Continuous invertibility of minimal Sturm–Liouville operators in Lebesgue spaces

Abstract

Using a standard theory of differential operators in Lebesgue spaces, we re-prove and generalize some results of Chernyavskaya and Shuster, giving (mostly sufficient) conditions that minimal operators determined by expressions of the form −(ry′)′ + qy with domain and range in possibly different Lp spaces on intervals with at least one singular endpoint have bounded inverses.

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

“…Let us now prove (2). Since in this case h(x) ≡ 1 2 , x ∈ R, theorem 3.1 implies that the operator G :…”

“…In the following, for brevity, this is referred to as 'problem (I), (II)' or the 'question on (I) and (II)'. It is easy to see that the problem (I), (II) can be reformulated in different terms (see [1,7]).…”

“…(1.6) (Here AC loc (R) is the set of functions absolutely continuous on every finite segment.) The linear operator L p is called the maximal Sturm-Liouville operator and problem (I), (II) is obviously equivalent to the problem on existence and boundedness of the operator L −1 p : L p → L p (see [1]). We can now give a precise statement of the problem studied in this paper.…”

Embedding theorems corresponding to correct solvability of a linear differential equation of the first order

“…We say that the space (1) (ℝ, ) ( (1) (ℝ, )) is embedded into the space (ℝ) (and write (1) (ℝ, ) → (ℝ) ( (1) (ℝ, ) → (ℝ))) if 1) (1) (ℝ, ) ⊆ (ℝ) ( (1) (ℝ, ) ⊆ (ℝ));…”

“…It is also clear that correct solvability of equation (1.1) in the space (ℝ), ∈ [1, ∞) is equivalent to continuous invertibility of the operator L : D (ℝ) → (ℝ) (see [1]). Definition 1.6.…”