Stability estimates with a priori bound for the inverse local Radon transform

We consider the inverse problem for the $2$-dimensional weighted local Radon transform $R_m[f]$, where $f$ is supported in $y\geq x^2$ and $R_m[f](ξ,η)=\int f(x, ξx + η) m(ξ, η, x)\,\text{d} x$ is defined near $(ξ,η)=(0,0)$. For weight functions satisfying a certain differential equation we give weak estimates of $f$ in terms of $R_m[f]$ for functions $f$ that satisfies an a priori bound.

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