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)

 
 

 

On weighted norm inequalities for the Lusin area integral


Authors: Carlos Segovia and Richard L. Wheeden
Journal: Trans. Amer. Math. Soc. 176 (1973), 103-123
MSC: Primary 31A05; Secondary 30A78, 42A40
DOI: https://doi.org/10.1090/S0002-9947-1973-0311921-0
MathSciNet review: 0311921
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that the Lusin area integral for the unit circle is a bounded operator on any weighted $ {L^p}$ space, $ 1 < p < \infty $, on which the conjugate function is a bounded operator. Results are also proved for the case $ 0 < p \leq 1$.


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

  • [1] 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 43 #527. MR 0274767 (43:527)
  • [2] A. P. Calderón and A. Zygmund, On the existence of certain singular integrals, Acta. Math. 88 (1952), 85-139. MR 14, 637. MR 0052553 (14:637f)
  • [3] L. Carleson, Interpolations by bounded analytic functions and the corona problem, Ann. of Math. (2) 76 (1962), 547-559. MR 25 #5186. MR 0141789 (25:5186)
  • [4] C. Fefferman and E. M. Steim, $ {H^p}$ spaces of several variables, Acta Math. 129 (1972), 137-193. MR 0447953 (56:6263)
  • [5] C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107-115. MR 0284802 (44:2026)
  • [6] V. F. Gapoškin, A generalization of the theory of M. Riesz on conjugate functions, Mat. Sb. 46 (88) (1958), 359-372. (Russian) MR 20 #6000. MR 0099561 (20:6000)
  • [7] H. Helson and G. Szegö, A problem in prediction theory, Ann. Mat. Pura Appl. (4) 51 (1960), 107-138. MR 22 #12343. MR 0121608 (22:12343)
  • [8] I. I. Hirschman, The decomposition of Walsh and Fourier series, Mem. Amer. Math. Soc. No. 15 (1955), 65 pp. MR 17, 257. MR 0072269 (17:257e)
  • [9] L. Hörmander, $ {L^p}$ estimates for (pluri-)subharmonic functions, Math. Scand. 20 (1967), 65-78. MR 38 #2323. MR 0234002 (38:2323)
  • [10] P. Krée, Sur les multiplicateurs dans $ \mathcal{F}\;{L^p}$ avec poids, Ann. Inst. Fourier (Grenoble) 16 (1966), fasc. 2, 91-121. MR 35 #7080. MR 0216245 (35:7080)
  • [11] J. Marcinkiewicz and A. Zygmund, A theorem of Lusin, Duke Math. J. 4 (1938), 473-485. MR 1546069
  • [12] B. Muckenhoupt, Weighted norm inequalities for the Hardy maximal function, Trans. Amer. Math. Soc. 165 (1972), 207-226. MR 0293384 (45:2461)
  • [13] R. A. Hunt, B. Muckenhoupt and R. L. Wheeden, Weighted norm inequalities for the conjugate function and Hilbert transform, Trans. Amer. Math. Soc. 176 (1973), 227-251. MR 0312139 (47:701)
  • [14] A. Zygmund, Trigonometrical series, 2nd rev. ed., Cambridge Univ. Press, New York, 1959. MR 21 #6498. MR 0107776 (21:6498)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 31A05, 30A78, 42A40

Retrieve articles in all journals with MSC: 31A05, 30A78, 42A40


Additional Information

DOI: https://doi.org/10.1090/S0002-9947-1973-0311921-0
Keywords: Lusin area integral, weighted norm inequalities, $ {H^p}$ spaces, Poisson integrals
Article copyright: © Copyright 1973 American Mathematical Society

American Mathematical Society