MULTIPLIERS AND SINGULAR INTEGRALS ON EXPONENTIAL GROWTH GROUPS

WALDEMAR HEBISCH AND TIM STEGER

Abstract. We propose a simple abstract version of Calderón– Zygmund theory, which is applicable to spaces with exponential volume growth, and then show that important specific operators can be treated within this framework.

1. Abstract Calderón–Zygmund theory

Throughout we use the usual convention that C stands for a positive constant, usually a large positive constant, whose precise value varies from occurrence to occurrence. However, to avoid writing such curiosities as “C + C + 3C ≤ C” we sometimes use C′, C′′, … instead of C. The precise values of C′, C′′, … also change from occurrence to occurrence.

Definition 1.1. We say that the space M with metric d and Borel measure μ has the Calderón–Zygmund property if there exists a constant C such that for every f in L^{1} and for every λ > C \frac{∥f∥_{L^{1}}}{μ(M)} (λ > 0 if μ(M) = ∞) we have a decomposition f = ∑ f_{i} + g, such that there exist sets Q_{i}, numbers r_{i}, and points x_{i} satisfying:

• f_{i} = 0 outside Q_{i},

• ∫ f_{i} dμ = 0,
• Q_{i} ⊂ B(x_{i}, Cr_{i}),
• ∑ μ(Q_{i}^{∗}) ≤ C \frac{∥f∥_{L^{1}}}{λ}, where Q_{i}^{∗} = \{x : d(x, Q_{i}) < r_{i}\},

Both authors are part of the European TMR network entitled “Harmonic Analysis”.

The first author was partially supported by KBN grant 5 P03A 050 20 and by the TMR network. The second author was partially supported by his university and by the Italian M.U.R.S.T.

2 WALDEMAR HEBISCH AND TIM STEGER

• ∑ ∥f_{i}∥_{L^{1}} ≤ C∥f∥_{L^{1}},

• |g| ≤ Cλ.

Since g = f−∑ f_{i}, we have ∥g∥_{L^{1}} ≤ C′∥f∥_{L^{1}}, hence ∥g∥_{L^{2}}^{2} ≤ C′′λ∥f∥_{L^{1}}.

If K(x, y) is any measurable kernel defined on M × M, then let K also denote the associated integral operator:

(Kf)(x) = ∫_{M} K(x, y)f(y) dy .

Theorem 1.2. If M has the Calderón–Zygmund property, if T = ∑_{n∈ℤ} K_{n} is bounded on L^{2}, and if for constants C, 0 < c < 1, a > 0, b > 0

∫ |K_{n}(x, y)|(1 + c^{n}d(x, y))^{a} dx ≤ C ,
∫ |K_{n}(x, y) − K_{n}(x, z)| dx ≤ C(c^{n}d(y, z))^{b} ,

then T is of weak type (1, 1) and bounded on L^{p}, 1 < p ≤ 2.

Remark 1.3. If ∑ K_{n} is strongly convergent on L^{2}, then easy extensions to our arguments show that the sum is also strongly convergent on L^{p}, 1 < p ≤ 2 and is convergent in measure for arguments in L^{1}. Similarly, almost everywhere convergence on L^{2} implies almost everywhere convergence on L^{p}, 1 ≤ p ≤ 2.

Remark 1.4. We find the given formulation very convenient. However, the assumption about K_{n} may be replaced by the weaker condition

∫_{d(x,y),d(x,z)>εd(y,x)} |K(x, y) − K(x, z)|dμ(x) ≤ C_{ε}

with ε > 0 small enough.

Lemma 1.5. Let f_{i}, Q_{i}, r_{i}, x_{i}, and Q_{i}^{∗} be as in 1.1. Then there exists C such that for all i

∑_{n:c^{n}r_{i}≥1} ∫_{(Q_{i}^{∗})^{c}} |K_{n}f_{i}|(x) dx ≤ C∥f_{i}∥_{L^{1}} .

MULTIPLIERS AND SINGULAR INTEGRALS 3

Proof.

∫_{(Q_{i}^{∗})^{c}} |K_{n}f_{i}|(x) dx

≤ ∥f_{i}∥_{L^{1}} sup_{y∈Q_{i}} ∫_{(Q_{i}^{∗})^{c}} |K_{n}(x, y)| dx ≤ ∥f_{i}∥_{L^{1}} sup_{y} ∫_{d(x,y)≥r_{i}} |K_{n}(x, y)| dx

≤ (c^{n}r_{i})^{−a}∥f_{i}∥_{L^{1}} sup_{y} ∫_{d(x,y)≥r_{i}} |K_{n}(x, y)|(1 + c^{n}d(x, y))^{a} dx

≤ C(c^{n}r_{i})^{−a}∥f_{i}∥_{L^{1}} .

Hence,

∑_{n:c^{n}r_{i}≥1} ∫_{(Q_{i}^{∗})^{c}} |K_{n}f_{i}|(x) dx ≤ C∥f_{i}∥_{L^{1}} ∑_{n:c^{n}r_{i}≥1}(c^{n}r_{i})^{−a} ≤ C′∥f_{i}∥_{L^{1}} . □

Lemma 1.6. Let f_{i}, Q_{i}, r_{i}, x_{i}, and Q_{i}^{∗} be as in 1.1. Then there exists C such that for all i

∑_{n:c^{n}r_{i}<1} ∫ |K_{n}f_{i}|(x) dx ≤ C∥f_{i}∥_{L^{1}} .

Proof. Since ∫ f_{i}(y)dy = 0,

∫ |K_{n}f_{i}|(x) dx = ∫ |∫_{Q_{i}} K_{n}(x, y)f_{i}(y)dy| dx

= ∫ |∫_{Q_{i}}(K_{n}(x, y) − K_{n}(x, x_{i}))f_{i}(y)dy| dx ≤ ∫ ∫_{Q_{i}} |K_{n}(x, y) − K_{n}(x, x_{i})||f_{i}(y)|dy dx

≤ ∥f_{i}∥_{L^{1}} sup_{y∈Q_{i}} ∫ |K_{n}(x, y) − K_{n}(x, x_{i})| dx

≤ C∥f_{i}∥_{L^{1}} sup_{y∈Q_{i}}(c^{n}d(y, x_{i}))^{b} ≤ C′(c^{n}r_{i})^{b}∥f_{i}∥_{L^{1}} .

Hence,

∑_{n:c^{n}r_{i}<1} ∫ |K_{n}f_{i}|(x) dx ≤ ∑_{n:c^{n}r_{i}<1} C′(c^{n}r_{i})^{b}∥f_{i}∥_{L^{1}} ≤ C′′∥f_{i}∥_{L^{1}} . □

Proof of 1.2. By the Marcinkiewicz interpolation theorem it is enough to show that T is of weak type (1, 1). Fix λ > 0. If λ ≤ C \frac{∥f∥_{L^{1}}}{μ(M)} , then

μ(\{x : |T f| > λ\}) ≤ μ(M) ≤ C \frac{∥f∥_{L^{1}}}{λ}

and we are done. So assume that λ > C \frac{∥f∥_{L^{1}}}{μ(M)} and fix a Calderón– Zygmund decomposition of f. Put E = \{x : ∑_{n,i} |K_{n}f_{i}| > \frac{λ}{2}\}, E_{1} = ⋃_{i} Q_{i}^{∗}. By 1.5 and 1.6,

∫_{E_{1}^{c}} ∑_{n,i} |K_{n}f_{i}| ≤ ∑_{i} ∫_{(Q_{i}^{∗})^{c}} ∑_{n} |K_{n}f_{i}| ≤ C ∑_{i} ∥f_{i}∥_{L^{1}} ≤ C′∥f∥_{L^{1}}

so

|E − E_{1}| ≤ \frac{2C′∥f∥_{L^{1}}}{λ} .

Now

|\{x : |T f(x)| > λ\}| ≤ |\{x : |T g(x)| > \frac{λ}{2}\}| + |E|

≤ \frac{4∥T g∥_{L^{2}}^{2}}{λ^{2}} + |E_{1}| + |E − E_{1}| ≤ \frac{C′′λ∥f∥_{L^{1}}}{λ^{2}} + C \frac{∥f∥_{L^{1}}}{λ} + \frac{2C′∥f∥_{L^{1}}}{λ}

≤ C′′′ \frac{∥f∥_{L^{1}}}{λ} . □

Remark 1.7. The above proof of 1.2 remains valid using the weaker hypotheses of sublinearity, namely

|(T f)(x)| ≤ ∑_{n} |(K_{n}f)(x)| ,

|(K_{n}(f_{1} + f_{2}))(x)| ≤ |(K_{n}f_{1})(x)| + |(K_{n}f_{2})(x)| .

