A localization property for $B^{s}_{pq}$ and $F^{s}_{pq}$ spaces

Hans Triebel

A localization property for $B_{p q}^{s}$ and $F_{p q}^{s}$ spaces

Hans Triebel

Studia Mathematica (1994)

Volume: 109, Issue: 2, page 183-195
ISSN: 0039-3223

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Abstract

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Let

f^{j} = \sum_{k} a_{k} f (2^{j + 1} x - 2 k)

, where the sum is taken over the lattice of all points k in

ℝ^{n}

having integer-valued components, j∈ℕ and

a_{k} \in ℂ

. Let

A_{p q}^{s}

be either

B_{p q}^{s}

or

F_{p q}^{s}

(s ∈ ℝ, 0 < p < ∞, 0 < q ≤ ∞) on

ℝ^{n} .

The aim of the paper is to clarify under what conditions

∥ f^{j} | A_{p q}^{s} ∥

is equivalent to

2^{j (s - n / p)} (\sum_{k} | a_{k} |^{p})^{1 / p} ∥ f | A_{p q}^{s} ∥

.

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Triebel, Hans. "A localization property for $B^{s}_{pq}$ and $F^{s}_{pq}$ spaces." Studia Mathematica 109.2 (1994): 183-195. <http://eudml.org/doc/216068>.

@article{Triebel1994,
abstract = {Let $f^\{j\} = ∑_\{k\} a_\{k\} f(2^\{j+1\}x - 2k)$, where the sum is taken over the lattice of all points k in $ℝ^n$ having integer-valued components, j∈ℕ and $a_k ∈ ℂ$. Let $A^\{s\}_\{pq\}$ be either $B^\{s\}_\{pq\}$ or $F^\{s\}_\{pq\}$ (s ∈ ℝ, 0 < p < ∞, 0 < q ≤ ∞) on $ℝ^n.$ The aim of the paper is to clarify under what conditions $∥f^\{j\} | A^\{s\}_\{pq\}∥$ is equivalent to $2^\{j(s-n/p)\} (∑_\{k\} |a_k|^p)^\{1/p\} ∥f | A^\{s\}_\{pq\}∥$.},
author = {Triebel, Hans},
journal = {Studia Mathematica},
keywords = {localization property},
language = {eng},
number = {2},
pages = {183-195},
title = {A localization property for $B^\{s\}_\{pq\}$ and $F^\{s\}_\{pq\}$ spaces},
url = {http://eudml.org/doc/216068},
volume = {109},
year = {1994},
}

TY - JOUR
AU - Triebel, Hans
TI - A localization property for $B^{s}_{pq}$ and $F^{s}_{pq}$ spaces
JO - Studia Mathematica
PY - 1994
VL - 109
IS - 2
SP - 183
EP - 195
AB - Let $f^{j} = ∑_{k} a_{k} f(2^{j+1}x - 2k)$, where the sum is taken over the lattice of all points k in $ℝ^n$ having integer-valued components, j∈ℕ and $a_k ∈ ℂ$. Let $A^{s}_{pq}$ be either $B^{s}_{pq}$ or $F^{s}_{pq}$ (s ∈ ℝ, 0 < p < ∞, 0 < q ≤ ∞) on $ℝ^n.$ The aim of the paper is to clarify under what conditions $∥f^{j} | A^{s}_{pq}∥$ is equivalent to $2^{j(s-n/p)} (∑_{k} |a_k|^p)^{1/p} ∥f | A^{s}_{pq}∥$.
LA - eng
KW - localization property
UR - http://eudml.org/doc/216068
ER -

References

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[1] I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Regional Conf. Ser. in Appl. Math. 61, SIAM, Philadelphia, 1992.
[2] D. E. Edmunds and H. Triebel, Eigenvalue distributions of some degenerate elliptic operators: an approach via entropy numbers, Math. Ann., to appear. Zbl0804.35097
[3] M. Frazier and B. Jawerth, Decomposition of Besov spaces, Indiana Univ. Math. J. 34 (1985), 777-799. Zbl0551.46018
[4] M. Frazier and B. Jawerth, A discrete transform and decomposition of distribution spaces, J. Funct. Anal. 93 (1990), 34-170.
[5] M. Frazier, B. Jawerth and G. Weiss, Littlewood-Paley Theory and the Study of Function Spaces, CBMS Regional Conf. Ser. in Math. 79, Amer. Math. Soc., Providence, 1991. Zbl0757.42006
[6] W. Sickel and H. Triebel, Hölder inequalities and sharp embeddings in function spaces of $B_{p} q^{s}$ and $F_{p} q^{s}$ type, submitted.
[7] R. H. Torres, Boundedness results for operators with singular kernels on distribution spaces, Mem. Amer. Math. Soc. 442 (1991). Zbl0737.47041
[8] H. Triebel, Theory of Function Spaces, Birkhäuser, Basel, 1983.
[9] H. Triebel, Theory of Function Spaces, II, Birkhäuser, Basel, 1992.
[10] H. Triebel, Approximation numbers and entropy numbers of embeddings of fractional Besov-Sobolev spaces in Orlicz spaces, Proc. London Math. Soc. 66 (1993), 589-618. Zbl0792.46024

Citations in EuDML Documents

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Edmunds, David E., Recent developments concerning entropy and approximation numbers

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