Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Hardy spaces and partial derivatives of conjugate harmonic functions


Authors: Anatoly Ryabogin and Dmitry Ryabogin
Journal: Proc. Amer. Math. Soc. 135 (2007), 2461-2470
MSC (2000): Primary 30E25; Secondary 42B25
Published electronically: April 5, 2007
MathSciNet review: 2302567
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we give necessary and sufficient conditions for a harmonic vector and all its partial derivatives to belong to $ H^p(\mathbf{R}^{n+1}_+)$ for all $ p>0$.


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

  • 1. A. A. Bonami, Integral inequalities for conjugate harmonic functions of serveral variables, Mat. Sb. (N.S.) 87(129) (1972), 188–203 (Russian). MR 0299818
  • 2. 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 0274767, 10.1090/S0002-9947-1971-0274767-6
  • 3. A.-P. Calderón and A. Zygmund, On higher gradients of harmonic functions, Studia Math. 24 (1964), 211–226. MR 0167631
  • 4. C. Fefferman and E. M. Stein, 𝐻^{𝑝} spaces of several variables, Acta Math. 129 (1972), no. 3-4, 137–193. MR 0447953
  • 5. T. M. Flett, Inequalities for the 𝑝th mean values of harmonic and subharmonic functions with 𝑝≤1, Proc. London Math. Soc. (3) 20 (1970), 249–275. MR 0257387
  • 6. V. I. Krylov, On functions regular in the half-plane, Math., Sb., (1939), 6 (48), pp.95-138.
  • 7. Ü. Kuran, Classes of subharmonic functions in 𝑅ⁿ×(0,+∞), Proc. London Math. Soc. (3) 16 (1966), 473–492. MR 0203059
  • 8. I. Privalov, Subharmonic functions, Moscow, 1937.
  • 9. A. K. Ryabogin, Conjugate harmonic functions of the Hardy class, Izv. Vyssh. Uchebn. Zaved. Mat. 9 (1991), 47–53 (Russian); English transl., Soviet Math. (Iz. VUZ) 35 (1991), no. 9, 46–51. MR 1169391
  • 10. A. K. Rjabogin, Boundary values of conjugate harmonic functions of several variables, Izv. Vyssh. Uchebn. Zaved. Mat. 12 (1980), 50–54 (Russian). MR 606677
  • 11. E. D. Solomentsev, On classes of subharmonic functions in the half-space, Notes of Moscow State Univ., 10, (1958).
  • 12. Elias M. Stein, Singular integrals and differentiability properties of functions, Princeton Mathematical Series, No. 30, Princeton University Press, Princeton, N.J., 1970. MR 0290095
  • 13. Elias M. Stein and Guido Weiss, On the theory of harmonic functions of several variables. I. The theory of 𝐻^{𝑝}-spaces, Acta Math. 103 (1960), 25–62. MR 0121579
  • 14. E. M. Stein and G. Weiss, Generalization of the Cauchy-Riemann equations and representations of the rotation group, Amer. J. Math. 90 (1968), 163–196. MR 0223492
  • 15. E. M. Stein and G. Weiss, An introduction to harmonic analysis on Euclidean spaces, Princeton University Press, Princeton NJ, 1969.
  • 16. Thomas H. Wolff, Counterexamples with harmonic gradients in 𝑅³, Essays on Fourier analysis in honor of Elias M. Stein (Princeton, NJ, 1991), Princeton Math. Ser., vol. 42, Princeton Univ. Press, Princeton, NJ, 1995, pp. 321–384. MR 1315554
  • 17. A. Zygmund, Trigonometric series: Vols. I, II, Second edition, reprinted with corrections and some additions, Cambridge University Press, London-New York, 1968. MR 0236587

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 30E25, 42B25

Retrieve articles in all journals with MSC (2000): 30E25, 42B25


Additional Information

Anatoly Ryabogin
Affiliation: Department of Mathematics, Ben Gurion University of the Negev, P.O.B. 653, Be’er Sheva 84105, Israel
Email: ryabs@math.ksu.edu

Dmitry Ryabogin
Affiliation: Department of Mathematics, Kansas State University, Manhattan, Kansas 66506-2602
Email: ryabs@math.ksu.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-07-08940-X
Keywords: Hardy spaces, subharmonic functions
Received by editor(s): January 31, 2006
Published electronically: April 5, 2007
Communicated by: Mei-Chi Shaw
Article copyright: © Copyright 2007 American Mathematical Society