Boundedness of the kernel B_{k,a} for the one-dimensional (k,a)-generalized Fourier transform

Determine whether the integral kernel B_{k,a}(x, X) of the one-dimensional (k,a)-generalized Fourier transform F_{k,a} is bounded on R×R for general parameters k and a; specifically, ascertain if there exists a constant C such that sup_{x,X∈R} |B_{k,a}(x, X)| ≤ C.

Background

The transform F_{k,a} has an integral representation involving a kernel B_{k,a}, which governs mapping properties such as L^p-bounds. In the classical case and certain deformations (e.g., a=2 under suitable constraints on k), boundedness of the kernel facilitates pointwise and norm estimates for the transform.

The authors highlight that, even in one dimension, the boundedness of B_{k,a} remains an unresolved issue in general, although for the specific case a=2 and appropriate k the kernel is known to be bounded. Resolving boundedness for arbitrary a would advance understanding of the operator’s uniform continuity and enable stronger functional inequalities analogous to those known for classical Fourier-type transforms.