Proof of Hypercontractivity of the Mehler Semigroup with Noise Weights (Nelson's Theorem)

Each result cited is universally quantified over the data in its own statement.

Throughout, for real $r\ge0$ we use from The Mehler Semigroup: Cylindrical Form, the Semigroup Law, Invariance, Positivity, Contraction, Symmetry, Hermite Eigenfunctions, Commutation with Partial Derivatives, and the Generator §cylindrical that $P_{r}$ maps $\mathcal{F}C_{\mathrm{pol}}(X)$ into itself, is linear on it, maps each constant function to itself, and that $P_{0}F=F$; and from The Mehler Semigroup: Cylindrical Form, the Semigroup Law, Invariance, Positivity, Contraction, Symmetry, Hermite Eigenfunctions, Commutation with Partial Derivatives, and the Generator §class that every function in $\mathcal{F}C_{\mathrm{pol}}(X)$ is Borel and lies in $L^{r'}(\gamma_{c})$ for every real $r'\ge1$, and that $\mathcal{F}C_{\mathrm{pol}}(X)$ contains the constants and is closed under sums and real multiples. For real $r'\ge1$ and a Borel $u\in L^{r'}(\gamma_{c})$, $\lVert u\rVert_{r'}=(\int_{X}|u|^{r'}\,d\gamma_{c})^{1/r'}$ by The Lebesgue Space of Power-Integrable Functions §norm and Power-Integrable Functions and the p-Seminorm §seminorm. Real powers $s^{\beta}=\exp(\beta\log s)$ of positive $s$ are those of Real Power of a Positive Real Number; for positive exponents they agree with the powers of Real Power of a Nonnegative Real Number §power, whose properties are those of Properties of Real Powers of Nonnegative Real Numbers. We use freely that $\exp$ is positive and strictly increasing with $\exp(u+v)=\exp(u)\exp(v)$ and $\exp(-u)=1/\exp(u)$ (Basic Properties of the Exponential Function), hence $\log$ is strictly increasing on $(0,\infty)$, and that $\log(st)=\log s+\log t$ and $\log'(s)=1/s$ (The Natural Logarithm); in particular $s^{\beta}s^{\beta'}=s^{\beta+\beta'}$, $(s^{\beta})^{2}=s^{2\beta}$ and $\log(s^{\beta})=\beta\log s$ for $s>0$. The function $x\mapsto1+|x|$ is continuous, hence Borel by claim 3 of Borel Measurability and Bounded Integration on a Metric Space, and it is integrable with respect to $\gamma_{c}$: $0\le1+|x|\le2+|x|^{2}$, so by Linearity and Monotonicity of the Lebesgue Integral §nonnegative, claim 6 of Borel Measurability and Bounded Integration on a Metric Space (for the constant $2$, $\gamma_{c}$ being a probability measure) and Moments of a Diagonal Gaussian Measure on a Hilbert Space: Coordinate Covariances, Finite Second Moment, and Exponential Moments of the Squared Norm §moment, $\int_{X}(1+|x|)\,\gamma_{c}(dx)\le\int_{X}(2+|x|^{2})\,\gamma_{c}(dx)=2+M_{2}(\gamma_{c})<\infty$, with the second moment $M_{2}$ of The Second Moment of a Borel Probability Measure on a Hilbert Space and the Probability Measures with Finite Second Moment §moment, and Integrable Function and the Lebesgue Integral applies. Consequently every Borel $u:X\to\mathbb{R}$ with $|u(x)|\le b(1+|x|)$ for all $x$ and some real $b\ge0$ is integrable with respect to $\gamma_{c}$, by the same two results.

Step 1 (Order). If $F,G\in\mathcal{F}C_{\mathrm{pol}}(X)$ satisfy $F(x)\le G(x)$ for every $x\in X$, then $P_{r}F(x)\le P_{r}G(x)$ for all $x\in X$ and real $r\ge0$: indeed $G-F\in\mathcal{F}C_{\mathrm{pol}}(X)$ is nonnegative, so $P_{r}G-P_{r}F=P_{r}(G-F)\ge0$ by linearity and The Mehler Semigroup: Cylindrical Form, the Semigroup Law, Invariance, Positivity, Contraction, Symmetry, Hermite Eigenfunctions, Commutation with Partial Derivatives, and the Generator §positivity.

