TRANSFERENCE FOR AMENABLE ACTIONS

WALDEMAR HEBISCH AND M. GABRIELLA KUHN

Abstract. Suppose G acts amenably on a measure space X with quasiinvariant σ-finite measure m. Let σ be an isometric representation of G on L^{p}(X, dm) and μ a finite Radon measure on G. We show that the operator σ_{μ}f(x) = ∫_{G}(σ(g)f)(x)dμ(g) has L^{p}(X, dm)-operator norm not exceeding the L^{p}(G)-operator norm of the convolution operator defined by μ. We shall also prove an analogue result for the maximal function M associated to a countable family of Radon measures μ_{n}.

1. Introduction

Let σ be a uniformly bounded representation of a locally compact group G on some Banach space B and K a convolution operator on L^{p}(G) given by

Kf(x) = ∫_{G} k(y)f(y^{−1}x)dy = k ∗ f(x)

with k ∈ L^{1}(G). The transferred operator K^{♯} is defined by letting

K^{♯}(F) = ∫_{G} k(y)(σ(y)F)dy .

The basic transference theory goes back to the earliest works of R. Coifman and G. Weiss around 1970 and says that if we have an action of an amenable group G on some (σ-finite) measure space X and we let B = L^{p}(X) then

(1)  ∥K^{♯}∥_{L^{p}(X)→L^{p}(X)} ≤ ∥K∥_{L^{p}(G)→L^{p}(G)} .

This theory has many different applications (it allows to convert L^{p}(𝕋)- Fourier multipliers into L^{p}(ℝ)- Fourier multipliers, to bound Riesz transform on SU(2) or, more generally, on compact Lie groups having a “nice” maximal torus) and has been generalized to different contexts: the central role is played by an amenable group.

2 WALDEMAR HEBISCH AND M. GABRIELLA KUHN

In 1978 R. Zimmer [8] introduced the notion of amenable action for locally compact topological groups G. When X is a homogeneous G space, say X = G/H, amenability of the action is equivalent to the amenability of H.

In this paper we shall prove that the above inequality (1) is true when G acts amenably on a measure space X with quasi-invariant σ-finite measure. We shall also prove an analogue result for the maximal function M_{σ} with respect to a countable family of Radon measures μ_{n} on G.

2. Notation and Results

Let G be a second countable locally compact group acting on a Borel space X with quasi invariant σ-finite measure m. We shall assume that G is acting on the right, that is (g, x) ∈ G × X → g^{−1}x. Let r(g, x) be the Radon-Nikodym derivative of the G action:

(2)  r(g, x) = \frac{dm_{g}(x)}{dm}  where m_{g}(E) = m(g^{−1}E)

equivalently

∫_{X} |f(g^{−1}x)|r(g, x)dm(x) = ∫_{X} |f(x)|dm(x)

for every measurable function f. Recall that r can be chosen to be a strict Borel cocycle, that is r(gs, x) = r(g, x)r(s, g^{−1}x) for all g , s ∈ G and x ∈ X.

We shall assume that m is a probability measure, so that

∫_{X} r(g, x)dm(x) = 1 for all g ∈ G .

This can be done by choosing appropriately a representative in the measure class of m.

Let α be a cocycle, that is a Borel function from G × X → C such that:

α(gs, x) = α(g, x)α(s, g^{−1}x) .

In the sequel we will assume that α is unitary, that is |α(g, x)| = 1.

For any given g ∈ G define a norm preserving map σ(g) : L^{p}(X) → L^{p}(X) by letting

σ(g)f(x) = f(g^{−1}x)(r(g, x))^{\frac{1}{p}} α(g, x) .

TRANSFERENCE FOR AMENABLE ACTIONS 3

Denote by L^{p}(G × X) the usual L^{p} space with respect to the product measure dg × dm where dg is the left invariant Haar measure on G.

