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THE UNCERTAINTY PRINCIPLE FOR GELFAND PAIRS

Joseph A. Wolf

Department of University of California Berkeley, California 94720 e-mail: [email protected]

Department of Mathematics, University of California, Berkeley California 94720; [email protected]

Abstract. We extend the classical Uncertainty Principle to the context of Gelfand pairs. The setting includes riemannian symmetric spaces, compact topo- logical groups, and locally compact abelian groups. If the locally compact abelian is Rn we recover a sharp form of the classical Heisenberg uncertainty principle.

Section 1. Introduction.

The classical uncertainty principle says that a function and its Fourier transform cannot both be mostly concentrated on short intervals: if f(t) has has most of its in an interval of length ℓ, and its Fourier tranform fˆ(τ) has has most of its support in an interval of length ℓˆ then ℓ ℓˆ ≧ 1 − η where η is specified by the precise meaning of “most of its support”. In quantum mechanics this says that the position and momentum of a particle cannot be determined simultaneously [9]. In signal processing it says that instantaneous frequency cannot be measured precisely [8]. And of course the Heisenberg and its commutation relations are the basic building block in on nilpotent Lie groups [10], and thus in the associated theories of Cauchy–Riemann spaces, Heisenberg manifolds, and hypoellilptic operators ([6], [1], [7]).

Research partially supported by N.S.F. Grant DMS 91 00578.

Typeset by AMS-TEX JOSEPH A. WOLF

A few years ago, D. L. Donoho and P. B. Stark proved [4] a sharp classical extension of the uncertainty principle: |T ||W | ≧ 1 − η whenever f(t) has most of its support in a set (not necessarily an interval) T of |T | and fˆ(τ) has most of its support in a set (not necessarily an interval) W of measure |W |.

Let T be a measurable set, 1T its indicator (= characteristic) function, and |||| = ||||p, an Lp norm. We say that

f ∈ Lp is ǫ − concentrated on T if ||f − 1T f|| ≦ ǫ||f||. (1.1)

The precise form of Donoho and Stark’s stronger L2 version mentioned above is [4]

Let f, fˆ ∈ L2(R), let ǫ,δ ≧ 0, and let T,W ⊂ R be measurable sets. Suppose that f is ǫ-concentrated on T and fˆ is δ-concentrated on W. (1.2) Then Lebesgue measures satisfy |T ||W | ≧ (1 − ǫ − δ)2.

The key point in the Donoho–Stark L2 argument is the operator norm inequality

2 ∨ ||QP || ≦ |T ||W | where Pf = 1T f and Qf = (1W fˆ) (1.3) where denotes Fourier transform as usual and ∨ is its inverse. K. T. Smith [11] extended the uncertainty principle (1.2) to locally compact abelianb groups by extending the inequality (1.3). In this paper we modify Smith’s arguments so that they apply more generally to Gelfand pairs, thus to riemannian symmetric spaces and compact topological groups as well as locally compact abelian groups. In the notation to be specified just below, we have two apparently different gen- eralizations to Smith’s extension of the uncertainty principle. Fix a Gelfand pair (G, K). Theorem 4.3 extends the uncertainty principle to functions on K\G/K. It depends on the spherical transform for the Gelfand pair (G, K) and the result- ing decomposition of L2(K\G/K) by positive definite zonal spherical functions. Theorem 6.7 extends the uncertainty principle to functions on G/K; it depends on the vector valued transform corresponding to a decomposition of L2(G/K). These two extensions are in fact equivalent because the underlying L2 decompositions use the same . In Section 2 we summarize function theory on K\G/K for a Gelfand pair (G, K). We start with the basic definitions, describe the spherical transform, and discuss the consequences for L2(K\G/K). Then in Section 3 we prove analogs of (1.3) for K\G/K. The corresponding uncertainty principle is in Section 4. UNCERTAINTY PRINCIPLE FOR GELFAND PAIRS

In Section 5 we describe the direct integral decomposition of L2(G/K) and the corresponding vector valued transform on G/K. Then in Section 6 we verify the analogs of the operator norm inequalities of §3 and derive the uncertainty principle for G/K. The reader will notice that we spend more space reviewing Gelfand pairs than we spend proving our uncertainty principles. That is the nature of the situation: we are showing that certain classical considerations are valid without essential change in a much broader context. Of course the reader familiar with Gelfand pairs need only glance at Sections 2 and 5 to catch the notation.

Section 2. Commutative Function Theory of Gelfand Pairs.

