On the correctness of the problem of inverting the finite Hilbert transform in certain aeroelastic models (Q914107)

The finite Hilbert transform of \(\gamma\), \(f(x)=\int^{1}_{- 1}\frac{\gamma (y)dy}{y-x}dy,\quad -1<y<1,\quad -1<x<1;\) arises in the theory of aeroelastic control systems wherein one solves for \(\gamma\) (y) in terms of f(x). The function f is assumed to be the sum of w and g, where \(g(x)=\int^{\infty}_{0}\frac{G(s)ds}{1-x+s}.\) Here w and G are constant multiples of the downwash function and the derivative of the circulation function, respectively. Recall that if J is an interval and p is a positive number then \(L^ p(J)\) is said to be the class of those Lebesgue measurable functions for which \(\int_{J}| G(x)|^ p dx\) is finite. When \(p=\infty\), \(L^{\infty}(J)\) is the class of essentially bounded functions on J. In view of the preceding and the fact that the summand w of f can usually be taken to be any class \(L^ p(-1,1)\), it is of importance to answer the question: What conditions on G ensure that g is in \(L^ p(-1,1)?\) The paper sets out to obtain such conditions which are presented in Propositions 2 and 3.