Denote by λ the left regular representation of G acting on L^{p}(G). Define a representation λ ⊗ σ on L^{p}(G × X) by the rule

(λ ⊗ σ)(g)F (h, x) = F (g^{−1}h, g^{−1}x)(r(g, x))^{\frac{1}{p}} α(g, x) .

Choose any finite Radon measure μ on G. Define operators (λ ⊗ σ)_{μ}, λ_{μ} and σ_{μ} acting respectively on L^{p}(G × X), L^{p}(G) and L^{p}(X) according to the following rules:

(λ⊗σ)_{μ}F (h, x) = ∫_{G} λ⊗σ(g)F (h, x)dμ(g) ∫_{G} F (g^{−1}h, g^{−1}x)(r(g, x))^{\frac{1}{p}} α(g, x)dμ(g)

λ_{μ}f(h) = ∫_{G} λ(g)f(h)dμ(g) = ∫_{G} f(g^{−1}h)dμ(g)

σ_{μ}f(x) = ∫_{G} σ(g)f(x)dμ(g) = ∫_{G} f(g^{−1}x)(r(g, x))^{\frac{1}{p}} α(g, x)dμ(g) .

We shall denote by ∥ · ∥_{p} the L^{p} norm in any of the above defined L^{p} spaces and by by ∥ · ∥_{p,p} the norm of any of the above defined operators.

When G is amenable it is known (see e.g. [2][3]) that ∥σ_{μ}∥_{p,p} ≤ ∥λ_{μ}∥_{p,p}. This phenomenon is called transference.

When p = 2 and G acts amenably on X the same inequalities are true: see [1] Theorem 3.2.3 (for discrete groups see also [5] for ergodic actions and [6] for general amenable actions including semisimple algebraic groups).

In this paper we shall prove transference for all p (1 ≤ p < ∞).

We refer to Zimmer’s book [9] for the general definition of amenable action.

For the purposes of this paper we need the following property of amenable Gspaces:

Lemma 2.1. [C. Anantharaman-Delaroche][1] Theorem 3.1.5

Assume that G is acting amenably on a standard Borel space with quasi-invariant measure m.

Then there exists a sequence ϕ_{i} of nonnegative functions ϕ_{i}(g, x) in L^{1}(G × X) such that for all f ∈ L^{1}(X) and any compact subset K ⊂ G we have a) ∫_{G} ϕ_{i}(g, x)dg = 1 for almost all x ∈ X

b) lim_{i}(sup_{k∈K} ∫_{G} ∫_{X} |ϕ_{i}(k^{−1}g, k^{−1}x) − ϕ_{i}(g, x)| |f(x)|dm(x) dg) = 0

Remark 2.2. When G is discrete and ergodic on X the existence of such a sequence was essentially the proof (given in [5]) that the quasi-regular representation of G on X is weakly contained into the regular representation of G. In [1] it is proved that the above condition is in fact equivalent to the amenabilty of the G action on X for any locally compact second countable group.

Fix once and for all a sequence ϕ_{i} of nonnegative functions satisfying (a) and (b) of Lemma 2.1.

For every f ∈ L^{p}(X) and i ∈ ℕ define a function f̃_{i} ∈ L^{p}(G × X) according to the following rule:

f̃_{i}(g, x) = (ϕ_{i}(g, x))^{\frac{1}{p}} f(x)

so that

σ̃_{μ}f_{i}(g, x) = (ϕ_{i}(g, x))^{\frac{1}{p}} (σ_{μ}f)(x) .

Lemma 2.3. Let μ be any finite Radon measure and f ∈ L^{p}(X). Let f̃_{i} and σ̃_{μ}f_{i} be as above.

Then we have

lim_{i} ∥(λ ⊗ σ)_{μ}f̃_{i} − σ̃_{μ}f_{i}∥_{p} = 0.

