Calderón's reproducing formula and singular integral operators on a real line
Calderón's reproducing formula has been useful in harmonic analysis, e.g., in wavelet theory. It says that for suitable functions \(\phi\), \(u\), \(v\) with \[ u_t(x)= t^{- 1} u(t^{- 1} x),\quad v_t(x)= t^{- 1} v(t^{- 1} x), \] and with \(*\) denoting convolution, \[ \phi= \frac 1{c_{u, v}} \int_0^{\infty} \phi * u_t * v_t\frac{dt} t. \] The authors examine what happens when \(u * v\) is replaced by a finite Borel measure, \(\mu\in M\), where it is assumed that \(\mu(\{\pm\infty\})= 0\). Their main result assumes that \(\mu\) has a Fourier transform with even and odd parts that \(\mu_{\pm}(\eta)= \frac 12(\widehat\mu(\eta) \pm \widehat\mu(- \eta))\), and we set \(\alpha_{\pm}= \int_0^{\infty} \widehat \mu_{\pm}(\eta) \frac{d\eta}\eta\). If \(\phi\in L^2\) and \(\alpha_{\pm}\) are finite, \[ I(\mu, \phi)= \lim_{\varepsilon\to 0, \rho\to \infty} \int_{\varepsilon}^{\rho} \phi * \mu_t\frac{dt}t= \alpha_+ \phi+ \alpha_{-} H\phi, \] where the convergence is in \(L^2\) and \(H\phi\) denotes the Hilbert transform of \(\phi\), \[ H\phi(x)= \text{p.v. } \frac 1{\phi i} \int_{-\infty}^{\infty} \phi(y)/(x- y) dy. \] A variety of sufficient conditions are given to assure that the above conditions hold and to determine the interesting cases \(\alpha_+= 1\), \(\alpha_{-}= 0\) or \(\alpha_+= 0\), \(\alpha_{-}= 1\). For example, if \(\text{supp } \mu\subseteq (0, + \infty)\), \(\mu(0, +\infty)= 0\), and \(\int_0^{\infty} |\log |x||d|\mu|(x)< \infty\), then \[ I(\mu, \phi)= c\phi\quad \text{with} \quad c= \int_0^{\infty} \log 1/|x|\;d\mu(x). \] The function \(\log 1/|x|\) is in BMO, which is the dual of \(H^1\). The authors remark that in a later paper, one of the authors gives results when \(\mu\) is an absolutely continuous density in \(H^1\).
scientific article; zbMATH DE number 764049