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Pointwise multiplication of Besov and Triebel-Lizorkin spaces. (English) Zbl 0839.46026
Summary: It is shown that para-multiplication applies to a certain product \(\pi(u, v)\) defined for appropriate \(u\) and \(v\) in \({\mathcal S}'(\mathbb{R}^n)\). Boundedness of \(\pi(\cdot, \cdot)\) is investigated for the anisotropic Besov and Triebel-Lizorkin spaces – i.e., for \(B^{M, s}_{p, q}\) and \(F^{M, s}_{p, q}\) with \(s\in \mathbb{R}\) and \(p\) and \(q\) in \(]0, \infty]\) (though \(p< \infty\) in the \(F\)-case) – with a treatment of the generic as well as of various borderline cases.
For \(\max(s_0, s_1)> 0\) the spaces \(B^{M, s_0}_{p_0, q_0}\oplus B^{M, s_1}_{p_1, q_1}\) and \(F^{M, s_0}_{p_0, q_0}\oplus F^{M, s_1}_{p_1, q_1}\) to which \(\pi(\cdot, \cdot)\) applies are determined. For generic \(F^{s_0}_{p_0, q_0}\oplus F^{s_1}_{p_1, q_1}\) the receiving \(F^s_{p, q}\) spaces are characterized.
It is proved that \(\pi(f, g)= f\cdot g\) holds for functions \(f\) and \(g\) when \(f\cdot g\in L_{1,\text{loc}}\) roughly speaking. In addition, \(\pi(f, u)= fu\) when \(f\in {\mathcal O}_M\) and \(u\in {\mathcal S}'\).
Moreover, for an arbitrary open set \(\Omega\subset \mathbb{R}^n\), a product \(\pi_\Omega(\cdot, \cdot)\) is defined by lifting to \(\mathbb{R}^n\). Boundedness of \(\pi\) on \(\mathbb{R}^n\) is shown to carry over to \(\pi_\Omega\) is general.

MSC:
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46F05 Topological linear spaces of test functions, distributions and ultradistributions
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