Cauchy Transforms and Bi-Axial Monogenic Power Functions

Abstract

We consider the transform of suitable Clifford valued functions or distributions $f\left( {{{{\vec{x}}}_{1}},{{{\vec{x}}}_{2}}} \right)$ defined on bi-axially symmetric domains in R ^p+q with p+q = m. Such functions have the expansion

$$f\left( {{{{\vec{x}}}_{1}},{{{\vec{x}}}_{2}}} \right) = \sum\limits_{{k,l}} {{{f}_{{k,l}}}} \left( {\left| {{{{\vec{x}}}_{1}}} \right|,\left| {{{{\vec{x}}}_{2}}} \right|} \right){{p}_{{k,l}}}\left( {{{{\vec{x}}}_{1}},{{{\vec{x}}}_{2}}} \right), $$

Where ${p_{k,l}}\left( {{{\vec x}_1},{{\vec x}_2}} \right)$ are bi-axial spherical monogenics of degree k in ${\vec x_1}$ and degree l in ${\vec x_2}$. We define generalised Cauchy transforms for the case when the f _k,l are independent of $\left| {{{\vec x}_2}} \right|$:

$$\Lambda_{k,l}^{\left( 1 \right)}\left( f \right)\left( {\vec x} \right) = - \frac{1}{{{\omega_m}}}\int_{{R^P}} {\frac{{\left[ {{{\vec x}_1} + {{\vec x}_2} - \vec u} \right]}}{{{{\left| {{{\vec x}_1} + {{\vec x}_2} - \vec u} \right|}^{m + 2l}}}}} {p_{k,l}}\left( {\vec \eta,{{\vec x}_2}} \right){f_{k,l}}\left( \lambda \right){d^p}\vec u$$

$${\Lambda ^{\left( 2 \right)}}\left( f \right)\left( {\vec x} \right) = - \frac{1}{{{\omega _m}}}\int_{{R^P}} {\frac{{\left[ {{{\vec x}_1} + {{\vec x}_2} - \vec u} \right]}}{{{{\left| {{{\vec x}_1} + {{\vec x}_2} - \vec u} \right|}^{m + 2l}}}}} \vec \eta {P_{k,l}}\left( {\vec \eta,{{\vec x}_2}} \right){f_{k,l}}\left( \lambda \right){d^p}\vec u$$

Where $\vec u = \lambda \vec \eta,\left| {\vec \eta } \right| = 1$.

The angular integrations are explicitly carried out and conditions for the existence of $\Lambda _{{k,l\alpha }}^{{\left( j \right)}} \equiv \Lambda _{{k,l}}^{{\left( j \right)}}\left( {{{\lambda }^{\alpha }}} \right)$ are derived. Finally INNER and OUTER bi-axial power functions are defined in analogy to the axial case and are related to the Cauchy transforms $\Lambda _{k,l,\alpha }^{\left( j \right)}$.