A singular integral inequality on a bounded interval

Abstract:

An inequality of the form (1.1) is established, where $p,q$ are real-valued functions on an interval $[a,b)$ of the real line, with $- \infty < a < b < \infty ,p(x) > 0$ on $[a,b),{\mu _0}$ is a real number and $f$ is a complex-valued function in a linear manifold so chosen that all three integrals in (1.1) are absolutely convergent. The problem is singular in that while ${p^{ - 1}} \in L(a,b)$ we require $q$ to have a behavior at $b$ of such a form that $q \notin L(a,b)$.