Extremals for a class of convolution operators (Q2705832)
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scientific article
| Language | Label | Description | Also known as |
|---|---|---|---|
| English | Extremals for a class of convolution operators |
scientific article |
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19 March 2001
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convolution operator
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fractional integral
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Sobolev inequality
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Extremals for a class of convolution operators (English)
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Suppose that \(q>1\) and that \(g\) on \({\mathbb R}^n\) is symmetrically decreasing. If there is an \(\varepsilon>0\) such that \(g\in L^{q-\varepsilon}\cap L^{q+\varepsilon}\), the author then proves that the convolution operator \(T_g:\;L^p\to L^r\), \(1/r=1/p+1/q-1\), attains its norm on the unit ball of \(L^p\) for any \(p>1\). Applications to and extensions of the well-known results for fractional integrals and Sobolev inequalities are also discussed.
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