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For proper lower semicontinuous convex functions, the Moreau envelope has gradient $\nabla f_{\tau }=(I-{\mathrm{prox}}_{\tau f})/\tau$. This follows by subgradient inversion under convex conjugacy and the quadratic conjugate. Firm nonexpansiveness of the proximal residual makes the gradient $1/\tau$-Lipschitz continuous.

Use the conventionThis proximal map is also the resolvent of a monotone operator $(I+\tau \partial f{)}^{-1}$. A proper lower semicontinuous convex function has an affine minorant, so the quadratic term makes this minimization coercive and strongly convex. A unique minimizer exists for every $x$. Its subgradient optimality condition isThe Moreau–Yosida regularisation is $f_{\tau }=f\square g^{\tau }$. The conjugate of an infimal convolution and the quadratic conjugate giveThe factor $\tau$ here is essential. Apply subgradient inversion under convex conjugacy, followed by the subdifferential sum rule with the everywhere differentiable quadratic:The last equivalence is precisely the unique proximal minimization condition. ThusThis proves both existence and uniqueness of the subgradient, rather than only identifying a possible element. The finite convex function $f_{\tau }$ is therefore differentiable, with $\nabla f_{\tau }=(I-P_{\tau })/\tau$, the gradient of a Moreau envelope.

For completeness, monotonicity of $\partial f$ applied to the two proximal conditions givesHence $P_{\tau }$ is firmly nonexpansive. Expanding the same inequality shows that $I-P_{\tau }$ is firmly nonexpansive too. In particular, $\nabla f_{\tau }$ is $1/\tau$-Lipschitz continuous. None of this requires a bounded effective domain; the result applies to the next example as well.

Write $h=f\square g$ for the infimal convolution. It is proper by the permitted hypothesis. If $h(x_0)$ and $h(x_1)$ are finite, choose $y_0,y_1$ within $\epsilon$ of their respective infima. For $0<t<1$, put $x_t=(1-t)x_0+t{x}_1$ and $y_t=(1-t)y_0+t{y}_1$. Convexity of $f$ and $g$ givesLet $\epsilon \downarrow 0$. If either endpoint value is infinite, the desired inequality is automatic. Thus the infimal convolution is convex, without assuming the infimum is attained.

For its convex conjugate, replace a negative infimum by a supremum and then change variables $u=x-y$:The two suprema separate because $u$ and $y$ are independent. Properness of $f$ and $g$ makes each supremum strictly greater than $-\infty$, so this separation remains valid when one or both are $+\infty$. Therefore $(f\square g{)}^{\ast }=f^{\ast }+g^{\ast }$. This is the conjugate of an infimal convolution. The bounded-domain assumption is unnecessary for these two calculations; the assumed properness and lower semicontinuity of $h$ will be used when applying subgradient inversion under convex conjugacy.

By the definition of the subdifferential, $p\in \partial J(u)$ meansRearrangement bounds $⟨v,p⟩-J(v)$ by $⟨u,p⟩-J(u)$, and equality is attained at $v=u$. Taking the supremum in the definition of the convex conjugate gives $J^{\ast }(p)=⟨u,p⟩-J(u)$. Conversely this equality bounds every member of that supremum and rearranges to the subgradient inequality. ThereforeApply the same argument to $J^{\ast }$ and use the Fenchel-Moreau theorem $J^{\ast \ast }=J$. With the canonical Hilbert identification of the bidual, the equality is also equivalent to $u\in \partial J^{\ast }(p)$. This proves subgradient inversion under convex conjugacy:The conditions include finiteness at the points in question; expressions involving $+\infty$ are not subgradients merely by formal subtraction.

Let $v={\mathrm{prox}}_J(u)$ and $p=u-v$. The proximal operator optimality condition gives $p\in \partial J(v)$. By subgradient inversion under convex conjugacy, $v\in \partial J^{\ast }(p)$. Since $v=u-p$, this is exactly the optimality condition defining $p={\mathrm{prox}}_{J^{\ast }}(u)$. Uniqueness of both Hilbert proximal minimizers proves Moreau decomposition:Both terms belong to the same Hilbert space after dual identification. No orthogonality of the two terms is asserted for a general convex $J$; that stronger property pertains to special indicator/cone cases.