In this note, we study how the well-known equivalence between spectral gaps, Poincaré inequalities and exponential rates of decay to equilibrium extends to systems which lack a spectral gap but have a bounded density of states near 0. Our main result relies solely on our ability to “differentiate” the resolution of the identity of a given operator. It is thus quite general and covers important examples such as Markov semigroups.
Our setup is as follows: let M be a manifold with Borel measure \(\mathrm {d}\mu \), \(\mathcal {H}=L^2(M,\mathrm {d}\mu ;\mathbb {R})\) equipped with scalar product \((\cdot ,\cdot )_\mathcal {H}\). We assume that \(H:D(H)\subset \mathcal {H}\rightarrow \mathcal {H}\) is a self-adjoint, nonnegative operator, so that \(-H\) is the infinitesimal generator of a Markov semigroup \((P_t)_{t\ge 0}\), whose invariant measure is \(\mathrm {d}\mu \), i.e., for every u that is bounded and nonnegative \(\int _M P_tu\,\mathrm {d}\mu =\int _M u\,\mathrm {d}\mu \) for any \(t\ge 0.\) Let \(\{E(\lambda )\}_{\lambda \ge 0}\) be the resolution of the identity of H, and let the associated Dirichlet form be
$$\begin{aligned} \mathcal {E}(u):=\int _{M}(H^{1/2}u)^2\,\mathrm {d}\mu . \end{aligned}$$
As stated above, instead of assuming a spectral gap, we assume the opposite: H has continuous spectrum in a neighborhood of 0. (And 0 itself is possibly an eigenvalue.) We show that an appropriate estimate of the density of the spectrum near 0 leads to a weaker version of the Poincaré inequality (also known as a weak Poincaré inequality, defined in Definition 1.3). This, in turn, leads to an algebraic decay rate for the associated semigroup.
In this paper, we employ the following definition for the variance of a given function \(u\in \mathcal {H}\):
$$\begin{aligned} {{\,\mathrm{Var}\,}}(u):=\int _M(u-E(\{0\})u)^2\,\mathrm {d}\mu \end{aligned}$$
where \(E(\{0\})\) is the projection onto the kernel of H. In the case where the kernel only consists of constant functions and \(\mu \) is a probability measure, this definition coincides with the standard definition, see [3, §4.2.1]. We discuss the significance of the resolution of the identity of H (and in particular the projection onto its kernel) and its relationship with functional inequalities and decay rates in Sect. 2.3.
We can now recall the classical Poincaré inequality (again, see [3, §4.2.1]):
Definition 1.1
(Poincaré Inequality). We say that H satisfies a Poincaré inequality if there exists \(C>0\) such that
$$\begin{aligned} {{\,\mathrm{Var}\,}}(u) \le C \mathcal {E}(u),\qquad \forall u\in D(\mathcal {E}), \end{aligned}$$
where C does not depend on u.
Remark 1.2
The topology of \(D(\mathcal {E})\) is the graph norm topology generated by \(\Vert \cdot \Vert _\mathcal {H}^2+\mathcal {E}(\cdot )\), see [3, §3.1.4].
The definition of a “weak Poincaré inequality” is somewhat ambiguous. This is addressed in further detail in Sect. 2.3. We adopt the following definition, motivated by Liggett [13, Equation (2.3)]:
Definition 1.3
(Weak Poincaré Inequality). Let \(\Phi :\mathcal {H}\rightarrow [0,\infty ]\) satisfy \(\Phi (u)<\infty \) on a dense subset of \(D(\mathcal {E})\). Let \(p\in (1,\infty )\). We say that H satisfies a \((\Phi ,{p})\)-weak Poincaré inequality (\((\Phi ,{p})\)-WPI) if there exists \(C>0\) such that
$$\begin{aligned} {{\,\mathrm{Var}\,}}(u)\le C\mathcal {E}(u)^{1/p} \Phi (u)^{1/q},\qquad \forall u\in D(\mathcal {E}), \end{aligned}$$
(1.1)
where C does not depend on u and where \(1/p+1/q=1\).
Remark 1.4
Note that (1.1) is meaningful only on a dense subset of \(D(\mathcal {E})\) where \(\Phi <+\infty \).
