Solving dual integral equations on Lebesgue spaces (Q2717539)

The authors study the dual integral equations of Titchmarsh type NEWLINE\[NEWLINE\begin{cases}\int_0^\infty t^\beta f(t)J_\alpha(xt) dt = g(x), &\text{for\( \quad 0 < x < 1\)}\\ \int_0^\infty f(t)J_\alpha(xt) dt =0,& \text{for\(\quad x> 1\)}, \end{cases}NEWLINE\]NEWLINE where \(J_\alpha\) stands for the Bessel function of order \(\alpha\), \(g\) is a given function and \(f\) is an unknown function. These equations are reformulated giving a better description in terms of continuous operators on \(L^p\) spaces, and are solved in these spaces. The solution is given both as an integral operator and as a Fourier-Neumann series whose \(L^p\) and almost everywhere convergence is studied. Uniqueness of the solution is also investigated.