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Description:
We prove new $L^p$-$L^q$-estimates for solutions to elliptic differential operators with constant coefficients in $\mathbb{R}^3$. We use the estimates for the decay of the Fourier transform of particular surfaces in $\mathbb{R}^3$ with vanishing Gaussian curvature due to Erdős–Salmhofer to derive new Fourier restriction– extension estimates. These allow for constructing distributional solutions in $L^q(\mathbb{R}^3)$ for $L^p$-data via limiting absorption by well-known means.