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

A. Pełczyński; M. Wojciechowski

Studia Mathematica (1993)

  • Volume: 107, Issue: 1, page 61-100
  • ISSN: 0039-3223

Abstract

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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 ∥ ∇ f ∥ E d x = ∥ f ∥ ₁ + ∥ ∇ f ∥ ₁ . It is proved that if f ∈ L ¹ ( 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 ) ↪ 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.

How to cite

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Pełczyński, A., and Wojciechowski, M.. "Molecular decompositions and embedding theorems for vector-valued Sobolev spaces with gradient norm." Studia Mathematica 107.1 (1993): 61-100. <http://eudml.org/doc/216022>.

@article{Pełczyński1993,
abstract = {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.},
author = {Pełczyński, A., Wojciechowski, M.},
journal = {Studia Mathematica},
keywords = {Sobolev space; Sobolev type embedding; embedding into Besov spaces},
language = {eng},
number = {1},
pages = {61-100},
title = {Molecular decompositions and embedding theorems for vector-valued Sobolev spaces with gradient norm},
url = {http://eudml.org/doc/216022},
volume = {107},
year = {1993},
}

TY - JOUR
AU - Pełczyński, A.
AU - Wojciechowski, M.
TI - Molecular decompositions and embedding theorems for vector-valued Sobolev spaces with gradient norm
JO - Studia Mathematica
PY - 1993
VL - 107
IS - 1
SP - 61
EP - 100
AB - 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.
LA - eng
KW - Sobolev space; Sobolev type embedding; embedding into Besov spaces
UR - http://eudml.org/doc/216022
ER -

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