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)



Maximal inequalities related to generalized a.e. continuity

Authors: W. B. Jurkat and J. L. Troutman
Journal: Trans. Amer. Math. Soc. 252 (1979), 49-64
MSC: Primary 46E30; Secondary 26D15
MathSciNet review: 534110
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: An integral inequality of the classical Hardy-Littlewood type is obtained for the maximal function of positive convolution operators associated with approximations of the identity in $ {R^n}$. It is shown that the (formally) rearranged maximal function can in general be estimated by an elementary integral involving the decreasing rearrangements of the kernel of the approximation and the function being approximated. (The estimate always holds when the kernel has compact support or a decreasing radial majorant integrable in a neighborhood of infinity; a one-dimensional counterexample shows that integrability alone may not suffice.)

The finiteness of the integral determines a Lorentz space of functions which are a.e. continuous in the generalized sense of the approximation. Conversely, in dimension one it is established that this space is the largest strongly rearrangement invariant Banach space of such functions. In particular, the new inequality provides access to the study of Cesàro continuity of order less than one.

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

  • [1] D. L. Burkholder, One-sided maximal functions and $ {H^p}$, J. Functional Analysis 18 (1975), 429-454. MR 0365693 (51:1945)
  • [2] L. A. Caffarel and A. P. Calderón, Weak-type estimates for Hardy-Littlewood maximal functions, Studia Math. 49 (1974), 217-223. MR 0335729 (49:509)
  • [3] A. P. Calderón and A. Zygmund, On the existence of certain singular integrals, Acta Math. 88 (1952), 85-139. MR 0052553 (14:637f)
  • [4] K. M. Chong and N. M. Rice, Equimeasurable rearrangements of functions, Queens Papers in Pure and Appl. Math. 28 (1971). MR 0372140 (51:8357)
  • [5] A. Cordoba, A radial multiplier and Kakeya maximal function, Bull. Amer. Math. Soc. 81 (1975), 428-430. MR 0365016 (51:1269)
  • [6] C. Fefferman and E. M. Stein, Some maximal inequalities, Amer. J. Math. 93 (1971), 107-115. MR 0284802 (44:2026)
  • [7] A. Garsia, Topics in almost everywhere convergence, Markham, Chicago, 1970. 8. G. H. Hardy, J. E. Littlewood and G. Polya, Inequalities, Cambridge Univ. Press., Cambridge, England, 1934. MR 0261253 (41:5869)
  • [9] A. Kolmogorov, Sur les fonctions harmoniques conjuguées et les séries de Fourier, Fund. Math. 7 (1925), 24-29.
  • [10] G. G. Lorentz, On the theory of the spaces $ \Lambda $, Pacific J. Math. 1 (1951), 411-429. MR 0044740 (13:470c)
  • [11] G. O. Okikiolu, Aspects of the theory of bounded integral operators in $ {L^p}$-spaces, Academic Press, New York, 1971. MR 0445237 (56:3581)
  • [12] S. Sawyer, Maximal inequalities of weak-type, Ann. of Math. (2) 84 (1966), 157-173. MR 0209867 (35:763)
  • [13] E. M. Stein, On limits of sequences of operators, Ann. of Math. (2) 74 (1961), 140-170. MR 0125392 (23:A2695)
  • [14] -, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, N. J., 1970. MR 0290095 (44:7280)
  • [15] P. L. Walker, Rearranging maximal functions in $ {R^n}$, Proc. Edinburgh Math. Soc. 19 (1975), 363-369. MR 0385047 (52:5917)
  • [16] N. Wiener, The ergodic theorem, Duke Math. J. 5 (1939), 1-18. MR 1546100
  • [17] A. Zygmund, Trigonometric series, 2nd ed., Vol. 2, Cambridge Univ. Press, Cambridge, England, 1959. MR 0107776 (21:6498)

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC: 46E30, 26D15

Retrieve articles in all journals with MSC: 46E30, 26D15

Additional Information

Article copyright: © Copyright 1979 American Mathematical Society

American Mathematical Society