2. Main theorems

Let T be a homogeneous tree of order q + 1 (so each vertex has q + 1 neighbours). On T we consider the natural distance d (length of the shortest path between two vertices). We fix an infinite geodesic g in T. We fix a numeration of the vertices in g (so we get a mapping N : g ↦ ℤ such that |N(x) − N(y)| = d(x, y) for x, y ∈ g).

MULTIPLIERS AND SINGULAR INTEGRALS 5

Think of the tree as hanging down from the point at infinity defined by the ray of g where N → +∞. With that picture in mind we define the level function, l(x), by the formula l(x) = N(x′) − d(x, x′), where x′ is the unique vertex of g closest to x. Let (Af)(x) = \frac{1}{2\sqrt{q}} ∑_{d(x,y)=1} q^{(l(y)−l(x))/2}f(y). For a complex function f on the tree we define the gradient ∇f by the formula

(∇f)(x) = ∑_{y:d(x,y)=1} |f(y) − f(x)| .

We define the measure μ on T by the formula

∫ f dμ = ∑ f(x)q^{l(x)} .

Note that A is self-adjoint on L^{2}(dμ). Also note that ∥A∥_{L^{1}(dμ)→L^{1}(dμ)} ≤ 1, hence that ∥A∥_{L^{p}(dμ)→L^{p}(dμ)} ≤ 1 for 1 ≤ p ≤ ∞. Thus the L^{2}- spectrum of A lies in [−1, 1]. For any θ ∈ ℝ the function q^{iθl(x)} is “almost” in L^{2}(dμ). More precisely q^{iθl(x)}e^{−εd(x,x_{0})} ∈ L^{2}(dμ) for any ε > 0 and x_{0} ∈ T. One calculates that A(q^{iθl(x)}) = cos θ q^{iθl(x)} and a slightly more involved calculation using q^{iθl(x)}e^{−εd(x,x_{0})} shows that cos θ is in the L^{2}-spectrum of A. Hence:

Remark 2.1. The L^{2}-spectrum of A is precisely [−1, 1].

Notation 2.2. If t > 0 and if H(λ) is any function defined for real λ,

(D_{t}H)(λ) = H(tλ) .

Theorem 2.3. Fix any nonzero ϕ ≥ 0 in C_{c}^{∞}([1/4, 2]). Suppose that F (λ) = H(1 + λ) (or F (λ) = H(1 − λ)) where supp H ⊂ [0, 2). If for some s > 3/2

sup_{t>0} ∥(D_{t}H)ϕ∥_{H(s)} < ∞ ,

then F (A) is of weak type (1, 1) and bounded on L^{p}(dμ), 1 < p < ∞. Moreover ∇(I − A)^{−1/2} is of weak type (1, 1) and bounded on L^{p}, 1 < p ≤ 2.

Let G = ℝ ⋉ ℝ^{Q} with s ∈ ℝ acting on ℝ^{Q} by n ↦ e^{−s}n. Multiplication is given by

(s_{1}, n_{1})(s_{2}, n_{2}) = (s_{1} + s_{2}, e^{s_{2}}n_{1} + n_{2}) .

This G is the Iwasawa (or AN) group corresponding to the real rank 1 symmetric space SO_{+}(Q+1, 1)/SO(Q). There is a distinguished Laplacian on G, which is up to a first order term (or up to conjugation) equal to the Laplace–Beltrami operator on the symmetric space. Rightinvariant vector fields are

X_{0} = ∂_{a} ,  X_{i} = e^{a}∂_{n_{i}}  for i = 1, …, Q.

Then we put

L = − ∑_{0≤i≤Q} X_{i}^{2} .

We consider L on L^{p}(G) with respect to left-invariant Haar measure dx (which is equal to Lebesgue measure in our coordinates).

Theorem 2.4. Fix any nonzero ϕ ≥ 0 in C_{c}^{∞}([1/2, 2]). If s_{0}, s_{1} > \frac{3}{2}, s_{1} > \frac{Q+1}{2},

sup_{t≥1} ∥(D_{t}F)ϕ∥_{H(s_{0})} < ∞ ,
sup_{0<t≤1} ∥(D_{t}F)ϕ∥_{H(s_{1})} < ∞ ,

then F (L) is of weak type (1, 1) and bounded on L^{p}, 1 < p < ∞. Moreover ∇L^{−1/2} is of weak type (1, 1) and bounded on L^{p}, 1 < p ≤ 2.

3. Covering lemma on the weighted tree

We defined the measure μ on T by the formula

∫ f dμ = ∑ f(x)q^{l(x)} .

We will write |R| as a shorthand for the measure μ(R). Say that x lies below y (and that y lies above x) if l(x) = l(y) − d(x, y). Say that

MULTIPLIERS AND SINGULAR INTEGRALS 7

R ⊂ T is an admissible trapezoid iff R consists of a single point x_{R}, or if there is x_{R} ∈ T and an integer h > 0, such that R consists of all x below x_{R} such that h ≤ l(x_{R}) − l(x) < 2h. Put h(R) = h in this second case and h(R) = 1 in the one point case. In either case one finds |R| = h(R)q^{l(x_{R})}. We call w(R) = q^{l(x_{R})} the width of R.

For an admissible trapezoid R we define its envelope R̃ as follows: if R consists of one point, then R̃ = R, otherwise R̃ consists of all x below x_{R} such that h/2 ≤ l(x_{R}) − l(x) < 4h. Note that if R_{1} ∩ R_{2} ≠ ∅ and w(R_{1}) ≥ w(R_{2}), then R_{2} ⊂ R̃_{1}. It is also easy to check that |R̃| ≤ 4|R|. We define a maximal function M by the formula

(Mf)(x) = ∑_{x∈R} |R|^{−1} ∫ |f(y)| dμ(y) = sup_{x∈R} |R|^{−1} ∑_{y∈R} |f(y)|q^{l(y)}

where the sup is taken over admissible trapezoids R.

Theorem 3.1. T with measure μ and distance d has the Calderón– Zygmund property. Furthermore M is of weak type (1, 1).

Proof. Let f ∈ L^{1}(dμ). Let S_{0} be the family of all admissible trapezoids R such that

∑_{x∈R} |f(x)|q^{l(x)} ≥ λ|R| .

Start by listing S_{0} in some arbitrarily chosen order. Choose R_{0} to be an admissible trapezoid in S_{0} of largest width (possible since the width is bounded by |R|, and |R| ≤ ∥f∥_{L^{1}}/λ). In case of ties, choose that trapezoid of largest width which occurs earliest in the listing of S_{0}. Now we proceed inductively: S_{i+1} consists of all R ∈ S_{i} disjoint from R_{0}, …, R_{i} and R_{i+1} is an admissible trapezoid in S_{i+1} of largest width. Since the R_{i} are disjoint, and since each R_{i} ∈ S we have

∑_{i} |R_{i}| ≤ \frac{1}{λ} ∑_{i} ∑_{x∈R_{i}} |f(x)|q^{l(x)} ≤ \frac{∥f∥_{L^{1}}}{λ} .

8 WALDEMAR HEBISCH AND TIM STEGER

Consequently

∑_{i} |R̃_{i}| ≤ 4 ∑_{i} |R_{i}| ≤ \frac{4∥f∥_{L^{1}}}{λ} .

If ∑_{x∈R} |f(x)|q^{l(x)} ≥ λ|R|, then by construction R intersects some R_{i} with width not smaller then the width of R, hence with R ⊂ R̃_{i}. So, putting E = ⋃ R̃_{i}, we have Mf ≤ λ outside E, and |E| ≤ \frac{4∥f∥_{L^{1}}}{λ}, so the second claim is proved.

To get the f_{i} in the Calderón–Zygmund decomposition, we first define auxiliary sets U_{i} and functions h_{i} as follows:

U_{i} = R̃_{i} − ⋃_{j<i} R̃_{j} ,

h_{i}(x) = f(x) for x ∈ U_{i} and h_{i}(x) = 0 for x ∉ U_{i}. We claim that

∑_{x} |h_{i}(x)|q^{l(x)} ≤ 6qλ|R̃_{i}| .

