A Generalized Characterization of Commutators of Parabolic Singular Integrals

Hofmann, Steve ; Li, Xinwei ; Yang, Dachun

Canadian mathematical bulletin, Tome 42 (1999) no. 4, pp. 463-477

Résumé

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 commutator 1 $$\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,{{T}_{A}}f\left( x \right)\equiv \text{p}\text{.v}\text{.}\int_{{{\mathbb{R}}^{n}}}{K\left( x-y \right)\left[ A\left( x \right)-A\left( y \right) \right]}f\left( y \right)dy$$ 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.

Détail

DOI : 10.4153/CMB-1999-054-0

Mots-clés : 42B20, parabolic singular integral, commutator, parabolic BMO sobolev space, homogeneous space, T1-theorem, symbol