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

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