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| A new generalized Radon transform $R_{alpha,,beta}$ on the plane
for functions even in each variable is defined, which has natural
connections with the bivariate Hankel transform, the generalized
biaxially symmetric potential operator $Delta_{alpha,,beta}$ and
the Jacobi polynomials $P_k^{(beta,,alpha)}(t)$. The transform
$R_{alpha,,beta}$ and its dual $R_{alpha,,beta}^ast$ are
studied in a systematic way, and in particular, the generalized
Fuglede formula and some inversion formulas for $R_{alpha,,beta}$
for functions in $L_{alpha,,beta}^p(RR^2_+)$ are obtained in
terms of the bivariate Hankel-Riesz potential. Moreover, the
transform $R_{alpha,,beta}$ is used to represent the solutions of
the partial differential equations $Lu:=sum_{j=1}^m
a_jDelta_{alpha,,beta}^ju=f$ with constant coefficients $a_j$\ |
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| Keywords:generalized Radon transform;Hankel transform;Jacobi polynomial;inversion formula;support theorem |
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