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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