Premium
The Approximation Numbers of Hardy‐Type Operators on Trees
Author(s) -
Evans W. D.,
Harris D. J.,
Lang J.
Publication year - 2001
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/plms/83.2.390
Subject(s) - mathematics , combinatorics , operator (biology) , type (biology) , interval (graph theory) , tree (set theory) , mathematics subject classification , constant (computer programming) , discrete mathematics , ecology , biochemistry , chemistry , repressor , biology , computer science , transcription factor , gene , programming language
The Hardy operator T a on a tree Γ is defined by ( T a f ) ( x ) : = v ( x ) ∫ a x f ( t ) u ( t ) d t for a , x ∈ Γ . Properties of T a as a map from L p (Γ) into itself are established for 1 ⩽ p ⩽ ∞. The main result is that, with appropriate assumptions on u and v , the approximation numbers a n ( T a ) of T a satisfy( ∗ )lim n → ∞ n a n ( T a ) = α p ∫ Γ| u v |d tfor a specified constant α p and 1 p < ∞. This extends results of Naimark, Newman and Solomyak for p = 2. Hitherto, for p ≠ 2, (*) was unknown even when Γ is an interval. Also, upper and lower estimates for the l q and weak‐ l q norms of a n ( T a ) are determined. 2000 Mathematical Subject Classification : 47G10, 47B10.