Traces of vector-valued Sobolev spaces (Q2892954)

Let \(E\) be an arbitrary Banach space and let \(\{ \phi_j \}^\infty_{j=0}\) be a standard resolution of unity in \(\mathbb{R}^n\). As usual \(\hat{g}\) stands for the Fourier transform of \(g\) and \(g^\vee\) for the inverse Fourier transform. Let \(s \in \mathbb{R}\) and \(1\leq p,q \leq \infty\). Then \(B^s_{p,q} (\mathbb{R}^n, E)\) collects all \(f \in S' (\mathbb{R}^n, E)\) such that NEWLINE\[NEWLINE \|f \, | B^s_{p,q} (\mathbb{R}^n, E) \| = \Big( \sum^\infty_{j=0} 2^{jsq} \big\| (\phi_j \hat{f} \big)^\vee \, | L_p (\mathbb{R}^n, E) \big\|^q \Big)^{1/q} NEWLINE\]NEWLINE is finite. Similarly \(F^s_{p,q} (\mathbb{R}^n, E)\) (now \(p<\infty\)) is defined replacing \(\ell_q \big( L_p (\mathbb{R}^n, E) \big)\) by \(L_p \big(\mathbb{R}^n, \ell_q (E) \big)\). These are the vector-valued generalizations of the well-known scalar spaces \(B^s_{p,q} (\mathbb{R}^n)\) and \(F^s_{p,q} (\mathbb{R}^n)\). The Sobolev spaces \(W^m_p (\mathbb{R}^n, E)\) with \(1<p<\infty\), \(m \in \mathbb{N}\), collect all \(f \in S' (\mathbb{R}^n, E)\) such that \(D^\alpha f \in L_p (\mathbb{R}^n, E)\), \(|\alpha| \leq m\). The authors develop the theory of these spaces without the usual additional assumption that \(E\) has the \(\mathrm{UMD}\) property. They prove atomic and (in particular) subatomic decompositions. On this basis they prove the trace theorems NEWLINE\[NEWLINE \text{tr} \, B^s_{p,q} (\mathbb{R}^n, E) = B^{s- \frac{1}{p}}_{p,q} (\mathbb{R}^{n-1}, E) NEWLINE\]NEWLINE (Theorem 4.9) and NEWLINE\[NEWLINE \text{tr} \, F^s_{p,q} (\mathbb{R}^n, E) = B^{s- \frac{1}{p}}_{p,p} (\mathbb{R}^{n-1}, E) NEWLINE\]NEWLINE (Theorem 4.17), where \(1\leq p \leq \infty\) (\(p<\infty\) for \(F\)-spaces), \(1\leq q \leq \infty\), \(s >1/p\), and the existence of universal extension operators. By inclusions one has corresponding assertions for the trace of \(W^m_p (\mathbb{R}^n, E)\). These are interesting final characterizations. Comprehensive references are given.