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Weighted norm inequalities for the Hankel- and -transformations

Published online by Cambridge University Press:  14 November 2011

S. A. Emara  and
H. P. Heinig
S. A. Emara
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, L8S 4K1, Canada
H. P. Heinig
Affiliation:
Department of Mathematics and Statistics, McMaster University, Hamilton, Ontario, L8S 4K1, Canada
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Synopsis

We give conditions on pairs of non-negative weight functions u and v which are sufficient that, for 1<p, q <∞,

where T is the Hankel-or the K-transformation.

The proofs are based on a weighted Marcinkiewicz interpolation theorem for linear operators. In the case that T is the Hankel transformation and 1<p≦q <∞, the result is similar to a weighted estimate of Heywood and Rooney [9], but with different weight conditions.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 103 , Issue 3-4 , 1986 , pp. 325 - 333
Copyright
Copyright © Royal Society of Edinburgh 1986

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References

1Benedetto, J. J., Heinig, H. P. and Johnson, R.. Weighted Hardy spaces and the Laplace transform II. Math. Nachr. (to appear).Google Scholar
2Calderón, A. P.. Spaces between L1 and L∞ and the theorem of Marcinkiewicz. Studio Math. 26 (1966), 273–299.Google Scholar
3Erdelyi, A. (ed.). Tables of integral transforms, Vol. II (New York: McGraw Hill, 1954).Google Scholar
4Gradshteyn, I. G. and Ryzhik, I. M.. Tables of Integrals, Series and Products (New York: Academic Press, 1980).Google Scholar
5Hardy, G. H., Littlewood, J. E. and Polya, G.. Inequalities (Cambridge University Press, 1952).Google Scholar
6Heinig, Hans P.. Weighted norm inequalities for classes of operators. Indiana Univ. Math. J. 33 (1984), 573–582.CrossRefGoogle Scholar
7Heinig, Hans P.. Estimates for operators in mixed weighted Lp-spaces. Trans. Amer. Math. Soc. 287 (1985), 483–493.Google Scholar
8Heinig, Hans P.. Weighted norm inequalities for certain integral operators II. Proc. Amer. Math. Soc. 95 (3) (1985), 387–396.CrossRefGoogle Scholar
9Heywood, P. and Rooney, P. G.. A weighted norm inequality for the Hankel transformation. Proc. Roy. Soc. Edinburgh Sect. A 99 (1984), 45–50.CrossRefGoogle Scholar
10Mazja, Wladimir. Einbettungssätze fur Sobolevsche Räume, Teil 1. Teubner Texte zur Math (Leipzig: Teubner, 1979).Google Scholar
11Muckenhoupt, B.. A note on the two weight function condition for the Fourier transform norm inequality. Proc. Amer. Math. Soc. 88 (1983), 97–100.CrossRefGoogle Scholar
12Muckenhoupt, B.. Weighted norm inequalities for the Fourier transform. Trans. Amer. Math. Soc. 276 (1983), 729–742.CrossRefGoogle Scholar
13Schwartz, Laurent. Mathematics for the Physical Sciences (Paris: Herman, 1966).Google Scholar