VOL. 83 2000 NO. 2
SOME REMARKS ON BOCHNER–RIESZ MEANS
BY
S. T H A N G A V E L U (BANGALORE)
Abstract. We studyLp norm convergence of Bochner–Riesz meansSRδf associated with certain non-negative differential operators. When the kernelSmR(x, y) satisfies a weak estimate for large values ofmwe proveLpnorm convergence ofSRδfforδ > n|1/p−1/2|, 1< p <∞, wherenis the dimension of the underlying manifold.
1. Introduction and main results. The aim of this note is to make some remarks concerning the Lp mapping properties of the Bochner–Riesz means associated with certain differential operators. To set up the notation, letΩ be a Riemannian manifold andP a differential operator of orderdon Ω which is self-adjoint and formally non-negative. Let
P f =
∞
\
0
λ dEλf
be the spectral resolution of P.The Bochner–Riesz mean of order δ≥0 of a function f is defined by
SRδf =
R\
0
1− λ
R δ
dEλf.
Our aim is to study the convergence of SRδf to f in Lp(Ω) as R tends to infinity.
It is clear that SRδf converges to f in the L2 norm iff ∈L2(Ω). How- ever, if 1 ≤ p < 2 we can expect the convergence in the Lp norm only for large values of δ. In fact, there is a necessary condition: let δ(p) = max{n|1/p−1/2| −1/2,0} be the critical index for the Lp summability.
Then by a transplantation theorem of Mityagin [16] (see Kenig, Stanton and Tomas [13]) it is known that δ > δ(p) is a necessary condition for the convergence ofSRδf to f in the Lp norm.
2000 Mathematics Subject Classification: Primary 42B99; Secondary 42C10.
Key words and phrases: Bochner–Riesz means, summability, nilpotent groups, Rock- land operators, unitary representations, Schr¨odinger operators, Heisenberg group.
[217]
It is therefore natural to conjecture that δ > δ(p) is also sufficient for the Lp convergence of SRδf. Let us call this the Bochner–Riesz conjecture.
In some special cases the conjecture has been proved for a certain range of p. For example, when Ω = Tn is the n-torus and P is the standard Laplacian on Tn thenSRδf converges to f inLp(Tn) forδ > δ(p) provided
|1/p−1/2| ≥1/(n+ 1) forn ≥3 and all p if n = 2.The same is true for the standard Laplacian onRn. For these results see Carleson–Sj¨olin [2] and Fefferman [4].
In the general situation the best known results are due to H¨ormander [8] and Peetre [19]. Their results are that we have Lp convergence when δ >2(n−1)|1/p−1/2|providedΩis compact. In 1987, H¨ormander’s result was greatly improved by Sogge [21]; he showed that the Bochner–Riesz conjecture holds for compact Riemannian manifolds when P is of degree 2 and |1/p−1/2| ≥ 1/(n+ 1).When n = 2 the conjecture has been proved for all p.
Once we leave the premises of compact manifolds and consider non- compact situations, not much is known. In the special cases of Hermite and special Hermite expansions which are associated with the operators H=−∆+|x|2on Rn and
L=−∆+ 1
4|z|2−i Xn j=1
xj
∂
∂yj −yj
∂
∂xj
on Cn respectively, the conjecture has been settled for a certain range of p. See the works of the author [24, 25], Karadzhov [12] and Stempak–
Zienkiewicz [23]. When P is the sublaplacian on a stratified nilpotent Lie group a weaker form of the conjecture is known to be true (see Mauceri [14], Mauceri–Meda [15], M¨uller–Stein [17] and the references there).
Returning to the general situation we recall the following estimate, due to H¨ormander [8] and Peetre [19], on the kernel of the Riesz mean associated with adth order elliptic differential operator. IfSRδ(x, y) is the kernel ofSRδf, that is, if
SδRf(x) =
\
SRδ(x, y)f(y)dy then forx, y belonging to a compact subsetB of Ω,
|SδR(x, y)| ≤CBRn/d(1 +R1/d|x−y|)−δ−1
where CB is independent of R. From this estimate it follows that the op- erators χBSRδχB, where χBf(x) =χB(x)f(x) is the operator of multiplica- tion by the indicator function of B,are uniformly bounded on Lp(Ω) when δ >2(n−1)|1/p−1/2|.This is the best one can get from the above kernel estimates which is local in nature.
