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Boundedness for multilinear commutator of Littlewood-Paley operator on Hardy and Herz-Hardy spaces. (English) Zbl 1134.42011

Let
\[ S_{\psi}^{b}(f)(x) = \Biggl\{\iint_{\{| x-y|< t\}}| F_t^{b}(f)(x,y)|^2\, \frac{dy\,dt}{t^{n+1}} \Biggr\}^{1/2}, \]
where
\[ F_t^{b}(f)(x,y)= \int_{\mathbb{R}^n} (b(x)-b(y)) \psi_t (y-t)f(z)\,dz \]
and \(\psi\) is a Littlewood-Paley function. The operator \(S_{\psi}^{b}\) is not bounded from \(H^p(\mathbb{R}^n)\) to \(L^p(\mathbb{R}^n)\) where \(b \in \text{BMO}(\mathbb{R}^n)\). If \(H^p(\mathbb{R}^n)\) is replaced by a suitable atomic Hardy space \(H_b^p(\mathbb{R}^n)\), this operator is bounded from \(H_b^p(\mathbb{R}^n)\) to \(L^p(\mathbb{R}^n)\). A function \(a(x)\) is called a \((p,b)\)-atom, if
\[ \operatorname{supp} a \subset B, \quad \| a \|_{L^{\infty}} \leq | B |^{-1/p}, \quad \int_B a(x)\,dx = \int_B a(x)b(x)\,dx=0, \]
and \(f\) is said to belong to \(H_b^p(\mathbb{R}^n)\) if \(f\) can be written as
\[ f(x) = \sum_{j=1}^{\infty}\lambda_j a_j(x), \]
where \(a_j\) are \((p,b)\)-atoms and \(\sum_{j=1}^{\infty} | \lambda_j |^p < \infty.\)
Liu, Lu and Xu [L. Liu, Lobachevskii J. Math. 12, 63–71 (2003; Zbl 1022.42010)] proved the following.
Let \(n/(n+1) < p \leq 1\). If \(b \in \text{BMO}(R^n)\), then \(S_{\psi}^{b}\) is bounded from \(H_b^p(\mathbb{R}^n)\) to \(L^p(\mathbb{R}^n)\). The authors generalize this theorem to multilinear commutators and Herz-Hardy spaces.

MSC:

42B25 Maximal functions, Littlewood-Paley theory
42B20 Singular and oscillatory integrals (Calderón-Zygmund, etc.)
42-02 Research exposition (monographs, survey articles) pertaining to harmonic analysis on Euclidean spaces

Citations:

Zbl 1022.42010
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