The Hilbertian Case
We start our discussion by considering the purely Hilbertian case, i.e., we consider generators with density of states that are defined on subspaces which respect the Hilbert structure of \(\mathcal {H}\), such as Sobolev spaces or weighted spaces. Our basic assumption is:
Assumption A1
There exists a dense subspace \(\mathcal {X}\subset \mathcal {H}\) such that
- (1)
\(\mathcal {X}\cap D(\mathcal {E})\) is dense in \(D(\mathcal {E})\) (in the topology of \(D(\mathcal {E})\)),
- (2)
for some constants \(r>0\), \(C_1>0\) and \(\alpha >-1\),
the mapping \(\lambda \mapsto \frac{\mathrm {d}}{\mathrm {d}\lambda }(E(\lambda )u,v)_{\mathcal {H}}\) is continuous on (0, r) for every \(u,v\in \mathcal {X}\) and satisfies
$$\begin{aligned} \left| \frac{\mathrm {d}}{\mathrm {d}\lambda }(E(\lambda )u,v)_{\mathcal {H}}\right| \le C_1\lambda ^\alpha \Vert u\Vert _\mathcal {X}\Vert v\Vert _\mathcal {X},\qquad \forall u,v\in \mathcal {X},\,\forall \lambda \in (0,r). \end{aligned}$$
(1.2)
Remark 1.5
We refer to the bilinear form \(\frac{\mathrm {d}}{\mathrm {d}\lambda }(E(\lambda )\cdot ,\cdot )_{\mathcal {H}}\) as the density of states (DoS) of H at \(\lambda \). Note that if the DoS satisfies a bound as in (1.2) and \(\mathcal {X}\) has a norm compatible with (and stronger than) the norm on \(\mathcal {H}\), then it induces an operator \(\mathcal {X}\rightarrow \mathcal {X}^*\) by the Riesz representation theorem.
We can finally state our main results on how (1.2) leads to a \((\Phi ,p)\)-WPI (Theorem 1.6) and, in turn, an explicit rate of decay (Theorem 1.7). Theorem 1.6 will be further generalized in Theorem 1.9 and then again in Proposition 1.13 where a precise constant in the WPI is obtained. The decay rates presented in Theorem 1.7 apply to the Markov semigroup generated by H.
Theorem 1.6
If Assumption A1 holds, then H satisfies a \((\Phi ,p)\)-weak Poincaré inequality with \(\Phi (u)=\Vert u\Vert _{\mathcal {X}}^2\) (and \(\Phi (u)=+\infty \) if \(u\in \mathcal {H}{\setminus }\mathcal {X}\)) and \(p=\frac{2+\alpha }{1+\alpha }\).
Theorem 1.7
Let Assumption A1 hold. Let \(u\in \mathcal {X}\) and suppose that there exist \(C_2=C_2(u)\ge 0\) and \(\beta \in \mathbb {R}\), such that the Markov semigroup satisfies
$$\begin{aligned} \Vert P_tu\Vert ^2_{\mathcal {X}}\le \Vert u\Vert ^2_{\mathcal {X}}+C_2t^\beta ,\qquad \forall t\ge 0. \end{aligned}$$
(1.3)
Then
$$\begin{aligned} {{\,\mathrm{Var}\,}}(P_t u) \le \left( {{\,\mathrm{Var}\,}}(u) ^{\frac{-1}{1+\alpha }}+C_3 \int _{0}^t {({\Vert u\Vert ^2_{\mathcal {X}}+C_2s^\beta })^{\frac{-1}{1+ \alpha }}{\,\mathrm {d}s}}\right) ^{-({1+\alpha })} \end{aligned}$$
(1.4)
where \(C_3\) is given explicitly (and only depends on \(\alpha ,\, C_1\)). In particular, \({{\,\mathrm{Var}\,}}(P_t u)\) satisfies the following decay rates as \(t\rightarrow +\infty \):
$$\begin{aligned} {{\,\mathrm{Var}\,}}({P_t u}) \le {\left\{ \begin{array}{ll} O({(\log t) ^{-(1+\alpha )}}) &{} \beta =1+\alpha .\\ O({t^{\beta -(1+\alpha )}}) &{} 0<\beta <1+\alpha .\\ O({t^{-(1+\alpha )}}) &{} C_2=0 \text { or }\beta \le 0. \end{array}\right. } \end{aligned}$$
Remark 1.8
-
1.
The choice of space \(\mathcal {X}\) is motivated by (1.3): it is beneficial to choose \(\mathcal {X}\) that is invariant under the Markov semigroup (i.e., if \(u\in \mathcal {X}\) then \(P_tu\in \mathcal {X}\) for all \(t\ge 0\)).
-
2.
Clearly, \(C_2(u)\) is subject to quadratic scaling, for example it can be \(C\Vert u\Vert _\mathcal {H}^2\) or \(C\Vert u\Vert _\mathcal {X}^2\), but the explicit form is not important.
