  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 Erd\H{o}s--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.
