Two nonequivalent conditions for weight functions

Abstract:

A nonnegative function on the real line satisfies the condition ${{\mathbf {A}}_\infty }$ if, given $\varepsilon > 0$, there exists a $\delta > 0$ such that if $I$ is an interval, $E \subset I$, and $|E| < \delta |I|$, then $\int _E {W \leq \varepsilon \int _I W }$. A nonnegative function on the real line satisfies the condition ${\mathbf {A}}$ if for every interval $I,\int _{2I} {W \leq C} \int _I W$, where $2I$ is the interval with the same center as $I$ and twice as long, and $C$ is independent of $I$. An example is given of a function that satisfies ${\mathbf {A}}$ but not ${{\mathbf {A}}_\infty }$.