Weighted norm inequalities for the quasi-derivatives of ordinary differential operators

Synopsis

For a compact interval I of the real line and for 1 ≤ p < ∞ the classical inequality

$$\int_{I}\mid y^{(k)}\mid^{p}\ dx \leq \varepsilon\int_{I}\mid y^{(n)}\mid^{p}\ dx +K(\varepsilon)\int_{I}\mid y\mid^{p} dx$$

is extended by replacing y⁽ⁿ⁾ by the action of an n-th order quasi-differential operator and y^(k) by the corresponding k-th quasi-derivative for 0 ≤+ k < n. Also quite general weight functions are allowed.