The theory of Gelfand pairs includes many of the general aspects of hamonic analysis on locally compact abelian groups and on symmetric spaces. There are some excellent treatments available, for example [2] and [5]. But we only need the basic structural facts. In this Section we summarize those aspects of Gelfand pair theory that correspond most closely to the Pontrjagin–Plancherel theory for locally compact abelian groups and to the analysis of central functions on compact groups. For brevity, we defer to [2] and [5] for references to original sources.

A. Some Basic Definitions.

Let G be a locally compact and K a compact . Let mG be on G, subject to the usual convention: counting measure if G is infinite and discrete, total mass 1 if G is compact. Let mK be Haar measure of total mass 1 on K. on G is the action of functions under the left :

−1 f1 ∗ f2(y)= f1(x)f2(x y)dmG(x). (2.1) ZG As in the case G = R, Young’s Inequality

||f ∗ h||p ≦ ||f||1||h||p for f ∈ L1(G) and h ∈ Lp(G) (2.2) is immediate.

A function f on G is called K–bi– if f(k1xk2)= f(x) for all x ∈ G and ki ∈ K. We say that a function f : G → C vanishes at ∞ if, given ǫ> 0, there is a compact subset Cǫ ⊂ G such that |f(x)| <ǫ for x∈ / Cǫ . Each of the spaces

C0(G) : continuous functions G → C with compact support C∞(G) : continuous functions G → C vanishing at ∞ Lp(G) : standard Lebesgue space of measurable functions G → C, 1 ≦ p ≦ ∞ JOSEPH A. WOLF projects onto its subspace

C0(K\G/K), C∞(K\G/K), Lp(K\G/K), 1 ≦ p ≦ ∞

♮ ♮ of K–bi–invariant functions, by f → f where f (x)= K K f(k1xk2)dmK (k1)dmK (k2).

C0(G) is an associative algebra under convolution.R NoteR that C0(K\G/K) is a subalgebra. Similarly, this time using Young’s Inequality, L1(K\G/K) is a subal- gebra of the convolution algebra L1(G). If x ∈ G then G carries a unique K–bi– invariant probability measure mx, defined by

f(y)dmx(y)= f(k1xk2)dmK (k1)dmK (k2) (2.3) ZG ZK ZK with support KxK.

2.4. Lemma. The following conditions are equivalent:

(1) The convolution algebra C0(K\G/K) is commutative. (2) The convolution algebra L1(K\G/K) is commutative. (3) If x,y ∈ G then KxK KyK = KyK KxK. (4) If x,y ∈ G then mx ∗ my = my ∗ mx. If those conditions hold, then G is unimodular.

2.5. Definition. (G, K) is commutative, i.e., is a Gelfand pair if the condi- tions of Lemma 2.4 obtain.

If G is a locally compact abelian group, and we set K = {1}, then of course (G, K) is commutative. But there are other interesting examples, riemannian symmetric spaces and compact groups, as follows. The name “Gelfand pair” comes from I. Gelfand’s result

2.6. Theorem. Let G be a locally and θ an involutive of G such that G = SK where (1) K is a compact subgroup of G, (2) if k ∈ K then θ(k)= k, and (3) if s ∈ S then θ(s)= s−1. Then G is unimodular and (G, K) is commutative.

2.7. Corollary. Let M be a connected riemannian , let G be any group of isometries of M that contains the identity component of the group of all isometries, and let K be the isotropy subgroup of G at some point of M. Then (G, K) is commutative. UNCERTAINTY PRINCIPLE FOR GELFAND PAIRS

2.8. Corollary. Let M be a compact topological group. Let G = M × M, so G acts on M by (x,y) : m → xmy−1. Let K be the stablizer of the identity element of M, so K = {(x,x−1) | x ∈ M} = ∆M, the diagonal M in G. Then (G, K) is commutative.

The special case of Corollary 2.7 is due to E.´ Cartan. In modern language, Cartan proved that the commuting algebra for the left regular representation of G on L2(G/K) is commutative, i.e. that this left regular representation is multiplicity– free. Corollary 2.7 includes the case. B. The Spherical Transform.

In the special case of a locally compact abelian group G, the spherical transform is the map f → fˆ,

−1 fˆ(ω)= f(x)ω(x )dmG(x), ω ∈ G (2.9) ZG

′ 1 1 b from Lp(G) to Lp (G) with the usual p + p′ = 1. This generalizes the classical Fourier transform. Here G is the Pontrjagin dual of G. By definition, G consists of all unitary charactersb on G, i.e., all continuous homomorphisms ω : G → {z ∈ C | |z| = 1} with group compositionb (ω1ω2)(x)= ω1(x)ω2(x). In a certain topology,b the case K = {1} of the one described in (2.18) – (2.22) below, G is a locally compact abelian group. b Zonal spherical functions are the analog of continuous homomorphisms

G → {z ∈ C | z = 0}, and positive zonal spherical functions are the analog of unitary characters. Now fix a Gelfand pair (G, K).

