Birkhoff regularity in terms of the growth of the norm for the Green function

We consider the ordinary differential operator L generated on 0, 1] by the differential expression and n linearly independent, homogeneous boundary conditions at the endpoints. We assume that the coefficients p _k (x) are Lebesgue-integrable complex functions. If the boundary conditions are Birkhoff regular, then the Green function G(λ), being the kernel of the operator (L − λ)⁻¹, admits the asymptotic estimate (for sufficiently large |λ| > c ₀) , where M = M(c ₀) is a certain constant. In the present paper, we prove the converse assertion: the fulfillment of this estimate on some rays implies the regularity of the operator L. __________ Translated from Fundamentalnaya i Prikladnaya Matematika, Vol. 12, No. 6, pp. 231–239, 2006.