Abstract
In this paper, we prove pointwise convergence of heat kernels for mGH-convergent sequences of \({{\mathrm{RCD}}}^{*}(K,N)\)-spaces. We obtain as a corollary results on the short-time behavior of the heat kernel in \({{\mathrm{RCD}}}^*(K,N)\)-spaces. We use then these results to initiate the study of Weyl’s law in the \({{\mathrm{RCD}}}\) setting.
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Acknowledgements
The first and third author acknowledge the support of the PRIN2015 MIUR Project “Calcolo delle Variazioni”. The second author acknowledges the support of the JSPS Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers, the Grant-in-Aid for Young Scientists (B) 16K17585 and the Scuola Normale Superiore for warm hospitality.
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Appendix: refinements of Karamata’s theorem
Appendix: refinements of Karamata’s theorem
In this section we prove Theorem 2.5 and its one-sided versions mentioned in Remark 2.6. We follow the proofs in Theorems 10.2 and 10.3 of [46], borrowing also the terminology “Abelian”, “Tauberian” from there.
Throughout this section \(\nu \) is a nonnegative and \(\sigma \)-finite Borel measure on \([0,+\infty )\). The results will then be applied to the case when \(\nu :=\sum _i\delta _{\lambda _i}\).
Lemma 5.1
For all \(t>0\) one has
Proof
By Cavalieri’s formula and the change of variables \(r=e^{-ty}\) we get
and we conclude, since \(\{x:\ e^{-tx}\ge e^{-ty}\}=[0,y]\). \(\square \)
We start with the Abelian case, easier when compared to the Tauberian one.
Theorem 5.2
(Abelian theorem) Assume that there exist \(\gamma \in [0,+\infty )\) and \(C \in [0,+\infty )\) such that
Then
More generally,
and
Proof
Let \(F(a):=\nu ([0, a])\) and \(G(a):=(a+1)^{-\gamma }F(a)\). Then (5.2) yields
In particular \(\sup _a G(a)<\infty \). Then Lemma 5.1 gives
Since for any \(t \in (0, 1]\)
applying the dominated convergence theorem to (5.7) as \(t \downarrow 0\) shows (5.3) because \(G(y/t) \rightarrow C\) as \(t \downarrow 0\) by (5.6).
The one-sided versions (5.4), (5.5) follow by an analogous argument, using Fatou’s lemma and noticing that in the \(\limsup \) case the functions in (5.8) are dominated as \(t\rightarrow 0^+\) by an integrable function. \(\square \)
Now we deal with the Tauberian case.
Theorem 5.3
(Tauberian theorem) Assume that there exist \(\gamma \in [0,+\infty )\) and \(D \in [0,+\infty )\) such that
Then
Proof
If \(\gamma =0\), then applying the monotone convergence theorem to (5.9) shows (5.10), hence we can assume \(\gamma >0\). For any \(t \in (0, 1]\) let \(\nu _t,\, \mu \) be Borel measures on \([0,+\infty )\) be, respectively, defined by
for any Borel subset A. Then (5.10) is equivalent to
because
In order to prove (5.12), we will show
for any \(f \in C_c([0,+\infty ))\) as follows.
Let \(\hat{\nu }_t:=e^{-x}\mathsf {d}\nu _t(x)\) and \(\hat{\mu }:=e^{-x}\mathsf {d}\mu (x)\) be the corresponding weighted measures on \([0,+\infty )\). Then (5.9) with Lemma 5.1 yields
In particular
More strongly, (5.9) yields
for any polynomial g(x) in \(e^{-x}\) (i.e., \(g(x)=\sum _{i=1}^Na_ie^{-ix}\)). Because
Let \(C_0([0,+\infty ))\) be the set of continuous functions f on \([0,+\infty )\) such that \(f(x) \rightarrow 0\) as \(x \rightarrow +\infty \). Then since the set of polynomials in \(e^{-x}\) is dense in \(C_0([0,+ \infty ))\) with respect to the norm \(\sup |f|\), applying the Stone–Weierstrass theorem to \((C_0([0,+\infty )), \sup |\cdot |)\) with (5.16) shows that (5.17) is satisfied for any \(g \in C_0([0,+\infty ))\), which implies (5.14).
