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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Necessary and sufficient conditions for the derivation of integrals of $L_{\psi }$-functions
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by C. A. Hayes PDF
Trans. Amer. Math. Soc. 223 (1976), 385-395 Request permission

Abstract:

It has been shown recently that a necessary and sufficient condition for a derivation basis to derive the $\mu$-integrals of all functions in ${L^{(q)}}(\mu )$, where $1 < q < + \infty$, and $\mu$ is a $\sigma$-finite measure, is that the basis possess Vitali-like covering properties, with covering families having arbitrarily small ${L^{(p)}}(\mu )$-overlap, where ${p^{ - 1}} + {q^{ - 1}} = 1$. The corresponding theorem for the case $p = 1,q = + \infty$ was established by R. de Possel in 1936. The present paper extends these results to more general dual Orlicz spaces. Under suitable restrictions on the dual Orlicz functions $\Phi$ and $\Psi$, it is shown that a necessary and sufficient condition for a basis to derive the $\mu$-integrals of all functions in ${L_\Psi }(\mu )$ is that the basis possess Vitali-like covering families whose ${L_\Phi }(\mu )$-overlap is arbitrarily small. Certain other conditions relating ${L_\Phi }(\mu )$-strength and derivability are also discussed.
References
  • Antonio Cordoba, On the Vitali covering properties of a differentiation basis, Studia Math. 57 (1976), no. 1, 91–95. MR 419714, DOI 10.4064/sm-57-1-91-95
  • Miguel de Guzmán, On the derivation and covering properties of a differentiation basis, Studia Math. 44 (1972), 359–364. MR 330373, DOI 10.4064/sm-44-4-359-364
  • C. A. Hayes Jr., Derivation in Orlicz spaces, J. Reine Angew. Math. 253 (1972), 162–174. MR 306431, DOI 10.1515/crll.1972.253.162
  • —, Derivation of the integrals of ${L^{(q)}}$-functions, Pacific J. Math. (to appear). C. Hayes and C. Y. Pauc, Derivation and martingales, Ergebnisse Math. Grenzgebiete, Bd. 49, Springer-Verlag, Berlin, 1970. R. de Posses, Sur la derivation abstraite des fonctions d’ensemble, J. Math. Pures Appl. 15 (1936), 391-409.
  • Adriaan Cornelis Zaanen, Linear analysis. Measure and integral, Banach and Hilbert space, linear integral equations, Interscience Publishers Inc., New York; North-Holland Publishing Co., Amsterdam; P. Noordhoff N. V., Groningen, 1953. MR 0061752
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Additional Information
  • © Copyright 1976 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 223 (1976), 385-395
  • MSC: Primary 26A24; Secondary 28A15
  • DOI: https://doi.org/10.1090/S0002-9947-1976-0427554-4
  • MathSciNet review: 0427554