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ANNALES
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L’INSTITUT FOURIER
Christopher MEANEY, Detlef MÜLLER & Elena PRESTINI
A.e. convergence of spectral sums on Lie groups
Tome 57, no 5 (2007), p. 1509-1520.
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Ann. Inst. Fourier, Grenoble
57, 5 (2007) 1509-1520
A.E. CONVERGENCE OF SPECTRAL SUMS ON LIE
GROUPS
by Christopher MEANEY,
Detlef MÜLLER & Elena PRESTINI
Abstract. — Let L be a right-invariant sub-Laplacian on a connected Lie
group G, and let SRf :=
∫ R
0
dEλf, R > 0, denote the associated “spherical partial
sums,” where L =
∫∞
0
λ dEλ is the spectral resolution of L. We prove that SRf(x)
converges a.e. to f(x) as R→∞ under the assumption log(2 + L)f ∈ L2(G).
Résumé. — Soit L un sous-Laplacien invariant à droite sur un groupe de Lie
G, et soit SRf :=
∫ R
0
dEλf, R > 0, l’opérateur “sommes sphériques partielles”
associé, où L =
∫∞
0
λ dEλ dénote la résolution spectrale de L. Nous prouvons que
SRf(x) converge vers f(x) p.p. quand R→∞, si log(2 + L)f ∈ L2(G).
1. Introduction
In [2] Carbery and Soria proved a.e. convergence of the spherical partial
sums (or better integrals) of functions belonging to the logarithmic Sobolev
space {f : log(2 + ∆)f ∈ L2(Rn)}, where −∆ denotes the Laplacian.
Recently the same result has been obtained [4] by a very short and simple
proof which works not only over dilates of the sphere, but more generally
over dilates of any fixed closed bounded region which is star shaped with
respect to the origin and has the origin in its interior. The proof in [4] is
based on the Rademacher-Menshov theorem. Since it makes use of very
basic principles, it lends itself for generalisations. In this paper we extend
it to arbitrary connected Lie groups, the “spherical partial sums” being
defined in terms of the spectral resolution of a given sub-Laplacian L. A
Keywords: Rademacher-Menshov theorem, sub-Laplacian, spectral theory.
Math. classification: 22E30, 43A50.
1510 Christopher MEANEY, Detlef MÜLLER & Elena PRESTINI
main tool is a Plancherel theorem for functions of L, which had been proved
by Hulanicki and Jenkins [6] in the case of stratified nilpotent Lie groups,
with a modified proof by Christ in [3], which extends easily to the case of
arbitrary connected Lie groups.
1.1. Preliminaries
Let G be a connected Lie group, with left-invariant Haar measure dx and
right invariant Haar measure ∆(x)−1dx , where ∆(x) denotes the modular
function. For 1 6 p 6 ∞, Lp(G) will denote the Lebesgue space with respect
to the left-invariant Haar measure dx. The following relations involving the
modular function ∆ will be used without further mentioning:∫
G
f(xg)dx = ∆(g)−1
∫
G
f(x)dx ,∫
G
f(x)dx =
∫
G
f(x−1)∆(x)−1dx .
Assume X1, . . . , Xk to be a family of right-invariant vector fields on
G satisfying Hörmander’s condition, which in this context means that
X1, . . . , Xk generate the Lie algebra g of G. Let
L := −
k∑
j=1
X2j
be the associated sub-Laplacian. L is right-invariant and essentially self-
adjoint on D(G) ⊂ L2(G). With a slight abuse of notation, its closure will
also be denoted by L. Moreover it is known (see, e.g., [7]) that there exist
ht ∈ L1(G), t > 0, the heat kernels, such that for all f ∈ L2(G)
(1.1) e−tLf(x) = ht ∗ f(x) =
∫
G
ht(y)f(y−1x) dy.
The {ht}t>0 form a one-parameter semigroup of smooth probability mea-
sures with respect to convolution.
Proposition 1.1. — Let G be as above, and denote by e the identity
element of G. Then there exists a constant c > 0 such that
(1.2) 0 < ht(e) 6 ct−α , 0 < t < 1 ,
where α = α(G) > 0 is the “local homogeneous dimension” of G associated
to the vector fields Xj .
ANNALES DE L’INSTITUT FOURIER
CONVERGENCE OF SPECTRAL SUMS 1511
Proof. — (1.2) follows from Theorem V.4.3 and Theorem IX.1.3 in [8].

