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We study horospherical Radon transforms that integrate functions on the$n$-dimensional real hyperbolic space over horospheres of arbitrary fixeddimension $1\le d\le n-1$. Exact existence conditions and new explicitinversion formulas are obtained for these transforms acting on smooth functionsand functions belonging to $L^p$. The case $d=n-1$ agrees with the well-knownGelfand-Graev transform.

Radon Transforms over Lower-Dimensional Horospheres in Real Hyperbolic Space