An inequality for the distribution of the Brownian gradient function

Davis, Burgess

doi:10.1090/S0002-9939-1973-0373036-0

An inequality for the distribution of the Brownian gradient function
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by Burgess Davis

Proc. Amer. Math. Soc. 37 (1973), 189-194

DOI: https://doi.org/10.1090/S0002-9939-1973-0373036-0

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Abstract:

The Brownian gradient function of a harmonic function u in the unit disc is shown to be distributionally about as large as the classical area function of u. This distribution function inequality strengthens some integral inequalities of Burkholder and Gundy.

References

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), 137–153. MR 274767, DOI 10.1090/S0002-9947-1971-0274767-6
D. L. Burkholder and R. F. Gundy, Distribution function inequalities for the area integral, Studia Math. 44 (1972), 527–544. MR 340557, DOI 10.4064/sm-44-6-527-544
J. L. Doob, Semimartingales and subharmonic functions, Trans. Amer. Math. Soc. 77 (1954), 86–121. MR 64347, DOI 10.1090/S0002-9947-1954-0064347-X
J. L. Doob, Conditional Brownian motion and the boundary limits of harmonic functions, Bull. Soc. Math. France 85 (1957), 431–458. MR 109961
G. A. Hunt, Some theorems concerning Brownian motion, Trans. Amer. Math. Soc. 81 (1956), 294–319. MR 79377, DOI 10.1090/S0002-9947-1956-0079377-3
A. Zygmund, Trigonometric series. 2nd ed. Vols. I, II, Cambridge University Press, New York, 1959. MR 107776