Addendum to the paper “On singular integrals”. (English) Zbl 0263.44003

From the authors’ abstract: Let \(x\in E^n\), \(\tilde f_{\varepsilon}(x) = (K_{\varepsilon}*f)(x)\), where \(K_{\varepsilon}(x) = \Lambda(x) | x|^{-n}\) for \(| x|>\varepsilon\) and is 0 otherwise, and \(\Lambda(x)\) is a homogeneous function of degree \(0\), whose mean value over the unit sphere \((\Sigma)\) \(| x| = 1\) is \(0\). Let \(\omega_1(\delta)\) be the “rotational” modulus of continuity of \(\Lambda\) in the metric \(L^1\): \[ \omega_1(\delta)= \sup_{| p|<\delta}\int_{\Sigma} | \Lambda(\rho x)-\Lambda(x)\,d\sigma, \] where \(\rho\) is an arbitrary rotation on \(\Sigma\), \(|\rho|\) its rnagnitude, and \(d\sigma\) the element of surface area. The note ciarifies a proof of the following theorem concerning section 13 (a) of the paper “On the existence of singular integrals” [the authors and M. Weiss, Proc. Symp. Pure Math. 10, 56–73 (1967; Zbl 0167.12202)]: If \(\omega_1(\delta)\) satisfies the Dini condition, i.e., \[ \int_0^1 \delta^{-1}\omega_1(\delta)\,d\delta<\infty, \]
then the operation \(\sup_{\varepsilon} |\tilde f_{\varepsilon}(x)|=Tf\) is of weak type \((1,1)\).
Reviewer: Yung Ming Chen


42B25 Maximal functions, Littlewood-Paley theory


Zbl 0167.12202
Full Text: DOI EuDML