Notice also that the Gaussian estimates for ht in [8] imply in particular
that ht ∈ L2(G) for t > 0.
Proposition 1.2. — The following relations hold true:
h2t(e) = ht ∗ ht(e) =
∫
G
ht(y)ht(y−1) dy;(1.3)
ht(y) = ht(y−1)∆(y)−1;(1.4) ∫
G
ht(y)2∆(y) dy =
∫
G
ht(y)2 dy.(1.5)
Proof. — (1.3) is clear. Since e−tL is a self-adjoint operator, we have
(e−tLf, g) = (f, e−tLg), that is, by (1.1),
(1.6) (ht ∗ f, g) = (f, ht ∗ g), f, g ∈ D(G) .
It is well-known that for ϕ ∈ L1(G)
(ϕ ∗ f, g) = (f, ϕ∗ ∗ g) ,
if one defines
ϕ∗(x) := ∆(x)−1ϕ(x−1) .
Thus ht = h∗t , which yields (1.4).
There remains to establish (1.5). By (1.3) and (1.4), we know that
h2t(e)=
∫
G
ht(y)ht(y−1) dy =
∫
G
ht(y)ht(y)∆(y) dy =
∫
G
ht(y)2∆(y) dy <∞ .
Replacing y by y−1, we obtain∫
G
ht(y)2∆(y) dy =
∫
G
ht(y−1)2∆(y)−1∆(y)−1dy .
In combination with (1.4), this is (1.5). 
The operator L, being self-adjoint, admits a spectral decomposition on
L2(G),
(1.7) L =
∫ ∞
0
λ dEλ .
Associated to L we consider the following “spherical” partial sum operators
SR, R > 0, given by
(1.8) SRf :=
∫ R
0
dEλf, f ∈ L2(G) .
TOME 57 (2007), FASCICULE 5
1512 Christopher MEANEY, Detlef MÜLLER & Elena PRESTINI
Clearly, since L is right-invariant, SR is right-invariant and bounded on
L2(G). Denote by χA the characteristic function of a set A. From the
Schwartz kernel theorem it follows that there exists a unique convolution
kernel KR ∈ D′(G) associated to the spectral multiplier χ[0,R] so that
(1.9) SRf = KR ∗ f, f ∈ D(G) .
1.2. The main result
We shall prove the following main result:
Theorem 1.3. — Let G and SR be as above, and assume that
log(2+L)f ∈ L2(G). Then limR→∞ SRf(x) = f(x) for almost every x ∈ G.
Moreover, for every compact subset K of G there exists a constant c > 0
such that
(1.10)
∫
K
∣∣∣ sup
R>0
|SRf(x)|
∣∣∣2dx 6 c ‖ log(2 + L)f ‖22 .
The strategy of proof follows the one developed in [4] in the case G = Rn
and L = −∆, the standard Laplacian, which in return is based on the
following classical result (see, e.g., [1], [9]):
Theorem 1.4 (Rademacher-Menshov). — Suppose that (X,µ) is a pos-
itive measure space. There is a positive constant c with the following prop-
erty:
For each orthogonal subset {fn : n ∈ N} in L2(X,µ) satisfying
(1.11)
∞∑
n=0
(log(n+ 2))2 ‖ fn ‖22<∞ ,
the maximal function F ∗(x) := supN∈N |
∑N
n=0 fn(x)| is in L2(X,µ), and
(1.12) ‖ F ∗ ‖26 c
( ∞∑
n=0
(log(n+ 2))2 ‖ fn ‖22
)1/2
.
In particular, when (1.11) holds, then the series
∑∞
n=0 fn(x) converges al-
most everywhere on X.
