Weight summability of solutions of the Sturm-Liouville equation

We consider an equation -y "(x) + q(x)y(x) = f(x), x is an element of R; (1) where f(x) is an element of L-p(R), p is an element of [1, infinity] (L-inf inity(R):= C(R)), 1 less than or equal to q(x) is an element of L-l(loc)(R) . We study requirements for a weight function r(x) is an element of L-p(loc )(R) and for q(x) under which, for a given p is an element of [1, infinity] , regardless of f(x) is an element of L-p(R), the solution y(x) is an eleme nt of L-p(R) of Eq. (1) satisfies the inequalities: parallel to r(x) y(x)parallel to(p) less than or equal to c parallel to f(x )parallel to(p), parallel to y "(x)parallel to(p) + parallel to q(x) y(x)pa rallel to(p) less than or equal to c parallel to f(x)parallel to(p), c = co nst. (C) 1999 Academic Press.