Lecture 8. Sobolev inequalities on Dirichlet spaces: The Varopoulos approach

Posted on February 17, 2019 by Fabrice Baudoin

In this lecture, we study Sobolev inequalities on Dirichlet spaces. The  approach we develop is related to  Hardy-Littlewood-Sobolev theory

The link between the Hardy-Littlewood-Sobolev theory and heat kernel upper bounds is due to Varopoulos, but the proof I give below I learnt it from my colleague Rodrigo Bañuelos. It bypasses the Marcinkiewicz interpolation theorem, that was originally used by Varopoulos by using instead the Stein’s maximal ergodic lemma. The advantage of the method is to get an explicit (non sharp) constant for the Sobolev inequality

Let (X,\mathcal{B}) be a good measurable space equipped with a \sigma-finite measure \mu. Let \mathcal E be a Dirichlet form on X. As usual, we denote by P_t the semigroup generated by P_t and we assume P_t 1=1.

We have the following so-called maximal ergodic lemma, which was first proved by Stein. We give here a probabilistic proof since it comes with a nice constant but you can for instance find the original (non probabilistic) proof here.

Lemma:(Stein’s maximal ergodic theorem) Let p > 1. For f \in L^p (X,\mu), denote f^*(x)=\sup_{t \ge 0} |P_t f(x)|. We have
\| f^* \|_{L^p (X,\mu)} \le \frac{p}{p-1} \| f \|_{L^p (X,\mu)}.

Proof: For x \in X, we denote by (X_t^x)_{t \ge 0} the Markov process associated with the semigroup P_t and started at x (we assume that such process exists without commenting on the exact assumptions). We fix T > 0. By construction, for t \le T, we have,
P_{T-t}f (X_T^x) =\mathbb{E} \left( f (X_{2T-t}^x) | X_T^x \right),
and thus
P_{2(T-t)}f (X_T^x) =\mathbb{E} \left( (P_{T-t} f) (X_{2T-t}^x) | X_T^x \right).
As a consequence, we obtain
\sup_{0 \le t \le T} | P_{2(T-t)}f (X_T^x) | \le \mathbb{E} \left(\sup_{0 \le t \le T} | (P_{T-t} f) (X_{2T-t}^x) | \mid X_T^x\right) .
Jensen’s inequality yields then
\sup_{0 \le t \le T} | P_{2(T-t)}f (X_T^x) |^p \le \mathbb{E} \left(\sup_{0 \le t \le T} | (P_{T-t} f) (X_{2T-t}^x) |^p \mid X_T^x\right).
We deduce
\mathbb{E} \left( \sup_{0 \le t \le T} | P_{2(T-t)}f (X_T^x) |^p \right) \le \mathbb{E} \left(\sup_{0 \le t \le T} | (P_{T-t} f) (X_{2T-t}^x) |^p \right).
Integrating the inequality with respect to the measure \mu, we obtain
\left\| \sup_{0 \le t \le T} | P_{2(T-t)}f | \right\|_p \le \left( \int_X \mathbb{E} \left(\sup_{0 \le t \le T} | (P_{T-t} f) (X_{2T-t}^x) |^p \right)d\mu(x)\right)^{1/p}.
By reversibility, we get then
\left\| \sup_{0 \le t \le T} | P_{2(T-t)}f | \right\|_p \le \left( \int_X \mathbb{E} \left(\sup_{0 \le t \le T} | (P_{T-t} f) (X_t^x) |^p \right)d\mu(x)\right)^{1/p}.
We now observe that the process (P_{T-t} f) (X_t^x) is martingale and thus Doob’s maximal inequality gives
\mathbb{E} \left(\sup_{0 \le t \le T} | (P_{T-t} f) (X_t^x) |^p \right)^{1/p} \le \frac{p}{p-1} \mathbb{E} \left( | f(X_T^x)|^p \right)^{1/p}.
The proof is complete. \square

We now turn to the theorem by Varopoulos. In the sequel, we assume that the semigroup P_t admits a measurable heat kernel p(x,y,t).

Theorem: Let n > 0, 0 < \alpha < n, and 1 < p< \frac{n}{\alpha}. If there exists C > 0 such that for every t > 0, x,y \in X,
p(x,y,t) \le \frac{C}{t^{n/2}},
then for every f \in L^p (X,\mu),
\| (-L)^{-\alpha/2} f \|_{\frac{np}{n-p\alpha}} \le \left( \frac{p}{p-1} \right)^{1-\alpha/n} \frac{ 2n C^{\alpha / n}}{ \alpha (n-p\alpha) \Gamma(\alpha /2)} \|f \|_p,

where L is the generator of \mathcal{E}.