Indeed it is easy to find three admissible trapezoids P_{1}, P_{2}, P_{3}, such that w(P_{k}) > w(R_{i}) (k = 1, 2, 3), |P_{k}| ≤ 2q|R̃_{i}|, and R̃_{i} ⊂ P_{1} ∪ P_{2} ∪ P_{3}. If

∑_{x∈P_{k}} |f(x)|q^{l(x)} ≥ λ|P_{k}| ,

then there is j < i such that P_{k} ∩ R_{j} ≠ ∅, hence P_{k} ⊂ R̃_{j}, hence h_{i} is zero on P_{k}. One way or the other ∑_{x∈P_{k}} |h_{i}(x)|q^{l(x)} ≤ λ|P_{k}|, which means that

∑_{x} |h_{i}(x)|q^{l(x)} ≤ ∑_{k=1}^{3} ∑_{x∈P_{k}} |h_{i}(x)|q^{l(x)} ≤ ∑_{k=1}^{3} λ|P_{k}| ≤ 6qλ|R̃_{i}|

as claimed. We put

f_{i} = h_{i} − (∑_{x} h_{i}(x)q^{l(x)})\frac{χ_{R_{i}}}{|R_{i}|} , g = f − ∑ f_{i} .

Now we put Q_{i} = R̃_{i}, r_{i} = h(R_{i})/4 and we choose arbitrary x_{i} ∈ R_{i}. The conditions f = g + ∑ f_{i}, f_{i} = 0 outside Q_{i}, and ∫ f_{i} dμ = 0 hold by definition. It is easy to check that Q_{i} ⊂ B(x_{i}, 32r_{i}). It is also easy to check that |Q_{i}^{∗}| ≤ 2|Q_{i}| so

∑ |Q_{i}^{∗}| ≤ 2|R̃_{i}| ≤ \frac{8∥f∥_{L^{1}}}{λ} .

Clearly

∑_{i} ∫ |f_{i}| ≤ 2 ∑_{i} ∫ |h_{i}| ≤ 2 ∫ |f| = 2∥f∥_{L^{1}} .

Now, g is equal to f outside E = ⋃ R̃_{i}, and

g = ∑_{i} (∑_{x} h_{i}(x)q^{l(x)}) \frac{χ_{R_{i}}}{|R_{i}|}

on E. Since (Mf)(x) ≤ λ outside E, a fortiori g(x) = f(x) ≤ λ outside E. On E we observe that the R_{i} are disjoint so

sup_{x∈E} |g|(x) ≤ sup_{i} |\frac{∑_{x} h_{i}(x)q^{l(x)}}{|R_{i}|} | ≤ sup_{i} \frac{6qλ|R̃_{i}|}{|R_{i}|} ≤ 24qλ

which ends the proof. □

4. Integral kernels on the weighted tree

Let (Lf)(x) = ∑_{d(x,y)=1} f(y). The spectral resolution of L is given by the formula:

(F (L)f)(x) = \frac{q}{2π(q + 1)} ∫_{0}^{π} F (2\sqrt{q} cos θ) ∑_{y} \frac{ϕ_{θ}(x, y)}{|c_{θ}|^{2}}f(y) dθ

where

ϕ_{θ}(x, y) = q^{−d(x,y)/2} (c_{s}e^{iθd(x,y)} + c̄_{s}e^{−iθd(x,y)}) = q^{−d(x,y)/2}2ℜ(c_{s}e^{iθd(x,y)})

and

c_{s} = \frac{−qi}{q + 1} \frac{1}{2 sin θ} (e^{iθ} − \frac{1}{q}e^{−iθ}) .

When q is odd (so the tree may by identified with a free group) the formula is given in [8] as Theorem 4.1 (one must change notation: q ↦ 2r − 1, θ ↦ log(2r − 1)t). For even q the formula (and the proof) in [8] still works. Let q^{l/2} stand for the operator f(x) ↦ q^{l(x)/2}f(x).

We have 2\sqrt{q}q^{l/2}A = Lq^{l/2} so

(F (A)f)(x) = \frac{q}{2π(q + 1)} ∫_{0}^{π} F (cos θ)q^{−l(x)/2}

× ∑_{y} q^{l(y)/2}q^{−d(x,y)/2}2ℜ(c_{s}e^{iθd(x,y)})\frac{1}{|c_{θ}|^{2}} f(y) dθ .

To simplify the sequel we put Ã = I − A. For our purpose we may treat the real and imaginary parts of F separately, so we may assume that F is real. Then

F (Ã)(x, y) = ℜ (K(x, y) ∫_{0}^{π} F (1 − cos θ)e^{iθd(x,y)}η(θ) sin θ dθ)

= ℜ(K(x, y)E_{F} (d(x, y)))

where

K(x, y) = q^{(−l(y)−l(x)−d(x,y))/2} ,

η(θ) = \frac{q}{2π(q + 1)} \frac{2}{c̄_{s} sin θ} = \frac{q}{2π(q + 1)} \frac{4(q + 1)}{qi (e^{−iθ} − \frac{1}{q}e^{iθ})}

= \frac{2}{πi (e^{−iθ} − \frac{1}{q}e^{iθ})}

and

(1)  E_{F} (k) = ∫_{0}^{π} F (1 − cos θ)e^{iθk}η(θ) sin θ dθ .

Lemma 4.1. Let s ≥ 0 and let m be the integer satisfying s ≤ m. Let a < b, c < d be fixed constants. Suppose ϕ : ℝ → ℝ is C(m) and increasing with ϕ(c) < a, ϕ(d) > b, ϕ′ > 0. Then there is C depending only on a, b, c, d, s, ∥1/(ϕ′)∥_{L^{∞}} and ∥ϕ∥_{C(m)} such that

∥F ∘ ϕ∥_{H(s)} ≤ C∥F ∥_{H(s)} .

whenever supp F ⊂ [a, b].

Proof. This lemma only formulates a well-known fact. □

Lemma 4.2. Fix ε, 0 < ε ≤ 1. If s > \frac{3}{2} + ε, then there exists C > 0 such that if for any nonnegative integer n we have

supp F ⊂ [2^{−2n−1}, 2^{−2n+2}] ∩ [0, 3/2],

then

∑_{k=0}^{∞} |E_{F} (k)|(1 + k)(1 + 2^{−n}k)^{ε} ≤ C∥D_{2^{−2n}}F ∥_{H(s)} ,

∑_{k=0}^{∞} |E_{F} (k)|(1 + 2^{−n}k)^{ε} ≤ C2^{−n}∥D_{2^{−2n}}F ∥_{H(s)} ,

∑_{k=0}^{∞} |E_{F} (k + 1) − E_{F} (k)| (1 + k)(1 + 2^{−n}k)^{ε} ≤ C2^{−n}∥D_{2^{−2n}}F ∥_{H(s)} .

Proof. Put

F (x) = (D_{2^{2n}}G)(x) = G(2^{2n}x) .

Changing variables in equation 1 to t = 2^{n}θ we get

E_{F} (k) = 2^{−n} ∫_{1}^{π} G(2^{2n+1} sin^{2}(2^{−n−1}t))η(2^{−n}t) sin(2^{−n}t)e^{i2^{−n}tk} dt

= 2^{−2n} ∫_{1}^{π} H(t)ψ_{l}(t)e^{itm} dt = 2^{−2n} ∫_{1}^{π} H̃_{l}(t)e^{itm} dt

where k = 2^{n}m + l, 0 ≤ l < 2^{n}

H̃_{l}(t) = H(t)ψ_{l}(t) ,
H(t) = G(2^{2n+1} sin^{2}(2^{−n−1}t)) ,
ψ_{l}(t) = e^{i2^{−n}tl}2^{n} sin(2^{−n}t)η(2^{−n}t) .

One easily sees that ψ_{l}(t) and its derivatives are bounded uniformly in n, l and t ∈ [0, π]. Also, each of the derivatives of ϕ_{n} = 2^{2n+1} sin^{2}(2^{−n−1}t) is uniformly bounded in n and ϕ′_{n} is uniformly bouded from below on counterimage of supp G_{n}, so by 4.1

∥H̃_{l}∥_{H(s)} ≤ C∥H∥_{H(s)}∥ψ_{l}∥_{C(s)} ≤ C′∥G∥_{H(s)} .

Hence

∑_{k=0}^{∞} |E_{F} (k)| (1 + k)(1 + 2^{−n}k)^{ε}

≤ 2^{n+1} ∑_{l=0}^{2^{n}−1} ∑_{m=0}^{∞} |E_{F} (2^{n}m + l)| (1 + m)^{1+ε}

≤ 2^{n+1} ∑_{l=0}^{2^{n}−1} (∑_{m=0}^{∞}(|E_{F} (2^{n}m + l)| (1 + m)^{s})^{2})^{1/2} (∑_{m=0}^{∞}(1 + m)^{2(1+ε−s)})^{1/2}

≤ C2^{n+1}2^{−2n} ∑_{l=0}^{2^{n}−1} ∥H̃_{l}∥_{H(s)} ≤ C′′∥G∥_{H(s)} = C′′∥D_{2^{−2n}}F ∥_{H(s)} .

