= Solution
The source uses the forward Fourier phase $e^{ikx}$, so $\widehat{\partial_x A}=-ik\widehat A$. For a nonzero <wavenumber> put $q=|k|$. Biharmonicity becomes $(\partial_y^2-q^2)^2\widehat A=0$. The decaying solutions are $(C+Dy)e^{-qy}$. The normal-<stress> condition gives $-k^2\widehat A(k,0)=-\widehat P(k)$, hence $C=\widehat P/k^2$. The shear condition gives $\widehat A_y(k,0)=0$, so $D=qC$. Consequently
$$
\boxed{\widehat A(k,y)=\frac{1+qy}{k^2}\widehat P(k)e^{-qy},\qquad k\ne0.}
$$
Its second derivatives are $\widehat{A_{xx}}=-\widehat P(1+qy)e^{-qy}$ and $\widehat{A_{yy}}=\widehat P(qy-1)e^{-qy}$. The given <strain>/<stress> inversion therefore yields
$$
2\mu\widehat v_y=\widehat P(2\nu-1-qy)e^{-qy}.
$$
Integrating in $y$ with the stated decaying-mode condition gives
$$
\boxed{2\mu\widehat v=\left(y+\frac{2(1-\nu)}q\right)\widehat P(k)e^{-qy},\qquad k\ne0.}
$$
This is the <Fourier solution of a loaded elastic half-space>.
The zero <wavenumber> needs separate treatment: these displayed divisions by $k^2$ and $|k|$ are not formulas at $k=0$. In particular a nonzero resultant $\widehat P(0)=\int P\,dx$ is incompatible with an ordinary decaying zero-mode <displacement> of the stipulated kind; two-dimensional net loading produces a logarithmic <displacement> reference issue. Balanced loads or an appropriate distributional/reference-<displacement> formulation are needed for a global transform solution. The nonzero modes and the <displacement>-gradient kernel below remain the correct formal calculation; the singular zero mode must not be silently assigned a finite value.
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