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Weighted estimates for singular integral operators on \(CMO\) spaces. (English) Zbl 1210.42026

Let \(E\subset {\mathbb R}^n\) and let \(|E|\) be its Lebesgue measure. Define \(B(0,R)\equiv\{x\in {\mathbb R}^n: |x|\leq R\}\) for any \(R>0\). For a weight function (namely, a locally integrable nonnegative function) \(\omega\), let \(\omega(E)\equiv\int_E\omega(x)\,dx\). For \(p\in[1,\infty)\), \(q\in(\frac{n}{n+1},1]\) and a weight function \(\omega\), the weighted \(CMO\) space \(CMO_q^p(\omega)({\mathbb R}^n)\) and the weighted weak \(CMO\) space \(WCMO_q^1(\omega)({\mathbb R}^n)\) are defined, respectively, by \[ CMO_q^p(\omega)({\mathbb R}^n)\equiv\{f\in L^p_{loc}({\mathbb R}^n): \|f\|_{CMO_q^p(\omega)}<\infty\} \] and \[ WCMO_q^1(\omega)({\mathbb R}^n)\equiv\{f\in L^1_{loc}({\mathbb R}^n): \|f\|_{WCMO_q^1(\omega)}<\infty\}, \] where \[ \|f\|_{CMO_q^p(\omega)}\equiv\sup_{R \geq1}\inf_c|B(0,R)|^{1-\frac1p-\frac1q} \left\{\int_{B(0,R)}|f(x)-c|^p\omega(x)\,dx\right\}^{\frac1p} \] and \[ \|f\|_{WCMO_q^1(\omega)}\equiv\sup_{R \geq1}|B(0,R)|^{-\frac{1}{q}}\inf_c\sup_{\lambda>0}\lambda \omega\left(\{x\in B(0,R):\;|f(x)-c|>\lambda\}\right). \] For \(p\in(1,\infty)\), a weight function \(\omega\) is called to belong to \(A_p\) if there exists a positive constant \(C\) such that for all balls \(Q\subset\mathbb{R}^n\), \[ \left(\frac{1}{|Q|}\int_Q\omega(x)\,dx\right) \left(\frac{1}{|Q|}\int_Q\omega(x)^{\frac{1}{1-p}}\,dx\right)^{p-1}\leq C; \] a weight function \(\omega\) is called to belong to \(A_1\) if \[ \frac{1}{|Q|}\int_Q\omega(x)\,dx \leq C\text{essinf}_{x\in Q}\omega(x). \] For \(\delta>0\), a weight function \(\omega\) is called to belong to \(RD(\delta)\) if there exists a positive constant \(C\) such that for all \(R>0\) and \(j>0\), \(\frac{\omega(B(0,2^jR))}{\omega(B(0,R))}\geq C2^{\delta j}\). An operator \(T\) is called a standard singular integral operator, if there exists a function \(K\) and a positive constant \(C_K\) such that \[ T(f)(x)\equiv \text{p.\,v.}\int_{\mathbb{R}^n}K(x-y)f(y)\,dy \] exists almost everywhere for all \(f\in L^2(\mathbb{R}^n)\), where \(|K(x)|\leq\frac{C_K}{|x|^n}\) and \(|\nabla K(x)|\leq\frac{C_K}{|x|^{n+1}}\) for all \(x\neq 0\) and \(\int_{\varepsilon<|x|<N}K(x)\,dx=0\) for all \(0<\varepsilon<N<\infty\). The modified singular integral operator \(\widetilde{T}\) is defined by \[ \widetilde{T}(f)(x)\equiv \text{p.\,v.}\int_{\mathbb{R}^n} \{K(x-y)-K(-y)\chi_{\{|y|\geq 1\}}\}f(y)\,dy \] for suitable \(f\) and all \(x\in \mathbb{R}^n\).
The authors show that if \(p\in(1,\infty), q\in(\frac{n}{n+1},1], \omega\in A_p\) and \(\omega\in RD(\delta)\), where \(\frac{\delta}{p}>n(\frac{1}{p}+\frac{1}{q}-1)-1\), then \(\widetilde{T}\) is bounded on \(CMO^p_q(\omega)(\mathbb{R}^n)\). Moreover, the authors show that for any \(q\in(\frac{n}{n+1},1]\), if \(\omega\in A_1\) and \(\omega\in RD(\delta)\), where \(\delta>\frac{n}{q}-1\), then \(\widetilde{T}\) is bounded from \(CMO^1_q(\omega)(\mathbb{R}^n)\) to \(WCMO^1_q(\omega)(\mathbb{R}^n)\).

MSC:

42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42B35 Function spaces arising in harmonic analysis
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