Step 2 (Three facts on bounded $C^{2}$ cylindrical functions). Let $H\in\mathcal{F}C^{2}_{b}(X)$, with $H=h\circ p_{n}$ for some $n\in\mathbb{N}$ and $h\in C^{2}_{b}(\mathbb{R}^{n})$, as in Bounded C^2 Cylindrical Functions on a Hilbert Space with an Orthonormal Basis §cylindrical. By The Ornstein-Uhlenbeck Operator with Noise Weights on Bounded C^2 Cylindrical Functions §second-derivatives, $H\in\mathcal{F}C^{1}_{b}(X)$ with representation $(n,h)$, and for $k\le n$, $\partial_{k}h\in C^{1}_{b}(\mathbb{R}^{n})$, $\partial_{k}H=(\partial_{k}h)\circ p_{n}\in\mathcal{F}C^{1}_{b}(X)$, and $\partial_{k}\partial_{k}H=(\partial_{k}\partial_{k}h)\circ p_{n}$ by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §partial.

(a) $L^{a}H$ has a representation of growth exponent $1$. By The Ornstein-Uhlenbeck Operator with Noise Weights on Bounded C^2 Cylindrical Functions §operator, $L^{a}H=\ell\circ p_{n}$ with $\ell(y)=\sum_{k=1}^{n}a_{k}(\partial_{k}\partial_{k}h(y)-y_{k}\,\partial_{k}h(y)/c_{k})$, since $x_{k}$ is the $k$-th component of $p_{n}(x)$. The functions $\partial_{k}h$ and $\partial_{k}\partial_{k}h$ are continuous by clauses 1 and 2 of C^k Maps on a Euclidean Open Set, and $y\mapsto y_{k}$ is continuous since $|y_{k}-y'_{k}|\le\lVert y-y'\rVert$ by Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §square; so $\ell$ is continuous by Continuity of Sums and Products of Real-Valued Functions on a Metric Space §on-set. With $C\ge0$ bounding $|\partial_{k}h|$ and $|\partial_{k}\partial_{k}h|$ for $k\le n$ (Bounded Twice Continuously Differentiable Functions with Bounded First and Second Partial Derivatives on Euclidean Space §bounded) and $|y_{k}|\le\lVert y\rVert$, we get $|\ell(y)|\le B_{1}(1+\lVert y\rVert)$ with $B_{1}=C\sum_{k=1}^{n}a_{k}(1+c_{k}^{-1})$, so $(n,\ell,B_{1},1)$ is a representation of $L^{a}H\in\mathcal{F}C_{\mathrm{pol}}(X)$ (Continuous Cylindrical Functions of Polynomial Growth on a Hilbert Space §class). Consequently, by The Mehler Semigroup: Cylindrical Form, the Semigroup Law, Invariance, Positivity, Contraction, Symmetry, Hermite Eigenfunctions, Commutation with Partial Derivatives, and the Generator §cylindrical, there is a real $B'\ge0$ with $|P_{r}(L^{a}H)(x)|\le B'(1+\lVert p_{n}(x)\rVert)\le B'(1+|x|)$ for all real $r\ge0$ and $x\in X$, using $\lVert p_{n}(x)\rVert\le|x|$: by Elementary Properties of the Euclidean Norm on $\mathbb{R}^n$ §square, $\lVert p_{n}(x)\rVert^{2}=\sum_{k=1}^{n}x_{k}^{2}\le\sum_{k=1}^{\infty}x_{k}^{2}\le|x|^{2}$ by Series of Nonnegative Real Numbers, Comparison, and the Geometric Series §dominates and Orthonormal Expansions in a Real Hilbert Space §bessel, and square roots preserve this by claim 5 of Real Powers Through the Exponential, and Elementary Asymptotic Tools: Monotonicity, Null Sequences of Negative Powers, Exponential Domination, Integer Rounding, and Square-Root and Exponential Inequalities.

