Bilinear decompositions for products of Orlicz-Hardy and Orlicz-Campanato spaces (Q6611143)

The authors study the Orlicz-Hardy space \( H^\varphi(\mathbb{R}^n) \) and its dual space, the Orlicz-Campanato space \( \mathcal{L}_\varphi(\mathbb{R}^n) \), for an Orlicz function \( \varphi \) with specific critical lower and upper types. They establish a bilinear decomposition for the product of functions in these spaces, showing that such a product can be expressed as the sum of two bilinear operators: \( S(f, g) \), which maps \( H^\varphi(\mathbb{R}^n) \times \mathcal{L}_\varphi(\mathbb{R}^n) \) to \( L^1(\mathbb{R}^n) \), and \( T(f, g) \), which maps \( H^\varphi(\mathbb{R}^n) \times \mathcal{L}_\varphi(\mathbb{R}^n) \) to the Musielak-Orlicz-Hardy space \( H^\Phi(\mathbb{R}^n) \).\N\NThis decomposition is shown to be sharp, as it characterizes the multiplier space of \( \mathcal{L}_\varphi(\mathbb{R}^n) \) as \( L^\infty(\mathbb{R}^n) \cap H^\Phi(\mathbb{R}^n)^\ast \). Applications include a priori estimate for the div-curl product involving \( H^\Phi(\mathbb{R}^n) \), as well as the boundedness of the Calderón-Zygmund commutator on related function spaces under suitable cancellations.