Even in the case of compact manifolds where the above estimate is
“global” it cannot be improved further and, therefore, it is not good enough to prove the Bochner–Riesz conjecture. In order to get around this difficulty, in [21] Sogge used a Fourier transform side argument to prove certainLp-L2 estimates for the projections associated with the spectral resolution. To be more specific, whenP is a second order operator, let
Pkf =
k\2
(k−1)2
dEλf.
Then Sogge used the following estimates, known as therestriction theorem:
kPkfk2≤Ckδ(p)kfkp, 1 p −1
2 ≥ 1
n+ 1.
The above estimates were proved by Sogge in [20] for second order elliptic differential operators on compact Riemannian manifolds. By adapting an argument of Fefferman–Stein [4] and Bonami–Clerc [1], he was able to show that the weak kernel estimates and restriction theorems are sufficient to prove summation results.
Unfortunately, we do not have good restriction theorems even on Rn for general elliptic differential operators. Since it is difficult to establish restriction theorems, we look for an alternative which can be used in the study of Bochner–Riesz means. Consider fractional powers of the operator P given by the spectral theorem as
(1 +P)−α/2f =
∞
\
0
(1 +λ)−α/2dEλf.
Once we have the restriction theorem it then follows that k(1 +P)−α/2fk22≤
X∞ k=1
k−2αkPkfk22,
which is dominated by CX∞
k=1
k−2α+2n(1/p−1/2)−1 kfk2p and therefore
k(1 +P)−α/2fk2≤Ckfkp
providedα > n(1/p−1/2).
In many situations, the Lp-L2 estimate for the operator (1 +P)−α/2 is easy to establish. We propose to use this in place of the restriction theorem.
As we show below, the Lp-L2 estimate for (1 +P)−α/2 will follow from a weak estimate for the Riesz kernel. Then by using the method of Sogge we
can establish a positive result for the Bochner–Riesz means which improves the known results, though falling short of being optimal.
Now we state our main results. Let P = P(x, D) be a (not necessarily elliptic) differential operator of degreedonRnwith smooth coefficients. We assume that the Riesz kernel associated with P satisfies the estimate (1.1) |SRδ(x, y)| ≤CRn/d(1 +R1/d|x−y|)−δ+β
for all x, y ∈ Rn where β is a fixed constant. Without loss of generality, we assume that the spectral projection P0 onto the kernel ofP is trivial on Lp(Ω).
Theorem 1.1. Let P be as above with Riesz kernel satisfying (1.1).
Then the Bochner–Riesz means SRδ are uniformly bounded on Lp(Rn) for 1< p <∞ whenever δ > δ(p) + 1/2.
As we have remarked earlier, for an arbitrary differential operator on a non-compact manifold the estimate (1.1) is available only locally. However, the local estimate is good enough to prove a local estimate for the Bochner–
Riesz means.
Theorem 1.2. Let P(x, D) be an elliptic differential operator of degree d with smooth coefficients on Rn and let B be any compact subset of Rn. Then χBSδRχB are uniformly bounded on Lp(Rn), 1 < p < ∞, whenever δ > δ(p) + 1/2.
By using the transplantation theorem of Mityagin [16], we can deduce global estimates for constant coefficient differential operators.
Corollary 1.3. Let P(D) be a homogeneous elliptic differential opera- tor onRn and let SRδ be the associated Bochner–Riesz means. Then SRδ are uniformly bounded on Lp(Rn) for 1< p <∞ whenever δ > δ(p) + 1/2.
Apart from the realm of constant coefficient differential opeartors there is at least one more class of differential operators, namely, Rockland operators on stratified nilpotent groups, for which global estimates for the Bochner–
Riesz kernel can be proved. So, for such operators we obtain the following result.
Corollary 1.4. Let L be a non-negative Rockland operator of homo- geneous degree d on a stratified nilpotent Lie group G and let SRδ be the associated Bochner–Riesz means. Let Q be the homogeneous dimension of the group and define the critical index byδ(p) = max{Q|1/p−1/2| −1/2,0}.
Then SRδ are uniformly bounded on Lp(G), 1 < p < ∞, provided δ >
δ(p) + 1/2.
Suppose H is a connected normal subgroup of G and π a unitary rep- resentation of Ginduced from a unitary character of H. Then following an
idea of Hulanicki and Jenkins [11] we can get summability results for oper- ators of the form π(L) whereL is a Rockland operator. This covers some Schr¨odinger operators with polynomial potential (see Corollaries 3.3 and 3.4).
