Estimates for derivatives and integrals of eigenfunctions and associated functions of the nonselfadjoint Sturm-Liouville operator with discontinuous coefficients. III. (English) Zbl 0949.34017

Lažetić, N.

Mat. Vesn. 48, No. 1-2, 29-46 (1996).

The author considers the formal Sturm-Liouville operator \[ Ly = -(p y^{'})^{'}+q y \] on a finite or infinite interval \(G\subset \mathbb{R}.\) He supposes that the complex-valued function \(q=q(x)\) belongs to the class \(L_1^{{\text{loc}}}(G)\) and \(p=p(x)\) is a piecevise continuously differentiable function on \(G.\) Sharp upper order estimates are obtained for the moduli of the \(k\)th-order derivative of the eigenfunctions and associated functions of the operator \(L\) (on compact subset of \(G\)) in terms of their \(L^2\) norms (on compact subset of \(G\)).
Similar estimates are obtained for repeated integrals of the eigenfunctions and associated functions of \(L.\)
The author gives an example which shows that these estimates are really best possible with respect to the order of the spectral parameter.
For part I see [Sov. Math., Dokl. 23, 558-560 (1981); translation from Dokl. Akad. Nauk SSSR 258, 541-544 (Russian) (1981; Zbl 0511.34015)] and for part II see [Publ. Inst. Math., Nouv. Ser. 60, 31-44 (1996; Zbl 0930.34062)].

Reviewer: Milutin Dostanić (Beograd)

MSC:

34B24

Sturm-Liouville theory

Keywords:

formal differential operator; eigenfunction; associated function; nonselfadjoint Sturm-Liouville operator; discontinuous coefficients; formal Sturm-Liouville operator

Citations:

Zbl 0511.34015; Zbl 0930.34062