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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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