We would like to thank the referee for his careful reading of the manu- script and for some useful suggestions.
2. Elliptic operators onRn.In this section we will prove Theorems 1.1, 1.2 and Corollary 1.3. We start with the following estimate for (1 +P)−α/d. Proposition 2.1. Let P(x, D) be a differential operator of degree d whose Riesz kernel satisfies the estimate (1.1). Then for 0 < α < n, 1< p < q <∞ and 1/q= 1/p−α/n we have
k(1 +P)−α/dfkq ≤Ckfkp. P r o o f. By the spectral theorem
(1 +P)−α/df =
∞\
0
(1 +λ)−α/ddEλf and so the kernel of (1 +P)−α/d is given by
Kα(x, y) =
∞\
0
(1 +λ)−α/ddEλ(x, y).
We want to make use of the estimate (1.1) for large values ofδ.AsEλ(x, y) = Sλ0(x, y), integrating by parts and making use of the identity
d
dλ(λmSλm(x, y)) =mλm−1Sλm−1(x, y) we obtain the expression
Kα(x, y) =Cα,m
∞
\
0
(1 +λ)−α/d−mλm−1Sλm−1(x, y)dλ.
If we use the estimate (1.1) we get
|Kα(x, y)| ≤C
∞
\
0
λ−α/d+n/d−1(1 +λ1/d|x−y|)−m+β+1dλ, which is easily seen to be bounded by
C|x−y|α−n
∞
\
0
λ−α/d+n/d−1(1 +λ1/d)−m+β+1dλ.
The last integral converges if m is large since 0< α < n and we obtain the estimate
|Kα(x, y)| ≤C|x−y|α−n
for the kernel of the operator (1 +P)−α/d. Now it is a routine matter to show that this operator has the required mapping properties. See, e.g., the proof of Theorem 1, Chapter V of Stein [22]. This completes the proof of the proposition.
We now proceed to the proof of Theorem 1.1. Since we closely fol- low Sogge [21] we will not give details. Choose ϕ ∈ C0∞(1/2,2) so that P∞
j=−∞ϕ(2jt) = 1 for t6= 0.Let
ϕδR,j(t) =ϕ(2j(1−t/R))(1−t/R)δ and for j= 1,2, . . . define
SR,jδ f =
∞
\
0
ϕδR,j(λ)dEλf.
Forj = 0 we define SR,0δ f =
∞\
0
ϕ0
1− λ R
1− λ
R δ
dEλf whereϕ0(t) = 1−P∞
j=1ϕ(2jt).
We can easily handle SR,0δ in the following way.
Proposition 2.2. kSR,0δ fkp≤Ckfkp, 1≤p≤ ∞.
P r o o f. As in the proof of Proposition 2.1 we can get SR,0δ (x, y) =
∞
\
0
λm−1Sλm−1(x, y)∂λmϕδR,0(λ)dλ.
Note thatϕδR,0is supported in (−∞, R/2) and satisfies the estimate
|∂λmϕδR,0(λ)| ≤CR−m. Therefore,
|SR,0δ (x, y)| ≤CR−m
R/2
\
0
λm−1|Sλm−1(x, y)|dλ.
Ifmis large enough,Sλm−1(x, y) is uniformly integrable and hence the propo- sition follows.
Proceeding with the proof of Theorem 1.1 we will show that given δ >
δ(p) + 1/2 there exists an ε >0 such that
kSR,jδ fkp≤C2−εjkfkp
for j = 1,2, . . . As in [21], using the kernel estimate we can show that for each γ >0 there is an ε >0 such that
\
R1/d|x−y|≥2(1+γ)j
|SR,jδ (x, y)|dy ≤C2−εj.
This will take care of the global part of the Riesz kernel. Then we prove the following.
Proposition 2.3. kSR,jδ fk2≤C2−jδ(R1/d)δ(p)+1/2kfkp. P r o o f. By the spectral theorem
kSR,jδ fk22=
∞
\
0
|ϕδR,j(λ)|2d(Eλf, f).