The second estimate follows similarly. For the third estimate one should note that e^{i(k+1)θ} − e^{ikθ} = (1 − e^{iθ})e^{ikθ} and then the first factor contributes 2^{−n} to the final result. □

Recall that for a complex function f on the tree we define the gradient ∇f by the formula

(∇f)(x) = ∑_{w:d(x,w)=1} |f(w) − f(x)| .

The gradient so defined is a sublinear operator.

Lemma 4.3. Suppose d(y, z) = 1 and l(z) = l(y) − 1. Then

∑_{x:d(x,y)=k} K(x, y)q^{l(x)} = \begin{cases}1 & for k = 0 \\ 2 + \frac{q−1}{q}(k − 1) & for k > 0 \end{cases} ,

∑_{x:d(x,y)=k} |K(x, y) − K(x, z)|q^{l(x)} ≤ 2 ,

∑_{x:d(x,y)=k} |∇_{x}K(x, y)|q^{l(x)} = 1 − \frac{1}{q} ≤ 2 .

Proof. Every vertex at distance k from y is reached starting from y making p steps up and then k − p steps down. There is only one possibility for each step up, while we have q choices when making a step down. However, we should not go back on our steps, so if p, k − p > 0, then we have only q − 1 possibilities for the first step down. Also, note that l(x) − l(y) = 2p − k. Hence

∑_{x:d(x,y)=k} q^{(−l(y)−l(x)−d(x,y))/2}q^{l(x)} = ∑_{x:d(x,y)=k} q^{(−l(y)+l(x)−k)/2}

= q^{k}q^{(−k−k)/2} + (∑_{p=1}^{k}(q − 1)q^{k−p−1}q^{(2p−k−k)/2}) + q^{(2k−k−k)/2}

= 1 + \frac{q − 1}{q}(∑_{p=1}^{k−1} 1) + 1 .

In the second sum, values of K(x, y) and K(x, z) are equal, unless x lies below z, so we get only the p = 0 term in the sum over p.

Similarly, in the third sum only the p = k term gives a nonzero contribution. Moreover, for the unique x contributing to the p = k term, the contribution |K(w, y) − K(x, y)| to ∇_{x}K(x, y) is zero for all but one w. For that one w, which is the unique vertex at distance one to x which lies above x, we have l(w) = l(x) + 1, d(w, y) = d(x, y) + 1, and K(w, y) = \frac{1}{q}K(x, y). □

Lemma 4.4. Fix ε, 0 < ε ≤ 1. If s > \frac{3}{2} + ε, then there is C such that if n is a nonnegative integer and supp F ⊂ [2^{−2n−1}, 2^{−2n+2}] ∩ [0, 3/2], then

∑_{x} |F (Ã)(x, y)| (1 + 2^{−n}d(x, y))^{ε}q^{l(x)} ≤ C∥D_{2^{−2n}}F ∥_{H(s)} ,

∑_{x} |F (Ã)(x, y) − F (Ã)(x, z)|q^{l(x)} ≤ C2^{−n}d(y, z)∥D_{2^{−n}}F ∥_{H(s)} ,

∑_{x} |∇_{x}F (Ã)|(x, y) (1 + 2^{−n}d(x, y))^{ε}q^{l(x)} ≤ C2^{−n}∥D_{2^{−n}}F ∥_{H(s)} .

Proof. We will prove only the second inequality (the first is easier, the third is similar to the second). It is enough to prove the lemma for real F. We may assume that d(y, z) = 1 and that l(z) = l(y) − 1.

Then

∑_{x} |F (Ã)(x, y) − F (Ã)(x, z)|q^{l(x)}

= ∑_{x} |ℜ(K(x, y)E_{F} (d(x, y)) − K(x, z)E_{F} (d(x, z)))|q^{l(x)}

≤ ∑_{x} |K(x, y) − K(x, z)||E_{F} (d(x, y))|q^{l(x)}

+ ∑_{x} K(x, z)|E_{F} (d(x, y)) − E_{F} (d(x, z))|q^{l(x)} = S_{1} + S_{2} .

Now, by 4.2 and 4.3

S_{1} = ∑_{k} |E_{F} (k)| ∑_{x:d(x,y)=k} |K(x, y) − K(x, z)|q^{l(x)}

≤ 2 ∑_{k} |E_{F} (k)| ≤ C2^{−n}∥D_{2^{−n}}F ∥_{H(s)} .

To estimate the second sum note that if d(x, z) = k then d(x, y) = k±1, so |E_{F} (d(x, y))−E_{F} (d(x, z))| ≤ |E_{F} (k+1)−E_{F} (k)|+|E_{F} (k)−E_{F} (k−1)| (formally putting E_{F} (−1) = E_{F} (0)). Then (again using 4.2 and 4.3)

S_{2} ≤ ∑_{k}(|E_{F} (k+1)−E_{F} (k)|+|E_{F} (k)−E_{F} (k−1)|) ∑_{x:d(x,z)=k} K(x, z)q^{l(x)}

≤ 4 ∑_{k} |E_{F} (k + 1) − E_{F} (k)| (1 + k)

≤ C2^{−n}∥D_{2^{−n}}F ∥_{H(s)} . □

Proof of 2.3. Let (−1)^{l} represent the operator f(x) ↦ (−1)^{l(x)}f(x). Observe that −(−1)^{l}Af = A((−1)^{l}f). That is, A is conjugated to −A by an operator which preserves all L^{p}(dμ)-norms as well as the weak-L^{1}(dμ) measure of the size of a function. Consequently, it is sufficient to prove the first claim of the theorem in the case F (λ) = H(1 − λ).

By standard estimates, the principal hypothesis on H is independent of the choice of ϕ. Now fix nonnegative ϕ, ψ ∈ C_{c}^{∞}([1/2, 4]) such that

MULTIPLIERS AND SINGULAR INTEGRALS 15

ϕ = ψ^{2}, and for all x > 0 ∑_{n=−∞}^{∞} ϕ(2^{2n}x) = 1. We have

H(x) = ∑_{n=−∞}^{∞} G_{n}(x) = ∑_{n=0}^{∞} G_{n}(x) where G_{n}(x) = ϕ(2^{2n}x)H(x) .

Of course

H(Ã) = ∑_{n=0}^{∞} G_{n}(Ã) .

We are going to show that the G_{n}(Ã) satisfy the assumptions of 1.2. By hypothesis, there is s > \frac{3}{2} and C independent of n such that

∥D_{2^{−2n}}G_{n}∥_{H(s)} ≤ C .

Fix ε, 0 < ε ≤ 1, such that s > \frac{3}{2} + ε. By 4.4 ∫ |G_{n}(Ã)|(x, y) (1 + 2^{−n}d(x, y))^{ε} dμ(x)

= ∑_{x} |G_{n}(Ã)(x, y)| (1 + 2^{−n}d(x, y))^{ε}q^{l(x)} ≤ C∥D_{2^{−2n}}G_{n}∥_{H(s)} ≤ C′

so the first assumption of 1.2 is satisfied. Similarly the second assumption of 1.2 follows directly from 4.4 so we get first claim of 2.3.

To get the second claim write

\frac{1}{\sqrt{t}} = ∑_{n} \frac{ϕ(2^{2n}t)}{\sqrt{t}} = ∑_{n} U_{n}(t) = ∑_{n} \frac{ψ(2^{2n}t)}{\sqrt{t}}ψ(2^{2n}t) = ∑_{n} V_{n}(t)W_{n}(t) .

It is easy to see that (for any s > 0)

∥D_{2^{−2n}}U_{n}∥_{H(s)} = ∥2^{n}U_{0}∥_{H(s)} = C2^{n} ,

∥D_{2^{−2n}}V_{n}∥_{H(s)} = ∥2^{n}V_{0}∥_{H(s)} = C2^{n} ,

and

∥D_{2^{−2n}}W_{n}∥_{H(s)} = ∥W_{0}∥_{H(s)} = C .

Using the third part of 4.4 we have for some C = C(s)

∫ |∇_{x}V_{n}(Ã)(x, y)| dμ(x) ≤ C so using the second part of 4.4 we get

∫ |∇U_{n}(Ã)(x, y) − ∇U_{n}(Ã)(x, z)| dμ(x)

≤ (sup_{w} ∫ |∇V_{n}(Ã)(x, w)| dμ(x))

× (∫ |W_{n}(Ã)(w, y) − W_{n}(Ã)(w, z)| dμ(w)) ≤ C′2^{−n}d(y, z) .

