    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:
  N. Chernyavskaya and L. Shuster, On the WKB-method, Diff. Uravnenija, 25 (1989), 1826-1829.  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.  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.  N. Chernyavskaya and L. Shuster, Classification of initial data for the Riccati equation, Boll. Unione Mat. Ital., 8 (2002), 511-525.  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.  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.  R. Courant, Differential and Integral Calculus, Vol. Ⅱ, Blackie and Son, Glasgow and London, 1936.  E. B. Davies and E. M. Harrell, Conformally flat Riemannian metrics, Schrödinger operators and semiclassical approximation, J. Diff. Eq., 66 (1987), 165-188.  E. Goursat, A Course in Mathematical Analysis, Vol. 1, Ch. IV, $\S$75, New York, Dover Publications, 1959.  L. W. Kantorovich and G. P. Akilov, Functional Analysis, Nauka, Moscow, 1977.  A. Kufner and L. E. Persson, Weighted Inequalities of Hardy Type, World Scientific Publishing Co., 2003.  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.  K. Mynbaev and M. Otelbaev, Weighted Function Spaces and the Spectrum of Differential Operators, Nauka, Moscow, 1988.  M. Otelbaev, A criterion for the resolvent of a Sturm-Liouville operator to be a kernel, Math. Notes, 25 (1979), 296-297.  C. C. Titchmarsh, The Theory of Functions, Oxford University Press, 1939.  show all references

##### References:
  N. Chernyavskaya and L. Shuster, On the WKB-method, Diff. Uravnenija, 25 (1989), 1826-1829.  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.  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.  N. Chernyavskaya and L. Shuster, Classification of initial data for the Riccati equation, Boll. Unione Mat. Ital., 8 (2002), 511-525.  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.  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.  R. Courant, Differential and Integral Calculus, Vol. Ⅱ, Blackie and Son, Glasgow and London, 1936.  E. B. Davies and E. M. Harrell, Conformally flat Riemannian metrics, Schrödinger operators and semiclassical approximation, J. Diff. Eq., 66 (1987), 165-188.  E. Goursat, A Course in Mathematical Analysis, Vol. 1, Ch. IV, $\S$75, New York, Dover Publications, 1959.  L. W. Kantorovich and G. P. Akilov, Functional Analysis, Nauka, Moscow, 1977.  A. Kufner and L. E. Persson, Weighted Inequalities of Hardy Type, World Scientific Publishing Co., 2003.  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.  K. Mynbaev and M. Otelbaev, Weighted Function Spaces and the Spectrum of Differential Operators, Nauka, Moscow, 1988.  M. Otelbaev, A criterion for the resolvent of a Sturm-Liouville operator to be a kernel, Math. Notes, 25 (1979), 296-297.  C. C. 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