Since ϕδR,j is supported in
R(1−2−j+1)≤λ≤R(1−2−j−1) and bounded byC2−jδ it follows that
kSR,jδ fk22≤C2−2jδ(R1/d)2δ(p)+1
∞
\
0
(1 +λ)−(2δ(p)+1)/dd(Eλf, f), which is dominated by
C2−2jδ(R1/d)2δ(p)+1k(1 +P)−α/dfk22
withα=n|1/p−1/2|.Using the result of Proposition 2.1 we obtain kSR,jδ fk2≤C2−jδ(R1/d)δ(p)+1/2kfkp,
which completes the proof of the proposition.
Finally, if V is any ball of radius 2(1+γ)jR−1/d, then kSR,jδ fkLp(V) ≤C(2(1+γ)jR−1/d)δ(p)+1/2kSR,jδ fk2, which by the result of the previous proposition is dominated by
C2−jδ2j(1+γ)(δ(p)+1/2)kfkp.
Therefore, ifδ > δ(p) + 1/2 we can chooseγ >0 so that δ >(1 +γ)(δ(p) + 1/2),which will then show that
kSR,jδ fkp≤C2−εjkfkp
for someε >0. The rest of the proof proceeds as in Sogge [21].
We will now indicate how Theorem 1.2 is proved. Suppose P =P(x, D) is an elliptic differential operator of order dwith smooth coefficients. The following local estimate for the associated Riesz kernel has been proved in Peetre [19] (see also H¨ormander [8]).
Proposition 2.4. Let P be as above. Then
|SRδ(x, y)| ≤CRn/d(1 +R1/d|x−y|)−δ+β
where C is uniform on compact subsets of Rn×Rn and β is a universal constant.
From this proposition it follows that the kernel of χBSRδχB satisfies a uniform estimate of the form (1.1) which can be used to take care of the
“part at infinity” of the operator. To deal with the local part we need a local version of Proposition 2.1.
Proposition 2.5. Let P be as above. Then for 0< α < n, 1< p < q <∞, and 1/q= 1/p−α/n we have
k(1 +P)−α/dfkLp(B)≤CBkfkLp(B) for any compact subset B of Rn.
P r o o f. We only have to show that the kernel of χB(1 +P)−α/dχB is bounded by a constant times |x−y|α−n. But this follows from the local estimate for the Riesz kernel given in Proposition 2.4.
To complete the proof Theorem 1.2 we only need to make the following observation. If V is any ball, then as before
kχBSR,jδ χBfkLp(V) ≤ |V|1/p−1/22−jδ(R1/d)δ(p)+1/2k(1 +P)−α/dχBfk2 whereα=n(1/p−1/2).But now, by the spectral theorem,
k(1 +P)−α/dχBfk22
= ((1 +P)−2α/dχBf, χBf)
≤\
B
|f(x)|pdx1/p\
B
|(1 +P)−2α/dχBf(x)|p′dx1/p′
. Using Proposition 2.5 we get the estimate
k(1 +P)−α/dχBfk2≤CBkfkp.
This estimate can be used to complete the proof of Theorem 1.2.
To prove Corollary 1.3 we make use the following transplantation theo- rem due to Mityagin [16] a proof of which can be found in [13].
Theorem 2.6. Let P(x, D) be a self-adjoint differential operator whose principal symbol is p(x, ξ). Suppose for some p, 1≤p≤ ∞,and a set B of positive measure the operators χBSRδχB are uniformly bounded on Lp(Rn) for a sequence of values of Rtending to infinity. Let x0be a point of density of B.Then χ(−∞,λ)(p(x0, ξ)) is a Fourier multiplier onLp(Rn).
Given a homogeneous differential operatorP(D) we can apply the above theorem to P(D) +|x|2 to obtain Corollary 1.3.
We conclude this section with an example. Consider the operator P(D) where
P(ξ) =ξm1 +. . .+ξnm
with m an even integer. Then Peetre [18] has proved that the associated Riesz kernel satisfies the estimate
|SRδ(x, y)| ≤CRn/m(1 +R1/m|x−y|)−δ′−1
whereδ′=δ+ (n−1)/m. He has also shown that this estimate is optimal.
Using Peetre’s estimate we can prove thatSRδ are uniformly bounded on Lp(Rn) for δ > 2(n−1)(1−1/m)|1/p−1/2| whereas by Corollary 1.3 we get the same for δ > n|1/p−1/2|. This is still far from the optimal result which is known only in the case when the “cospheres” {ξ : P(ξ) = 1} are strictly convex.
3. Rockland operators on nilpotent groups. In this section we study Bochner–Riesz means associated with positive Rockland operators on a stratified group. We employ standard notations and terminology. A general reference for this section is the monograph of Folland and Stein [5].