In this way we have checked that

∇U_{n}(Ã)

satisfies the second assumption of 1.2. We verify the first assumption of 1.2 by direct application of 4.4 to U_{n}(t). This ends the proof. □

5. A covering lemma on certain AN groups

Let G = ℝ⋉N, N = ℝ^{n}. We assume that the multiplication is given by the formula

(t, x_{1}, …, x_{n})(s, y_{1}, …, y_{n}) = (t + s, exp(a_{1}s)x_{1} + y_{1}, …, exp(a_{n}s)x_{n} + y_{n})

where the a_{i} ≠ 0 are real numbers. In the sequel we will pretend that a_{i} are positive, negative a_{i} are easy but tedious to handle. One easily checks that Lebesgue measure on ℝ^{n+1} is left invariant, so we may take it as Haar measure. On G we consider the natural right-invariant riemanianian distance given by ds^{2} = dt^{2} + ∑ exp(−2ta_{i}) dx_{i}^{2}. Put M = 2 max(1, a_{1}, …, a_{n}). Note that for large balls (say r > 1) we have

{(t, x) : |x| < ce^{c}r, |t| < cr} ⊂ B(0, r) ⊂ {(t, x) : |x| < e^{Mr}, |t| < r}

Lemma 5.1. G has the Calderón–Zygmund property.

We need some preparation before the proof. We say that a parallelopiped R is admissible iff (t, x) ∈ R if and only if

t ∈ [m_{0}2^{k_{0}}, (m_{0} + 1)2^{k_{0}}) ,
x_{i} ∈ [m_{i}2^{k_{i}}, (m_{i} + 1)2^{k_{i}}) for i = 1, …, n

where m_{i}, k_{i} for i = 0, …, n are integers and for k_{0} < 0 we have

e^{2M}2^{k_{0}} ≤ exp(−a_{i}(m_{0} + \frac{1}{2})2^{k_{0}})2^{k_{i}} ≤ 4e^{8M}2^{k_{0}}

i = 1, …, n and for k_{0} ≥ 0 we have

exp(M2^{k_{0}+1}) ≤ exp(−a_{i}(m_{0} + \frac{1}{2})2^{k_{0}})2^{k_{i}} ≤ 4 exp(M2^{k_{0}+3}) .

We will write R = R(k_{0}, …, k_{n}, m_{0}, …, m_{n}).

Note that there is C such that if R = R(k_{0}, …, k_{n}, m_{0}, …, m_{n}) is an admissible parallelopiped, x_{R} is the centerpoint of R and r_{R} = 2^{k_{0}}, then R ⊂ B(x_{R}, Cr_{R}) and |R^{∗}| = |{x : d(x, R) < r_{R}}| ≤ 32^{Q}.

Lemma 5.2. If Q is an admissible parallelopiped, then there exists a sequence of partitions P_{j} of Q such that

• each P_{j} consists of admissible parallelopipeds,

• for each j all R ∈ P_{j} have a common k_{0}

• for each R ∈ P_{j} either R ∈ P_{j+1} or R is a sum of two members of P_{j+1} (of equal volumes).

• the parallelopipeds in P_{j} are arbitrarily small for large j.

Proof. We write

P_{j} = {R(k_{0}, …, k_{n}, m_{0}, …, m_{n}) ⊂ Q : k_{0} = f_{0}(j) , k_{i} = f_{i}(j, m_{0}), i = 1, …, n}

where f_{i} are to be specified. Now, the second condition is satisfied by definition. To satisfy the third condition we require that either

18 WALDEMAR HEBISCH AND TIM STEGER

f_{0}(j + 1) = f_{0}(j) and f_{i}(j, m_{0}) − 1 ≤ f_{i}(j + 1, m_{0}) ≤ f_{i}(j, m_{0}) or f_{0}(j +1) = f_{0}(j)−1 and f_{i}(j +1, 2m_{0}) = f_{i}(j +1, 2m_{0} +1) = f_{i}(j, m_{0}). The first case correspond to splitting (some of) parallelopipeds in x coordinates, the second corresponds to splitting all parallelopipeds into half in t coordinate. As a normalization we also require that f_{0}(0) = k_{0}, that when splitting in x we perform as many splittings as allowed by admissibility condition, and that P_{j} ≠ P_{j+1}.

To finish the proof we should show how to divide P_{j}. If f_{0}(j) ≤ 0, then in each step we just substract one from f_{i} (cycling over i, i = 0, …, n). So in the sequel we need only to handle f_{0}(j) > 0. Our rules make as keep f_{0}(j) unchanged and decrease f_{i}(j, m_{0}), i = 1, …, n as long as admissibility allows. So, we may assume that splitting in x coordinates is forbidden. Hence, we have

exp(M2^{f_{0}(j)+1}) ≤ exp(−a_{i}(m_{0} + \frac{1}{2})2^{f_{0}(j)})2^{f_{i}(j,m_{0})} ≤ 2 exp(M2^{f_{0}(j)+1})

Now according to our rules, we make f_{0}(j + 1) = f_{0}(j) − 1, f_{i}(j + 1, 2m_{0}) = f_{i}(j + 1, 2m_{0} + 1) = f_{i}(j, m_{0}) and we should check that the result is admissible. However

exp(−a_{i}(2m_{0} + \frac{1}{2})2^{f_{0}(j+1)}) = exp(−a_{i}(m_{0} + \frac{1}{4})2^{f_{0}(j)})

≥ exp(−a_{i}(m_{0} + \frac{1}{2})2^{f_{0}(j)}) exp(−M2^{f_{0}(j)})

so

exp(−a_{i}(2m_{0} + \frac{1}{2})2^{f_{0}(j+1)})2^{f_{i}(j+1,2m_{0})}

≥ exp(−M2^{f_{0}(j)}) exp(−a_{i}(m_{0} + \frac{1}{2})2^{f_{0}(j)})2^{f_{i}(j,m_{0})}

≥ exp(−M2^{f_{0}(j)}) exp(M2^{f_{0}(j)+1})

= exp(M2^{f_{0}(j)}) = exp(M2^{f_{0}(j+1)+1})

MULTIPLIERS AND SINGULAR INTEGRALS 19

which gives one of admissibility conditions (lower bound for 2m_{0}). Similar, computation gives upper bound for 2m_{0} +1, and two other bounds (upper bound for 2m_{0} and lower bound for 2m_{0} + 1). □ Proof of 5.1. Fix f and λ > 0. We should decompose f. First, note that there is a partition P of G into dyadic parallelopipeds such that for each Q ∈ P we have |Q| > ∥f∥_{L^{1}}/λ. Namely, let ψ_{i}(l, m_{0}) be the largest integers so that R(l, ψ_{1}(l, m_{0}), …, ψ_{n}(l, m_{0}), m_{0}, m_{1}, …, m_{n}) is admissible. We take l large enough and enough and write

G = ⋃_{m_{i}∈ℤ,m_{0}∈{0,−1}}R(l, ψ_{1}(l, m_{0}), …, ψ_{n}(l, m_{0}), …, m_{0}, m_{1}, …, m_{n})

∪ ⋃_{k≥l,m_{i}∈ℤ,m_{0}∈{1,−2}}R(k, ψ_{1}(k, m_{0}), …, ψ_{n}(k, m_{0}), m_{0}, m_{1}, …, m_{n}) .

Next, on each Q ∈ P we apply 5.2 and use standard stopping time argument. □

6. Integral kernels on Lie groups

On a Lie group with right-invariant distance function d(·, ·), let d(x) = d(x, e), so d(x, y) = d(xy^{−1}).

Theorem 6.1. Assume G and L are as in [22]. If ε > 0, if s_{0}, s_{1} > \frac{3}{2} + ε, if s_{1} > \frac{Q+1}{2} + ε, then there is C such that for supp F ⊂ [1/2, 2] and t ≥ 1

∫ |F (tL)|(x) (1 + t^{−1/2}d(x))^{ε} ≤ C∥F ∥_{H(s_{0})}

while for 0 < t ≤ 1

∫ |F (tL)|(x) (1 + t^{−1/2}d(x))^{ε} ≤ C∥F ∥_{H(s_{1})} .

Proof. This follows using the methods of [22]. □

20 WALDEMAR HEBISCH AND TIM STEGER

In particular, 6.1 applies to the group in the statement of Theorem 2.4, namely G = ℝ ⋉ ℝ^{Q} where s ∈ ℝ acts on ℝ^{Q} via n ↦ e^{−s}n. For our next result, 6.2, our attention will be exclusively on that group.