(b) Powers. Suppose $\delta\le H(x)\le K$ for all $x\in X$, with reals $0<\delta\le K$, and let $\beta\in\mathbb{R}$; write $H^{\beta}$ for $x\mapsto H(x)^{\beta}$. Then $H^{\beta}\in\mathcal{F}C^{1}_{b}(X)$ with representation $(n,w\circ h)$, where $w(s)=s^{\beta}$ on $(0,\infty)$, and $\partial_{k}(H^{\beta})=\beta H^{\beta-1}\partial_{k}H$ for every $k\in\mathbb{N}$. Indeed, for $y\in\mathbb{R}^{n}$ the vector $z=p_{n}^{*}(y)=\sum_{k=1}^{n}y_{k}e_{k}$, with the synthesis map $p_{n}^{*}$ of Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §coordinates, has $p_{n}(z)=y$, by linearity of the inner product and orthonormality of $(e_{k})$ (Borel Probability Measures on a Real Hilbert Space with an Orthonormal Basis: Standing Notation §space), so $h(y)=H(z)\in[\delta,K]$. By Chain Rule for One-Dimensional Derivatives, $w$ is differentiable at every $s>0$ with $w'(s)=\exp(\beta\log s)\,\beta/s=\beta s^{\beta-1}$; so $w$ and likewise $w'$ (a multiple of $s\mapsto s^{\beta-1}$) are continuous on $(0,\infty)$ by Differentiability at an Interior Point Implies Continuity There. For $y\in\mathbb{R}^{n}$ and $k\le n$, the map $\sigma\mapsto h(y_{1},\dots,y_{k}+\sigma,\dots,y_{n})$ is differentiable at $\sigma=0$ with derivative $\partial_{k}h(y)$ by Partial Derivative on a Euclidean Open Set, so by Chain Rule for One-Dimensional Derivatives the partial derivative $\partial_{k}(w\circ h)(y)$ exists and equals $w'(h(y))\,\partial_{k}h(y)=\beta h(y)^{\beta-1}\partial_{k}h(y)$. The functions $w\circ h$ and $(w'\circ h)\,\partial_{k}h$ are continuous, by Continuity Between Metric Spaces is Equivalent to Sequential Continuity §on-subset for the composites and Continuity of Sums and Products of Real-Valued Functions on a Metric Space §on-set for the product, so $w\circ h$ is of class $C^{1}$ on $\mathbb{R}^{n}$ by clauses 1 and 3 of C^k Maps on a Euclidean Open Set. Since $h$ takes values in $[\delta,K]$ and $\beta\log s$ lies between $\beta\log\delta$ and $\beta\log K$ for $s\in[\delta,K]$, the function $w\circ h$ is bounded by $\max(\delta^{\beta},K^{\beta})$, and $|\partial_{k}(w\circ h)|\le|\beta|\max(\delta^{\beta-1},K^{\beta-1})\,C$, with $C$ as in (a); so $w\circ h\in C^{1}_{b}(\mathbb{R}^{n})$ (Bounded Continuously Differentiable Functions with Bounded Partial Derivatives on Euclidean Space §bounded) and $H^{\beta}=(w\circ h)\circ p_{n}\in\mathcal{F}C^{1}_{b}(X)$ (Bounded C^1 Cylindrical Functions on a Hilbert Space with an Orthonormal Basis §cylindrical). By Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §partial, applied to the representations $(n,w\circ h)$ and $(n,h)$, $\partial_{k}(H^{\beta})=\beta H^{\beta-1}\partial_{k}H$ for $k\le n$, and both sides vanish for $k>n$.

(c) Integration by parts. For every $\varphi\in\mathcal{F}C^{1}_{b}(X)$, the function $\varphi\,L^{a}H$ is integrable with respect to $\gamma_{c}$ and

$$\int_{X}\varphi\,L^{a}H\,d\gamma_{c}=-\sum_{k=1}^{n}a_{k}\int_{X}\partial_{k}H\,\partial_{k}\varphi\,d\gamma_{c}.\tag{1}$$

Indeed, let $k\le n$. By Gaussian Integration by Parts for Products of Cylindrical Functions, and Closability of the Noise Gradient in the Gaussian Lebesgue Space §ibp, applied with $\partial_{k}H\in\mathcal{F}C^{1}_{b}(X)$ in place of $F$, the functions $\partial_{k}\partial_{k}H\,\varphi$ and $\partial_{k}H\,(x_{k}\varphi/c_{k}-\partial_{k}\varphi)$ are integrable and $\int\partial_{k}\partial_{k}H\,\varphi\,d\gamma_{c}=\int\partial_{k}H\,(x_{k}\varphi/c_{k}-\partial_{k}\varphi)\,d\gamma_{c}$. The function $\partial_{k}H\,\partial_{k}\varphi$ is Borel and bounded by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §bounded-borel and Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, hence integrable by claim 6(b) of Borel Measurability and Bounded Integration on a Metric Space; so $x\mapsto x_{k}\,\partial_{k}H(x)\,\varphi(x)/c_{k}$, the sum of the last two functions, is integrable, and by Linearity and Monotonicity of the Lebesgue Integral §integrable, $\int\varphi\,(\partial_{k}\partial_{k}H-x_{k}\partial_{k}H/c_{k})\,d\gamma_{c}=-\int\partial_{k}H\,\partial_{k}\varphi\,d\gamma_{c}$. Multiplying by $a_{k}$, summing over $k\le n$ and using The Ornstein-Uhlenbeck Operator with Noise Weights on Bounded C^2 Cylindrical Functions §operator gives (1).

