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A Generalized Characterization of Commutators of Parabolic Singular Integrals
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- Journal:
- Canadian Mathematical Bulletin / Volume 42 / Issue 4 / 01 December 1999
- Published online by Cambridge University Press:
- 20 November 2018, pp. 463-477
- Print publication:
- 01 December 1999
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- Article
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Let
$x\,=\,\left( {{x}_{1}},\,.\,.\,.\,,\,{{x}_{n}} \right)\,\in \,{{\mathbb{R}}^{n}}$ and 
${{\delta }_{\text{ }\!\!\lambda\!\!\text{ }}}x\,=\,\left( {{\text{ }\!\!\lambda\!\!\text{ }}^{{{\alpha }_{1}}}}{{x}_{1}},\,.\,.\,.\,,\,{{\text{ }\!\!\lambda\!\!\text{ }}^{{{\alpha }_{n}}}}{{x}_{n}} \right)$, where 
$\text{ }\lambda \,>\text{0}$ and 
$1\,\le \,{{\alpha }_{1}}\,\le \,\cdot \,\cdot \,\cdot \,\le \,{{\alpha }_{n}}$. Denote 
$\left| \alpha \right|\,=\,{{\alpha }_{1}}+\,\cdot \,\cdot \,\cdot \,+{{\alpha }_{n}}$. We characterize those functions 
$A\left( x \right)$ for which the parabolic Calderón commutator1
$$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{T}_{A}}f\left( x \right)\equiv \text{p}\text{.v}\text{.}\int_{{{\mathbb{R}}^{n}}}{K\left( x-y \right)\left[ A\left( x \right)-A\left( y \right) \right]}f\left( y \right)dy$$is bounded on
${{L}^{2}}\left( {{\mathbb{R}}^{n}} \right)$, where 
$K\left( {{\delta }_{\text{ }\!\!\lambda\!\!\text{ }}}x \right)\,=\,{{\text{ }\!\!\lambda\!\!\text{ }}^{-\,\left| \alpha \right|\,-\,1}}K\left( x \right)$, 
$K$ is smooth away fromthe origin and satisfies a certain cancellation property.