Proof. First, we will assume that μ has compact support F. Choose ε > 0 and fix f. By Lemma 2.1 (with K = F^{−1}) we can find N such that for all i ≥ N

∫_{G} ϕ_{i}(g, x)dg = 1

sup_{k∈F}(∫_{G} ∫_{X} |ϕ_{i}(kg, kx) − ϕ_{i}(g, x)||f(x)|^{p} dm(x) dg) < ε.

We need to compute the norm in L^{p}(G × X) of the quantity

((λ ⊗ σ)_{μ}f̃_{i} − σ̃_{μ}f_{i})(g, x) =

∫_{F} \{(ϕ_{i}(k^{−1}g, k^{−1}x))^{\frac{1}{p}} − (ϕ_{i}(g, x))^{\frac{1}{p}} \}\{r(k, x)\}^{\frac{1}{p}} α(k, x) f(k^{−1}x) dμ(k) .

Denote by |μ| the measure total variation of μ. Using Minkowski’s integral inequality we get

\{∫_{G×X} |(λ ⊗ σ)_{μ}f̃_{i} − σ̃_{μ}f_{i}|^{p}(g, x) d(g) dm(x)\}^{\frac{1}{p}} ≤

∫_{F} d|μ|(k)

\{∫_{G×X} |(ϕ_{i}(k^{−1}g, k^{−1}x))^{\frac{1}{p}} − (ϕ_{i}(g, x))^{\frac{1}{p}} |^{p}r(k, x)|f(k^{−1}x)|^{p} d(g) dm(x)\}^{\frac{1}{p}}

= ∫_{F} \{∫_{G×X} |ϕ_{i}(kg, kx)^{\frac{1}{p}} − ϕ_{i}(g, x)^{\frac{1}{p}} |^{p}|f(x)|^{p} d(g)dm(x)\}^{\frac{1}{p}} d|μ|(k)
(3)  ≤ ∫_{F} \{∫_{G×X} |ϕ_{i}(kg, kx) − ϕ_{i}(g, x)||f(x)|^{p} d(g)dm(x)\}^{\frac{1}{p}} d|μ|(k)
≤ ε^{\frac{1}{p}} ∥μ∥

where for the last inequality (3) we used |a^{\frac{1}{p}} − b^{\frac{1}{p}} |^{p} ≤ |a − b| (valid for positive a and b ) and ∥μ∥ stays for |μ|(G).

So, we showed that for fixed compactly supported μ and given f we can choose N such that for all i ≥ N

∥(λ ⊗ σ)_{μ}f̃_{i} − σ̃_{μ}f_{i}∥_{p} ≤ ε^{\frac{1}{p}} ∥μ∥

which gives our claim for compactly supported μ.

Now, let K_{n} be an increasing sequence of compact sets such that G = ∪_{n}K_{n}. Let μ_{n} denote the restriction of μ to K_{n}.

∥μ − μ_{n}∥ = ∫_{K_{n}^{c}} d|μ| → 0

becouse |μ| is finite and countably additive. For fixed n

lim sup_{i} ∥(λ ⊗ σ)_{μ}f̃_{i} − σ̃_{μ}f_{i}∥_{p} ≤

lim sup_{i} (∥(λ ⊗ σ)_{μ_{n}} f̃_{i} − { \atop σ_{μ_{n}}f}_{i}∥_{p} + ∥f∥_{p}∥μ − μ_{n}∥)
= ∥f∥_{p}∥μ − μ_{n}∥.

Since ∥μ − μ_{n}∥ is arbitrarly small when n is large we get the claim. □

We are now able to prove our result:

Theorem 2.4. Let G be a second countable locally compact group acting amenably on a standard Borel space with quasi-invariant σ-finite measure m. Fix a finite Radon measure μ on G. Define, as before,

σ_{μ}f(x) = ∫_{X} f(g^{−1}x)(r(g, x))^{\frac{1}{p}} α(g, x)dμ(g) .

