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.