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Hardy–Stein identities and square functions for semigroups
Author(s) -
Bañuelos Rodrigo,
Bogdan Krzysztof,
Luks Tomasz
Publication year - 2016
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/jdw042
Subject(s) - mathematics , square (algebra) , pure mathematics , identity (music) , lévy process , type (biology) , jump , semigroup , function (biology) , martingale (probability theory) , fourier transform , mathematical analysis , philosophy , physics , geometry , ecology , quantum mechanics , evolutionary biology , biology , aesthetics
We prove a Hardy–Stein‐type identity for the semigroups of symmetric, pure‐jump Lévy processes. Combined with the Burkholder–Gundy inequalities, it gives the L p two‐way boundedness, for 1 < p < ∞ , of the corresponding Littlewood–Paley square function. The square function yields a direct proof of the L p ‐boundedness of Fourier multipliers obtained by transforms of martingales of Lévy processes.