Weak type \((1,1)\) bounds for a class of the Littlewood-Paley operators (Q1775419)

Let \(\Omega\in L^1(S^{n-1}) \) be homogeneous of degree zero and have cancellation property \[ \int_{S^{n-1}}\Omega(x')\,d\sigma(x')=0. \] The authors consider the parametrized Littlewood-Paley function \(\mu_{\lambda}^{*,\rho}\) of Marcinkiewicz type, defined by \[ \mu_{\lambda}^{*,\rho}(f)(x)=\left(\iint_{\mathbb R_+^{n+1}} \left( \frac {t}{t+| x-y| }\right)^{\lambda n}\big| (\varphi_t^\rho\ast f)(y)\big| ^2 \frac {dydt}{t^{n+1}}\right)^{1/2} \] for \(\lambda>1\), where \(\varphi_t^\rho(x) =\frac 1{t^n}\Omega(x)| x/t| ^{-n+\rho}\chi_{| x| \leq1}(x/t)\). Let \(\omega_q(\delta)\) be the \(L^q\) modulus of continuity, defined by \(\omega_q(\delta) =\sup_{\| \gamma\| \leq \delta}\bigl(\int_{S^{n-1}} | \Omega(\gamma x')-\Omega(x')| ^q\,d\sigma(x') \bigr)^{1/q}\), where \(\gamma\) is a rotation on \(S^{n-1}\) and \(\| \gamma\| =\sup_{x'\in S^{n-1}}{\| \gamma x'-x'\| }\). The authors' main result is as follows: Let \(\Omega\in L^2(S^{n-1}) \) satisfy the cancellation condition and \( \int_0^1 \frac {\omega _2(\delta)}{\delta } (1+| \log \delta | )^\sigma d\delta <\infty\) for some \(\sigma>1\). Then, if \(\rho>n/2\) and \(\lambda>2\rho/n+1/n+1\), \( \mu_{\lambda}^{*,\rho}\) has the weak \((1, 1)\) property. From this, it follows the weak \((1,1)\) boundedness of the area integral of the Marcinkiewicz type. The weak \((1,1)\) boundedness of Marcinkiewicz integral was given by \textit{D. Fan} and \textit{S. Sato} [Tohoku Math. J., II. Ser. 53, 265--284 (2001; Zbl 1021.42005)].