Then the operator norm of σ_{μ} acting on L^{p}(X) is bounded by the operator norm of λ_{μ} acting on L^{p}(G). In other words

(4)  ∥σ_{μ}∥_{p,p} ≤ ∥λ_{μ}∥_{p,p} .

Proof. Define

(UF)(g, x) = F (g, gx)(r(g^{−1}, x))^{\frac{1}{p}} α(g^{−1}, x) = σ(g^{−1})F (g, x)

for every F ∈ L^{p}(G × X). Observe that U intertwines λ ⊗ σ to λ ⊗ i where

(λ ⊗ i)(g)F (h, x) = F (g^{−1}h, x) .

Observe that U^{−1} is also well defined:

U^{−1}F (g, x) = σ(g)F (g, x)

and ∥UF ∥_{p} = ∥F ∥_{p} = ∥U^{−1}F ∥_{p}. So that ∥(λ ⊗ σ)_{μ}∥_{p,p} = ∥(λ ⊗ i)_{μ}∥_{p,p}. Since obviously ∥(λ ⊗ i)_{μ}∥_{p,p} ≤ ∥λ_{μ}∥_{p,p}, it is enough to compare the norm of σ_{μ} with the norm of (λ ⊗ σ)_{μ}.

Fix f ∈ L^{p}(X). Let f̃_{i} and σ̃_{μ}f_{i} be as in Lemma 2.3. Observe that

∥f∥_{p} = ∥f̃_{i}∥_{p} and ∥σ_{μ}f∥_{p} = ∥σ̃_{μ}f_{i}∥_{p} .

Since

∥σ̃_{μ}f_{i}∥_{p} ≤ ∥(λ ⊗ σ)_{μ}f̃_{i}∥_{p} + ∥(λ ⊗ σ)_{μ}f̃_{i} − σ̃_{μ}f_{i}∥_{p} ≤
∥(λ ⊗ σ)_{μ}∥_{p,p} ∥f̃_{i}∥_{p} + ∥(λ ⊗ σ)_{μ}f̃_{i} − σ̃_{μ}f_{i}∥_{p} =
∥(λ ⊗ σ)_{μ}∥_{p,p}∥f∥_{p} + ∥(λ ⊗ σ)_{μ}f̃_{i} − σ̃_{μ}f_{i}∥_{p}

we have also

∥σ_{μ}f∥_{p} ≤ ∥(λ ⊗ σ)_{μ}∥_{p,p}∥f∥_{p} + ∥(λ ⊗ σ)_{μ}f̃_{i} − σ̃_{μ}f_{i}∥_{p} .

Taking the limit as i → ∞ we get

∥σ_{μ}f∥_{p} ≤ ∥(λ ⊗ σ)_{μ}∥_{p,p} ∥f∥_{p}

which gives

∥σ_{μ}f∥_{p} ≤ ∥(λ ⊗ σ)_{μ}∥_{p,p} ∥f∥_{p} = ∥(λ ⊗ i)_{μ}∥_{p,p} ∥f∥_{p} ≤ ∥λ_{μ}∥_{p,p} ∥f∥_{p}

.

□

Remark 2.5. As an application take G = SL(2, R) and X = P^{1}, the projec-

tive line. Since X ≃ SL(2, R)/P where P = ( \begin{matrix}a & 0 \\ b & a^{−1} \end{matrix} ), X is an amenable

G space. The G action is given by fractional linear transformations g^{−1}x =

\frac{ax+b}{cx+d} for g^{−1} = ( \begin{matrix}a & b \\ c & d \end{matrix} ) and the Radon–Nikodym derivative r(g, x) is

given by r(g, x) = \frac{1}{(cx+d)^{2}} . For every f ∈ L^{p}(X) define

R_{t}f(x) = \frac{f(g^{−1}x)}{(cx + d)^{\frac{2}{p} +it}}

or, more generally

R_{μ}f(x) = ∫_{G} \frac{f(g^{−1}x)}{(cx + d)^{\frac{2}{p} +it}} dμ(g) .

