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

show all references

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