DOUBLY WEIGHTED SHARP WIRTINGER INEQUALITIES ON ℝ+

Abstract

We consider the $\rho$-weighted $p$-norm of functions $f:{ℝ}_{+}\to ℝ$ with finite $q$-norm $\parallel f^{(n)}\psi {\parallel }_q,1\le q\le +\infty$. A sharp Wirtinger type inequality

$$\parallel f\rho {\parallel }_p\le C_{n,p,q}\parallel f^{(n)}\psi {\parallel }_q\text{ for all}1\le p,q\le +\infty$$

is established for functions $f$ such that $f^{(j)}(x_i)=0$ for all $0\le j\le {\alpha }_i-1$, $i=1,\dots ,r$, $n={\sum }_{i=1}^r{\alpha }_i$, where $\psi$, $\rho$ and $\omega =\rho ∕\psi$ are nonincreasing on ${ℝ}_{+}$, and ${\omega }^{1∕\alpha }$ is integrable for $\alpha =n-1∕q+1∕p$. Using Hermite interpolation, we express $C_{n,p,q}$ in terms of the norm of a certain integral type operator. Then we calculate $C_{n,1,1}$ and $C_{n,\infty ,\infty }$ in two specific cases.