Characterization of weak Lebesgue spaces - MathOverflow [closed] most recent 30 from http://mathoverflow.net 2013-05-22T06:40:18Z http://mathoverflow.net/feeds/question/80246 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/80246/characterization-of-weak-lebesgue-spaces Characterization of weak Lebesgue spaces unknown (google) 2011-11-06T21:24:40Z 2011-11-08T08:43:56Z <p>I would be interested to know whether the following is true:</p> <p>Let $\Omega$ be a bounded open set in $\mathbf{R}^n$. Let $g$ be a nonnegative function $g : \Omega \to \mathbf{R}$. If there is a constant $C > 0$ such that $$\frac{1}{|A|^{1-1/p}} \int_A g \leq C$$ for all measurable subset $A \subset \Omega$, then $g$ is in weak-$L^p(\Omega)$.</p> <p>If the above inequality holds only for open balls in $\Omega$, is $g$ still in weak-$L^p(\Omega)$ ?</p> <p>Edit: I changed the question to make it more relevant and less naive. Most comments below are out of date.</p>