We recall several conventions relative to convolution and the modular function. As always, convolution of measures is defined by

∫_{G} f(x) d(μ_{1} ∗ μ_{2})(x) = ∫_{G×G} f(yz) dμ_{1}(y) dμ_{2}(z) .

The ∗-operation on measures is dμ^{∗}(x) = dμ̄(x^{−1}). One has (μ_{1}∗μ_{2})^{∗} = μ_{2}^{∗}∗μ_{1}^{∗}. For Dirac measures δ_{x}∗δ_{y} = δ_{xy} and (δ_{x})^{∗} = δ_{x^{−1}}. If ∥μ∥ stands for the total variation, then ∥μ_{1} ∗ μ_{2}∥ ≤ ∥μ_{1}∥ ∥μ_{2}∥, ∥μ^{∗}∥ = ∥μ∥.

The base measure on G, denoted simply by dx, is taken as leftinvariant Haar measure, and thus d^{∗}x is right-invariant Haar measure. The modular function is defined by d^{∗}x = δ(x) dx. It follows that

d(xx_{0}) = d^{∗}(x_{0}^{−1}x^{−1}) = δ(x_{0}^{−1}x^{−1}) d(x_{0}^{−1}x^{−1}) = δ(x_{0}^{−1}x^{−1}) d(x^{−1}) = δ(x_{0}^{−1}x^{−1}) d^{∗}(x) = δ(x_{0}^{−1}) d(x) .

For the group under consideration, ℝ⋉ℝ^{Q}, one finds that dx is ordinary Lebesgue measure and δ(s, n) = e^{−Qs}.

For the purposes of convolution and the ∗-operation, we identify the function f(x) with the measure f(x) dx. One then finds:

(f_{1} ∗ f_{2})(x) = ∫_{G} f_{1}(y)f_{2}(y^{−1}x) dy , (δ_{x} ∗ f)(z) = f(x^{−1}z) ,
f^{∗}(x) = δ(x)f̄(x^{−1}) ,  (f ∗ δ_{x})(z) = δ(x)f(zx^{−1}) .

Having fixed this identification it is automatic that (f_{1} ∗ f_{2})^{∗} = f_{2}^{∗} ∗ f_{1}^{∗}, (f ∗ δ_{x})^{∗} = δ_{x^{−1}} ∗ f^{∗}, etc. Since L^{1}-norm (relative to dx) corresponds to total variation, one has ∥f_{1} ∗ f_{2}∥_{L^{1}} ≤ ∥f_{1}∥_{L^{1}}∥f_{2}∥_{L^{1}}, ∥f^{∗}∥_{L^{1}} = ∥f∥_{L^{1}}. The inner product for L^{2}(dx) may be written as ⟨f_{1}, f_{2}⟩ = (f_{2}^{∗} ∗ f_{1})(e).

As explained in Section 2 we work with the right-invariant vector fields X_{i}, and the right-invariant Laplacian L given by

X_{0} = ∂_{s} ,  X_{i} = e^{s}∂_{n_{i}}, i = 1, …, Q ,  L = − ∑_{0≤i≤Q} X_{i}^{2} .

On L^{2}(dx) this Laplacian is symmetric and may be extended to a positive self-adjoint operator. Let p_{t} be the heat kernel for (G, L), meaning that exp(−tL)(f) = p_{t} ∗ f. Since exp(−tL) is self-adjoint, one has p_{t} = p_{t}^{∗}.

We use the (right-invariant) distance function adapted to the vector fields X_{i}, 0 ≤ i ≤ Q. Here follows the usual definition. For fixed x, y ∈ G consider all smooth curves γ : [0, 1] → G such that γ(0) = x, γ(1) = y. For such a curve write γ′(t) = ∑_{i} a_{i}(t)X_{i}. Then

d(x, y) = inf_{γ}(∫_{0}^{1} ∑_{i} |a_{i}(s)|^{2} ds)^{1/2} .

It follows that |(X_{i}d)(x)| ≤ 1 in the Lipschitz sense. For the group under consideration d(x) is smooth (except at x = e), so this inequality is valid in the naive sense (except at x = e).

Theorem 6.2. Assume G = ℝ⋉ℝ^{Q} and L are as in 2.4. There exist C and ε > 0 such that uniformly in t > 0

∑_{i} ∫ |X_{i}p_{t}(x)| exp(εt^{−1/2}d(x)) dx ≤ Ct^{−1/2} .

Proof. As long as t is bounded from above the estimate follows easily from well-known pointwise bounds on the heat kernel, see for example [18] (in fact [18] is an overkill, since for elliptic operators the estimates we need where already known in the sixties). So it is enough to prove our claim for t > 1.

We may assume that Q = 2l is even. Indeed G_{Q} is a quotient of G_{Q+1}, the heat kernel on G_{Q} is the push-forward of the heat kernel on G_{Q+1}, and for 0 ≤ i ≤ Q the vector fields (X_{i}) on G_{Q} and G_{Q+1} respectively

22 WALDEMAR HEBISCH AND TIM STEGER

match up under the quotient map. Consequently our estimate on G_{Q+1} implies the same estimate on G_{Q}.

With Q = 2l one gets

δ(s, n) = e^{−2ls} , δ(s, n)^{1/2} = e^{−ls} .

The distance on G is given by the formula

(2)  d((s, n), e) = arc cosh(\frac{1}{2}(e^{s} + e^{−s}(1 + |n|^{2})) .

The following explicit formula for p_{t}(s, n) is taken from [5].

p_{t}(x) = e^{ct}δ^{1/2}q_{t}(d(x, e)) ,
q_{t}(r) = Ce^{−ct}t^{−1/2}D_{r}^{l} exp(−r^{2}/(4t))

where

D_{r} = \frac{−1}{sinh(r)}∂_{r} .

The reduction to even Q avoids the use of fractional derivatives in the above formula.

The first stage of our proof follows along the lines of [5]. Put

Φ_{1}(r) = \frac{r}{sinh(r)} , Φ_{j+1} = D_{r}Φ_{j} .

Easy induction shows that one can write

CD_{r}^{l} exp(−r^{2}/(4t)) = exp(−r^{2}/(4t)) ∑_{k=1}^{l} t^{−k}ψ_{l,k}(r) ,

ψ_{l,k} = ∑_{|α|=l} c_{α} ∏_{i=1}^{k} Φ_{α_{i}}

where the c_{α} are positive.

One then checks that when r → ∞

Φ_{j}(r) = re^{−jr} + O(e^{−jr}) ,
∂_{r}Φ_{j}(r) = ∂_{r}(re^{−jr}) + O(e^{−jr})

so

ψ_{l,k}(r) = r^{k}e^{−lr} + O((r + 1)^{k−1}e^{−lr}) ,
∂_{r}ψ_{l,k}(r) = ∂_{r}(r^{k}e^{−lr}) + O((r + 1)^{k−1}e^{−lr}) .

It is known that the Φ_{j} are all positive. So, likewise, all the ψ_{l,k} are positive. Now the formula for the heat kernel reads

(3)  p_{t} = ∑_{k=1}^{l} t^{−(2k+1)/2} exp(−d^{2}/(4t))δ^{1/2}ψ_{l,k}(d) .

Hence

(4) X_{i}p_{t} = t^{−3/2} exp(−d^{2}/(4t))X_{i}(δ^{1/2}ψ_{l,1}(d))

+ (X_{i}d/2)t^{−5/2}d exp(−d^{2}/(4t))δ^{1/2}ψ_{l,1}(d)

+ X_{i} (∑_{k=2}^{l} t^{−(2k+1)/2} exp(−d^{2}/(4t))δ^{1/2}ψ_{l,k}(d)) .

We will deal with these three terms separately. The first term (which we will deal with last) is the really delicate one. To bound the second term, we use the inequality |X_{i}d| ≤ 1:

|(X_{i}d/2)t^{−5/2}d exp(−d^{2}/(4t))δ^{1/2}ψ_{l,1}(d)|

≤ Ct^{−2}(d^{2}/(4t))^{1/2} exp(−d^{2}/(4t))δ^{1/2}ψ_{l,1}(d)

≤ C′t^{−2} exp(−d^{2}/(8t))δ^{1/2}ψ_{l,1}(d)
≤ C′′t^{−1/2}p_{2t} .

Now use the fact that ∥p_{2t}∥_{L^{1}} = 1.

