Weighted norm inequalities for convolution operators and links with the Weiss Conjecture |
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Authors: | Zen Harper |
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Institution: | (1) Kenoron, West Lynne, Cheddar, Somerset, BS27 3JL, England |
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Abstract: | We study convolution operators on weighted Lebesgue spaces and obtain weight characterisations for boundedness of these operators
with certain kernels. Our main result is Theorem 3 which enables us to obtain results for certain kernel functions supported
on bounded intervals; in particular we get a direct proof of the known characterisations for Steklov operators in Section
3 by using the weighted Hardy inequality. Our methods also enable us to obtain new results for other kernel functions in Section
4. In Section 5 we demonstrate that these convolution operators are related to operators arising from the Weiss Conjecture
(for scalar-valued observation functionals) in linear systems theory, so that results on convolution operators provide elementary examples of nearly bounded semigroups
not satisfying the Weiss Conjecture. Also we apply results on the Weiss Conjecture for contraction semigroups to obtain boundedness
results for certain convolution operators. |
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Keywords: | Primary 44A35 Secondary 47G10 47B35 |
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