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Molecular decompositions and embedding theorems for vector-valued Sobolev spaces with gradient norm. (English) Zbl 0811.46028

Summary: Let \(E\) be a Banach space. Let \(L_{(1)}^ 1 (\mathbb{R}^ d, E)\) be the Sobolev space of \(E\)-valued functions on \(\mathbb{R}^ d\) with the norm \[ \int_{\mathbb{R}^ d} \| f\|_ E dx+\int_{\mathbb{R}^ d} \|\nabla f\|_ E dx= \| f\|_ 1+\| \nabla f\|_ 1. \] It is proved that if \(f\in L_{(1)}^ 1 (\mathbb{R}^ d, E)\) then there exists a sequence \((g_ m)\subset L_{(1)}^ 1 (\mathbb{R}^ d, E)\) such that \(f= \sum_ m g_ m\); \(\sum_ m (\| g_ m\|_ 1+ \|\nabla g_ m\|_ 1) <\infty\); and \(\| g_ m \|_ \infty^{1/d} \| g_ m \|_ 1^{(d- 1)/d}\leq b\|\nabla g_ m\|_ 1\) for \(m=1,2,\dots\), where \(b\) is an absolute constant independent of \(f\) and \(E\). The result is applied to prove various refinements of the Sobolev type embedding \(L_{(1)}^ 1 (\mathbb{R}^ d, E)\hookrightarrow L^ 2(\mathbb{R}^ d, E)\). In particular, the embedding into Besov spaces \[ L_{(1)}^ 1 (\mathbb{R}^ d, E)\hookrightarrow B_{p,1}^{\theta (p,d)} (\mathbb{R}^ d, E) \] is proved, where \(\theta(p,d)= d(p^{-1}+ d^{-1} -1)\) for \(1<p\leq d/(d-1)\), \(d=1,2,\dots\;\). The latter embedding in the scalar case is due to Bourgain and Kolyada.

MSC:

46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46E40 Spaces of vector- and operator-valued functions
42B15 Multipliers for harmonic analysis in several variables
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