Step 3 (The case $t=0$). If $t=0$, then $q-1\le\exp(0)(p-1)=p-1$, so $q=p$, and $\lVert P_{0}F\rVert_{q}=\lVert F\rVert_{p}$ since $P_{0}F=F$. From now on $t>0$.

Step 4 (Main estimate). Let $G\in\mathcal{F}C_{\mathrm{pol}}(X)$ and reals $0<\delta\le K$ satisfy $\delta\le G(x)\le K$ for every $x\in X$. We show $\lVert P_{t}G\rVert_{q}\le\lVert G\rVert_{p}$.

(i) The functions $f_{r}$. For real $r\ge0$ put $f_{r}=P_{r}G$. By Step 1, applied to $\delta\le G\le K$, and since $P_{r}$ fixes constants, $\delta\le f_{r}(x)\le K$ for all $x$. By The Mehler Semigroup: Cylindrical Form, the Semigroup Law, Invariance, Positivity, Contraction, Symmetry, Hermite Eigenfunctions, Commutation with Partial Derivatives, and the Generator §continuity, $r\mapsto f_{r}(x)$ is continuous on $[0,\infty)$ for each $x$. For $r>0$, $f_{r}\in\mathcal{F}C^{2}_{b}(X)$ by The Mehler Semigroup: Cylindrical Form, the Semigroup Law, Invariance, Positivity, Contraction, Symmetry, Hermite Eigenfunctions, Commutation with Partial Derivatives, and the Generator §smoothing, $G$ being bounded. Let $0<u<r$ and $H_{u}=f_{u}\in\mathcal{F}C^{2}_{b}(X)$; then $f_{r}=P_{r-u}H_{u}$ by The Mehler Semigroup: Cylindrical Form, the Semigroup Law, Invariance, Positivity, Contraction, Symmetry, Hermite Eigenfunctions, Commutation with Partial Derivatives, and the Generator §semigroup, so The Mehler Semigroup: Cylindrical Form, the Semigroup Law, Invariance, Positivity, Contraction, Symmetry, Hermite Eigenfunctions, Commutation with Partial Derivatives, and the Generator §generator, applied to $H_{u}$ at the positive time $r-u$, gives $L^{a}f_{r}=P_{r-u}(L^{a}H_{u})$ and, the limit there being taken over all real nonzero increments of absolute value less than $r-u$, among others,

$$r\mapsto f_{r}(x)\ \text{is differentiable at every }r>0\ \text{with derivative}\ L^{a}f_{r}(x),\qquad |L^{a}f_{r}(x)|\le B'_{u}(1+|x|)\ \ (r>u),\tag{2}$$

where $B'_{u}$ is the number of Step 2(a) for $H=H_{u}$.

(ii) The functions $Q$, $\Psi$, $\Lambda$. Put $Q(r)=1+(p-1)\exp(2r/\kappa)$ for real $r$; then $Q(0)=p$, $Q$ is strictly increasing with $Q(r)\ge p>1$ for $r\ge0$, and by Chain Rule for One-Dimensional Derivatives and Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives, $Q$ is differentiable with $Q'(r)=\frac{2}{\kappa}(p-1)\exp(2r/\kappa)=\frac{2}{\kappa}(Q(r)-1)>0$. For real $r\ge0$ the function $x\mapsto f_{r}(x)^{Q(r)}=|f_{r}(x)|^{Q(r)}$ is Borel by Power-Integrable Functions and the p-Seminorm §measurable-power and takes values in $[\delta^{Q(r)},K^{Q(r)}]$, so it is integrable by claim 6(b) of Borel Measurability and Bounded Integration on a Metric Space; put

$$\Psi(r)=\int_{X}f_{r}^{Q(r)}\,d\gamma_{c}\in[\delta^{Q(r)},K^{Q(r)}],\qquad\Lambda(r)=\frac{\log\Psi(r)}{Q(r)} .$$

By Power-Integrable Functions and the p-Seminorm §seminorm and Real Power of a Nonnegative Real Number §power, $\lVert f_{r}\rVert_{Q(r)}=\Psi(r)^{1/Q(r)}=\exp(\Lambda(r))$. Put $K_{*}=\max(K,1)$; for $0\le r\le T$ we have $f_{r}^{Q(r)}\le\exp(Q(r)\log K_{*})\le K_{*}^{Q(T)}$, as $\log K_{*}\ge0$.

