Weighted estimates for solutions of a Sturm–Liouville equation in the space L₁(ℝ)

Abstract

We consider the equationwhereUnder these conditions, (1) is correctly solvable in L1(ℝ), i.e.(i) for any function f ∈ L1(ℝ), there exists a unique solution of (1), y ∈ L1(ℝ);(ii) there is an absolute constant c1 ∈ (0, ∞) such that the solution of (1),In this work we strengthen the a priori inequality (1). We find minimal requirements for a given weight function θ ∈ Lloc1(ℝ) under which the solution of (1), y ∈ L1(ℝ), satisfies the estimatewhere c2 is some absolutely positive constant.

Weighted Estimates for Solutions of the General Sturm-Liouville Equation and the Everitt-Giertz Problem. I

“…In [35,36], the first examples of inseparable operators and the first study of sufficient conditions for separability (of the Sturm-Liouville operator in L 2 ) were given. These results were then strengthened and developed by the authors themselves [37][38][39][40][41][42][43][44], and in successive papers [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][17][18][19][20][21][22][23][24][28][29][30]33,34,46,47,[49][50][51][52][53][54][55][56][58][59]…”

“…(The first investigation in the second group is due to Otelbaev; see [51,56].) Each of these approaches has its advantages and disadvantages, which will be made clear after obtaining unconditional criteria for the separability of (1.1) in L p for p ∈ (1, ∞) (for p = 1, (1.7) holds automatically; see [22,30,45,53]).…”

“…Since we are studying (1.1), according to our classification, this research belongs to the second group of papers. In particular, we study conditions for the separability of (1.1) in L p , remaining in the framework of the approach to (1.7) that was developed in the papers [3,19,20,23,30,45,53,61].…”

Methods of analysis of the condition for correct solvability in L p (ℝ) of general Sturm-Liouville equations

“…Note that in this paper some results from [4], [5] are strengthened. Moreover, in the special case r ≡ 1, 0 q ∈ L loc (R), another more effective form for the solution of the considered problem is obtained (for details see [1], [3])). The structure of the paper is as follows.…”