Weight estimates for solutions of linear singular differential equations of the first order and the Everitt-Giertz problem. (English) Zbl 1265.34135

Chernyavskaya, N. A.; Shuster, L. A.

Differ. Integral Equ. 25, No. 5-6, 467-504 (2012).

Summary: We consider the equation \[ -y'(x)+q(x)y(x)=f(x),\; x\in \mathbb {R}, \] where \(f\in L_p(\mathbb {R})\), \(p\in [1,\infty ]\) (\(L_{\infty }(\mathbb {R}):=C(\mathbb {R})\)) and \(0\leq q\in L_1^{\text{loc}}(\mathbb {R})\). We assume that the above equation is correctly solvable in \(L_p(\mathbb {R})\). Let \(y\in L_p(\mathbb {R})\) be a solution of the above equation.
In the present paper, we find minimal requirements for the weight function \(r\in L_p^{\text{loc}}(\mathbb {R})\) under which the estimate \[ \| ry\| _p\leq c(p)\| f\| _p \] holds for all \(f\in L_p(\mathbb {R})\) with an absolute constant \(c(p)\in (0,\infty )\).

MSC:

34C11	Growth and boundedness of solutions to ordinary differential equations
34A30	Linear ordinary differential equations and systems

Keywords:

weight estimate; linear singular differential equation; separable operator