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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

The distribution function in the Morrey space


Author: Josefina Álvarez Alonso
Journal: Proc. Amer. Math. Soc. 83 (1981), 693-699
MSC: Primary 46E30; Secondary 26B35
MathSciNet review: 630039
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Abstract: For $ 1 \leqslant p \leqslant \infty $, we consider $ p$-integrable functions on a finite cube $ {Q_0}$ in $ {{\mathbf{R}}^n}$, satisfying

$\displaystyle {\left( {\frac{1} {{\vert Q\vert}}\int_Q {\vert f(x) - {f_Q}{\vert^p}dx} } \right)^{1/p}} \leqslant C\varphi (\vert Q\vert)$

for every parallel subcube $ Q$ of $ {Q_0}$, where $ \vert Q\vert$ 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.

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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9939-1981-0630039-0
PII: S 0002-9939(1981)0630039-0
Keywords: Mean oscillation, distribution function, Morrey space
Article copyright: © Copyright 1981 American Mathematical Society