Holder inequality (reverse or equality?)

Question

For bounded $\Omega\in\mathbb{R}^n$, it is easy to see by the Holder inequality that $\int_{\Omega} u\,dx\leq (\int_{\Omega} 1^2\,dx)^{\frac{1}{2}} (\int_{\Omega} u^2\,dx)^{\frac{1}{2}}=|\Omega|^{\frac{1}{2}}\,(\int_{\Omega} u^2\,dx)^{\frac{1}{2}}$ or $(\int_{\Omega} u\,dx)^2\leq |\Omega| \,(\int_{\Omega} u^2\,dx)$ . Then my question is: Do there exist certain classes of $u$ and $\Omega$ such that the inequality $(\int_{\Omega} u\,dx)^2\leq |\Omega| \,(\int_{\Omega} u^2\,dx)$ becomes equality (or even becomes the reverse inequality)?

Answer 1

Functions $u$ that are positive almost everywhere and satisfy the reverse Holder inequality $$ \int_B u^2\ dx\le c|B|^{-1}\left(\int_Bu\ dx\right)^2$$ with some constant $c$ belong to the so called Muckenhoupt $A_\infty$ class. In fact one can replace the exponent $2$ by any $p$ and require $$ \int_B u^p\ dx\le c|B|^{1-p}\left(\int_Bu\ dx\right)^p$$ and the same conclusion holds. (In fact $u$ satisfies the above inequality for all balls $B$ for some $p>1$ if and only if it is a $A_\infty$ weight.)

There is also a notion of Muckenhoupt $A_p$-weights and in general, their theory is closely related to the boundedness of the maximal operator $\mathcal M: L^p(\omega\ dx)\to L^p(\omega \ dx)$ and other topics in harmonic analysis.