Proof: We first observe that the bound
p(x,y,t) \le \frac{C}{t^{n/2}},
implies that |P_t f(x)| \le \frac{C^{1/p}}{t^{n/2p}} \| f \|_p. Denote I_\alpha f (x)=(-L)^{-\alpha/2} f (x). We have
I_\alpha f (x)=\frac{1}{\Gamma(\alpha /2) } \int_0^{+\infty} t^{\alpha /2 -1 }P_t f (x) dt
Pick \delta > 0, to be later chosen, and split the integral in two parts:
I_\alpha f (x)=J_\alpha f(x) +K_\alpha f (x),
where J_\alpha f (x)=\frac{1}{\Gamma(\alpha /2) } \int_0^{\delta} t^{\alpha /2 -1 }P_t f (x) dt and K_\alpha f (x)=\frac{1}{\Gamma(\alpha /2) } \int_\delta^{+\infty} t^{\alpha /2 -1 }P_t f (x) dt. We have
| J_\alpha f (x) | \le \frac{1}{\Gamma(\alpha /2) } \int_0^{+\infty} t^{\alpha /2 -1 }dt | f^* (x) | =\frac{2}{\alpha \Gamma(\alpha /2) } \delta^{\alpha /2} | f^* (x) |.
On the other hand,
| K_\alpha f(x)| \le \frac{1}{\Gamma(\alpha /2) } \int_\delta^{+\infty} t^{\alpha /2 -1 } | P_t f (x)| dt
\le \frac{C^{1/p}}{\Gamma(\alpha /2) } \int_\delta^{+\infty} t^{\frac{\alpha} {2}-\frac{n}{2p} -1 } dt \| f \|_p
\le \frac{C^{1/p}}{\Gamma(\alpha /2) } \frac{1}{-\frac{\alpha} {2}+\frac{n}{2p} }\delta^{\frac{\alpha} {2}-\frac{n}{2p} } \| f \|_p .
We deduce
| I_\alpha f (x) | \le \frac{2}{\alpha \Gamma(\alpha /2) } \delta^{\alpha /2} | f^* (x) |+ \frac{C^{1/p}}{\Gamma(\alpha /2) } \frac{1}{-\frac{\alpha} {2}+\frac{n}{2p} }\delta^{\frac{\alpha} {2}-\frac{n}{2p} } \| f \|_p.
Optimizing the right hand side of the latter inequality with respect to \delta yields
| I_\alpha f (x) |\le \frac{ 2n C^{\alpha / n}}{ \alpha (n-p\alpha) \Gamma(\alpha /2)} \|f \|^{\alpha p /n}_p |f^*(x)|^{1-p\alpha/n}.
The proof is then completed by using Stein’s maximal ergodic theorem \square

A special case, of particular interest, is when \alpha =1 and p=2. We get in that case the following Sobolev inequality:

Theorem: Let n > 2. If there exists C > 0 such that for every t > 0, x,y \in X,
p(x,y,t) \le \frac{C}{t^{n/2}},
then for every f \in\mathcal{F},
\| f \|_{\frac{2n}{n-2}} \le 2^{1-1/n} \frac{ 2n C^{1 / n}}{ (n-2) \sqrt{\pi}} \sqrt{\mathcal{E}(f) }.

We mention that the constant in the above Sobolev inequality is not sharp even in the Euclidean case.

In many situations, heat kernel upper bounds with a polynomial decay are only available in small times the following result is thus useful:

Theorem: Let n > 0, 0 < \alpha < n, and 1 < p < \frac{n}{\alpha}. If there exists C > 0 such that for every 0 < t \le 1, x,y \in X,
p(x,y,t) \le \frac{C}{t^{n/2}},
then, there is constant C' such that for every f \in L^p(X,\mu),
\| (-L+1)^{-\alpha/2} f \|_{\frac{np}{n-p\alpha}} \le C' \|f \|_p

Proof: We apply the Varopoulos theorem to the semigroup Q_t=e^{-t} P_t. Details are let to the reader \square

The following corollary shall be later used:

Corollary: Let n > 2. If there exists C > 0 such that for every 0 < t \le 1, x,y \in X,
p(x,y,t) \le \frac{C}{t^{n/2}},
then there is constant C' such that for every f \in \mathcal{F},
\| f \|_{\frac{2n}{n-2}} \le C' \left( \sqrt{\mathcal{E}(f)} + \| f \|_2 \right)

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