(iii) Continuity. Let $r_{0}\ge0$ and let $(r_{j})_{j\in\mathbb{N}}$ be a sequence in $[0,\infty)$ converging to $r_{0}$; discarding finitely many terms, $r_{j}\le T_{0}=r_{0}+1$. For each $x$, $f_{r_{j}}(x)\to f_{r_{0}}(x)$ by (i) and Continuity Between Metric Spaces is Equivalent to Sequential Continuity §sequential, and $Q(r_{j})\to Q(r_{0})$, $Q$ being continuous by Differentiability at an Interior Point Implies Continuity There; since $\log$ and $\exp$ are continuous (Differentiability at an Interior Point Implies Continuity There), $f_{r_{j}}(x)^{Q(r_{j})}=\exp(Q(r_{j})\log f_{r_{j}}(x))\to f_{r_{0}}(x)^{Q(r_{0})}$. These functions are bounded by the constant $K_{*}^{Q(T_{0})}$, which is integrable by claim 6(a) of Borel Measurability and Bounded Integration on a Metric Space, so Dominated Convergence Theorem gives $\Psi(r_{j})\to\Psi(r_{0})$. By Continuity Between Metric Spaces is Equivalent to Sequential Continuity §criterion, $\Psi$ is continuous on $[0,\infty)$; as $\Psi>0$, $Q>0$, and $\log$ is continuous, $\Lambda$ is continuous on $[0,\infty)$ by Continuity Between Metric Spaces is Equivalent to Sequential Continuity §on-subset.

(iv) Differentiability. Let $r_{0}>0$, put $u=r_{0}/2$, $T=2r_{0}$ and $U=(u,T)$. For $r\in U$ and $x\in X$ put $\Theta(r,x)=f_{r}(x)^{Q(r)}=\exp(Q(r)\log f_{r}(x))$. By (2), Chain Rule for One-Dimensional Derivatives and Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives, $r\mapsto\Theta(r,x)$ is differentiable at every $r\in U$ with

$$D_{1}\Theta(r,x)=f_{r}(x)^{Q(r)}\Bigl(Q'(r)\log f_{r}(x)+Q(r)\,\frac{L^{a}f_{r}(x)}{f_{r}(x)}\Bigr).$$

With $\lambda=\max(|\log\delta|,|\log K|)\ge|\log f_{r}(x)|$, (2) and (ii) give, for $r\in U$, $|D_{1}\Theta(r,x)|\le K_{*}^{Q(T)}\bigl(Q'(T)\lambda+Q(T)B'_{u}(1+|x|)/\delta\bigr)$, a continuous function of $x$ of the form $b(1+|x|)$ with a real $b\ge0$, hence integrable by the preamble. By (ii), $x\mapsto\Theta(r,x)$ is integrable for each $r\in U$. So Differentiation under the Integral Sign shows that $x\mapsto D_{1}\Theta(r_{0},x)$ is measurable and integrable and that the restriction of $\Psi$ to $U$ is differentiable at $r_{0}$ with derivative $\int_{X}D_{1}\Theta(r_{0},x)\,\gamma_{c}(dx)$, and by Differentiability at an Interior Point is a Local Property so is $\Psi$ itself. Write $f=f_{r_{0}}$, $Q_{0}=Q(r_{0})$ and $Q_{0}'=Q'(r_{0})$; since $f^{Q_{0}}/f=f^{Q_{0}-1}$, $D_{1}\Theta(r_{0},x)=Q_{0}'f(x)^{Q_{0}}\log f(x)+Q_{0}f(x)^{Q_{0}-1}L^{a}f(x)$. The function $f^{Q_{0}-1}L^{a}f$ is Borel and integrable: $f\in\mathcal{F}C^{2}_{b}(X)$ by (i), with $\delta\le f\le K$, so $f^{Q_{0}-1}\in\mathcal{F}C^{1}_{b}(X)$ by Step 2(b) (with $H=f$ and $\beta=Q_{0}-1$) and is Borel by Bounded C^1 Cylindrical Functions: Linear Structure, the Gradient and the Partial Derivatives, Bounds and Integrability, and Density in the Square-Integrable Functions §bounded-borel, with $0<f^{Q_{0}-1}\le K_{*}^{Q_{0}-1}$ because $Q_{0}-1>0$ and $f\le K_{*}$ (Properties of Real Powers of Nonnegative Real Numbers §monotone); $L^{a}f\in\mathcal{F}C_{\mathrm{pol}}(X)$ by Step 2(a) (with $H=f$), hence is Borel by The Mehler Semigroup: Cylindrical Form, the Semigroup Law, Invariance, Positivity, Contraction, Symmetry, Hermite Eigenfunctions, Commutation with Partial Derivatives, and the Generator §class, and $|L^{a}f(x)|\le B'_{u}(1+|x|)$ by (2); so the product is Borel by Arithmetic, Absolute Values, and Pointwise Limits of Measurable Real-Valued Functions, is bounded in absolute value by $K_{*}^{Q_{0}-1}B'_{u}(1+|x|)$, and is integrable by the preamble. As $Q_{0}'>0$, the function $f^{Q_{0}}\log f=\bigl(D_{1}\Theta(r_{0},\cdot)-Q_{0}f^{Q_{0}-1}L^{a}f\bigr)/Q_{0}'$ is integrable, and the linearity of Linearity and Monotonicity of the Lebesgue Integral §integrable gives