24 WALDEMAR HEBISCH AND TIM STEGER

Although the third term of (4) is given by a complicated formula, it is really of lower order (in t) than the others, and can be bounded quite directly. Consider the term for some k ≥ 2.

|X_{i} (t^{−(2k+1)/2} exp(−d^{2}/(4t))δ^{1/2}ψ_{l,k}(d)) |

= |t^{−(2k+1)/2}((X_{i} exp(−d^{2}/(4t)))δ^{1/2}ψ_{l,k}(d)

+ exp(−d^{2}/(4t))(X_{i}δ^{1/2})ψ_{l,k}(d)
+ exp(−d^{2}/(4t))δ^{1/2}(X_{i}ψ_{l,k}(d))|
= |t^{−(2k+3)/2}(X_{i}d/2)d exp(−d^{2}/(4t))δ^{1/2}ψ_{l,k}(d)
+ t^{−(2k+1)/2}( exp(−d^{2}/(4t))(X_{i}δ(e))δ^{1/2}ψ_{l,k}(d)
+ exp(−d^{2}/(4t))δ^{1/2}(X_{i}d)∂_{r}ψ_{l,k}(d))|
≤ Ct^{−(2k+3)/2}d exp(−d^{2}/(4t))δ^{1/2}(d + 1)^{k}e^{−ld}
+ Ct^{−(2k+1)/2} exp(−d^{2}/(4t))δ^{1/2}(d + 1)^{k}e^{−ld}

≤ C′t^{−2}(1 + d^{k}t^{−k/2} + d^{k−1}t^{−(k−1)/2}) exp(−d^{2}/(4t))δ^{1/2}(d + 1)e^{−ld} ≤ C′′t^{−2} exp(−d^{2}/(8t))δ^{1/2}(d + 1)e^{−ld} ≤ C′′′t^{−1/2}p_{2t} .

At the next to the last line, we use k ≥ 2 to get t^{−2}.

For the first term of (4), where k = 1, we must give a more detailed argument. In that term occurs the factor X_{i}(δ^{1/2}ψ_{l,1}(d)) where δ^{1/2} = exp(−ls) and ψ_{l,1}(d) ≈ cd exp(−ld). Except for a relatively small piece of G, the two exponentials, exp(−ls) and exp(−ld), almost cancel one another out, and so the derivative X_{i}(δ^{1/2}ψ_{l,1}(d)) is very much smaller than one would naively expect.

A comparison with the analogous calculation on a tree is striking.

δ^{1/2} exp(−d)  ⟷ K(x, y) ,
δ^{1/2}  ⟷ q^{(−l(x)−l(y))/2} ,
exp(−d)  ⟷ q^{−d(x,y)/2} .

MULTIPLIERS AND SINGULAR INTEGRALS 25

In the case of the tree the two exponential factors cancel each other out perfectly (and consequently ∇_{x}K(x, y) = 0) unless x lies directly above y. In the present case, things are fuzzier, but essentially the same phenomenon controls the situation.

For T > 1, arc cosh(T) = log(2T) + (1/4)T^{−2} + …, the expansion continuing as a convergent power series in T^{−2}. Applying this to (2), that is to the exact formula for d(s, n) = d((s, n), e) gives

d(s, n) = log(e^{s} + e^{−s}(1 + |n|^{2})) + O((e^{s} + e^{−s}(1 + |n|^{2}))^{−2}) , ∂_{s}d(s, n) = ∂_{s} log(e^{s} + e^{−s}(1 + |n|^{2})) + O((e^{s} + e^{−s}(1 + |n|^{2}))^{−2}) ,

e^{s}∂_{n_{i}}d(s, n) = e^{s}∂_{n_{i}} log(e^{s} + e^{−s}(1 + |n|^{2})) + O((e^{s} + e^{−s}(1 + |n|^{2}))^{−2}) .

Next, since δ^{1/2}(s, n) = e^{−ls}, since ψ_{l,1}(r) = cre^{−lr} + O(e^{−lr}), and since ∂_{r}ψ_{l,1}(r) = −clre^{−lr} + O(e^{−lr}):

(δ^{1/2}ψ_{l,1})(s, n) = cd(s, n)(e^{2s} + 1 + |n|^{2})^{−l} + O(δ^{1/2}e^{−ld}) ,
∂_{s}(δ^{1/2}ψ_{l,1})(s, n) = cd∂_{s}(e^{2s} + 1 + |n|^{2})^{−l} + O(δ^{1/2}e^{−ld})

= cd\frac{−2le^{2s}}{(e^{2s} + 1 + |n|^{2})^{l+1}} + O(δ^{1/2}e^{−ld}) ,

and

e^{s}∂_{n_{i}}(δ^{1/2}ψ_{l,1})(s, n) = cd∂_{n_{i}}(e^{2s} + 1 + |n|^{2})^{−l} + O(δ^{1/2}e^{−ld})

= cd\frac{−2le^{s}n_{i}}{(e^{2s} + 1 + |n|^{2})^{l+1}} + O(δ^{1/2}e^{−ld}) .

26 WALDEMAR HEBISCH AND TIM STEGER

Now ∫ |∂_{s}p_{t}|(x) exp(εt^{−1/2}d(x)) dx ≤ C ∫ |t^{−3/2} exp(−d^{2}/(4t))d∂_{s}((e^{2s} + 1 + |n|^{2})^{−l})| exp(εt^{−1/2}d(x)) dx

+ Ct^{−1/2} ∫ p_{2t} exp(εt^{−1/2}d(x)) dx

+ t^{−3/2} ∫ exp(−d^{2}/(4t))O(δ^{1/2}e^{−ld}) exp(εt^{−1/2}d(x)) dx
= C(I_{1} + I_{2}) + I_{3} .

Similarly

∫ |e^{s}∂_{n_{i}}p_{t}|(x) exp(εt^{−1/2}d(x)) dx ≤ C(I_{4} + I_{2}) + I_{3}

where I_{4} = ∫ |t^{−3/2} exp(−d^{2}/(4t))de^{s}∂_{n_{i}}((e^{2s}+1+|n|^{2})^{−l})| exp(εt^{−1/2}d(x)) dx . So to finish the proof we need to estimate I_{i}, i = 1, 2, 3, 4. We can absorb exp(εt^{−1/2}d(x)) into the Gaussian factor, so

I_{2} ≤ Ct^{−1/2} ∫ p_{3t} = Ct^{−1/2} .

We can compare I_{3} with a linear combination of the p_{t}, using the bound O(δ^{1/2}e^{−ld}) ≤ C2^{2k}p_{2^{2k}} for 2^{k} ≤ d(x) < 2^{k+1}.

This follows from (3), the formula for p_{t}, recalling that all the terms there are positive. Hence

t^{−3/2} ∫ exp(−d^{2}/(4t))O(δ^{1/2}e^{−ld}) exp(εt^{−1/2}d(x)) dx

= t^{−3/2} ∫_{d(x)≥t^{1/2}} +t^{−3/2} ∫_{d(x)<t^{1/2}}
≤ Ct^{−1/2} ∫ p_{2t} + t^{−3/2} ∫_{d(x)<t^{1/2}} O(δ^{1/2}e^{−ld})

\begin{aligned}≤ Ct^{−1/2} & + t^{−3/2} ∫_{d(x)<1} O(δ^{1/2}e^{−ld}) \\ & + t^{−3/2} ∑_{k≥0,2^{k}<t^{1/2}} ∫_{2^{k}≤d(x)<2^{k+1}} O(δ^{1/2}e^{−ld}) \end{aligned}

≤ C′t^{−1/2} + ∑_{k≥0,2^{k}<t^{1/2}} C′′t^{−3/2}2^{2k} ≤ C′t^{−1/2} + C′′′t^{−1/2} .

To estimate I_{1} note that

∫_{ℝ^{2l}} \frac{2le^{2s}}{(e^{2s} + 1 + |n|^{2})^{l+1}} dn = ∫_{ℝ^{2l}} \frac{2l}{(1 + e^{−2s} + |n|^{2})^{l+1}} dn ≤ C

with constant C independent of s. Hence

I_{1} = ∫ |t^{−3/2} exp(−d^{2}/(4t))d∂_{s}((e^{2s} + 1 + |n|^{2})^{−l})| exp(εt^{−1/2}d(x)) dx

≤ Ct^{−1} ∫ | exp(−d^{2}/(8t))∂_{s}((e^{2s} + 1 + |n|^{2})^{−l})| dx

≤ Ct^{−1} ∫ exp(−s^{2}/(8t)) ∫_{ℝ^{2l}} \frac{2le^{2s}}{(e^{2s} + 1 + |n|^{2})^{l+1}} dn ds

≤ C′t^{−1} ∫ exp(−s^{2}/(8t)) ds = C′′t^{−1/2} .

