Estimates for derivatives and integrals of eigenfunctions and associated functions of nonself-adjoint Sturm-Liouville operator with discontinuous coefficients (III)

Abstract

In this paper we consider derivatives of higher order and certain "double'' integrals of the eigenfunctions and associated functions of the formal Sturm-Liouville operator $$\Cal L(u)(x)=-\bigl(p(x)\,u'(x)\bigr)'+q(x)\,u(x)$$defined on a finite or infinite interval $G\subseteq R$. We suppose that the complex-valued potential $q=q(x)$ belongs to the class $L_1^{loc}(G)$ and that piecewise continuously differentiable coefficient $p=p(x)$ has a finite number of the discontinuity points in $G$. Order-sharp upper estimates are obtained for the suprema of the moduli of the $k$-th order derivatives $(k\geq 2$) of the eigenfunctions and associated functions $\{\,\overset{i}\to{u}_{\lambda}(x)\,|\,i=0,1,\dots\,\}$ of the operator $\Cal L$ in terms of their norms in metric $L_2$ on compact subsets of $G$ (on the entire interval $G$). Also, order-sharp upper estimates are established for the integrals (over closed intervals $[y_1,y_2]\subseteq \overline G$)$$ınt_{y_1}^{y_2}\biggl(ınt_a^y\overset{i}\to{u}_{\lambda}(\xi)\,d\xi\biggr)dy,\qquad ınt_{y_1}^{y_2}\biggl(ınt_y^b\overset{i}\to{u}_{\lambda}(\xi)\,d\xi\biggr)dy$$in terms of $L_2$-norms of the mentioned functions when $G$ is finite. The corresponding estimates for derivatives $\overset{i}\to{u}_{\lambda}'(x)$ and integrals $ınt_{y_1}^{y_2}\overset{i}\to{u}_{\lambda}(y)\,dy$ were proved in [5]–[6].