# American Institute of Mathematical Sciences

May  2018, 17(3): 1023-1052. doi: 10.3934/cpaa.2018050

## Spaces admissible for the Sturm-Liouville equation

 1 Department of Mathematics, Ben-Gurion University of the Negev, P.O.B. 653, Beer-Sheva, 84105, Israel 2 Department of Mathematics, Bar-Ilan University, 52900 Ramat Gan, Israel

Received  August 2016 Revised  November 2017 Published  January 2018

We consider the equation
 $-{y}''(x)+q(x)y(x)=f(x),\ \ \ \ x\in \mathbb{R}\text{ }\ \ \ \ \ \ \ \ \ \ \left( 1 \right)$
where
 $f∈ L_p^{\text{loc}}(\mathbb R),$
 $p∈[1,∞)$
and
 $0≤ q∈ L_1^{\text{loc}}(\mathbb R).$
By a solution of (1) we mean any function
 $y,$
absolutely continuous together with its derivative and satisfying (1) almost everywhere in
 $\mathbb R.$
Let positive and continuous functions
 $μ(x)$
and
 $θ(x)$
for
 $x∈\mathbb R$
be given. Let us introduce the spaces
 \begin{align} & {{L}_{p}}(\mathbb{R},\mu )=\left\{ f\in L_{p}^{\text{loc}}(\mathbb{R}):\|f\|_{{{L}_{p}}(\mathbb{R},\mu )}^{p}=\int_{-\infty }^{\infty }{|}\mu (x)f(x){{|}^{p}}dx < \infty \right\}, \\ & {{L}_{p}}(\mathbb{R},\theta )=\left\{ f\in L_{p}^{\text{loc}}(\mathbb{R}):\|f\|_{{{L}_{p}}(\mathbb{R},\theta )}^{p}=\int_{-\infty }^{\infty }{|}\theta (x)f(x){{|}^{p}}dx <\infty \right\}. \\ \end{align}
In the present paper, we obtain requirements to the functions
 $μ,θ$
and
 $q$
under which
1) for every function
 $f∈ L_p(\mathbb R,θ)$
there exists a unique solution (1)
 $y∈ L_p(\mathbb R,μ)$
of (1);
2) there is an absolute constant
 $c(p)∈(0,∞)$
such that regardless of the choice of a function
 $f∈ L_p(\mathbb R,θ)$
the solution of (1) satisfies the inequality
 $\|y\|_{L_p(\mathbb R,μ)}≤ c(p)\|f\|_{L_p(\mathbb R,θ)}.$
Citation: N. A. Chernyavskaya, L. A. Shuster. Spaces admissible for the Sturm-Liouville equation. Communications on Pure and Applied Analysis, 2018, 17 (3) : 1023-1052. doi: 10.3934/cpaa.2018050
##### References:
 [1] N. Chernyavskaya and L. Shuster, On the WKB-method, Diff. Uravnenija, 25 (1989), 1826-1829. [2] N. Chernyavskaya and L. Shuster, Estimates for the Green function of a general Sturm-Liouville operator and their applications, Proc. Amer. Math. Soc., 127 (1999), 1413-1426. [3] N. Chernyavskaya and L. Shuster, A criterion for correct solvability of the Sturm-Liouville equation in the space Lp(R), Proc. Amer. Math. Soc., 130 (2002), 1043-1054. [4] N. Chernyavskaya and L. Shuster, Classification of initial data for the Riccati equation, Boll. Unione Mat. Ital., 8 (2002), 511-525. [5] N. Chernyavskaya and L. Shuster, Davies-Harrell representations, Otelbaev's inequalities and properties of solutions of Riccati equations, J. Math. Anal. Appl., 334 (2007), 998-1021. [6] N. Chernyavskaya and L. Shuster, A criteria for correct solvability in Lp(R) of a general Sturm-Liouville equation, J. London Math. Soc. (2), 80 (2009), 99-120. [7] R. Courant, Differential and Integral Calculus, Vol. Ⅱ, Blackie and Son, Glasgow and London, 1936. [8] E. B. Davies and E. M. Harrell, Conformally flat Riemannian metrics, Schrödinger operators and semiclassical approximation, J. Diff. Eq., 66 (1987), 165-188. [9] E. Goursat, A Course in Mathematical Analysis, Vol. 1, Ch. IV, $\S$75, New York, Dover Publications, 1959. [10] L. W. Kantorovich and G. P. Akilov, Functional Analysis, Nauka, Moscow, 1977. [11] A. Kufner and L. E. Persson, Weighted Inequalities of Hardy Type, World Scientific Publishing Co., 2003. [12] J. L. Masssera and J. J. Schaffer, Linear Differential Equations and Function Spaces, Pure and Applied Mathematics, Vol. 21, Academic Press, New York -London, 1966. [13] K. Mynbaev and M. Otelbaev, Weighted Function Spaces and the Spectrum of Differential Operators, Nauka, Moscow, 1988. [14] M. Otelbaev, A criterion for the resolvent of a Sturm-Liouville operator to be a kernel, Math. Notes, 25 (1979), 296-297. [15] C. C. Titchmarsh, The Theory of Functions, Oxford University Press, 1939.

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##### References:
 [1] N. Chernyavskaya and L. Shuster, On the WKB-method, Diff. Uravnenija, 25 (1989), 1826-1829. [2] N. Chernyavskaya and L. Shuster, Estimates for the Green function of a general Sturm-Liouville operator and their applications, Proc. Amer. Math. Soc., 127 (1999), 1413-1426. [3] N. Chernyavskaya and L. Shuster, A criterion for correct solvability of the Sturm-Liouville equation in the space Lp(R), Proc. Amer. Math. Soc., 130 (2002), 1043-1054. [4] N. Chernyavskaya and L. Shuster, Classification of initial data for the Riccati equation, Boll. Unione Mat. Ital., 8 (2002), 511-525. [5] N. Chernyavskaya and L. Shuster, Davies-Harrell representations, Otelbaev's inequalities and properties of solutions of Riccati equations, J. Math. Anal. Appl., 334 (2007), 998-1021. [6] N. Chernyavskaya and L. Shuster, A criteria for correct solvability in Lp(R) of a general Sturm-Liouville equation, J. London Math. Soc. (2), 80 (2009), 99-120. [7] R. Courant, Differential and Integral Calculus, Vol. Ⅱ, Blackie and Son, Glasgow and London, 1936. [8] E. B. Davies and E. M. Harrell, Conformally flat Riemannian metrics, Schrödinger operators and semiclassical approximation, J. Diff. Eq., 66 (1987), 165-188. [9] E. Goursat, A Course in Mathematical Analysis, Vol. 1, Ch. IV, $\S$75, New York, Dover Publications, 1959. [10] L. W. Kantorovich and G. P. Akilov, Functional Analysis, Nauka, Moscow, 1977. [11] A. Kufner and L. E. Persson, Weighted Inequalities of Hardy Type, World Scientific Publishing Co., 2003. [12] J. L. Masssera and J. J. Schaffer, Linear Differential Equations and Function Spaces, Pure and Applied Mathematics, Vol. 21, Academic Press, New York -London, 1966. [13] K. Mynbaev and M. Otelbaev, Weighted Function Spaces and the Spectrum of Differential Operators, Nauka, Moscow, 1988. [14] M. Otelbaev, A criterion for the resolvent of a Sturm-Liouville operator to be a kernel, Math. Notes, 25 (1979), 296-297. [15] C. C. Titchmarsh, The Theory of Functions, Oxford University Press, 1939.
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