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On some weighted norm inequalities for Littlewood-Paley operators. (English) Zbl 1177.42016
Define the dyadic square function \(S_d(f)\) by \(S_d(f)(x)=\Bigl(\sum_{Q}(f_Q-f_{\tilde Q})^2\chi_Q(x)\bigr)^{1/2}\), where \(Q\) moves over all dyadic cubes, \(f_Q=|Q|^{-1}\int_Qf(y)dy\), and \(\tilde Q\) is the smallest dyadic cube with \(\tilde Q\supsetneq Q\). Let \(\varphi\in C^\infty(\mathbb R^n)\), \(\text{supp}\varphi\subset \{|x| \leq1\}\), and \(\int\varphi dy=0\). Define the Littlewood-Paley \(g_\lambda^*\) function \(g_{\varphi,\lambda}^*(f)\) by \[ g_{\varphi,\lambda}^*(f)(x)=\int_{\mathbb R_+^{n+1}}|f*\varphi_t(y)|^2\Bigl( \frac{t}{t+|x-y|}\Bigr)^{\lambda n}\frac{dy\,dt}{t^{n+1}}\quad(\lambda>0). \] The author shows : (1) Let \(S(f)\) be either \(S_d(f)\) or \(g_{\varphi,\lambda}^*(f)\) for \(\lambda>2\). Then for any weights \(w\) and \(v\), and any function \(f\), \[ \|S(f)\|_{L_w^2} \leq c\sqrt{\|v^{-1}\|_{A_\infty}\,\|(w,v)\|_{A_2}}\|f\|_{L_v^2}, \] where \(c=c_n\) if \(S(f)=S_d(f)\) and \(c=c_{\varphi,\lambda,n}\) if \(S(f)= g_{\varphi,\lambda}^*(f)\).
(2) Suppose that for two functions \(f\) and \(g\), \[ \|f\|_{L_w^2} \leq c_0\sqrt{\|v^{-1}\|_{A_\infty}\,\|(w,v)\|_{A_2}}\|g\|_{L_v^2} \] for all weights \(w\) and \(v\). Then for any \(p>1\) and \(w\in A_p\), \[ \|f\|_{L_w^p} \leq c\|w\|_{A_p}^{\max\{1,p/2\}/(p-1)}\|g\|_{L_w^p}, \] where \(c\) depends only on \(p\) and \(n\).
Using the above tow results, the author deduces two results. One is : (3) For any \(p>1\), \(\|S(f)\|_{L_w^p}\leq c\|w\|_{A_p}^{\max\{1,p/2\}/(p-1)}\|g\|_{L_w^p}\). The other is: (4) Let \(p>1\). Let \(\Omega\in C^\infty(S^{n-1})\) with \(\int \Omega d\sigma(y')=0\). For the maximal Calderón-Zygmund singular integral \(T_*\) with kernel \(K(x)=\Omega(x/|x|)|x|^{-n}\), there is \(c>0\) such that \(\|T_*(f)\|_{L_w^p} \leq c\|w\|_{A_p}^{\frac{1}{2}+\max\{1,p/2\}/(p-1)}\|f\|_{L_w^p}\).

42B25 Maximal functions, Littlewood-Paley theory
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
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