Let G be a stratified group with a dilation structure δt, t > 0. The homogeneous dimension Qof Gis defined by the requirement
\
f(δtx)dx=t−Q
\
f(x)dx
wheredxis the Haar measure on the group. By|x|we mean a homogeneous norm onG.A left invariant differential operatorLonGis called aRockland operator if it is homogeneous of some degreed >0, that is,
L(f(δtx)) =tdLf(δtx), f ∈C∞(G),
and for every non-trivial unitary representationπ ofGthe operatorπ(L) is injective on C∞ vectors.
A positive Rockland operator L satisfies the following subelliptic esti- mate proved by Helffer and Nourrigat [7]: for every multi-index I there are constants C and ksuch that
kXIfk2≤C(kLkfk2+kfk2), f ∈C0∞(G).
ThenLis essentially self-adjoint and its closure is the infinitesimal generator of a semigroup of linear operators onL2(G) which is of the formTtf =f∗pt, t >0, where pt is a Schwartz class function (see Theorem 4.25 of [5]). The homogeneity of L implies that
pt(x) =t−Q/dp1(δt−1/dx).
In our analysis the following estimate established by Dziuba´nski, Hebisch and Zienkiewicz [3] plays an important role.
Theorem 3.1. Let L be a positive Rockland operator of homogeneous degreedand pt the associated heat kernel. Then there are positive constants a and C such that
|pt(x)| ≤Ct−Q/de−a|x|d/(d−1)t−1/(d−1).
Consider the Bochner–Riesz means SRδ associated with the operator L.
In order to prove Corollary 1.4 we require the following estimate on the Riesz kernel.
Theorem 3.2.Let Lbe homogeneous of degree d. There is a constant β such that for large positive values of δ we have the estimate
|SRδ(x)| ≤CRQ/d(1 +R1/d|x|)−δ+β where the constant C is independent of R.
WhenLis the sublaplacian on a stratified group, the above estimate has been proved in Hulanicki and Jenkins [10] by using the functional calculus for the commutative Banach subalgebra A of L1(G) generated by linear combinations of pt(x), t > 0. In a later paper [9] Hulanicki proved the weaker estimate
|SRδ(x)| ≤CRQ/d(1 +R1/d|x|)−δ/3+β
for any Rockland operator. At that time the sharp estimates on the heat kernel given in [3] were not known for general Rockland operators. A close examination of the proof in [10] reveals that once we have the estimates of Theorem 3.2, the main result of [10] (Theorem 1.12) can be proved for any Rockland operator of homogeneous degreed. We refer to [10] for the details.
Using estimates on the Bochner–Riesz kernel the authors in [10] and [11]
have obtained summability results for Rockland operators. Once we have Theorem 3.2 the Riesz kernel estimate can be used to prove Corollary 1.4 as in Section 1, which is a quantitative version of the corresponding theorems in [10] and [11]. However, our results are still not optimal (see e.g. Mauceri [14] for the case of the sublaplacian on the Heisenberg group).
Now let π be a representation of Ginduced from a unitary character of a normal connected subgroupH ofG. Then the operators π(x),x∈G, act on functions on G/H according to the formula
π(x)f(yH) =a(x, yH)f(yH.xH)
where ais a scalar function of modulus one. In [11] Hulanicki and Jenkins have shown that when Lis a Rockland operator of the form
L= Xk j=1
(−1)njXj2nj
where Xj, j = 1, . . . , k, generate the Lie algebra of G and π is as above, thenπ(L) is a positive self-adjoint operator. They have further shown that the kernel sδR of the Riesz means associated with π(L) can be expressed as an integral of the kernelSRδ forLand hence proved summability results for operators of the formπ(L).
We can combine Corollary 1.4 with their idea to get more precise quanti- tative versions of summability results for operators of the formπ(L).This is better explained in the case of Heisenberg group. Recall that the Heisenberg groupG=Cn×Ris a two-step nilpotent Lie group with group law
(z, t)(w, s) =
w+z, t+s+1
2Im(z.w)
. The vector fields Xj, j= 1, . . . ,2n, and T defined by
Xj = ∂
∂xj
−1 2yj
∂
∂t, Xj+n = ∂
∂yj
+1 2xj
∂
∂t
forj = 1, . . . , n and T =∂/∂t form a basis for the Heisenberg Lie algebra.
