Bilinear Decompositions for Products of Orlicz–Hardy and Orlicz–Campanato Spaces

For an Orlicz function \(\varphi \) $\phi$ with critical lower type \(i(\varphi )\in (0, 1)\) $i(\phi )\in (0,1)$ and upper type \(I(\varphi )\in (0,1)\) $I(\phi )\in (0,1)$ , set \(m(\varphi )=\lfloor n(1/i(\varphi )-1)\rfloor \) $m(\phi )=\lfloor n(1/i(\phi )-1)\rfloor$ . In this paper, the authors establish bilinear decomposition for the product of the Orlicz–Hardy space \(H^{\varphi }({\mathbb {R}}^{n})\) $H^{\phi }({\mathbb{R}}^n)$ and its dual space—the Orlicz–Campanato space \({\mathfrak {L}}_{\varphi }({\mathbb {R}}^{n})\) ${\mathfrak{L}}_{\phi }({\mathbb{R}}^n)$ . In particular, the authors prove that the product (in the sense of distributions) of \(f\in H^{\varphi }({\mathbb {R}}^{n})\) $f\in H^{\phi }({\mathbb{R}}^n)$ and \(g\in {\mathfrak {L}}_{\varphi }({\mathbb {R}}^{n})\) $g\in {\mathfrak{L}}_{\phi }({\mathbb{R}}^n)$ can be decomposed into the sum of S(f, g) and T(f, g), where S is a bilinear operator bounded from \(H^{\varphi }({\mathbb {R}}^{n})\times {\mathfrak {L}}_{\varphi }({\mathbb {R}}^{n})\) $H^{\phi }({\mathbb{R}}^n)\times {\mathfrak{L}}_{\phi }({\mathbb{R}}^n)$ to \(L^{1}({\mathbb {R}}^{n})\) $L^1({\mathbb{R}}^n)$ and T is another bilinear operator bounded from \(H^{\varphi }({\mathbb {R}}^{n})\times {\mathfrak {L}}_{\varphi }({\mathbb {R}}^{n})\) $H^{\phi }({\mathbb{R}}^n)\times {\mathfrak{L}}_{\phi }({\mathbb{R}}^n)$ to the Musielak–Orlicz–Hardy space \(H^{\Phi }({\mathbb {R}}^{n})\) $H^{\mathrm{\Phi }}({\mathbb{R}}^n)$ , with \(\Phi \) $\mathrm{\Phi }$ being a Musielak–Orlicz function determined by \(\varphi \) $\phi$ . The bilinear decomposition is sharp in the following sense: any vector space \({\mathcal {Y}}\subset H^{\Phi }({\mathbb {R}}^{n})\) $\mathcal{Y}\subset H^{\mathrm{\Phi }}({\mathbb{R}}^n)$ that adapted to the above bilinear decomposition should satisfy \( L^\infty ({\mathbb {R}}^{n})\cap {\mathcal {Y}}^{*}=L^\infty ({\mathbb {R}}^{n})\cap (H^{\Phi }({\mathbb {R}}^{n}))^{*} \) $L^{\infty }({\mathbb{R}}^n)\cap \mathcal{Y}_{}^{\ast }=L^{\infty }({\mathbb{R}}^n)\cap {(H^{\mathrm{\Phi }}({\mathbb{R}}^n))}^{\ast }$ . Indeed, \(L^\infty ({\mathbb {R}}^{n})\cap (H^{\Phi }({\mathbb {R}}^{n}))^{*}\) $L^{\infty }({\mathbb{R}}^n)\cap {(H^{\mathrm{\Phi }}({\mathbb{R}}^n))}^{\ast }$ is just the multiplier space of \({\mathfrak {L}}_{\varphi }({\mathbb {R}}^{n})\) ${\mathfrak{L}}_{\phi }({\mathbb{R}}^n)$ . As applications, the authors obtain not only a priori estimate of the div-curl product involving the space \(H^{\Phi }({\mathbb {R}}^{n})\) $H^{\mathrm{\Phi }}({\mathbb{R}}^n)$ , but also the boundedness of the Calderón–Zygmund commutator [b, T] from the Hardy type space \(H^{\varphi }_{b}({\mathbb {R}}^{n})\) $H_b^{\phi }({\mathbb{R}}^n)$ to \(L^{1}({\mathbb {R}}^{n})\) $L^1({\mathbb{R}}^n)$ or \(H^{1}({\mathbb {R}}^{n})\) $H^1({\mathbb{R}}^n)$ under \(b\in {\mathfrak {L}}_{\varphi }({\mathbb {R}}^{n})\) $b\in {\mathfrak{L}}_{\phi }({\mathbb{R}}^n)$ , \(m(\varphi )=0\) $m(\phi )=0$ and suitable cancellation conditions of T.