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A Hardy space for Fourier integral operators. (English) Zbl 1031.42020
The author introduces a new subspace of \(L^1(\mathbb{R}^n)\), denoted by \({\mathcal H}^1_{\text{FIO}}(\mathbb{R}^n)\), on which the algebra of Fourier integral operators of order \(0\), associated to local canonical transformations, acts continuously. This space is in many ways analogous to the local version of the real Hardy space, which can be characterized as the largest subspace of \(L^1(\mathbb{R}^n)\) that is preserved by order \(0\) pseudodifferential operators. A key role is played by the atomic and molecular decomposition theorems for \({\mathcal H}^1_{\text{FIO}}(\mathbb{R}^n)\). The boundedness of Fourier integral operators on \({\mathcal H}^1_{\text{FIO}}(\mathbb{R}^n)\) is established by showing that such an operator maps a molecule centered at one point in \(S^*(\mathbb{R}^n)= \mathbb{R}^n\times S^{n-1}\) to a molecule centered at the image of that point under the associated canonical transformation. The author also establishes the embedding theorem \[ {\mathcal H}^1(\mathbb{R}^n) @>\langle D\rangle^{-{n-1\over 2}}>>{\mathcal H}^1_{\text{FIO}}(\mathbb{R}^n)@> I>>{\mathcal H}^1(\mathbb{R}^n), \] where \[ \|f\|_{{\mathcal H}^1(\mathbb{R}^n)}= \|(1- r(D))f\|_{H^1(\mathbb{R}^n)}+ \|r(D) f\|_{L^1(\mathbb{R}^n)} \] and \(r\) is a function in \(C^\infty_c(\mathbb{R}^n)\) such that \(r(\xi)= 1\) if \(|\xi|\leq 1\).

MSC:
42B30 \(H^p\)-spaces
35S30 Fourier integral operators applied to PDEs
46E30 Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
47G30 Pseudodifferential operators
42B25 Maximal functions, Littlewood-Paley theory
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