We also need a Plancherel theorem for functions of the sub-Laplacian.
Denote by L∞B ([0,∞)) the space of all Borel measurable essentially bounded
“multipliers” on [0,∞). Arguing similarly as for the operators SR, we see
that to any spectral multiplier σ ∈ L∞B ([0,∞)) corresponds a unique dis-
tribution Kσ ∈ D′(G) such that
σ(L)f = Kσ ∗ f for all f ∈ D(G) .
ANNALES DE L’INSTITUT FOURIER
CONVERGENCE OF SPECTRAL SUMS 1513
2. Plancherel’s theorem for functions of L
In this section we prove a Plancherel theorem which essentially goes back
to Hulanicki and Jenkins (see [6], formula (1.8)), at least in the case of a
stratified nilpotent Lie group. The latter result had been reproved by Christ
in [3], by a different proof. What comes very handy for us about Christ’s
proof is that it extends in a straight-forward way to arbitrary connected
Lie groups.
Proposition 2.1. — There exists a unique sigma finite positive Borel
measure ω on [0,∞) such that the following holds:
If σ ∈ L∞B ([0,∞)), then Kσ ∈ L2(G) if and only if∫ ∞
0
|σ(λ)|2dω(λ) <∞ .
Moreover, then
‖ Kσ ‖22=
∫ ∞
0
|σ(λ)|2dω(λ) .
Proof. — The proof follows very closely the proof of Proposition 3 in
([3], pp. 79-80): Observe first that if σ, τ ∈ L∞B ([0, b]) with Kτ ∈ L2(G),
then
(2.1) σ(L)Kτ = Kστ = Kτσ .
This can be shown as in [3]. Next, fix b > 0. Following Hulanicki and Stein
(see [5]), we conclude with (2.1) that
Kσ ∈ L2(G) for every σ ∈ L∞B ([0, b]) .
Indeed, since σ(λ) = (σ exp)(λ)e−λ, we have Kσ = (σ exp)(L)h1 ∈ L2(G),
since h1 ∈ L2(G).
In particular Γ := Kχ ∈ L2(G), where χ := χ[0,b]. For t > 0, set
φt := e−tLΓ = Kχ exp(−t ·) = χ(L)ht .
Consider any σ ∈ L∞B ([0, b]). As in [3], by means of (2.1), one easily
shows that
(φt, |σ|2(L)φt) =
∫ ∞
0
e−2tλ|σ|2(λ) d(Γ, EλΓ) →
∫ ∞
0
|σ|2(λ) d(Γ, EλΓ)
as t→ 0, where d(Γ, EλΓ) is the positive, finite Borel measure that assigns
to each Borel set B ⊂ [0,+∞[ the mass (Γ, χB(L)Γ).
On the other hand, since φt = χ(L)ht, one easily shows that
(φt, |σ|2(L)φt) =‖ σ(L)ht ‖22 .
TOME 57 (2007), FASCICULE 5
1514 Christopher MEANEY, Detlef MÜLLER & Elena PRESTINI
And, by (2.1),
σ(L)ht = Kσ exp(−t ·) = e−tLKσ ,
so that σ(L)ht → Kσ in L2(G) as t → 0, by spectral theory. We conclude
that
‖ Kσ ‖22=
∫ b
0
|σ|2(λ) d(Γ, EλΓ)
for every σ ∈ L∞B ([0, b]).
At most one Borel measure on [0, b] can have this property, and b is
arbitrary, so there exists a unique sigma-finite Borel measure ω on [0,∞)
satisfying
‖ Kσ ‖22 =
∫ ∞
0
|σ(λ)|2 dω(λ)
for all σ with compact support on [0,+∞).
