Endpoint boundedness for multilinear integral operators of some sublinear operators on Herz and Herz type Hardy spaces.

Wang, Kewei; Liu, Lanzhe

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Let $b_{1}, b_{2} \in BMO (ℝ^{n})$ and $T_{σ}$ be a bilinear Fourier multiplier operator with associated multiplier $σ$ satisfying the Sobolev regularity that ${sup}_{κ \in ℤ} {∥ σ_{κ} ∥}_{W^{s_{1}, s_{2}} (ℝ^{2 n})} < \infty$ for some $s_{1}, s_{2} \in (n / 2, n]$ . In this paper, the behavior on $L^{p_{1}} (ℝ^{n}) \times L^{p_{2}} (ℝ^{n})$ $(p_{1}, p_{2} \in (1, \infty))$ , on $H^{1} (ℝ^{n}) \times L^{p_{2}} (ℝ^{n})$ $(p_{2} \in [2, \infty))$ , and on $H^{1} (ℝ^{n}) \times H^{1} (ℝ^{n})$ , is considered for the commutator $T_{σ, \vec{b}}$ defined by $\begin{matrix} T_{σ, \vec{b}} (f_{1}, f_{2}) (x) = & b_{1} (x) T_{σ} (f_{1}, f_{2}) (x) - T_{σ} (b_{1} f_{1}, f_{2}) (x) & + b_{2} (x) T_{σ} (f_{1}, f_{2}) (x) - T_{σ} (f_{1}, b_{2} f_{2}) (x) . \end{matrix}$ By kernel estimates of the bilinear Fourier multiplier operators and employing some techniques in the theory of bilinear singular integral operators, it is proved that these mapping properties are very similar to those...