A probabilistic approach to $H^{p}(R^{d})$
HTML articles powered by AMS MathViewer
- by D. Stroock and S. R. S. Varadhan PDF
- Trans. Amer. Math. Soc. 192 (1974), 245-260 Request permission
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
- D. L. Burkholder and R. F. Gundy, Extrapolation and interpolation of quasi-linear operators on martingales, Acta Math. 124 (1970), 249β304. MR 440695, DOI 10.1007/BF02394573
- D. L. Burkholder, R. F. Gundy, and M. L. Silverstein, A maximal function characterization of the class $H^{p}$, Trans. Amer. Math. Soc. 157 (1971), 137β153. MR 274767, DOI 10.1090/S0002-9947-1971-0274767-6
- C. Fefferman and E. M. Stein, $H^{p}$ spaces of several variables, Acta Math. 129 (1972), no.Β 3-4, 137β193. MR 447953, DOI 10.1007/BF02392215 A. Garsia, Notes on B.M.O. martingales and related topics (to appear).
- R. K. Getoor and M. J. Sharpe, Conformal martingales, Invent. Math. 16 (1972), 271β308. MR 305473, DOI 10.1007/BF01425714
- Lars HΓΆrmander, Estimates for translation invariant operators in $L^{p}$ spaces, Acta Math. 104 (1960), 93β140. MR 121655, DOI 10.1007/BF02547187
- Satoru Igari, An extension of the interpolation theorem of Marcinkiewicz, Proc. Japan Acad. 38 (1962), 731β734. MR 147815
- F. John and L. Nirenberg, On functions of bounded mean oscillation, Comm. Pure Appl. Math. 14 (1961), 415β426. MR 131498, DOI 10.1002/cpa.3160140317
- K. Murali Rao, On decomposition theorems of Meyer, Math. Scand. 24 (1969), 66β78. MR 275510, DOI 10.7146/math.scand.a-10920
- Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
- Elias M. Stein and Guido Weiss, Introduction to Fourier analysis on Euclidean spaces, Princeton Mathematical Series, No. 32, Princeton University Press, Princeton, N.J., 1971. MR 0304972
- Daniel W. Stroock and S. R. S. Varadhan, Diffusion processes with continuous coefficients. I, Comm. Pure Appl. Math. 22 (1969), 345β400. MR 253426, DOI 10.1002/cpa.3160220304
Additional Information
- © Copyright 1974 American Mathematical Society
- 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