We are now in a position to prove (5.12) by using (5.14). Indeed, it is well known that the weak convergence implies \(\nu _t(E)\rightarrow D\mu (E)/\Gamma (\gamma )\) for any compact set \(E\subset [0,+\infty )\) with \(\mu (\partial E)=0\). Choosing \(E=[0,1]\) we obtain (5.14). \(\square \)
Remark 5.4
The difficulty to obtain a one-sided version out of the previous proof, as we did for the Abelian case, can also be explained as follows: if we consider the push forward \(\sigma _t\) of the measures \(\hat{\nu _t}\) under the map \(x\mapsto e^{-x}\), the argument above shows that all moments of all weak limit points of \(\sigma _t\) are uniquely determined. Hence, since a finite Borel measure in [0, 1] is uniquely determined by its moments, uniqueness follows. If we replace the assumption (5.9) by a bound on the \(\liminf \) or the \(\limsup \), we find only an inequality between the moments of the measures, which does not seem to imply, in general, the corresponding inequality for the measures.
Proposition 5.5
Assume that for some \(\gamma \in [0, +\infty )\) one has
Then
Proof
Note that for any \(\lambda >0\) and any \(t>0\)
By (5.18), for any \(\epsilon >0\) there exists \(t_0>0\) such that \(\int _{[0,+\infty )}e^{-st}\mathsf {d}\nu (s)\le (C_0+\epsilon )t^{-\gamma }\) for any \(t<t_0\). Thus (5.20) yields \(\nu ([0,\lambda ])\le e^{\lambda t}(C_0+\epsilon )t^{-\gamma }\) for any \(\lambda >0\) and any \(t<t_0\). Letting \(\lambda :=t^{-1}\) and then letting \(t \downarrow 0\) shows (5.19). \(\square \)
Proposition 5.6
Assume that for some \(\gamma \in [0,+\infty )\) one has
Then
Proof
Call \(C_0>0\) the \(\liminf \) and \(C_1<+\infty \) the \(\limsup \) in (5.21). Note that for any \(\lambda >0\) and any \(t>0\)
In particular, letting \(\lambda :=t^{-1}\) yields
Thus there exists \(t_0>0\) such that for any \(t<t_0\)
Next let us discuss the right-hand side of (5.24). By (5.21)and Proposi tion 5.5 there exists \(\hat{\lambda }>0\) such that \(\nu ([0,\lambda ])\le (eC_1+1)\lambda ^{\gamma }\) for any \(\lambda \ge \hat{\lambda }\). Thus for any \(t>0\) with \(t^{-1} \ge \hat{\lambda }\) we get
In particular
for any \(k \in \mathbb {N}\) and any \(t>0\) with \(t^{-1}\ge \hat{\lambda }\).
For any \(\delta >0\) there exists \(k_0 \in \mathbb {N}\) such that \(\sum _{\ell =k_0+1}^{\infty }e^{-\ell }(\ell +1)^{\gamma }<\delta \). Then, combining (5.24) with (5.25) yields
for any \(t>0\) with \(t<t_0\) and \(t^{-1} \ge \hat{\lambda }\), which easily shows (5.22) choosing \(\delta >0\) so small that \((eC_1+1)\delta <C_0/2\). \(\square \)
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Ambrosio, L., Honda, S. & Tewodrose, D. Short-time behavior of the heat kernel and Weyl’s law on \({{\mathrm{RCD}}}^*(K,N)\) spaces. Ann Glob Anal Geom 53, 97–119 (2018). https://doi.org/10.1007/s10455-017-9569-x
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DOI: https://doi.org/10.1007/s10455-017-9569-x