A sharp Wirtinger inequality and some related functional spaces

We consider the generalized Wirtinger inequality $$({\int }_0^{T}a\mathrm{\mid }u{\mathrm{\mid }}^q{)}^{1\mathrm{/}q}\le C({\int }_0^T{a}^{1-p}\mathrm{\mid }{u}^{\mathrm{\prime }}{\mathrm{\mid }}^p{)}^{1\mathrm{/}p},$$ with $p,q>1$, $T>0$, $a\in L^1[0,T]$, $a\ge 0$, $a\equiv ̸0$ and where $u$ is a $T$-periodic function satisfying the constraint $${\int }_0^{T}a\mathrm{\mid }u{\mathrm{\mid }}^{q-2}u=0.$$ We provide the best constant $C>0$ as well as all extremals. Furthermore, we characterize the natural functional space where the inequality is defined.