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 $y\ge x^2$ and $R_m[f](\xi ,\eta )=\int f(x,\xi x+\eta )m(\xi ,\eta ,x)\text{d}x$ is defined near $(\xi ,\eta )=(0,0)$. 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.