$$\Psi'(r_{0})=Q_{0}'\int_{X}f^{Q_{0}}\log f\,d\gamma_{c}+Q_{0}\int_{X}f^{Q_{0}-1}L^{a}f\,d\gamma_{c}.\tag{3}$$

(v) Rewriting the derivative. The function $f\in\mathcal{F}C^{2}_{b}(X)$ has a representation $f=h\circ p_{n}$ with $h\in C^{2}_{b}(\mathbb{R}^{n})$ and satisfies $\delta\le f\le K$. By Step 2(b), $\varphi=f^{Q_{0}-1}$ and $g=f^{Q_{0}/2}$ lie in $\mathcal{F}C^{1}_{b}(X)$ with representations of length $n$, $\partial_{k}\varphi=(Q_{0}-1)f^{Q_{0}-2}\partial_{k}f$ and $\partial_{k}g=\frac{Q_{0}}{2}f^{Q_{0}/2-1}\partial_{k}f$. By (1) with $H=f$, the linearity of the integral, and The Noise Gradient of a Cylindrical Function and the Noise Tangent Space at a Probability Measure on a Hilbert Space §gradient, which gives $|\nabla_{a}g|_{a}^{2}=\sum_{k=1}^{n}a_{k}(\partial_{k}g)^{2}=\frac{Q_{0}^{2}}{4}f^{Q_{0}-2}\sum_{k=1}^{n}a_{k}(\partial_{k}f)^{2}$,

$$Q_{0}^{2}\int_{X}f^{Q_{0}-1}L^{a}f\,d\gamma_{c}=-Q_{0}^{2}(Q_{0}-1)\int_{X}f^{Q_{0}-2}\sum_{k=1}^{n}a_{k}(\partial_{k}f)^{2}\,d\gamma_{c}=-4(Q_{0}-1)\int_{X}|\nabla_{a}g|_{a}^{2}\,d\gamma_{c}.$$

Moreover $g^{2}=f^{Q_{0}}$ and, with $\phi$ the function of The Function $s\log s$: Continuity, Young's Inequality and Lower Bounds, with the Elementary Bounds for the Exponential and the Logarithm, $\phi(g^{2})=f^{Q_{0}}\log(f^{Q_{0}})=Q_{0}\,f^{Q_{0}}\log f$; also $\int_{X}g^{2}\,d\gamma_{c}=\Psi(r_{0})>0$, so $\phi(\int_{X}g^{2}\,d\gamma_{c})=\Psi(r_{0})\log\Psi(r_{0})$. Hence, by (3) and the definition of the entropy in The Entropy of a Nonnegative Function with Respect to a Probability Measure §entropy (whose hypotheses hold by The Gaussian Log-Sobolev, HWI and Poincare Inequalities in the Noise Geometry, in Entropy Form and in Gross's Functional Form §gross),

$$Q_{0}\,\Psi'(r_{0})-Q_{0}'\,\Psi(r_{0})\log\Psi(r_{0})=Q_{0}'\operatorname{Ent}_{\gamma_{c}}(g^{2})-4(Q_{0}-1)\int_{X}|\nabla_{a}g|_{a}^{2}\,d\gamma_{c}.$$

Since $1/Q(r)=\exp(-\log Q(r))$, Chain Rule for One-Dimensional Derivatives and Sum, Constant Multiple, and Product Rules for One-Dimensional Derivatives show that $\Lambda$ is differentiable at $r_{0}$ with

$$\Lambda'(r_{0})=\frac{\Psi'(r_{0})}{Q_{0}\,\Psi(r_{0})}-\frac{Q_{0}'\log\Psi(r_{0})}{Q_{0}^{2}}=\frac{1}{Q_{0}^{2}\Psi(r_{0})}\Bigl(Q_{0}'\operatorname{Ent}_{\gamma_{c}}(g^{2})-4(Q_{0}-1)\int_{X}|\nabla_{a}g|_{a}^{2}\,d\gamma_{c}\Bigr).$$