A Generalized Theorem: Departing from the Hilbert Structure
Theorems 1.6 and 1.7 demonstrate how estimates on the density of states near 0 imply a weak Poincaré inequality and a rate of decay to equilibrium. However, it is not essential to restrict oneself to a subspace \(\mathcal {X}\). In fact, it is often desirable to deal with functional spaces that are not contained in \(\mathcal {H}\), as it may provide improved estimates and decay rates. In particular, this makes sense when the operator in question is the generator of a Markov semigroup, and acts on a range of spaces simultaneously. Hence, we replace Assumption A1 by a more general one.
Assumption A2
There exist Banach spaces \(\mathcal {X},\mathcal {Y}\) of functions on M, a constant \(r>0\) and a function \(\psi _{\mathcal {X},\mathcal {Y}}\in L^1(0,r)\) that is strictly positive a.e. on (0, r), such that
- (1)
\(\mathcal {X}\cap \mathcal {Y}\cap D(\mathcal {E})\) is dense in \(D(\mathcal {E})\) (in the topology of \(D(\mathcal {E})\)).
- (2)
The mapping \(\lambda \mapsto \frac{\mathrm {d}}{\mathrm {d}\lambda }(E(\lambda )u,v)_{\mathcal {H}}\) is continuous on (0, r) for every \(u\in \mathcal {X}\cap \mathcal {H}\) and \(v\in \mathcal {Y}\cap \mathcal {H}\) and satisfies
$$\begin{aligned} \left| \frac{\mathrm {d}}{\mathrm {d}\lambda }(E(\lambda )u,v)_{\mathcal {H}}\right| \le \psi _{\mathcal {X},\mathcal {Y}}(\lambda )\Vert u\Vert _\mathcal {X}\Vert v\Vert _\mathcal {Y},\qquad \forall \lambda \in (0,r). \end{aligned}$$
(1.5)
We can now state the following more general theorem.
Theorem 1.9
Let the conditions of Assumption A2 hold, and define \(\Psi _{\mathcal {X},\mathcal {Y}}(\rho )=\int _0^\rho \psi _{\mathcal {X},\mathcal {Y}}(\lambda )\,\mathrm {d}\lambda \), \(\rho \in (0,r)\). Then:
- a.
There exists \(K_0\in (0,1)\) such that the following functional inequality holds:
$$\begin{aligned}&(1-K)\Psi _{\mathcal {X},\mathcal {Y}}^{-1}\left( K\frac{{{\,\mathrm{Var}\,}}(u)}{\Vert u\Vert _\mathcal {X}\Vert u\Vert _\mathcal {Y}}\right) {{\,\mathrm{Var}\,}}(u)\le \mathcal {E}(u),\nonumber \\&\qquad \forall K\in (0,K_0),\,\forall u\in D(\mathcal {E}) \end{aligned}$$
(1.6)
where \(\Vert u\Vert _\mathcal {X}=+\infty \) if \(u\notin \mathcal {X}\) and similarly for \(\mathcal {Y}\).
- b.
If \(\mathcal {X}=\mathcal {Y}\) and \(\psi _{\mathcal {X},\mathcal {Y}}(\lambda )=C_1\lambda ^\alpha \), \(\alpha >-1\), the estimate (1.6) reduces to the \((\Phi ,p)\)-WPI as in Definition 1.3 with \(\Phi (u)=\Vert u\Vert _\mathcal {X}^2\) and \(p=\frac{\alpha +2}{\alpha +1}\).
- c.
If, in addition, \(\mathcal {X}=\mathcal {Y}\subset \mathcal {H}\) then we obtain Theorem 1.6.
Remark 1.10
The inequality (1.6) can be viewed as an implicit form of the weak Poincaré inequality. Note that setting \(K=0\) (which is excluded in the theorem) leads to the Poincaré inequality.
The power of this result is demonstrated in the following corollary, where the celebrated Nash inequality is obtained as a simple consequence. This simple derivation is discussed in Remark 4.2.
