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

Abstract

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