and Carleson measures
Author:
Wayne Stewart Smith
Journal:
Trans. Amer. Math. Soc. 287 (1985), 107126
MSC:
Primary 42B30; Secondary 46E15
MathSciNet review:
766209
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Abstract: This paper concerns certain generalizations of , the space of functions of bounded mean oscillation. Let be a positive nondecreasing function on with . A locally integrable function on is said to belong to if its mean oscillation over any cube is , where is the edge length of . Carleson measures are known to be closely related to . Generalizations of these measures are shown to be similarly related to the spaces . For a cube in denotes its volume and is the set . A measure on is called a Carleson measure if , for all cubes . L. Carleson proved that a compactly supported function in can be represented as the sum of a bounded function and the balyage, or sweep, of some Carleson measure. A generalization of this theorem involving and Carleson measures is proved for a broad class of growth functions, and this is used to represent as a dual space. The proof of the theorem is based on a proof of J. Garnett and P. Jones of Carleson's theorem. Another characterization of using Carleson measures is a corollary. This result generalizes a characterization of due to C. Fefferman. Finally, an atomic decomposition of the predual of is given.
 [1]
Lennart
Carleson, An interpolation problem for bounded analytic
functions, Amer. J. Math. 80 (1958), 921–930.
MR
0117349 (22 #8129)
 [2]
Lennart
Carleson, Interpolations by bounded analytic functions and the
corona problem, Ann. of Math. (2) 76 (1962),
547–559. MR 0141789
(25 #5186)
 [3]
Lennart
Carleson, Two remarks on 𝐻¹ and BMO, Advances in
Math. 22 (1976), no. 3, 269–277. MR 0477058
(57 #16602)
 [4]
Ronald
R. Coifman and Guido
Weiss, Extensions of Hardy spaces and their
use in analysis, Bull. Amer. Math. Soc.
83 (1977), no. 4,
569–645. MR 0447954
(56 #6264), http://dx.doi.org/10.1090/S000299041977143255
 [5]
C.
Fefferman and E.
M. Stein, 𝐻^{𝑝} spaces of several variables,
Acta Math. 129 (1972), no. 34, 137–193. MR 0447953
(56 #6263)
 [6]
John
B. Garnett and Peter
W. Jones, BMO from dyadic BMO, Pacific J. Math.
99 (1982), no. 2, 351–371. MR 658065
(85d:42021)
 [7]
Lars
Hörmander, 𝐿^{𝑝} estimates for (pluri)
subharmonic functions, Math. Scand. 20 (1967),
65–78. MR
0234002 (38 #2323)
 [8]
Svante
Janson, Generalizations of Lipschitz spaces and an application to
Hardy spaces and bounded mean oscillation, Duke Math. J.
47 (1980), no. 4, 959–982. MR 596123
(83j:46037)
 [9]
, Lipschitz spaces and bounded mean oscillation, Rend. Circ. Mat. Palermo (2) Suppl. No. 1 (1981), 111114.
 [10]
Daniel
H. Luecking, Forward and reverse Carleson inequalities for
functions in Bergman spaces and their derivatives, Amer. J. Math.
107 (1985), no. 1, 85–111. MR 778090
(86g:30002), http://dx.doi.org/10.2307/2374458
 [11]
Carl
Mueller, A characterization of BMO and BMOᵨ, Studia
Math. 72 (1982), no. 1, 47–57. MR 665891
(84j:42032)
 [12]
Donald
Sarason, Function theory on the unit circle, Virginia
Polytechnic Institute and State University Department of Mathematics,
Blacksburg, Va., 1978. Notes for lectures given at a Conference at Virginia
Polytechnic Institute and State University, Blacksburg, Va., June
19–23, 1978. MR 521811
(80d:30035)
 [13]
David
A. Stegenga, Multipliers of the Dirichlet space, Illinois J.
Math. 24 (1980), no. 1, 113–139. MR 550655
(81a:30027)
 [14]
Elias
M. Stein, Singular integrals and differentiability properties of
functions, Princeton Mathematical Series, No. 30, Princeton University
Press, Princeton, N.J., 1970. MR 0290095
(44 #7280)
 [1]
 L. Carlson, An interpolation problem for bounded analytic functions, Amer. J. Math. 80 (1958), 921930. MR 0117349 (22:8129)
 [2]
 , Interpolation by bounded analytic functions and the corona problem, Ann. of Math. (2) 76 (1962), 547559. MR 0141789 (25:5186)
 [3]
 , Two remarks on and , Adv. in Math. 22 (1976), 269277. MR 0477058 (57:16602)
 [4]
 R. R. Coiffman and G. Weiss, Extensions of Hardy spaces, Bull. Amer. Math. Soc. 83 (1977), 569645. MR 0447954 (56:6264)
 [5]
 C. Fefferman and E. M. Stein, spaces of several variables, Acta Math. 129 (1972), 137193. MR 0447953 (56:6263)
 [6]
 J. Garnett and P. Jones, from dyadic , Pacific J. Math. 99 (1982), 351372. MR 658065 (85d:42021)
 [7]
 L. Hörmander, estimates for (pluri)subharmonic functions, Math. Scand. 20 (1967), 6578. MR 0234002 (38:2323)
 [8]
 S. Janson, Lipschitz spaces and bounded mean oscillation, Duke Math. J. 47 (1980), 959982. MR 596123 (83j:46037)
 [9]
 , Lipschitz spaces and bounded mean oscillation, Rend. Circ. Mat. Palermo (2) Suppl. No. 1 (1981), 111114.
 [10]
 D. Luecking, Forward and reverse Carleson inequalities for functions in Bergman spaces and their derivatives, preprint. MR 778090 (86g:30002)
 [11]
 C. Mueller, A characterization of and , Studia Math. 72 (1982), 4757. MR 665891 (84j:42032)
 [12]
 D. Sarason, Function theory on the unit circle, Virginia Polytechnic Institute and State Univ., Blacksburg, Va., 1979. MR 521811 (80d:30035)
 [13]
 D. Stegenga, Multipliers of the Dirichlet space, Illinois J. Math. 24 (1980), 113139. MR 550655 (81a:30027)
 [14]
 E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, N.J., 1970. MR 0290095 (44:7280)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947198507662099
PII:
S 00029947(1985)07662099
Article copyright:
© Copyright 1985 American Mathematical Society