The dilation structure is given by the automorphismδr(z, t) = (rz, r2t), and
|(z, t)|4=|z|4+t2 defines a homogeneous norm.
Let H={(0, t) :t∈Rn}be the center of the Heisenberg group which is a normal subgroup. The quotientG/H is then identified withCn. Consider the unitary representation π of Gon L2(Cn) given by
π(z, t)f(w) =eite(i/2)Im(w.z)f(z+w).
Suppose
L=X
i,j
aijXiXj
is a positive Rockland operator on G. Then π(L) =X
i,j
aijXeiXej
whereXei=π(Xi) which is obtained by replacing∂/∂tinXibyi.Note that L andπ(L) are related by
L(eitf(z)) =eitπ(L)f(z).
The Bochner–Riesz meansδRf associated withπ(L) can be expressed in terms of the kernel ofSRδ as
sδRf(w) =
\
G
SRδ(z, t)π(z, t)f(w)dz dt.
Thus, ifsδR(w, z) denotes the kernel of sδR then it is given by sδR(w, z) =e(i/2)Im(w.z)
\
G
SRδ(z−w, t)eitdt.
Now we can use the estimate on SRδ(z, t) to get an estimate for sδR(w, z).
Indeed,
|sδR(w, z)| ≤CRQ/2
∞\
−∞
(1 +R2|z−w|4+R2t2)(−δ+β)/4dt whereQ= 2n+ 2.By a change of variables, we get the estimate
|sδR(w, z)| ≤CRn+1(1 +R2|z−w|4)(−δ+β+2)/4
∞\
−∞
(1 +R2t2)(−δ+β)/4dt.
The last integral converges if δ is large and we get the estimate
|sδR(w, z)| ≤CRn(1 +R1/2|z−w|)−δ+β+2 for such values ofδ.
Thus we can get the following corollary to Corollary 1.4.
Corollary 3.3.Let π(L) be as above. ThensδR are uniformly bounded on Lp(Cn), 1< p <∞, whenever δ >2n|1/p−1/2|.
A similar transference technique can be used to study Bochner–Riesz means associated with certain Schr¨odinger operators on Rn. To do this we consider the Schr¨odinger representation̺of the Heisenberg group onL2(Rn) which is given by
̺(z, t)f(ξ) =eitei(x.ξ+(1/2)x.y)f(ξ+y).
Under this representation, the vector fieldsXj transform as
̺(Xj) =− ∂
∂ξj
, ̺(Xj+n) =iξj
for j = 1, . . . , n. Thus, if L = P
i,jaijXiXj is a Rockland operator, then
̺(L) is an operator of the form X
i,j
bij
∂2
∂ξi∂ξj
+X
j
cj(ξ) ∂
∂ξj
+p(ξ) wherecj(ξ) andp(ξ) are polynomials.
Let SeRδ be the Bochner–Riesz mean associated with the operator ̺(L).
Then it is given in terms of SRδ(z, t) by the equation SeRδf(ξ) =
\
G
SRδ(z, t)̺(z, t)f(ξ)dz dt.
If we letz=x+iythe kernel SeRδ(ξ, y) ofSeRδ is given by SeRδ(ξ, y) =
\
R
\
Rn
SRδ(x+i(y−ξ), t)eite(i/2)x.(y+ξ)dx dt.
Since
|SRδ(x+iy, t)| ≤CRn+1(1 +R2|x|4+R2|y|4+R2t2)(−δ+β)/4 when δ is large enough we obtain the estimate
|SeRδ(ξ, y)| ≤CRn/2(1 +R1/2|y−ξ|)−δ+β+n+2. Once we have this estimate the next corollary follows.
Corollary 3.4. Let L and ̺(L) be as above. Then SeRδ are uniformly bounded on Lp(Rn) for 1< p <∞ provided δ > n|1/p−1/2|.
An important and more difficult problem is to prove the summability theorems forδ > δ(p).In a subsequent paper we will show that the weak es- timates on the Bochner–Riesz kernel can be used to prove certain multiplier theorems.
REFERENCES
[1] A. B o n a m i et J. L. C l e r c, Sommes de Ces`aro et multiplicateurs des d´eveloppe- ments en harmoniques sph´eriques, Trans. Amer. Math. Soc. 183 (1973), 223–263.
