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

TL;DR

This paper establishes stability estimates with an a priori bound for the inverse of a weighted local Radon transform in two dimensions, under specific differential equation conditions on the weight.

Contribution

It provides new weak stability estimates for the inverse weighted local Radon transform with particular weight functions satisfying a differential equation.

Findings

Derived stability estimates for the inverse transform.

Applicable to functions with an a priori bound.

Focus on weights satisfying a specific differential equation.

Abstract

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 $R_m[f](\xi,\eta)=\int f(x, \xi x + \eta) m(\xi, \eta, x)\,\text{d} x$ 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.