Inversion formulae for the cosh-weighted Hilbert transform (Q2839355)

The so-called \(\cosh\)-weighted Hilbert transform \(H_\mu\), \(\mu\geq 0\), is the integral operator NEWLINE\[NEWLINE H_\mu(f)(y) : = \dfrac 1\pi \int_{-1}^1 \dfrac{\cosh(\mu(x-y))}{x-y} f(x) dx, NEWLINE\]NEWLINE defined on the \(L^p:=L^p([-1,1])\) space for \(p\in (1,+\infty)\). The authors deal with the problem of determining the function \(f\) from measurements of its transform \(H_\mu(f)\). They give several explicit formulae reconstructing \(f\) for Hölder-continuous functions on \([-1,1]\) as well as for \(L^p\) classes. Their obtained inversions formulae are analogous to those inverting the classical finite Hilbert transform \(H_0\) (\(\mu=0\)). Moreover, an analysis of the action of \(H_\mu\) in the \( L^p\) spaces with \( p>1\) is described and a characterization of its null space and its range are provided. Mainly, they show that the null-space turns out to be one-dimensional in \( L^p\) for any \( p\in (1,2)\) and trivial if \(p\geq 2\). This recovers the known result for \(H_0\). Also, the authors prove that the \(\cosh\)- weighted Hilbert transform is not Fredholm if and only if \(p=2\). In such a case, a range condition is given.