The claim then follows for arbitrary σ ∈ L∞B ([0,+∞)) by standard ap-
proximation arguments. 
Corollary 2.2. — The following identities hold true:
‖ ht ‖22 =
∫ ∞
0
e−2tλdω(λ) for every t > 0 ;
‖ KR ‖22 =
∫ R
0
dω(λ) .
3. On inverse Laplace transforms
Observe next that, by Proposition 1.2,
h2t(e) =
∫
G
ht(y)2 dy ,
so that, as a function of t,
ht(e) =
∫ ∞
0
e−tλ dω(λ), t > 0 ,
is the Laplace-transform of the measure ω.
Lemma 3.1. — Let ω be a positive sigma finite Borel measure on [0,∞),
and denote by
Ω(t) :=
∫ ∞
0
e−tλ dω(λ), t > 0 ,
its Laplace transform. Moreover, put
W (R) :=
∫ R
0
dω(λ) = ω([0, R]) .
ANNALES DE L’INSTITUT FOURIER
CONVERGENCE OF SPECTRAL SUMS 1515
(i) If there exist constants a,C > 0 such that
(3.1) Ω(t) 6 Ct−a for t 1,
then there is a constant C1 such that
(3.2) W (R) 6 C1Ra for R 1.
(ii) If in addition to (3.1), also an estimate from below
(3.3) Ω(t) > C0t−a, t 1 ,
holds for some C0 > 0, then there is a constant C2 > 0 such that
W also satisfies an inverse inequality
W (R) > C2Ra for R 1.
Proof. — Let R > 0. An integration by parts shows that
Ω(t) >
∫ R
0
e−tλ dω(λ) = e−tRW (R) + t
∫ R
0
e−tλW (λ) dλ .
Since ω > 0, this implies, by (3.1),
e−tRW (R) 6 Ct−a for t 1 ,
i.e.,
W (R) 6 Ct−aetR, for every t 1 .
The right-hand side is minimal for t = aR . Thus, for R sufficiently large,
we obtain
W (R) 6 C
( a
R
)−a
ea = C1Ra ,
with C1 = Ca−aea. This proves (i).
To prove (ii), assume that (3.1) and (3.3) hold. Then, by (3.3), for
0 < t 1,
(3.4) C0 6
∫ R
0
tae−tλ dω(λ) +
∫ ∞
R
tae−tλ dω(λ) = I + II .
Again, an integration by parts yields
II = tae−tλW (λ)
∣∣∣∞
R
+
∫ ∞
R
ta+1e−tλW (λ) dλ.
Because of (3.2), the boundary term at ∞ vanishes, and thus
II 6
∫ ∞
R
ta+1e−tλW (λ) dλ .
Using again (3.2), for R sufficiently large, we obtain
II 6 C1
∫ ∞
R
ta+1e−tλλa dλ = C1
∫ ∞
Rt
λae−λ dλ .
TOME 57 (2007), FASCICULE 5
1516 Christopher MEANEY, Detlef MÜLLER & Elena PRESTINI
Choose A > 0 so large that C1
∫∞
A
λae−λdλ 6 C0/2, and put t = AR .
Then we have II 6 C0/2, so that, by (3.4),
C0
2
6 I =
∫ R
0
(
A
R
)a
e−
A
R λ dω(λ) 6 AaR−aW (R) ,
hence W (R) > (C02 A
−a)Ra. 
Corollary 3.2. — Let ω be the “Plancherel measure” from Propo-
sition 2.1, and let W (R) :=
∫ R
0
dω(λ), R > 0. Then there is a constant
C1 > 0 such that
W (R) 6 C1Rα, R 1 ,
where α = α(G) is as in (1.2).
Proof. — By Corollary 2.2, Proposition 1.2 and Proposition 1.1 we have∫ ∞
0
e−2tλdω(λ) =‖ ht ‖22= h2t(e) 6 c(2t)−α ,
if 0 < t < 1/2. The estimate therefore follows from Lemma 3.1 (i). 
