$W^ p$-spaces and Fourier transform
HTML articles powered by AMS MathViewer
- by R. S. Pathak and S. K. Upadhyay PDF
- Proc. Amer. Math. Soc. 121 (1994), 733-738 Request permission
Abstract:
The spaces $W_M^p,W_{M,a}^p,{W^{\Omega ,p}},{W^{\Omega ,b,p}},W_M^{\Omega ,p},W_{M,a}^{\Omega ,b,p}$ generalizing the spaces of type W due to Gurevich (also given by Friedman, and Gelfand and Shilov) are investigated. Here M, $\Omega$ are certain continuous increasing convex functions, a, b are positive constants and $1 \leq p < \infty$. The Fourier transformation F is shown to be a continuous linear mapping as follows: $F:W_{M,a}^p \to {W^{\Omega ,1/a,r}},F:{W^{\Omega ,b,p}} \to W_{M,1/b}^r,F:W_{M,a}^{\Omega ,b,p} \to W_{M,1/b}^{\Omega ,1/a,r}$. These results will be used in investigating uniqueness classes of certain Cauchy problems in future work.References
- Avner Friedman, Generalized functions and partial differential equations, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1963. MR 0165388
- I. M. Gel′fand and G. E. Shilov, Generalized functions. Vol. 3: Theory of differential equations, Academic Press, New York-London, 1967. Translated from the Russian by Meinhard E. Mayer. MR 0217416 B. L. Gurevich, New types of test function spaces and spaces of generalized functions and the Cauchy problem for operator equations, dissertation, Kharkov, 1956. (Russian) S. K. Upadhyay, On certain weighted ${L^p}$-spaces and Fourier and Hankel transforms of distributions, Ph.D. thesis, Banaras Hindu University, Varanesi, 1993.
Additional Information
- © Copyright 1994 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 121 (1994), 733-738
- MSC: Primary 46F05; Secondary 42A38, 46E10
- DOI: https://doi.org/10.1090/S0002-9939-1994-1185272-8
- MathSciNet review: 1185272