By The Gaussian Log-Sobolev, HWI and Poincare Inequalities in the Noise Geometry, in Entropy Form and in Gross's Functional Form §gross, applied with the present $\kappa$, $\operatorname{Ent}_{\gamma_{c}}(g^{2})\le2\kappa\int_{X}|\nabla_{a}g|_{a}^{2}\,d\gamma_{c}$; as $Q_{0}'=2(Q_{0}-1)/\kappa>0$, we get $Q_{0}'\operatorname{Ent}_{\gamma_{c}}(g^{2})\le4(Q_{0}-1)\int_{X}|\nabla_{a}g|_{a}^{2}\,d\gamma_{c}$, hence $\Lambda'(r_{0})\le0$.

(vi) Conclusion. By (iii) the restriction of $\Lambda$ to $[0,t]$ is continuous, and by (iv)-(v) it is differentiable at every point of $(0,t)$ with nonpositive derivative: each such point is an interior point of both $[0,\infty)$ and $[0,t]$, and the difference quotients of the restriction at it are among those of $\Lambda$, so the condition of differentiability satisfied by $\Lambda$ there, with the same derivative, holds for the restriction. By Mean Value Theorem on a Closed Real Interval there is $\xi\in(0,t)$ with $\Lambda(t)-\Lambda(0)=t\,\Lambda'(\xi)\le0$. Since $Q(0)=p$ and $f_{0}=G$, this gives $\lVert P_{t}G\rVert_{Q(t)}=\exp(\Lambda(t))\le\exp(\Lambda(0))=\lVert G\rVert_{p}$. Finally $q\le Q(t)$, because $q-1\le\exp(2t/\kappa)(p-1)=Q(t)-1$. If $q<Q(t)$, put $\rho_{1}=Q(t)/q$ and $\rho_{2}=Q(t)/(Q(t)-q)$, conjugate exponents since $1/\rho_{1}+1/\rho_{2}=1$ and $\rho_{1}>1$. By Elementary Properties of the p-Seminorm §rescaling (with $r=q$, $s=\rho_{1}$), $|P_{t}G|^{q}$ is $\rho_{1}$-integrable with $\lVert\,|P_{t}G|^{q}\rVert_{\rho_{1}}=\lVert P_{t}G\rVert_{Q(t)}^{q}$, and the constant $1$ has $\lVert1\rVert_{\rho_{2}}=1$; so Hoelder's Inequality, for Two and for Finitely Many Factors §holder gives $\int_{X}|P_{t}G|^{q}\,d\gamma_{c}\le\lVert P_{t}G\rVert_{Q(t)}^{q}$, and taking powers $1/q$ (Properties of Real Powers of Nonnegative Real Numbers §monotone, Properties of Real Powers of Nonnegative Real Numbers §inverse) yields $\lVert P_{t}G\rVert_{q}\le\lVert P_{t}G\rVert_{Q(t)}$, which holds trivially if $q=Q(t)$. Hence $\lVert P_{t}G\rVert_{q}\le\lVert G\rVert_{p}$.

Step 5 (Bounded nonnegative functions). Let $F\in\mathcal{F}C_{\mathrm{pol}}(X)$ with $0\le F(x)\le L$ for all $x$, and let $\delta>0$. Then $G=F+\delta\in\mathcal{F}C_{\mathrm{pol}}(X)$ satisfies $\delta\le G\le L+\delta$, so $\lVert P_{t}G\rVert_{q}\le\lVert G\rVert_{p}$ by Step 4. Now $P_{t}G=P_{t}F+\delta$ and $P_{t}F\ge0$ by The Mehler Semigroup: Cylindrical Form, the Semigroup Law, Invariance, Positivity, Contraction, Symmetry, Hermite Eigenfunctions, Commutation with Partial Derivatives, and the Generator §positivity, so $0\le P_{t}F\le P_{t}G$ and $\lVert P_{t}F\rVert_{q}\le\lVert P_{t}G\rVert_{q}$ by Elementary Properties of the p-Seminorm §comparison. By Minkowski's Inequality and the Seminormed Space of Power-Integrable Functions §minkowski, $\lVert G\rVert_{p}\le\lVert F\rVert_{p}+\lVert\delta\rVert_{p}=\lVert F\rVert_{p}+\delta$, as $\int_{X}\delta^{p}\,d\gamma_{c}=\delta^{p}$ and $(\delta^{p})^{1/p}=\delta$ by Properties of Real Powers of Nonnegative Real Numbers §inverse. So $\lVert P_{t}F\rVert_{q}\le\lVert F\rVert_{p}+\delta$ for every $\delta>0$, whence $\lVert P_{t}F\rVert_{q}\le\lVert F\rVert_{p}$.

