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 probabilistic approach to $ H\sp{p}(R\sp{d})$


Authors: D. Stroock and S. R. S. Varadhan
Journal: Trans. Amer. Math. Soc. 192 (1974), 245-260
MSC: Primary 60G45; Secondary 42A36
DOI: https://doi.org/10.1090/S0002-9947-1974-0365696-0
MathSciNet review: 0365696
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: The relationship between $ {H^p}({R^d}),1 \leq p < \infty $, and the integrability of certain functionals of Brownian motion is established using the connection between probabilistic and analytic notions of functions with bounded mean oscillation. An application of this relationship is given in the derivation of an interpolation theorem for operators taking $ {H^1}({R^d})$ to $ {L^1}({R^d})$.


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

  • [1] D. Burkholder and R. Gundy, Extrapolation and interpolation of quasi-linear operators on martingales, Acta. Math. 124 (1970) 249-304. MR 0440695 (55:13567)
  • [2] D. Burkholder, R. Gundy and M. Silverstein, A maximal function characterization of $ {H^p}$, Trans. Amer. Math. Soc. 157 (1971), 137-153. MR 43 #527. MR 0274767 (43:527)
  • [3] C. Fefferman and E. Stein, $ {H^p}$ spaces of several variables (to appear). MR 0447953 (56:6263)
  • [4] A. Garsia, Notes on B.M.O. martingales and related topics (to appear).
  • [5] R. Getoor and M. Sharpe, Conformal martingales (to appear). MR 0305473 (46:4603)
  • [6] L. Hörmander, Estimates for translation invariant operators in $ {L^p}$ spaces, Acta. Math. 104 (1960) 93-140. MR 22 #12389. MR 0121655 (22:12389)
  • [7] S. Igari, An extension of the interpolation theorem of Marcinkiewicz, Proc. Japan Acad. 38 (1962), 731-734. MR 26 #5328. MR 0147815 (26:5328)
  • [8] F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415-426. MR 24 #A1348. MR 0131498 (24:A1348)
  • [9] M. Rao, On decomposition theorems of Meyer, Math. Scand. 24 (1969), 66-78. MR 43 #1264. MR 0275510 (43:1264)
  • [10] E. Stein, Singular integrals and differentiability properties of functions, Princeton Math. Series, no. 30, Princeton Univ. Press, Princeton, N.J., 1970. MR 44 #7280. MR 0290095 (44:7280)
  • [11] E. Stein and G. Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Univ. Press, Princeton, N.J., 1971. MR 0304972 (46:4102)
  • [12] D. Stroock and S. R. S. Varadhan, Diffusion processes with continuous coefficients. I, Comm. Pure Appl. Math. 22 (1969), 345-400. MR 40 #6641. MR 0253426 (40:6641)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 60G45, 42A36

Retrieve articles in all journals with MSC: 60G45, 42A36


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1974-0365696-0
Keywords: Martingale, Wiener measure, harmonic functions, interpolation
Article copyright: © Copyright 1974 American Mathematical Society

American Mathematical Society