A note on a modified Hilbert transform

The Hilbert transform (Formula presented.) is a useful tool in the mathematical analysis of time-dependent partial differential equations in order to prove coercivity estimates in anisotropic Sobolev spaces in case of a bounded spatial domain Ω, but an infinite time interval (Formula presented.). Instead, a modified Hilbert transform (Formula presented.) can be used if we consider a finite time interval (Formula presented.). In this note we prove that the classical and the modified Hilbert transformations differ by a compact perturbation, when a suitable extension of a function defined on a bounded time interval (Formula presented.) onto (Formula presented.) is used. This result is important when we deal with space–time variational formulations of time-dependent partial differential equations, and for the implementation of related space–time finite and boundary element methods for the numerical solution of parabolic and hyperbolic equations with the heat and wave equations as model problems, respectively.