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Let Γ be a convex curve in the plane and let μ ∈ M(R2) be the arc-length measure of Γ. Let us rotate Γ by an angle θ and let μθ be the corresponding measure. Let Tf(x, θ) = f ∗ μθ(x). Then ‖Tf‖L3(T×R2) ≤ c‖f‖L3/2(R2). This is optimal for an arbitrary Γ. Depending on the curvature of Γ, this estimate can be improved by introducing mixed-norm estimates of the form ‖Tf‖ Ls ( T,Lp(R2) ) ≤ c ‖f‖Lp(R2) where p and p′ are conjugate exponents.

Convex curves, Radon transforms and convolution operators defined by singular measures