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Embedding Derivatives of Hardy Spaces into Lebesgue Spaces
Author(s) -
Luecking Daniel H.
Publication year - 1991
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/s3-63.3.595
Subject(s) - mathematics , hardy space , embedding , unit sphere , entire function , characterization (materials science) , monomial , combinatorics , lebesgue measure , order (exchange) , measure (data warehouse) , pure mathematics , ball (mathematics) , lebesgue integration , discrete mathematics , mathematical analysis , materials science , finance , database , artificial intelligence , computer science , economics , nanotechnology
A characterization is given of those measures μ on U , the upper half plane R + 2 or the unit disk, such that differentiation of order m maps H p boundedly into L p (μ), where 0< p <∞ and 0< q <∞. The cases where 0< p = q <2 and 0< q < p are the only two not previously known. The solution is presented in the n real variable setting R + n+1 of Fefferman and Stein [ 7 ] with an arbitrary differential monomial of order m replacing complex differentiation. Defining the function k ( x , y ) as the μ‐measure of a hyperbolic ball of fixed radius centred at ( x , y ), we may describe the characterization here briefly, if opaquely, in terms of membership of k in a ‘weighted tent space’ or an L ∞ ‐analogue of one (depending on the size of q ). In the course of the proof there is developed a theory of ‘tent spaces’ with respect to arbitrary measures on R + n+1 . A consequence of the theory is an interpolation theorem for the values of derivatives of H p ‐functions at the points of an ‘ n ‐lattice’. This may be of independent interest.