Remarks 3.3. — a) The results in [8] even show that ht(e) ∼ t−α, 0 <
t < 1. Lemma 3.1 (i)(ii) therefore implies that in fact W (R) ∼ Rα as
R→∞. In particular,
(3.5) lim
R→∞
W (R) = +∞.
The latter fact can also be derived easily from Proposition 2.1:
Suppose that supR>0W (R) < ∞. Then ω([0,∞)) < ∞, so that, by
Proposition 2.1, the multiplier σ = 1 is represented by a convolution kernel
K1 ∈ L2(G). This means that δe = K1 ∈ L2(G). But this cannot happen
on a connected Lie group.
b) We also note that W (R) = ω([0, R]) is increasing and right-continuous
in R.
4. Proof of the main theorem
By the spectral theorem, the space of functions f ∈ L2(G) such that
SRf = f for some R > 0 is dense in L2(G), and for these functions obvi-
ously limR→∞ SRf(x) = f(x). We therefore only have to prove the estimate
(1.10).
To this end, we choose a sequence 0 =: R0 < R1 < R2 < . . . recursively
as follows:
Rk+1 := sup{R > Rk : W (R) < W (Rk) + 1}, k > 0.
ANNALES DE L’INSTITUT FOURIER
CONVERGENCE OF SPECTRAL SUMS 1517
By Remark 3.3 a), this recursion leads to an infinite sequence {Rk}k∈N
tending to infinity. Moreover, since W (R) = ω([0, R]) is increasing and
right-continuous in R, we have
(4.1) W (Rk+1) > W (Rk) + 1,
for every k.
By (4.1) and Corollary 3.2, we have
k 6 W (Rk) 6 C1Rαk .
This easily implies
(4.2) log(3 + k) 6 c log(2 +Rk), k > 1 .
for some constant c = c(α).
For k ∈ N we let ψk := χ(Rk,Rk+1] denote the characteristic function of
the interval (Rk, Rk+1], and define the pairwise orthogonal projections Pk
on L2(G) by P0 := S0 = χ{0}(L), and
Pk := SRk − SRk−1 = ψk−1(L), k > 1 .
Then
SRn =
n∑
k=0
Pk.
Assume that log(2+L)f ∈ L2(G). Clearly, the functions Pkf , k ∈ N, are
pairwise orthogonal. Moreover, by (4.2), for k > 1
(log(2 + k))2 ‖ Pkf ‖22 =
∫
(Rk−1,Rk ]
(log(3 + k − 1))2d(f,Eλf)
6 c2
∫
(Rk−1,Rk ]
(log(2 +Rk−1))2 d(f,Eλf)
6 c2
∫
(Rk−1,Rk ]
(log(2 + λ))2 d(f,Eλf),
and clearly (log(2+0))2 ‖ P0f ‖22= c′
∫
{0}(log(2+λ))
2 d(f,Eλf). Therefore
∞∑
k=0
(log(2 + k))2 ‖ Pkf ‖22 6 C1
∫ ∞
0
(log(2 + λ))2 d(f,Eλf)
= C1 ‖ log(2 + L)f ‖22<∞ .
If we define the discrete maximal operator M by
Mf(x) := sup
n>0
|
n∑
k=0
Pkf(x)| = sup
n>0
|SRnf(x)| ,
TOME 57 (2007), FASCICULE 5
1518 Christopher MEANEY, Detlef MÜLLER & Elena PRESTINI
the Rademacher-Menshov theorem thus implies that
(4.3) ‖ Mf ‖2 6 C ‖ log(2 + L)f ‖2 ,
where the constant C is independent of f .
We can dominate the maximal function over arbitrary R > 0 by the
maximal function over the sequence (Rn)∞n=0 plus a remainder:
sup
R>0
|SRf(x)| 6 Mf(x) + sup
