-boundedness of the Hilbert transform

The Hilbert transform is essentially the only singular operator in dimension 1. This undoubtedly makes it one of the the most important linear operators in harmonic analysis. The Hilbert transform has had a profound bearing on several theoretical and physical problems across a wide range of disciplines; this includes problems in Fourier convergence, complex analysis, potential theory, modulation theory, wavelet theory, aerofoil design, dispersion relations and high-energy physics, to name a few. In this monograph, we revisit some of the established results concerning the global behavior of the Hilbert transform, namely that it is is weakly bounded on L(R), and strongly bounded on Lp(R) for 1 < p < ∞, and provide a self-contained derivation of the same using real-variable techniques. This note is partly based on the expositions on the Hilbert transform in [1, 3]. Context The Hilbert transform of a sufficiently well-behaved function f(x) is defined to be H f(x) = 1 π lim ε→0 ∫ |t|>ε f(x− t) dt t . (1) It is not immediately not clear that H f(x) is well-defined even for nice functions f(x). Though (1) “almost” looks like an ordinary convolution, there are however certain technical subtleties associated with the transform. The primitive idea behind the definition of the transform is quite simple, namely to transform f(x) by convolving with the kernel 1/πx. It is doing so rigorously that one encounters technical difficulties since the kernel fails to be absolutely integrable owing to its slow decay and, more importantly, the singularity at the origin. The limiting argument in (1) is used to avoid the singularity by truncating the kernel around the origin in a systematic fashion; as will be shown shortly, this indeed works for sufficiently regular functions. The other pathology, namely the slow decay of the [email protected]. Biomedical Imaging Group, École Polytechnique Fédérale de Lausanne (EPFL), Switzerland.