Estimates for the Green function of a general Sturm-Liouville operator and their applications

Abstract

Abstract. For a general Sturm-Liouville operator with nonnegative coefficients, we obtain two-sided estimates for the Green function, sharp by order on the diagonal.

A criterion for compactness in Lp(ℝ) of the resolvent of the maximal general form Sturm–Liouville operator

“…Theorem 2.1 (Chernyavskaya and Shuster [2]). Suppose that conditions (1.2) and has a fundamental system of solutions (FSS) with the following properties:…”

“…Theorem 2.3 (Chernyavskaya and Shuster [2]; Davies and Harrell [10]). For the FSS {u, v} we have the Davies-Harrell representations…”

“…Remark 2.4. Representations (2.7) and (2.9) are given in [10] for r ≡ 1, and in [2] for r ≡ 1. Henceforth, conditions (1.2), (1.3) are tacitly assumed to be satisfied unless stated otherwise.…”

“…Lemma 2.5 (Chernyavskaya and Shuster [2]). For every given x ∈ R each of the following equations in d 0 has a unique finite positive solution: For x ∈ R we introduce the following functions:…”

“…Remark 2.7 (Chernyavskaya and Shuster [2]). Two-sided sharp-by-order a priori estimates of the function ρ first appeared in Otelbaev's paper, [16] (under some additional requirements for r and q).…”