## 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

### MSC:

 42B25 Maximal functions, Littlewood-Paley theory

Zbl 0167.12202
Full Text: