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Atomic decomposition and weak factorization in generalized Hardy spaces of closed forms. (English) Zbl 1386.32008

The Hardy space of Musielak-Orlicz type \(H^{\log}(\mathbb{R}^n)\) appears naturally when considering products of functions in \(H^1(\mathbb{R}^n)\) and in \(\mathrm{BMO}(\mathbb{R}^n)\). Namely, \(H^1(\mathbb{R}^n)\cdot \mathrm{BMO}(\mathbb{R}^n) \subset L^1(\mathbb{R}^n)+H^{\log}(\mathbb{R}^n)\). One of the main results of this paper is a converse result which states that \(L^1(\mathbb{R}^n)+H^{\log}(\mathbb{R}^n)\) coincides with the weak product of \(H^1(\mathbb{R}^n)\) and \(\mathrm{BMO}(\mathbb{R}^n)\).
The most difficult part of the proof of the above factorization is to decompose a function in \( H^{\log}(\mathbb{R}^n)\) as an infinite sum of products of functions in \(H^1(\mathbb{R}^n)\) and in \(\mathrm{BMO}(\mathbb{R}^n)\). This decomposition follows as a consequence of a pair of interesting results which are also proved in this paper. The first is an atomic decomposition for closed forms with coefficients in some Hardy spaces of Musielak-Orlicz type \(H^\varphi(\mathbb{R}^n)\). The second is a weak decomposition for closed forms with coefficients in \(H^{\log}(\mathbb{R}^n)\) in terms of an infinite sum of wedge products of closed forms with coefficients in \(H^1(\mathbb{R}^n)\) and closed forms with coefficients in \(\mathrm{BMO}(\mathbb{R}^n)\).

MSC:

32A37 Other spaces of holomorphic functions of several complex variables (e.g., bounded mean oscillation (BMOA), vanishing mean oscillation (VMOA))
47B35 Toeplitz operators, Hankel operators, Wiener-Hopf operators
47B10 Linear operators belonging to operator ideals (nuclear, \(p\)-summing, in the Schatten-von Neumann classes, etc.)
46E22 Hilbert spaces with reproducing kernels (= (proper) functional Hilbert spaces, including de Branges-Rovnyak and other structured spaces)
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References:

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