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Use the scaled proximal operator ${\mathrm{prox}}_{\gamma E}$; take $\gamma =1$ for each unscaled resolvent.

For the quadratic case, write $D^{T}D=\mathrm{diag}(d_1,\dots ,d_n)$ and $\xi =Qx$. Since $Q$ is an orthogonal matrix, the quadratic proximal term becomes $\Vert \xi -Qz{\Vert }^2/2$. The subgradient optimality condition givesAll $d_j\ge 0$, so each diagonal entry is positive, including when $d_j=0$. Multiplying by the inverse diagonal matrix givesThe proximal operator of Poisson data fidelity separates pointwise. At each $t$, the scalar optimality equation on $x>0$ isSince $y>0$, the two roots have opposite signs. Only the positive root is admissible in the logarithm:The negative root is outside the effective domain. The second derivative $1+\gamma y/x^2$ of the proximal objective is positive, establishing the unique pointwise minimizer. The integral formula is understood on the function spaces where these quantities are admissible.

Finally, for $E(x)=|x|$, positive and negative minimizers obey $x-z+\gamma =0$ and $x-z-\gamma =0$, respectively. At zero the subdifferential condition is $z\in [-\gamma ,\gamma ]$. Combining these cases yields soft thresholding:Thus $\gamma =1$ supplies all three requested closed forms.