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Weighted Norm Inequalities for Singular Integral Operators
Author(s) -
Pérez C.
Publication year - 1994
Publication title -
journal of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.441
H-Index - 62
eISSN - 1469-7750
pISSN - 0024-6107
DOI - 10.1112/jlms/49.2.296
Subject(s) - singular integral , maximal operator , mathematics , iterated function , combinatorics , singular integral operators , norm (philosophy) , integer (computer science) , operator (biology) , mathematical analysis , integral equation , chemistry , biochemistry , repressor , political science , computer science , transcription factor , law , bounded function , gene , programming language
For a Calderón‐Zygmund singular integral operator T , we show that the following weighted inequality holds∫ R n| T f ( y ) | p w ( y ) dy ⩽ C ∫ R n| f ( y ) | pM [ p ] + 1w ( y ) dy ,where M k is the Hardy‐Littlewood maximal operator M iterated k times, and [ p ] is the integer part of p . Moreover, the result is sharp since it does not hold for M [ p ] . We also give the following endpoints results: w ( { y ∈ R n : | Tf ( y ) | > λ } ) ⩽ C λ ∫ R n| f ( y ) | M 2 w ( y ) dy , and∫ R n| T f ( y ) | w ( y ) dy ⩽ C ∥ f ∥ H 1 ( M w ), where H 1 (μ) is the atomic Hardy space with respect to μ.