Corollary 1.11
(Nash inequality). When \(H=-\Delta :H^2(\mathbb {R}^d)\subset L^2(\mathbb {R}^d)\rightarrow L^2(\mathbb {R}^d)\) and \(\mathcal {Y}=\mathcal {X}=L^1(\mathbb {R}^d)\) the inequality (1.6) is precisely Nash’s inequality [15]:
$$\begin{aligned} \Vert u\Vert _{L^2}^2 \le C \left( \Vert \nabla u\Vert _{L^2}^2\right) ^{\frac{d}{d+2}} \left( \Vert u\Vert _{L^1}^2\right) ^{\frac{2}{d+2}},\qquad \forall u\in L^1(\mathbb {R}^d)\cap H^1(\mathbb {R}^d), \end{aligned}$$
where \(C>0\) does not depend on u. Furthermore, using Proposition 1.13 an explicit constant may be computed to yield \(C=\left( \frac{|\mathbb {S}^{d-1}|}{2} \right) ^{\frac{2}{2+d}}\frac{2+d}{d}\).
Proof
The (simple) proof of this corollary is done by applying our results to the heat semigroup. More details are provided in the examples (Sect. 4), in particular see Remark 4.2. \(\square \)
Remark 1.12
The requirement that \(\psi _{\mathcal {X},\mathcal {Y}}\) is strictly positive a.e. on (0, r), for some \(r>0\) (perhaps very small), is quite natural as we are interested in operators that lack a spectral gap. However, one can easily generalize our result even if that is not the case by defining
$$\begin{aligned}\Psi _{\mathcal {X},\mathcal {Y}}^{-1}(y)=\sup \left\{ x\in (0,r) \;|\; \Psi _{\mathcal {X},\mathcal {Y}}(x) \le y\right\} .\end{aligned}$$
Precise Constants
Under additional mild assumptions, one can improve Theorem 1.9 by replacing the inequality (1.6) which contains an arbitrary constant K with an inequality that has an explicit constant. The question of how far this constant is from being sharp is the topic of ongoing research.
Proposition 1.13
Let the conditions of Assumption A2 hold. Assume in addition that \(\psi _{\mathcal {X},\mathcal {Y}}\) can be extended to a continuous function on (0, R), where \(R\in [r,+\infty ]\) is such that if \(\Psi _{\mathcal {X},\mathcal {Y}}(\rho ):=\int _0^\rho \psi _{\mathcal {X},\mathcal {Y}}(\lambda )\,\mathrm {d}\lambda \), \(\rho \in (0,R)\) and
$$\begin{aligned} g_{\mathcal {X},\mathcal {Y}}(\rho ):=\Psi _{\mathcal {X},\mathcal {Y}}(\rho )+\rho \psi _{\mathcal {X},\mathcal {Y}}(\rho ) \end{aligned}$$
then g is non-decreasing and \(\lim _{\rho \rightarrow 0^{+}}g_{\mathcal {X},\mathcal {Y}}(\rho )=0\), \(\lim _{\rho \rightarrow R^{-}}g_{\mathcal {X},\mathcal {Y}}(\rho )=+\infty \). Then:
- a.
The following functional inequality holds:
$$\begin{aligned}&\left( g_{\mathcal {X},\mathcal {Y}}^{-1}\left( \frac{{{\,\mathrm{Var}\,}}(u)}{\Vert u\Vert _\mathcal {X}\Vert u\Vert _\mathcal {Y}} \right) \right) ^2 \psi _{\mathcal {X},\mathcal {Y}}\left( g_{\mathcal {X},\mathcal {Y}}^{-1}\left( \frac{{{\,\mathrm{Var}\,}}(u)}{\Vert u\Vert _\mathcal {X}\Vert u\Vert _\mathcal {Y}} \right) \right) \Vert u\Vert _\mathcal {X}\Vert u\Vert _\mathcal {Y}\le \mathcal {E}(u),\nonumber \\&\quad \forall u\in D(\mathcal {E}), \end{aligned}$$
(1.7)
where \(\Vert u\Vert _\mathcal {X}=+\infty \) if \(u\notin \mathcal {X}\) and similarly for \(\mathcal {Y}\).
- b.
If \(\mathcal {X}=\mathcal {Y}\), and \(\psi _{\mathcal {X},\mathcal {Y}}(\lambda )=C_1\lambda ^\alpha \), \(\alpha >-1\) then the estimate (1.7) reduces to the \((\Phi ,p)\)-WPI as in Definition 1.3 with \(\Phi (u)=\Vert u\Vert _\mathcal {X}^2\), \(p=\frac{\alpha +2}{\alpha +1}\) and \(C=C_1^{\frac{1}{2+\alpha }}\frac{2+\alpha }{1+\alpha }\).
Organization of the paper. Before proceeding to prove our theorems, we first discuss both the classical and the weak Poincaré inequalities, and their connection to Markov semigroups in Sect. 2. The proofs will follow in Sect. 3, and we then present various applications of these theorems in Sect. 4, where we shall also prove Corollary 1.11.