We establish a mixed norm estimate for the Radon transform in $\mathbb{R}^2$ when the set of directions has fractional dimension. This estimate is used to prove a result about an exceptional set of directions connected with projections of planar sets. That leads to a conjecture analogous to a well-known conjecture of Furstenberg.

MSC Classifications:

44A12, 28A78 show english descriptions Radon transform [See also 92C55]
Hausdorff and packing measures 44A12 - Radon transform [See also 92C55]
28A78 - Hausdorff and packing measures

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