Traces of multipliers in pairs of weighted Sobolev spaces (Q1769937)

This paper deals with spaces \(M(S_1\to S_2)\) of multipliers \(\mu\) from a Banach space \(S_1\) into a Banach space \(S_2\), equipped with the norm \(| | \mu| | =\underset{| | u| | _{S_1}\leq 1} {\sup} | | \mu u| | _{S_2}.\) Let \(\mathbb{R}^n_+\) denote the half-space \( \{z=(x,y),\;x=(x_1,..., x_{n-1})\in \mathbb{R}^{n-1},\;y>0\}.\) A large part of the paper concerns usual fractional Sobolev spaces in \(\mathbb{R}^{n-1}\) and weighted Sobolev spaces \(W_p^{s,\alpha}(\mathbb{R}_+^n)\), where \(s\) is a nonnegative integer, \(p\in (1,\infty)\) and \(-1<\alpha p<p-1\) (cf. \textit{S. V. Uspenskii} [Trudy Mat. Inst. Steklov, 60, 1961, 282--303, English Translation: Am. Math. Soc. Transl. 87, 121--145 (1970; Zbl 0198.46106)]). The authors show that, for every \(\gamma \in M(W^m_p(\mathbb{R}^{n-1}) \to W^l_p(\mathbb{R}^{n-1}))=M_1,\) where \(m\) and \(l\) are positive nonintegers with \(m\geq l\), the Dirichlet problem \[ \Delta \Gamma =0\;\;\text{in } \mathbb{R}^n_+,\;\;\Gamma| _{\mathbb{R}^{n-1}} =\gamma \] has a unique solution in \( M( W_p^{t,\beta}(\mathbb{R}^n_+)\to W_p^{s,\alpha}(\mathbb{R}^n_+))=M_2,\) with \(t=[m]+1\), \(\beta=1-\{m\}-\frac1p\), \(s=[l]+1\), \(\alpha = 1-\{l\}-\frac1p\), and \(| | \Gamma| | _{M_2}\leq c | | \gamma| | _{M_1}\). The proof is based on properties of multiplier spaces, the introduction of an extension operator \(T\), and pointwise and \(L^p\)-estimates for \(T \gamma\) and \(\nabla T \gamma.\) At the end of the paper, it is mentioned that, by means of local maps, a similar result can be obtained for bounded domains in \(\mathbb R ^n\) with a smooth boundary.