Generalized Hilbert transforms in tempered distributions (Q583553)

The author establishes a method for defining the ``generalized Hilbert transform'' on the whole space of tempered distributions. For that purpose he introduces the space \({\mathcal D}_{L^ p_ k}({\mathbb{R}})\), which is the projective limit of the reflexive Banach spaces \(L^ p_{k,\ell}({\mathbb{R}}):\) \[ L^ p_{k,\ell}({\mathbb{R}})=\{\phi \in {\mathcal S}':\quad q^ p_{k,\ell}(\phi):=\max_{0\leq \alpha \leq k,0\leq \beta \leq 1}\| x^{\alpha}D^{\beta}\phi (x)\|_{L^ p}<\infty \}, \] where \(1<p<\infty\) and k, \(\ell\) are nonnegative integer. Let \(a=(a_ 1,...,a_ k)\) be a k-tuple of complex numbers, with nonzero imaginary part. Then the author shows that the limit \[ H_ a\phi =\lim_{\epsilon \to 0^+,N\to \infty}H_ a^{\epsilon,N} \phi, \] in \({\mathcal D}_{L^ p_ k}({\mathbb{R}})\), where \[ (H_ a^{\epsilon,N} \phi)(x)=\frac{1}{\pi (x-a_ 1)...(x-a_ k)}\int_{\epsilon <| t| <N}(x-t-a_ 1),..(x-t-a_ k)\frac{\phi (x-t)}{t}dt, \] exists for any \(\phi\) in \({\mathcal D}_{L^ p_ k}({\mathbb{R}})\), and he calls it the ``generalized Hilbert transform''. Finally, for f in \({\mathcal D}'_{L_ k}({\mathbb{R}})\) he defines the functional \(H^*_ af\) via the formula \(<H^*_ af,\phi >:=<f,H_ a\phi >.\) It turns out that then \(H^*_ af\) is the \({\mathcal D}'_{L^ p_ k}({\mathbb{R}})\) and has all the ``expected'' properties of the Hilbert transform. Moreover, as the author proves, the inductive limit of \({\mathcal D}'_{L^ p_ k}({\mathbb{R}})\) as \(k\to \infty\) is just \({\mathcal S}'\), hence this generalized Hilbert transform can be applied to any tempered distribution and it is independent of k and a up to a polynomial. In view of that, this definition of the Hilbert transform contains some others such as, e.g., those of H. G. Tillman (1953), S. Koizumi (1959-60) and E. J. Beltrami and M. R. Wohlers (1965-66).