Weighted norm inequalities for the multilinear Calderón-Zygmund operators
The purpose of this paper is to establish some weighted estimates for the multilinear Calderón-Zygmund operator and its corresponding commutator. Let \(m\geq 1\), let \(K(x;y_1,\dots, y_m)\) be a locally integrable function defined away from the diagonal \(x=y_1=y_2=\cdots=y_m\) in \((\mathbb R^n)^{m+1}\), let \(A>0\) and \(\gamma \in (0,1]\) be two constants. \(K\) is a kernel in \(m\text{-CZK}(A, \gamma)\) if it satisfies the size condition that for all \((x, y_1, \dots, y_m) \in (\mathbb R^n)^{m+1}\) with \(x \neq y_j\) for some \(1 \leq j \leq m\), \[ |K(x;y_1,\dots,y_m)| \leq \frac{A}{(|x-y_1|+\cdots+|x-y_m|)^{mn}}, \] and satisfies the regularity condition that \[ K(x;y_1,\dots,y_m) - K(x';y_1,\dots, y_m)| \leq \frac{A|x-x'|^{\gamma}}{(|x-y_1|+\cdots+|x-y_m|)^{mn+\gamma}} \] whenever \(\max_{1\leq k \leq m}|x-y_k| \geq 2|x-x'|\), and also that for each fixed \(k\) with \(1\leq k \leq m\), \[ |K(x;y_1,\dots,y_k,\dots,y_m)-K(x;y_1,\dots,y'_k,\dots,y_m)| \leq \frac{A|y_k-y'_k|^\gamma}{(|x-y_1|+\cdots+|x-y_m|)^{mn+\gamma}} \] whenever \(\max_{1\leq j \leq m}|x-y_j| \geq 2|y_k-y'_k|\). An operator \(T\) defined on the \(m\)-fold product of Schwartz spaces and taking values in the space of tempered distributions, is said to be an \(m\)-linear Calderón-Zygmund operator with kernel \(K\) if {\parindent=6,5mm \begin{itemize}\item[(a)] \(T\) is \(m\)-linear, \item[(b)] for some \(q_1,\dots, q_m \in [1,\infty]\) and some \(q \in (0,\infty)\) with \(1/q = \sum_{k=1}^{m}1/q_k\), \(T\) can be extended to be a bounded operator from \(L^{q_1}(\mathbb R^n) \times L^{q_2}(\mathbb R^n) \times \cdots \times L^{q_m}(\mathbb R^n)\) to \(L^{q}(\mathbb R^n)\), \item[(c)] for \(f_1,\dots, f_m \in L^2(\mathbb R^n)\) with compact supports, and for \(x \not\in \bigcap_{k=1}^{m}\operatorname{supp}f_k\), \[ T(f_1,\dots, f_m)(x)=\int_{(\mathbb R^n)^m} K(x;y_1,\dots, y_m) \prod_{k=1}^{m}f_k(y_k)\,dy_1\cdots dy_m, \] and \(K\) is in \(m\text{-CZK}(A,\gamma)\) for some constants \(A\) and \(\gamma\). \end{itemize}} Let \(b_1, b_2, \dots, b_m\) be BMO\((\mathbb R^n)\) functions and set \(\vec{b}=(b_1, b_2, \dots, b_m)\). Define the commutator of the operator \(T\) by \[ T_{\vec{b}}(f_1, \dots, f_m)(x) = \sum_{j=1}^{m}T_{\vec{b}}^{j}(f_1, \dots, f_m)(x), \] where \[ T_{\vec{b}}^{j}(f_1, \dots, f_m)(x) = b_j(x)T(f_1, \dots, f_j, \dots, f_m)(x) - T(f_1, \dots, b_jf_j, \dots, f_m)(x). \] In the paper the author establishes some weighted norm inequalities with general weights for the multilinear Calderón-Zygmund operator \(T\) and the commutator \(T_{\vec{b}}\), which can be stated as follows. Let \(m \geq 1\), let \(T\) be an \(m\)-linear Calderón-Zygmund operator. {\parindent=6,5mm \begin{itemize}\item[(i)] If \(p_1\in [1, \infty)\), \(p_2, \dots, p_m \in [1, \infty] \) and \(p \in (0, \infty)\) with \(1/p = \sum_{k=1}^{m}1/{p_k}\), then for any \(\delta >0\), there exists a positive constant \(C\), such that for all weights \(\omega\) and all bounded functions \(f_1, \dots, f_m\) with compact supports, \[ \|T(f_1, \dots, f_m)\|_{L^{p, \infty}(\mathbb R^n, \omega)} \leq C\|f_1\|_{L^{p_1}(\mathbb R^n, M_{L(\log L)^{p_1 -1 +\delta}} \omega)} \prod_{k=2}^{m}\|f_k\|_{L^{p_k}(\mathbb R^n, M\omega)}, \] where as in the sequel, when \(j_0 \geq j\), the term \(\prod_{k=j_0}^{j-1}\) will disappear. Furthermore, if \(p_1 \in (1, \infty)\) and \(p_1, \dots, p_m \in (1, \infty]\), then \[ \|T(f_1, \dots, f_m)\|_{L^{p}(\mathbb R^n, \omega)} \leq C\|f_1\|_{L^{p_1}(\mathbb R^n, M_{L(\log L)^{p_1 -1 +\delta}} \omega)} \prod_{k=2}^{m}\|f_k\|_{L^{p_k}(\mathbb R^n, M\omega)}. \] \item[(ii)] If \(p_1, \dots, p_m \in (1, \infty]\) and \(p \in (0, \infty) \) with \(1/p = \sum_{k=1}^{m}1/{p_k}\), then for any \(\delta >0\), there exists a positive constant \(C\), such that for all weights \(\omega\) and all bounded functions \(f_1, \dots, f_m\) with compact supports, \[ \|T(f_1, \dots, f_m)\|_{L^{p}(\mathbb R^n, \omega)} \leq C\prod_{k=1}^{m}\|f_k\|_{L^{p_k}(\mathbb R^n, M_{L(\log L)^{p^* -1 +\delta}} \omega)}, \] where as in the sequel, \(p^* = \max\{1, p\}\). \end{itemize}} Let \(m \geq 1\), let \(T\) be an \(m\)-linear Calderón-Zygmund operator. If \(p_1, \dots, p_m \in (1, \infty]\) and \(p \in (0, \infty) \) with \(1/p = \sum_{k=1}^{m}1/{p_k}\), then for any \(\delta >0\), there exists a positive constant \(C\) such that for all weights \(\omega\) and all bounded functions \(f_1, \dots, f_m\) with compact supports, \[ \|T_{\vec{b}}(f_1, \dots, f_m)\|_{L^{p}(\mathbb R^n, \omega)} \leq C\|\vec{b}\|_{\text{BMO}(\mathbb R^n)}\prod_{k=1}^{m}\|f_k\|_{L^{p_k}(\mathbb R^n, M_{L(\log L)^{2p^* -1 +\delta}} \omega)}, \] with \(\|\vec{b}\|_{\text{BMO}(\mathbb R^n)} = \sum_{1 \leq k \leq m}\|b_k\|_{\text{BMO}(\mathbb R^n )}\).
scientific article; zbMATH DE number 5857640