n>0
(
sup
Rn6r<Rn+1
|Srf(x)− SRnf(x)|
)
.
To control the remainder term, fix a compact subset K in G.
Fix n ∈ N, and assume that Rn 6 r < Rn+1. By our definition of Rn,
then W (r) < W (Rn) + 1. Moreover, Srf − SRnf = η(L)f = Kη ∗ f , if we
put η := χ(Rn,r]. Proposition 2.1 then implies that
‖ Kη ‖22=
∫ ∞
0
η2(λ) dω(λ) = ω((Rn, r]) = W (r)−W (Rn) < 1,
i.e.,
‖ Kη ‖26 1 .
Moreover, since η = ηψn, we have
Kη ∗ f = Kη ∗ (ψn(L)f) = Kη ∗ (Pn+1f) .
By Cauchy-Schwarz’ inequality, we therefore obtain
|Srf(x)− SRnf(x)| =
∣∣∣ ∫
G
Kη(y)Pn+1f(y−1x) dy
∣∣∣
=
∣∣∣ ∫
G
Kη(xy)Pn+1f(y−1) dy
∣∣∣
=
∣∣∣ ∫
G
Kη(xy−1)∆(y)−1Pn+1f(y) dy
∣∣∣
6 Aη(x) ‖ Pn+1f ‖2 ,
where
A2η(x) :=
∫
G
|Kη(xy−1)∆(y)−1|2 dy
= ∆(x)−1
∫
G
|Kη(y−1)∆(y)−1|2 dy
= ∆(x)−1 ‖ K∗η ‖22 .
But, since the operator η(L) is self-adjoint, we have K∗η = Kη, so that
Aη(x) = ∆(x)−1/2 ‖ Kη ‖26 ∆(x)−1/2 ,
ANNALES DE L’INSTITUT FOURIER
CONVERGENCE OF SPECTRAL SUMS 1519
hence
|Srf(x)− SRnf(x)| 6 ∆(x)−1/2 ‖ Pn+1f ‖2 .
Since K is compact and ∆−1/2 is continuous, we thus get for x ∈ K
|Srf(x)− SRnf(x)| 6 CK ‖ Pn+1f ‖2 ,
hence
sup
n>0
(
sup
Rn6r<Rn+1
|Srf(x)− SRnf(x)|
)
6 CK
( ∞∑
n=0
‖ Pn+1f ‖22
)1/2
6 CK ‖ f ‖2 ,
for every x ∈ K. We thus obtain
sup
R>0
|SRf(x)| 6 Mf(x) + CK ‖ f ‖2
for every x ∈ K. In combination with (4.3) this implies estimate (1.10),
which completes the proof of Theorem 1.3. 
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TOME 57 (2007), FASCICULE 5
1520 Christopher MEANEY, Detlef MÜLLER & Elena PRESTINI
Christopher MEANEY
Macquarie University
Department of Mathematics
North Ryde NSW 2109 (Australia)
chrism@maths.mq.edu.au
Detlef MÜLLER
C.A.-Universität Kiel
Mathematisches Seminar
Ludewig-Meyn-Str.4
D-24098 Kiel (Germany)
mueller@math.uni-kiel.de
Elena PRESTINI
Università di Roma “Tor Vergata"
Dipartimento di Matematica
Via della Ricerca Scientifica
00133 Roma (Italie)
prestini@mat.uniroma2.it
ANNALES DE L’INSTITUT FOURIER


A.e. convergence of spectral sums on Lie groups

Djament, Aurélien

Foncteurs de division et structure de $I^{\otimes 2}\otimes \Lambda ^n$ dans la catégorie $\mathcal{F}$

Heinloth, Franziska

A note on functional equations for zeta functions with values in Chow motives

Guillemin, Victor

Sternberg, Shlomo

Riemann sums over polytopes

Ycart, Bernard

Cut-off for large sums of graphs

Eremenko, Alexandre

Jakobson, Dmitry

Nadirashvili, Nikolai

On nodal sets and nodal domains on $\mathbf{S^2}$ and ${\mathbb{R}}^{\mathbf{2}}$

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