| Autor: |
Diestel, Geoff, Grafakos, Loukas, Honzik, Peter, Si, Zengyan, Terwilleger, Erin |
| Jazyk: |
angličtina |
| Rok vydání: |
2011 |
| Předmět: |
|
| Zdroj: |
Commun. Math. Anal. Conference 3 (2011), 99-107 |
| Popis: |
Suppose that $\Omega$ lies in the Hardy space $H^1$ of the unit circle $\mathbf S^{1}$ in $\mathbf R^2$. We use the Calderón-Zygmund method of rotations and the uniform boundedness of the bilinear Hilbert transforms to show that the bilinear singular operator with the rough kernel $\mathrm{p.v.} \, \Omega(x/|x|) |x|^{-2}$ is bounded from $L^p(\mathbf R)\times L^q(\mathbf R)$ to $L^r(\mathbf R)$, for a large set of indices satisfying $1/p+1/q=1/r$. We also provide an example of a function $\Omega$ in $L^q(\mathbf S^{ 1})$ with mean value zero to show that the singular integral operator given by convolution with $\mathrm{p.v.} \, \Omega(x/|x|) |x|^{-2}$ is not bounded from $L^{p_1}(\mathbf R)\times L^{p_2} (\mathbf R )$ to $ L^{p}(\mathbf R )$ for $1/22.$ |
| Databáze: |
OpenAIRE |
| Externí odkaz: |
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