Step 6 (Nonnegative functions). Let $F\in\mathcal{F}C_{\mathrm{pol}}(X)$ be nonnegative, with representation $(n,\psi,B,q_{0})$. For $L\in\mathbb{N}$ let $F_{L}=\min(F,L)=\min(\psi,L)\circ p_{n}$. The function $\min(\psi,L)$ is continuous by Continuity of the Absolute Value, Maximum and Minimum of Real-Valued Functions on a Metric Space and Continuity of Sums and Products of Real-Valued Functions on a Metric Space §constants, and $|\min(\psi(y),L)|\le|\psi(y)|+L\le(B+L)(1+\lVert y\rVert^{q_{0}})$; so $F_{L}\in\mathcal{F}C_{\mathrm{pol}}(X)$ with $0\le F_{L}\le L$, $F_{L}\le F_{L+1}\le F$. By Step 5 and Elementary Properties of the p-Seminorm §comparison, $\lVert P_{t}F_{L}\rVert_{q}\le\lVert F_{L}\rVert_{p}\le\lVert F\rVert_{p}$. For $x\in X$, $P_{t}F_{L}(x)=\int_{X}F_{L}(M_{t}(x,y))\,\gamma_{c}(dy)$ by The Mehler Maps and the Mehler Semigroup with Noise Weights on Continuous Cylindrical Functions of Polynomial Growth §semigroup; the integrands are nonnegative, nondecreasing in $L$, and equal to $F(M_{t}(x,y))$ as soon as $L\ge F(M_{t}(x,y))$, so Monotone Convergence Theorem shows that the nondecreasing sequence $(P_{t}F_{L}(x))_{L}$ converges to $P_{t}F(x)$. Hence, by Properties of Real Powers of Nonnegative Real Numbers §monotone and Properties of Real Powers of Nonnegative Real Numbers §continuity, $(P_{t}F_{L}(x)^{q})_{L}$ is nondecreasing with limit, hence supremum, $P_{t}F(x)^{q}$, where $P_{t}F\ge P_{t}F_{L}\ge0$ by Step 1 and The Mehler Semigroup: Cylindrical Form, the Semigroup Law, Invariance, Positivity, Contraction, Symmetry, Hermite Eigenfunctions, Commutation with Partial Derivatives, and the Generator §positivity. By Monotone Convergence Theorem and Elementary Properties of the p-Seminorm §power,

$$\int_{X}(P_{t}F)^{q}\,d\gamma_{c}=\sup_{L}\int_{X}(P_{t}F_{L})^{q}\,d\gamma_{c}=\sup_{L}\lVert P_{t}F_{L}\rVert_{q}^{q}\le\lVert F\rVert_{p}^{q},$$

using Properties of Real Powers of Nonnegative Real Numbers §monotone; taking powers $1/q$ (Properties of Real Powers of Nonnegative Real Numbers §inverse) gives $\lVert P_{t}F\rVert_{q}\le\lVert F\rVert_{p}$.

Step 7 (General functions). Let $F\in\mathcal{F}C_{\mathrm{pol}}(X)$ with representation $(n,\psi,B,q_{0})$. Then $|F|=|\psi|\circ p_{n}$ with $|\psi|$ continuous by Continuity of the Absolute Value, Maximum and Minimum of Real-Valued Functions on a Metric Space and $|\psi|(y)=|\psi(y)|\le B(1+\lVert y\rVert^{q_{0}})$, so $|F|\in\mathcal{F}C_{\mathrm{pol}}(X)$. Step 1 applied to $-|F|\le F\le|F|$, with $P_{t}(-|F|)=-P_{t}|F|$, gives $|P_{t}F|\le P_{t}|F|$ pointwise. By Elementary Properties of the p-Seminorm §comparison and Step 6, $\lVert P_{t}F\rVert_{q}\le\lVert P_{t}|F|\rVert_{q}\le\lVert\,|F|\,\rVert_{p}=\lVert F\rVert_{p}$, the last equality because $|F|$ and $F$ have the same absolute value. Together with Step 3 this proves the claim.