Weighted estimates for solutions of a Sturm-Liouville equation in the space
N.A. Chernyavskaya
Department of Mathematics and Computer Science,Ben-GurionUniversity of the Negev,Beer-Sheva, Israel
N. El-Natanov
Department of Computer Sciences and Mathematics, ArielUniversityCenter of Samaria, Ariel, Israel
L.A. Shuster
Department of Mathematics, Bar-IlanUniversity,Ramat Gan, Israel
Abstract.We consider an equation
(1)
where and for some Under these conditions, (1) is correctly solvable in
1. Introduction
In this work, we consider an equation
(1)
where and
(2)
Moreover, we always assume that equation (1) is correctly solvable in This means (see [4, Ch.III, §6, no.2] that the following assertions hold:
- for any function , there exists a unique solution of (1),
- there is an absolute constant such that the solution of (1), satisfies the inequality
(3)
Throughout the sequel, stands for absolute positive constants which are not essential for exposition.
Our general goal is to study the possibility of strengthening the a priori inequality (3). More precisely, we find requirements for a given weight function under which the solutions of (1), satisfy the estimate
(4)
2. Prelminaries
In the sequel, we assume that condition (2) holds and we do not mention it again in the statements.
Theorem 1. [2] Equation (1) is correctly solvable in if and only if there exists such that Here
(5)
Lemma 2. [3] Under condition (5), for a given consider the equations in
(6)
Each of the equations of (6) has a unique finite solution. Denote the solutions of equation (6) by , and d, respectively. Thus for a given the functions , (introduced in [1]) and (introduced by M. Otelbaev, see [5]) are well-defined.
3. Results
We now give the following definition.
Definition 1. Let and be continuous positive functions defined in We say that they are weakly equivalent in (and write if there is a constant such that for all , the following inequalities hold:
(7)
Remark 2. It is useful to note that if (7) (with constant ) are established only for
Theorem 3. For the validity of estimate (4) it is necessary, and under the condition also sufficient, that the inequality holds. Here
(8)
In the next two theorems, we find requirements to the function under which
Theorem 4. The functions and are weakly equivalent in if where
(9)
To state Theorem7, we need some parametrization of the set of functions
Definition 5. If there exist constants and such that for all and every the inequalities
(10)
hold, we say that the function belongs to the class (and write where
Lemma 6. For every we have
Theorem 7. Let , Then
(11)
4. Example
We consider the equation
(12)
where We show that the solution of the equation (12), , satisfies the inequality
(13)
if and only if
References
[1] N. Chernyavskaya and L. Shuster, On the WKB-method, Diff. Uravnenija 25 (1989), no.10, 1826-1829.
[2] N. Chernyavskaya and L. Shuster, A criterion for correct solvability of the Sturm-Liouville equation in the space , Proc. Amer. Math. Soc. 130 (2001), no.4, 1043-1054.
[3] N. Chernyavskaya and L. Shuster, Classification of initial data for the Riccati equation, Bollettino dela Unione Matematica Italiana 8, 5-B (2002), 511-525.
[4] R. Courant Partial Differential Equations, John Wiley & Sons, New York, 1962.
[5] K. Mynbaev and M. Otelbaev, Weighted Function Spaces and the Spectrum of Differential Operators, Nauka, Moscow, 1988.
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