Fourier solution of a loaded elastic half-space ($\hat{A}=(1+|k|y)\hat{P}{e}^{-|k|y}/k^2$)

For plane strain in $y>0$ under surface normal traction $-P(x)$ and zero surface shear, use the forward Fourier transform phase $e^{ikx}$. At $k\ne 0$, biharmonicity gives $({\partial }_y^2-k^2{)}^2\hat{A}=0$, whose decaying part is $(C+Dy)e^{-|k|y}$. Boundary traction fixes $C=\hat{P}/k^2$, $D=|k|C$, proving the displayed formula. Constitutive inversion and a decaying displacement mode give $2\mu \hat{v}=[y+2(1-\nu )/|k|]\hat{P}{e}^{-|k|y}$. Multiplication by $-ik$ produces an unambiguous displacement-gradient convolution kernel.

The zero mode is separate: a nonzero resultant load $\hat{P}(0)$ conflicts with an ordinary fully decaying displacement normalization in two dimensions and leads to a logarithmic reference issue. Balanced loading or a distributional/reference-displacement treatment is required before extending the transformed formulas globally. Dividing by $k^2$ and assigning a finite value at $k=0$ is invalid.