Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

A maximal function characterization of the class $ H\sp{p}$


Authors: D. L. Burkholder, R. F. Gundy and M. L. Silverstein
Journal: Trans. Amer. Math. Soc. 157 (1971), 137-153
MSC: Primary 30.67
DOI: https://doi.org/10.1090/S0002-9947-1971-0274767-6
MathSciNet review: 0274767
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ u$ be harmonic in the upper half-plane and $ 0 < p < \infty $. Then $ u =$   Re$ F$ for some analytic function $ F$ of the Hardy class $ {H^p}$ if and only if the nontangential maximal function of $ u$ is in $ {L^p}$. A general integral inequality between the nontangential maximal function of $ u$ and that of its conjugate function is established.


References [Enhancements On Off] (What's this?)

  • [1] F. Bagemihl and W. Seidel, Some boundary properties of analytic functions, Math. Z. 61 (1954), 186-199. MR 16, 460. MR 0065643 (16:460d)
  • [2] D. L. Burkholder and R. F. Gundy, Extrapolation and interpolation of quasi-linear operators on martingales, Acta Math. 124 (1970), 249-304. MR 0440695 (55:13567)
  • [3] J. L. Doob, Semimartingales and subharmonic functions, Trans. Amer. Math. Soc. 77 (1954), 86-121. MR 16, 269. MR 0064347 (16:269a)
  • [4] -, Conditional Brownian motion and the boundary limits of harmonic functions, Bull. Soc. Math. France 85 (1957), 431-458. MR 22 #844. MR 0109961 (22:844)
  • [5] -, Boundary limit theorems for a half-space, J. Math. Pures Appl. (9) 37 (1958), 385-392. MR 22 #845. MR 0109962 (22:845)
  • [6] G. H. Hardy and J. E. Littlewood, A maximal theorem with function theoretic applications, Acta Math. 54 (1930), 81-116. MR 1555303
  • [7] G. A. Hunt, Some theorems concerning Brownian motion, Trans. Amer. Math. Soc. 81 (1956), 294-319. MR 18, 77. MR 0079377 (18:77a)
  • [8] -, Markoff chains and Martin boundaries, Illinois J. Math. 4 (1960), 313-340. MR 23 #A691. MR 0123364 (23:A691)
  • [9] H. P. McKean, Jr., Stochastic integrals, Probability and Math. Statist., no. 5, Academic Press, New York, 1969. MR 40 #947. MR 0247684 (40:947)
  • [10] P. W. Millar, Martingale integrals, Trans. Amer. Math. Soc. 133 (1968), 145-166. MR 37 #2308. MR 0226721 (37:2308)
  • [11] R. E. A. C. Paley and A. Zygmund, A note on analytic functions in the unit circle, Proc. Cambridge Philos. Soc. 28 (1932), 266-272.
  • [12] A. Zygmund, Trigonometrical series, 2nd ed., Cambridge Univ. Press, Cambridge, 1959. MR 21 #6498. MR 0076084 (17:844d)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 30.67

Retrieve articles in all journals with MSC: 30.67


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1971-0274767-6
Keywords: Hardy class, harmonic function, conjugate harmonic function, nontangential maximal function, Brownian motion, martingale
Article copyright: © Copyright 1971 American Mathematical Society

American Mathematical Society