $(1)$ for every parallel subcube $Q$ of ${Q_0}$, such that ${\delta _f}(\lambda ) \geqslant {C_1}{\delta _g}(\lambda )$ for $\lambda \geqslant {\lambda _0}$, ${C_1} > 0$, where $\delta (\lambda )$ denotes the distribution function.