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The Cameron-Storvick function space integral: An L(Lp , Lp′ ) theory

Published online by Cambridge University Press:  22 January 2016

G. W. Johnson  and
D. L. Skoug
G. W. Johnson
Affiliation:
University of Nebraska
D. L. Skoug
Affiliation:
University of Nebraska
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In [3] Cameron and Storvick introduced a very general operator-valued function space “integral”. In [3-5, 8, 9, 11, 13-20] the existence of this integral as an operator from L2 to L2 was established for certain functions. Recently the existence of the integral as an operator from L1 to L∞, has been studied [6, 7, 21]. In this paper we study the integral as an operator from Lp to Lp′, where 1 < p ≤ 2. The resulting theorems extend the theory substantially and indicate relationships between the L2-L2 and L1-L∞ theories that were not apparent earlier. Even in the most studied case, p = p′ = 2, the results below strengthen the theory.

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 60 , February 1976 , pp. 93 - 137
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1976

References

[1] Boas, R. P., Entire Functions, Academic Press, New York, 1954.Google Scholar
[2] Butzer, P. L. and Nessel, R. J., Fourier Analysis and Approximation, Vol. 1, Academic Press, New York, 1971.CrossRefGoogle Scholar
[3] Cameron, R. H. and Storvick, D. A., An operator valued function space integral and a related integral equation, J. Math, and Mech., 18 (1968), 517–552.Google Scholar
[4] Cameron, R. H. and Storvick, D. A., An integral equation related to the Schroedinger equation with application to integration in function space, Problems in Analysis, Princeton U. Press, Princeton, 1970, 175–193.Google Scholar
[5] Cameron, R. H. and Storvick, D. A., An operator valued function space integral applied to integrals of functions of class L 2 , J. Math. Anal. Appl., 42 (1973), 330–372.Google Scholar
[6] Cameron, R. H. and Storvick, D. A., An operator valued function space integral applied to integrals of functions of class L1 , Proc. of the London Math. Soc., 27 (1973), 345–360.Google Scholar
[7] Cameron, R. H. and Storvick, D. A., An operator valued function space integral applied to multiple integrals of functions of class L1 , Nagoya Math. J., 51 (1973), 91–122.CrossRefGoogle Scholar
[8] Collins, R. J., An operator valued function space integral applied to analytic and non-analytic functions of integrals, Thesis, U. of Minnesota (1971).Google Scholar
[9] Ewan, Robert A., The Cameron-Storvick operator valued function space integral for a class of finite dimensional functionals, Thesis, U. of Nebraska (1972).Google Scholar
[10] Fuks, B. A., Analytic Functions of Several Complex Variables, A.M.S. Translations of Mathematical Monographs, 8 (1963).CrossRefGoogle Scholar
[11] Haugsby, Bernard O., An operator valued integral in a function space of continuous vector valued functions, Thesis, U. of Minnesota (1972).Google Scholar
[12] Hille, E. and Phillips, R. S., Functional Analysis and Semi-groups, A.M.S. Colloq. Publ., 31 (1957).Google Scholar
[13] Johnson, G. W. and Skoug, D. L., Operator-valued Feynman integrals of certain finite-dimensional functionals, Proc. Amer. Math. Soc., 24 (1970), 774–780.CrossRefGoogle Scholar
[14] Johnson, G. W. and Skoug, D. L., Operator-valued Feynman integrals of finite-dimensional functionals, Pacific J. of Math., 34 (1970), 774–780.CrossRefGoogle Scholar
[15] Johnson, G. W. and Skoug, D. L., An operator valued function space integral: a sequel to Cameron and Storvick’s paper, Proc. Amer. Math. Soc., 27 (1971), 514–518.Google Scholar
[16] Johnson, G. W. and Skoug, D. L., A Banach algebra of Feynman integrable functionals with application to an integral equation formally equivalent to Schroedinger’s equation, J. of Functional Analysis, 12 (1973), 129–152.CrossRefGoogle Scholar
[17] Johnson, G. W. and Skoug, D. L., Feynman integrals of non-factorable finite-dimensional functionals, Pacific J. of Math., 45 (1973), 257–267.Google Scholar
[18] Johnson, G. W. and Skoug, D. L., Cameron and Storvick’s function space integral for certain Banach spaces of functionals, J. of London Math. Soc., 9 (1974), 103–117.Google Scholar
[19] Johnson, G. W. and Skoug, D. L., Cameron and Storvick’s function space integral for a Banach space of functionals generated by finite-dimensional functionals, Annali Di Mathematica Pura Ed Applicata, 104 (1975), 67–83.CrossRefGoogle Scholar
[20] Johnson, G. W. and Skoug, D. L., A function space integral for a Banach space of functionals on Wiener space, Proc. Amer. Math. Soc., 43 (1974), 141–148.Google Scholar
[21] Johnson, G. W. and Skoug, D. L., The Cameron-Storvick function space integral: The L1 theory, to appear in J. of Math. Anal, and Appl., 50 (1975), 647–667.Google Scholar
[22] Kato, T., Perturbation Theory for Linear Operators, Springer-Verlag, New York, 1966.Google Scholar
[23] Naimark, M. A., Normed Rings, P. Noordhoff, Ltd., Groningen, 1964.Google Scholar
[24] Petersen, L., Thesis, U. of Nebraska, in preparation.Google Scholar
[25] Stein, E. M., Singular Integrals and Differentiability Properties of Functions, Princeton U. Press, Princeton, 1970.Google Scholar
[26] Stein, E. M. and Weiss, G., Introduction to Fourier Analysis on Euclidean Spaces, Princeton U. Press, Princeton, 1971.Google Scholar
[27] Zaanen, A. C., Integration, North-Holland Publishing Co., Amsterdam, 1967.Google Scholar