The argument for I_{4} is similar, and that ends the proof. □

The key estimate 6.2 has now been proved for the case of G = ℝ⋉ℝ^{Q}, as needed for the proof of 2.4. We will state our next two results in greater generality, using that key estimate as one of the hypotheses.

We recall the general set-up. Let G be a Lie group. On G use left-invariant Haar measure as the basic measure. Work always with

28 WALDEMAR HEBISCH AND TIM STEGER

right-invariant vector fields and differential operators. Assume that on G there are given vector fields X_{i} and that the distance d(·, ·) on G is the distance adapted to those vector fields. Assume also that there is a given self-adjoint Laplacian L ≥ 0 on G, and that its heat kernel is p_{t}(·).

Lemma 6.3. Let G, X_{i}, d, L, and p_{t} be as above. If, as in 6.2, we have ∥X_{i}p_{t}∥_{L^{1}} ≤ Ct^{−1/2} for each i, then there is C′ such that

∥p_{t} ∗ (δ_{x} − δ_{y})∥_{L^{1}} ≤ C′t^{−1/2}d(x, y) .

Proof. First, we need an auxiliary formula. Let γ : [0, 1] ↦ G be a smooth curve. Fix s. Assume that γ′(s) = Y (γ(s)), where Y is a right-invariant vector field. We have

∂_{u}(p_{t} ∗ δ_{γ(s+u)})|_{u=0} = ∂_{u}(p_{t} ∗ δ_{exp(uY)} ∗ δ_{γ(s)})|_{u=0}

so, applying the ∗-operation and using the fact that p_{t}^{∗} = p_{t},

∥∂_{u}(p_{t}∗δ_{γ(s+u)})|_{u=0}∥_{L^{1}} = ∥∂_{u}(δ_{(γ(s))^{−1}}∗δ_{exp(−uY)}∗p_{t})|_{u=0}∥_{L^{1}} = ∥Y p_{t}∥_{L^{1}} .

If γ′(s) = ∑_{i} a_{i}(s)X_{i}, then since s was arbitrary,

∥∂_{s}(p_{t} ∗ δ_{γ(s)})∥_{L^{1}} ≤ ∑_{i} |a_{i}(s)| ∥X_{i}p_{t}∥_{L^{1}} .

Now, assume that γ joins x and y. Then

∥p_{t} ∗ (δ_{x} − δ_{y})∥_{L^{1}}

≤ ∫_{0}^{1} ∥∂_{s}(p_{t} ∗ δ_{γ(s)})∥_{L^{1}} ds ≤ ∫_{0}^{1} ∑_{i} |a_{i}(s)|∥X_{i}p_{t}∥_{L^{1}} ds

≤ Ct^{−1/2}(∫_{0}^{1} ∑_{i} |a_{i}(s)|^{2} ds)^{1/2} .

Since d(x, y) = inf_{γ}(∫_{0}^{1} ∑_{i} |a_{i}(s)|^{2} ds)^{1/2} we get the claim. □

Theorem 6.4. Let G, X_{i}, d, L, and p_{t} be as above. Suppose that • G satisfies the Calderón–Zygmund condition,

• the conlusion of 6.2 holds.

Then the Riesz transforms X_{i}L^{−1/2} are of weak type (1, 1) and bounded on L^{p}, 1 < p ≤ 2.

Proof. We are going to use 1.2. We write

Γ(\frac{1}{2})X_{i}L^{−1/2} = ∫_{0}^{∞} t^{−1/2}X_{i}p_{t} dt = ∑_{n} ∫_{2^{n}}^{2^{n+1}} t^{−1/2}X_{i}p_{t} dt = ∑_{n} K_{n} .

For our next calculation we use the following natural convention. Suppose K is the operator given by the kernel K(x, y), namely (Kf)(x) = ∫ K(x, y)f(y) dy. Then for any measure μ we define

(Kμ)(x) = ∫ K(x, y) dμ(y).

Note, that ((f · g) ∗ δ_{y})(x) = (f ∗ δ_{y})(x)g(xy^{−1}). Now, putting a = 1 and c = 2^{−1/2}

∫ |K_{n}(x, y)(1 + c^{−n}d(x, y))| dx

≤ ∫_{2^{n}}^{2^{n+1}} ∫ |(t^{−1/2}(X_{i}p_{t}) ∗ δ_{y})(x)(1 + c^{−n}d(xy^{−1}))| dx
= ∫_{2^{n}}^{2^{n+1}} ∥(t^{−1/2}(X_{i}p_{t})(1 + c^{−n}d)) ∗ δ_{y}∥_{L^{1}} dt
≤ C ∫_{2^{n}}^{2^{n+1}} ∥t^{−1/2}(X_{i}p_{t}) exp(εt^{−1/2}d)∥_{L^{1}} dt
≤ C′ ∫_{2^{n}}^{2^{n+1}} t^{−1} dt ≤ C′

so the first assumption about K_{n} in 1.2 holds.

∫ |K_{n}(x, y) − K_{n}(x, z)| dx = ∥K_{n}(δ_{y} − δ_{z})∥_{L^{1}}

≤ ∫_{2^{n}}^{2^{n+1}} ∥t^{−1/2}(X_{i}p_{t})(δ_{y} − δ_{z})∥_{L^{1}} dt
≤ ∫_{2^{n}}^{2^{n+1}} t^{−1/2}∥X_{i}p_{t/2}∥_{L^{1}} ∥p_{t/2} ∗ (δ_{y} − δ_{z})∥_{L^{1}} dt

≤ C ∫_{2^{n}}^{2^{n+1}} t^{−3/2}d(y, z) dt ≤ C′2^{−n/2}d(y, z) = C′c^{n}d(y, z)

so the second assumption of 1.2 is also satisfied, ending the proof. □

Proof of 2.4. The result will follow from 1.2. Indeed, by 5.1, G has the Calderón–Zygmund property. Fix ϕ ∈ C_{c}^{∞}(ℝ_{+}) such that supp ϕ ⊂ [1/2, 2] and ∑_{n} ϕ(2^{n}x) = 1 for all x > 0. Put F_{n}(x) = F (2^{−n}x)ϕ and G_{n}(x) = F_{n}(x) exp(x). We write

F (L) = ∑ F_{n}(2^{n}L) = ∑ G_{n}(2^{n}L) exp(−2^{n}L)

Now the first assumption of 1.2 follow from 6.1 (applied directly to F_{n}) and the second from 6.1 (applied to G_{n}) and 6.3, which ends the proof. □

This proof of the spectral multiplier part of 2.4 is also applicable to any G, X_{i}, d, L, and p_{t} as above such that

• G satisfies the Calderón–Zygmund condition,

• the conclusion of 6.1 holds,

• the conclusion of 6.3 holds.

7. Final remarks

The method used to obtain the Calderón–Zygmund decomposition on the tree is inspired by [32] (p. 309, Lemma XVII.3.2) and uses the same ideas as [17]. Working on trees made ideas simpler, and made clear which maximal function is relevant for singular integrals (on

MULTIPLIERS AND SINGULAR INTEGRALS 31

Lie groups there are many natural maximal functions, some bounded, some unbounded [9],[10]). We use a somewhat different method on Lie groups, which allows us to handle the case where the roots are of both signs (for instance, unimodular groups). Both proofs are related to the construction of Fölner sequences.

For simplicity, we restricted ourselves to rank 1 case. However, our arguments works the same in a product setting.

The first author can prove that if a (locally compact) group G with left-invariant Haar measure and right-invariant distance satisfies the Calderón–Zygmund property, then G is amenable.

In [27] Sjögren proves that X_{1}L^{−1/2} is of weak type (1, 1) in the Q = 1 case, G = ℝ⋉ℝ. The result in our Theorem 2.4 is stronger than that because it deals also with X_{0}. In [31] Wängeforos extends Sjögren’s results to AN groups associated to arbitrary rank 1 symmetric spaces. In [14] Gaudry and Sjögren prove, again for G = ℝ⋉ℝ, that L^{−1/2}X_{1} is of weak type (1, 1). That operator is −(X_{1}L^{−1/2})^{∗}, so their result gives p > 2 estimates for X_{1}L^{−1/2}. Although Gaudry and Sjögren do not mention trees explicitly, Theorem 3 in [14] does have a strong tree-like flavor.

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Institute of Mathematics, Wrocław University, and Institute of Mathematics, PAN

  E-mail address: hebisch@math.uni.wroc.pl

Istituto Di Matematica e Fisica, Università degli Studi di Sassari, Via Vienna 2, 07100 Sassari, Italia

  E-mail address: steger@ssmain.uniss.it