= Solution
On the polynomial subspace of $L^2(\gamma)$, the <Gaussian creation and annihilation operators> are $a^-=D$ and $a^+=x-D$. <Gaussian integration by parts> makes them adjoints there and gives $[a^-,a^+]=I$. The <Gaussian number operator> is
$$
\boxed{N=a^+a^-=-D^2+xD.}
$$
The <Probabilists' Hermite polynomials> are $\operatorname{He}_n(x)=(-1)^ne^{x^2/2}D^ne^{-x^2/2}=(a^+)^n1$. Their generating function is $e^{sx-s^2/2}=\sum_n\operatorname{He}_n(x)s^n/n!$, giving $D\operatorname{He}_n=n\operatorname{He}_{n-1}$. The commutator also yields
$$
\boxed{N\operatorname{He}_n=n\operatorname{He}_n,\qquad\int\operatorname{He}_m\operatorname{He}_n\,d\gamma=n!\,\mathbf1_{m=n}.}
$$
The orthogonality follows by repeated <integration by parts>; the leading coefficient is one, so these polynomials span every polynomial. By the allowed density assumption $h_n=\operatorname{He}_n/\sqrt{n!}$ is an <orthonormal basis>. Define the closed number operator by $N\sum c_nh_n=\sum nc_nh_n$ on $\sum n^2|c_n|^2<\infty$. This diagonal <multiplication operator> is self-adjoint and positive semidefinite; polynomial truncations show it is the closure of the polynomial operator.
The <Ornstein-Uhlenbeck semigroup> is $P_t=e^{-tN}$, or $P_tf=\sum_ne^{-nt}c_nh_n$. It also has the <Mehler formula for the Ornstein-Uhlenbeck semigroup>
$$
\boxed{P_tf(x)=\mathbb E\left[f\!\left(e^{-t}x+\sqrt{1-e^{-2t}}\,Z\right)\right],\qquad Z\sim N(0,1).}
$$
To verify the formula, apply its right side to $e^{sx-s^2/2}$: the <Gaussian moment-generating function> makes the result $e^{se^{-t}x-s^2e^{-2t}/2}$, proving the <eigenvalue> identity on every Hermite polynomial. <Positivity> and invariance of <Gaussian measure> give $L^2$ contraction by <Jensen inequality>, so polynomial density extends the equality to all $L^2(\gamma)$. The coefficient expansion gives
$$
\|P_tf-\mathbb E_\gamma f\|_2^2=\sum_{n\geq1}e^{-2nt}|c_n|^2\longrightarrow0.
$$
For continuously differentiable $f$ with bounded <derivative>, differentiate the Mehler expectation by the <dominated convergence theorem> to get the <Ornstein-Uhlenbeck gradient commutation identity>
$$
\boxed{(P_tf)'=e^{-t}P_t(f').}
$$
The same identity extends to the <Gaussian Sobolev form domain>. Mehler's formula also gives <pointwise convergence> to the Gaussian mean for such $f$, since bounded <derivative> permits at most linear growth.
The <Gaussian Dirichlet energy> is the energy form of $N$:
$$
\boxed{\mathcal E_\gamma(f)=\lim_{t\downarrow0}\frac{\|f\|_2^2-\langle f,P_tf\rangle}{t}=\sum_{n\geq1}n|c_n|^2=\int|f'|^2\,d\gamma.}
$$
For polynomials this follows from $N=(a^-)^*a^-$, and closure extends it to the form domain. For the given continuously differentiable $f$, <Gaussian integration by parts> yields $\langle f',h_{n-1}\rangle=\sqrt n c_n$, so <Parseval identity> proves the derivative-energy equality directly. Bounded <derivative> puts $f$ in the <Gaussian Sobolev space>, hence in that form domain; it need not be in the full <operator domain>, so $\langle f,Nf\rangle$ is not always a legitimate initial definition.
Use the <entropy functional> $\operatorname{Ent}_\gamma(h)=\int h\log h\,d\gamma-(\int h\,d\gamma)\log\int h\,d\gamma$. Begin with bounded smooth $h>0$ bounded away from zero, and put $h_t=P_th$. Invariance and <Gaussian integration by parts> give the <Ornstein-Uhlenbeck entropy dissipation identity>
$$
-\frac{d}{dt}\operatorname{Ent}_\gamma(h_t)=\int\frac{|h_t'|^2}{h_t}\,d\gamma.
$$
The gradient identity and the permitted weighted <Cauchy-Schwarz inequality>, applied with weight $h$, imply
$$
\frac{|h_t'|^2}{h_t}=e^{-2t}\frac{|P_t(h')|^2}{P_th}\leq e^{-2t}P_t\!\left(\frac{|h'|^2}{h}\right).
$$
The entropy tends to zero as $t\to\infty$ by Mehler's formula and bounded convergence. Integrating in time and using invariance yields $\operatorname{Ent}_\gamma(h)\leq\frac12\int|h'|^2/h\,d\gamma$. Set $h=f^2$ to obtain the sharp <Gaussian logarithmic Sobolev inequality>
$$
\boxed{\operatorname{Ent}_\gamma(f^2)\leq2\int|f'|^2\,d\gamma=2\mathcal E_\gamma(f).}
$$
For the stated possibly unbounded $f$, apply the argument to smooth positive clipped approximations with a common positive lower bound and uniformly bounded <derivatives>. They can converge pointwise together with their <derivatives> and have a common linear-growth bound. Since <Gaussian measure> integrates every polynomial moment, <dominated convergence theorem> passes both the entropy and the <derivative> energy to the limit. This also justifies use of the supplied inequality when $2ff'$ itself is unbounded. The normalization $\|f\|_1=1$ concerns $f$, and does not imply $\int f^2\,d\gamma=1$; the second term in the entropy definition must be retained.
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