Then R_{t} is an isometry for every real t and 1 ≤ p < ∞; while ∥R_{μ}∥ ≤ ∥λ_{μ}∥.

More generally we can apply our result to boundaries of symmetric spaces: X = G /MAN (so we can transfer to the corresponding principial series representations).

We can also handle symmetric spaces: X = G/K (however, since K is compact a more direct argument works better).

Remark 2.6. Assume that μ (not necessarly given by a Radon measure) is a bounded convolver of L^{p}(G) for some p (1 ≤ p < ∞). Suppose also that G is amenable: another phenomenon related to transference says that such a μ is also a bounded

8 WALDEMAR HEBISCH AND M. GABRIELLA KUHN

convolver of L^{2}(G) (see [4]). We want to remark that such phenomenon does not hold in the case of amenable actions: take the free group F_{r} on r generators (r ≥ 2) acting on itself, this action is amenable since the subgroup {e} is compact and X = F_{r}/e. It is well known that there exists a bounded convolver of ℓ^{p}(F_{r}) which is not a convolver of ℓ^{q}(F_{r}) for any q ≠ p: see [7]. Moreover, in the same paper it is shown that for every positive M there exists a function f with finite support such that ∥λ_{f}∥_{2,2} > M∥λ_{f}∥_{p,p} (λ denotes, as usual, the left regular representation of F_{r}).

Theorem 2.7. Let G and X as in Theorem 2.4. Fix any sequence {μ_{n}} of finite Radon measures on G. Define

M_{σ}f(x) = sup_{n} |σ_{μ_{n}}f|(x) = sup_{n} | ∫_{X} f(g^{−1}x)(r(g, x))^{\frac{1}{p}} α(g, x)dμ_{n}(g)| .

Analogously define

M_{λ}f(h) = sup_{n} |λ_{μ_{n}}f|(h) = sup_{n} | ∫_{G} f(g^{−1}h)dμ_{n}(g)| .

Then the operator norm of M_{σ} acting on L^{p}(X) is bounded by the operator norm of the corrsponding M_{λ} acting on L^{p}(G).

Sketch of the proof.

Notation as in Theorem 2.4. It is enough to show that, for every positive integer L

∥M_{{σ,L}}f∥_{p} ≤ ∥M_{{λ,L}}∥_{p,p}∥f∥_{p}

where

M_{{σ,L}}f(x) = sup_{1≤n≤L} |σ_{μ_{n}}f|(x)

and M_{{λ,L}} is defined analogously. Let

{ \atop M_{\{σ,L\}}f}_{i}(g, x) = (ϕ_{i}(g, x))^{\frac{1}{p}} M_{\{σ,L\}}f(x) =

(ϕ_{i}(g, x))^{\frac{1}{p}} sup_{1≤n≤L} | ∫_{X} f(g^{−1}x)(r(g, x))^{\frac{1}{p}} α(g, x)dμ_{n}(g)| =

sup_{1≤n≤L} |(ϕ_{i}(g, x))^{\frac{1}{p}} ∫_{X} f(g^{−1}x)(r(g, x))^{\frac{1}{p}} α(g, x)dμ_{n}(g)| =

sup_{1≤n≤L} |{ \atop σ_{μ_{n}}f}_{i}|(x) .

TRANSFERENCE FOR AMENABLE ACTIONS 9

Again we have

∥M_{\{σ,L\}}f∥_{p} = ∥\begin{matrix} \\ M_{\{σ,L\}}f & i \end{matrix}∥_{p}

Remember the defintion of U^{−1} : L^{p}(G × X) → L^{p}(G × X):

(U^{−1})F (g, x) = F (g, g^{−1}x)(r(g, x))^{\frac{1}{p}} α(g, x) = σ(g)F (g, x) .

Put

V F (g, x) = F (g, gx)(r(g^{−1}, x))^{\frac{1}{p}}

so that

(V^{−1})F (g, x) = F (g, g^{−1}x)(r(g, x))^{\frac{1}{p}}

and V is an isometry.

