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Hardy type spaces for the wave operator. (English) Zbl 0971.35045

The classic Hardy space \(H^{p}(\mathbb{R}^{n})\) is the space of boundary values \(f(x)=u(0,x)\) of the functions \(u(t,x)\), which are harmonic in the half-space \(\mathbb{R}_{+}^{n+1}\), with the norm \(||f||_{p}= \sup_{t>0}||u(t,\cdot)||_{L_{p}(\mathbb{R}^{n})}\), and the functions conjugate to \(f\) are \(f_{j}^{\ast }=R_{j}f(x)\), where \(R_{j}\) is the Riesz transform corresponding to the \(j\)-th variable. Instead of harmonic functions the authors consider the waves as the functions \(u(t,x), t>0, x\in \mathbb{R}^{n}\), which satisfy the wave equation \(\frac{\partial^{2}u} {\partial t^{2}}=\Delta u\), and introduce the half-waves as the solutions of the (pseudo-differential) equation \(\frac{\partial u}{\partial t}= \sqrt{\Delta }u\). The system of functions \(\{v_{j}\}\) is said to be conjugate to some half-wave \(u\) if \(v_{j}\) are half-waves and \(v_{j}(0,x)=R_{j}u(0,x)\). The authors introduce the Hardy spaces \(\widetilde{H}^{p}(\mathbb{R}^{n})\) of half-waves and conjugate functions and prove that for \(1\leq p<\infty\) there exists an injection \(\widetilde{H}^{p}(\mathbb{R}^{n})\rightarrow H^{p}(\mathbb{R}^{n})\) (Theorem 2), and these two spaces are isomorphic if and only if \(p=2\) for \(n>1\) or \(1\leq p<\infty\) for \(n=1\) (Theorem 3).

MSC:

35L05 Wave equation
42B30 \(H^p\)-spaces
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