Molecular decompositions and embedding theorems for vector-valued Sobolev spaces with gradient norm

Pełczyński, A.; Wojciechowski, M.

Studia Mathematica, Tome 104 (1993), p. 61-100 / Harvested from The Polish Digital Mathematics Library

Résumé

Let E be a Banach space. Let $L ¹_{(1)} (ℝ^{d}, E)$ be the Sobolev space of E-valued functions on $ℝ^{d}$ with the norm $ʃ_{ℝ^{d}} ∥ f ∥_{E} d x + ʃ_{ℝ^{d}} ∥ \nabla f ∥_{E} d x = ∥ f ∥ ₁ + ∥ \nabla f ∥ ₁$ . It is proved that if $f \in L ¹_{(1)} (ℝ^{d}, E)$ then there exists a sequence $(g_{m}) \subset L_{(1)} ¹ (ℝ^{d}, E)$ 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,..., where b is an absolute constant independent of f and E. The result is applied to prove various refinements of the Sobolev type embedding $L_{(1)} ¹ (ℝ^{d}, E) ↪ L ² (ℝ^{d}, E)$ . In particular, the embedding into Besov spaces $L ¹_{(1)} (ℝ^{d}, E) ↪ B_{p, 1}^{θ (p, d)} (ℝ^{d}, E)$ is proved, where $θ (p, d) = d (p^{- 1} + d^{- 1} - 1)$ for 1 < p ≤ d/(d-1), d=1,2,... The latter embedding in the scalar case is due to Bourgain and Kolyada.

Publié le : 1993-01-01

Zbl 0811.46028

EUDML-ID : urn:eudml:doc:216022

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     author = {A. Pe\l czy\'nski and M. Wojciechowski},
     title = {Molecular decompositions and embedding theorems for vector-valued Sobolev spaces with gradient norm},
     journal = {Studia Mathematica},
     volume = {104},
     year = {1993},
     pages = {61-100},
     zbl = {0811.46028},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv107i1p61bwm}
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Pełczyński, A.; Wojciechowski, M. Molecular decompositions and embedding theorems for vector-valued Sobolev spaces with gradient norm. Studia Mathematica, Tome 104 (1993) pp. 61-100. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv107i1p61bwm/

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