Matrix-weighted Besov-type and Triebel–Lizorkin-type spaces III: characterizations of molecules and wavelets, trace theorems, and boundedness of pseudo-differential operators and Calderón–Zygmund operators

• Published in: Mathematische Zeitschrift Vol. 308; no. 2
• Main Authors: Bu, Fan, Hytönen, Tuomas, Yang, Dachun, Yuan, Wen
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer Berlin Heidelberg 01-10-2024; Springer Nature B.V
• Subjects: Differential equations; Mathematics; Mathematics and Statistics; Operators (mathematics); Pseudo-differential operator; Calderón–Zygmund operator; 47A56; Wavelet; Trace; Molecule; Matrix weight; 42B35; 46E35; 42B25; Triebel–Lizorkin-type space; 42C40; 42B20; Besov-type space; dimension; 35S05
• ISSN: 0025-5874; 1432-1823
• DOI: 10.1007/s00209-024-03584-8

This is the last one of three successive articles by the authors on matrix-weighted Besov-type and Triebel–Lizorkin-type spaces
B
˙
p
,
q
s
,
τ
(
W
)
and
F
˙
p
,
q
s
,
τ
(
W
)
. In this article, the authors establish the molecular and the wavelet characterizations of these spaces. Furthermore, as applications, the authors obtain the optimal boundedness of trace operators, pseudo-differential operators, and Calderón–Zygmund operators on these spaces. Due to the sharp boundedness of almost diagonal operators on their related sequence spaces obtained in the second article of this series, all results presented in this article improve their counterparts on matrix-weighted Besov and Triebel–Lizorkin spaces
B
˙
p
,
q
s
(
W
)
and
F
˙
p
,
q
s
(
W
)
. In particular, even when reverting to the boundedness of Calderón–Zygmund operators on unweighted Triebel–Lizorkin spaces
F
˙
p
,
q
s
, these results are still better.