Molecular decompositions and embedding theorems for vector-valued Sobolev spaces with gradient norm
Studia Mathematica, Tome 107 (1993) no. 1, pp. 61-100
Voir la notice de l'article provenant de la source Institute of Mathematics Polish Academy of Sciences
Let E be a Banach space. Let $L¹_{(1)}(ℝ^d,E)$ be the Sobolev space of E-valued functions on $ℝ^d$ with the norm $ʃ_{ℝ^d} ∥f∥_E dx + ʃ_{ℝ^d} ∥∇f∥_E dx = ∥f∥₁ + ∥∇f∥₁$. It is proved that if $f ∈ L¹_{(1)}(ℝ^d,E)$ then there exists a sequence $(g_m) ⊂ L_{(1)}¹(ℝ^d,E)$ such that $f = ∑_m g_m$; $∑_m (∥g_m∥₁ + ∥∇g_m ∥₁) ∞$; and $∥g_m∥_∞^{1/d} ∥g_m∥₁^{(d-1)/d} ≤ b∥∇g_m∥₁$ 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 $L_{(1)}¹(ℝ^d,E) ↪ L²(ℝ^d,E)$. In particular, the embedding into Besov spaces $L¹_{(1)} (ℝ^d,E) ↪ B_{p,1}^{θ(p,d)}(ℝ^d,E)$ 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.
@article{10_4064_sm_107_1_61_100,
author = {A. Pe{\l}czy\'nski and },
title = {Molecular decompositions and embedding theorems for vector-valued {Sobolev} spaces with gradient norm},
journal = {Studia Mathematica},
pages = {61--100},
publisher = {mathdoc},
volume = {107},
number = {1},
year = {1993},
doi = {10.4064/sm-107-1-61-100},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/sm-107-1-61-100/}
}
TY - JOUR AU - A. Pełczyński AU - TI - Molecular decompositions and embedding theorems for vector-valued Sobolev spaces with gradient norm JO - Studia Mathematica PY - 1993 SP - 61 EP - 100 VL - 107 IS - 1 PB - mathdoc UR - http://geodesic.mathdoc.fr/articles/10.4064/sm-107-1-61-100/ DO - 10.4064/sm-107-1-61-100 LA - en ID - 10_4064_sm_107_1_61_100 ER -
%0 Journal Article %A A. Pełczyński %A %T Molecular decompositions and embedding theorems for vector-valued Sobolev spaces with gradient norm %J Studia Mathematica %D 1993 %P 61-100 %V 107 %N 1 %I mathdoc %U http://geodesic.mathdoc.fr/articles/10.4064/sm-107-1-61-100/ %R 10.4064/sm-107-1-61-100 %G en %F 10_4064_sm_107_1_61_100
A. Pełczyński; . Molecular decompositions and embedding theorems for vector-valued Sobolev spaces with gradient norm. Studia Mathematica, Tome 107 (1993) no. 1, pp. 61-100. doi: 10.4064/sm-107-1-61-100
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