Weight Summability of Solutions of the Sturm–Liouville Equation

Correct solvability, embedding theorems and separability for the Sturm–Liouville equation

“…(1.1) is also separable in L p . Hence, in this case S (2) p (R, q) ∼ W (2) p (R, q) for every p ∈ [1, ∞). Example 1.8 [2] Let p ∈ (1, ∞) and let α, β be positive numbers such that β < α < pβ.…”

“…(1.1) with coefficient q from (1.10) is correctly solvable in L p , p ∈ [1, ∞) (see Theorem 1.1 above). Since, as is shown in [2], such an equation is not separable in L p , we have…”

“…Contrary to the case p = 1, for every p ∈ (1, ∞) there exist Eq. (1.1) which are correctly solvable but not separable in L p (see [2]). We illustrate both possible situations by the next examples.…”

“…∞), if and only if θ ≥ 1. Note that for p ∈ [1, ∞), the embedding S (2) p (R, q) → L p exists if and only if θ ≥ 1. Since in this case q(x) ∈ [0, 2], x ∈ R, for θ ≥ 1 Eq.…”

“…Note that the approach to the problem (I)-(II) in [5] for p = 1 differs from that in the present work. Here, we make use of another weighted space S (2) p (R, q), p ∈ [1, ∞) :…”