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Boundary values of Cauchy transforms in weighted spaces of integrable distributions. (English) Zbl 1047.46033
L. Schwartz generalized the Hilbert transformation \[ H:L^p(\mathbb R)\longrightarrow L^p(\mathbb R),\;f\longmapsto f\ast\text{vp}\,\frac 1x,\;1<p<\infty, \] to the spaces \(\mathcal D'_{L^p}\) of distributions which are finite sums of derivatives of \(L^p\)-functions [Théorie des distributions, Nouv. éd., Hermann, Paris (1966; Zbl 0149.09501), p. 259]. In a short note [Anais Acad. Brasil. Ci. 34, 13–21 (1962; Zbl 0109.33903), p. 18] he showed that the definition of \(H\) can be extended to the weighted space \(w\mathcal D'_{L^1}=(1+x^2)^{1/2}\mathcal D_{L^1}'\) and that the condition \(T\in w\mathcal D_{L^1}'\) is the most general one which allows convolution with \(\text{vp}\, \frac 1x.\) Using H. G. Tillmann’s analytic representation of distributions, H. Bremermann and M. Orton defined \(H\) on all of \(\mathcal D'\) [cf., e.g., M. Orton, SIAM J. Math. Anal. 4, 656–670 (1973; Zbl 0236.46045)].
On the spaces \(w^k\mathcal D_{L^1}',\;k=2,3,\dots,\) a more constructive definition of \(H\) was given and investigated more closely by C. Carton-Lebrun [Appl. Anal. 29, 235–251 (1988; Zbl 0577.46039)]. Essentially, a Hilbert transform \(H_{\eta,k}\) is defined, depending on a cut-off function \(\eta\) and the power \(k,\) by putting \[ H_{\eta,k}f:=H(\eta f)+x^kH\bigl(x^{-k}(1-\eta)f\bigr). \] The paper under review repeats the main properties of \(H_{\eta,k}\) in the introductory section. In Section 5, a Cauchy transform \(C_{\eta,k}\) on \(w^k\mathcal D'_{L^1}\) is defined analogously to \(H_{\eta,k}.\) Theorem 5.2 states its analyticity, its mapping properties and its boundary values (i.e., the Plemelj relations which are also called Sohozki formulas). Note that similar statements are given in the above mentioned paper of Orton but not on continuity.
MSC:
46F12 Integral transforms in distribution spaces
46F20 Distributions and ultradistributions as boundary values of analytic functions
30E20 Integration, integrals of Cauchy type, integral representations of analytic functions in the complex plane
44A15 Special integral transforms (Legendre, Hilbert, etc.)
30E25 Boundary value problems in the complex plane
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