Observe that

V^{−1}(sup_{n} |F_{n}|) = sup_{n}(|U^{−1}F_{n}|) .

For every f ∈ L^{p}(G × X) write f = U^{−1}F and evaluate

M_{{λ⊗σ,L}}f(g, x) = sup_{1≤L≤n} |(λ ⊗ σ)_{μ_{n}}f|(g, x) = sup_{1≤L≤n} |(λ ⊗ σ)_{μ_{n}}U^{−1}F |(g, x) =

sup_{1≤L≤n} |U^{−1}(λ ⊗ i)_{μ_{n}}F |(g, x) = V^{−1} sup_{1≤L≤n} |(λ ⊗ i)_{μ_{n}}F |(g, x) =

V^{−1}(M_{{λ⊗i,L}})F (g, x) .

Hence

(5)  ∥M_{{λ⊗σ,L}}∥_{p,p} = ∥M_{{λ⊗i,L}}∥_{p,p} ≤ ∥M_{{λ,L}}∥_{p,p} .

Compute

(6) ∥M_{\{σ,L\}}f∥_{p} = ∥{ \atop M_{\{σ,L\}}f}_{i}∥_{p} ≤

∥{ \atop M_{\{σ,L\}}f}_{i} − M_{\{λ⊗σ,L\}}f̃_{i}∥_{p} + ∥M_{\{λ⊗σ,L\}}∥_{p,p}∥f∥_{p} .

Since

(7)| sup_{1≤n≤L} |{ \atop σ_{μ_{n}}f}_{i}|(g, x) − sup_{1≤n≤L} |(λ ⊗ σ)_{μ_{n}} f̃_{i}|(g, x)| ≤ ∑_{n=1}^{L} |{ \atop σ_{μ_{n}}f}_{i}−(λ⊗σ)_{μ_{n}} f̃_{i}|(g, x)

one has

∥{ \atop M_{\{σ,L\}}f}_{i} − M_{\{λ⊗σ,L\}}f̃_{i}∥_{p} ≤ ∑_{n=1}^{L} ∥{ \atop σ_{μ_{n}}f}_{i} − (λ ⊗ σ)_{μ_{n}} f̃_{i}∥_{p} → 0.

Conclude the proof as in Theorem 2.4. □

References

[1] C. Anantharaman-Delaroche; On spectral characterization of amenability, preprint available on the web site www.univ-orleans.fr

[2] R. Coifmann, G. Weiss ; Operators associated with representations of amenable groups singular integrals induced by ergodic flows, the rotation methods and multipliers. Studia Math. 47, (1973), 285–303.

[3] R. Coifmann, G. Weiss ; Transference methods in analysis. C.B.M.S. Regional Conference Series in Math. 31, AMS Providence 1–59 (1977) reprinted in 1986.

[4] C, Herz ; The theory of p-spaces with an application to convolution operators. Trans. Amer. Math. Soc. 154, (1971), 69–82

[5] M.G.Kuhn ; Amenable actions and weak containment of certain representations of discrete groups. Proc. Amer. Math. Soc. 122, (1994) 751–757.

[6] A. Nevo ; The spectral theory of amenable actions and invariants of discrete groups. to appear in Geometria Dedicata

[7] T. Pytlik ; A construction of convolution operators on free group. Studia Math. 74, (1984), 73–76

[8] R. Zimmer ; Amenable ergodic group actions and an application to Poisson boundaries of random walks. J. Func. Anal. 27 (1978), 350–372

[9] R. Zimmer ; Ergodic theory and semisimple groups. Birkhäuser Boston 1984

Mathematical Institute, Wrocław Uniwersity, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland

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

Dipartimento di Matematica, Università di Milano “Bicocca”, Viale Sarca 202, 20126 Milano, ITALIA

  E-mail address: kuhn@matapp.unimib.it