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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 $$\|T_*(f)\|_{L_w^p} \leq c\|w\|_{A_p}^{\frac{1}{2}+\max\{1,p/2\}/(p-1)}\|f\|_{L_w^p}$$.

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