[2] L. C a r l e s o n and P. S j ¨o l i n, Oscillatory integrals and multiplier problem for the disc, Studia Math. 44 (1972), 287–299.
[3] J. D z i u b a ´n s k i, W. H e b i s c h and J. Z i e n k i e w i c z,Note on semigroups generated by positive Rockland operators on graded homogeneous groups, ibid. 110 (1994), 115–
126.
[4] C. F e f f e r m a n, A note on spherical summation multipliers, Israel J. Math. 15 (1972), 44–52.
[5] G. F o l l a n d and E. S t e i n,Hardy Spaces on Homogeneous Groups, Princeton Univ., Princeton, 1982.
[6] W. H e b i s c h, Almost everywhere summability of eigenfunction expansions associ- ated to elliptic operators, Studia Math. 96 (1990), 263–275.
[7] B. H e l f f e r et J. N o u r r i g a t, Caract´erisation des op´erateurs hypoelliptiques ho- mog`enes `a gauche sur un groupe nilpotent gradu´e, Comm. Partial Differential Equa- tions 4 (1979), 899–958.
[8] L. H ¨o r m a n d e r,On the Riesz means of spectral functions and eigenfunction ex- pansions for elliptic differential operators, in: Some Recent Advances in the Basic Sciences, Vol. 2, Yeshiva Univ., New York, 1969, 155–202.
[9] A. H u l a n i c k i,A functional calculus for Rockland operators on nilpotent Lie groups, Studia Math. 78 (1984), 253–266.
[10] A. H u l a n i c k i and J. J e n k i n s, Almost everywhere summability on nilmanifolds, Trans. Amer. Math. Soc. 278 (1983), 703–715.
[11] —, —,Nilpotent Lie groups and summability of eigenfunction expansions of Schr¨o- dinger operators, Studia Math. 80 (1984), 235–244.
[12] G. K a r a d z h o v,Riesz summability of multiple Hermite series inLp spaces, C. R.
Acad. Bulgare Sci. 47 (1994), 5–8.
[13] C. E. K e n i g, R. S t a n t o n and P. T o m a s,Divergence of eigenfunction expansions, J. Funct. Anal. 46 (1982), 28–44.
[14] G. M a u c e r i,Riesz means for the eigenfunction expansions for a class of hypoelliptic differential operators, Ann. Inst. Fourier (Grenoble) 31 (1981), no. 4, 115–140.
[15] G. M a u c e r i and S. M e d a,Vector valued multipliers on stratified groups, Rev. Mat.
Iberoamericana 6 (1990), 141–154.
[16] B. S. M i t j a g i n [B. S. Mityagin],Divergenz von Spektralentwicklungen inLp-R¨au- men, in: Linear Operators and Approximation II, Internat. Ser. Numer. Math. 25, Birkh¨auser, Basel, 1974, 521–530.
[17] D. M ¨u l l e r and E. M. S t e i n, On spectral multipliers for Heisenberg and related groups, J. Math. Pures Appl. 73 (1994), 413–440.
[18] J. P e e t r e,Remarks on eigenfunction expansions for elliptic differential operators with constant coefficients, Math. Scand. 15 (1964), 83–97.
[19] —,Some estimates for spectral functions, Math. Z. 92 (1966), 146–153.
[20] C. D. S o g g e,Concerning theLpnorm of spectral clusters for second order elliptic operators on compact manifolds, J. Funct. Anal. 77 (1988), 123–134.
[21] —,On the convergence of Riesz means on compact manifolds, Ann. of Math. 126 (1987), 439–447.
[22] E. S t e i n,Singular Integrals and Differentiability Properties of Functions, Princeton Univ. Press, Princeton, 1971.
[23] K. S t e m p a k and J. Z i e n k i e w i c z,Twisted convolution and Riesz means, J. Anal.
Math. 76 (1998), 93–107.
[24] S. T h a n g a v e l u,Lectures on Hermite and Laguerre Expansions, Princeton Univ.
Press, Princeton, 1993.
[25] —, Hermite and special Hermite expansions revisited, Duke Math. J. 94 (1998), 257–278.
[26] —, Harmonic Analysis on the Heisenberg Group, Progr. Math. 159, Birkh¨auser, Boston, 1998.
Stat-Math Division Indian Statistical Institute 8th Mile, Mysore Road Bangalore-560 059 India E-mail: veluma@isibang.ac.in
Received 28 January 1999; (3693)
revised 18 October 1999
