Indefinite integrals of functions from $L^p(\mathbb{R})$

Abstract

In this paper we seek conditions under which the indefinite integrals of a function $\varphi$ from $L^p(\mathbb{R})$ belong to $L^p(\mathbb{R}) + \mathbb{C}$. We prove that if the spectrum sp($\varphi$) of $\varphi$ is isolated from zero, then it is improperly integrable for $(1 \leq p \lt \infty)$ and its indefinite integrals belong to $L^p(\mathbb{R}) + \mathbb{C}$. Also, we give applications to the differential equation $u^1(x) + \lambda u(x) = \varphi(x)$.