2.10. Definition. A nonzero Radon measure on G is spherical if −1 (1) it is K–bi–invariant: (k1Ek2 )= (E) for measurable E ⊂ G, and (2) : C0(K\G/K) → C is an algebra homomorphism, i.e., (f∗h)= (f)(h),

where (f) means G f(x)d(x). R 2.11. Definition. A continuous function ω : G → C is a zonal spherical func- −1 tion or zsf if d(x)= ω(x )dmG(x) defines a spherical measure.

All continuous homomorphisms ω : G → {z ∈ C | z = 0} (often called quasi- characters) are zsf. Every spherical measure is absolutely continuous with respect JOSEPH A. WOLF

−1 to Haar measure mG, so it has form d(x) = ω(x )dmG(x) where ω is K–bi– invariant. Here we may choose ω continuous, hence zsf, and check ω(1) = 1. So the notions of spherical measure and zsf are essentially equivalent. They correspond to quasi–characters on locally compact abelian groups. The following shows that they enjoy properties close to those of quasi–characters.

2.12. Proposition. These are equivalent for a function ω : G → C: (1) ω is a on G. (2) ω (i) is continuous, (ii) is K–bi–invariant, (iii) satisfies ω(1) = 1, and (iv) if f ∈ C0(K\G/K) there is a constant λf ∈ C such that f ∗ ω = λf ω.

(3) ω is not identically zero, and if x,y ∈ G then ω(x)ω(y)= K ω(xky)dmK (k). R The Fourier transform, whether on R or an arbitrary locally compact abelian group, only uses unitary characters. So we recall

C −1 2.13. Definition. A function φ : G → is positive definite if cicjφ(xi xj) ≧ 0 whenever n ≧ 1, {c1,...,cn}⊂ C, and {x1,...,xn}⊂ G. P Note that a positive definite function φ satisfies φ(1) ≧ |φ(x)| and φ(x−1)= φ(x) for all x ∈ G.

2.14. Theorem. Let φ be a continuous positive definite function on G with φ(1) = 1. Then there is a π of G, and a cyclic unit vector u in the representation space Hπ, such that φ(x)= u, π(x)u for all x ∈ G. The pair (π,u) is unique up to unitary equivalence.

The connection with Gelfand pairs is

2.15. Theorem. Let φ be a positive definite zonal spherical function and (π,u) the corresponding cyclic unitary representation, φ(x) = u, π(x)u. Then π is ir- K reducible and u spans the space Hπ of π(K)–fixed vectors. Conversely, if π is an K irreducible unitary representation of G and Hπ is spanned by a unit vector u then φ(x)= u, π(x)u is a positive definite zonal spherical function.

Write S = S(G, K) for the set of all zonal spherical functions f : G → C, and write P = P (G, K) for the set of all positive definite zonal spherical functions.

2.16. Definition. The spherical transform is the map f → fˆ, from K–bi– invariant functions on G to functions on S = S(G, K), given by

−1 fˆ(ω)= ω(f)= f(x)ω(x )dmG(x). (2.17) ZG UNCERTAINTY PRINCIPLE FOR GELFAND PAIRS

If f ∈ L1(K\G/K), in particular if f ∈ C0(K\G/K), then the integral is abso- lutely convergent. Map

S → Cf by ω → (fˆ(ω)) as f ranges over C0(K\G/K). (2.18)

This is injective, andY we now view S as a subspace of the topological product space Cf . The subspace topology on S is the weak topology for the functions fˆ with f ∈ C0(K\G/K). Since positive definite functions are bounded by their value at theQ identity element 1 ∈ G,

P = P (G, K) ⊂ Df where Df = {zf ∈ Cf ||zf | ≦ ||f||1}. (2.19) Y If ω ∈ S, a calculation shows that

∗ ω ∈ P if and only if ω(f ∗ f ) ≧ 0 for all f ∈ C0(K\G/K). (2.20)

Here f ∗(x) = f(x−1) because G is unimodular. Combining (2.19) and (2.20), one proves that

P has closure cl(P ) consisting of all z ∈ Df such that a) f → z is a linearY functional on C (K\G/K), f 0 (2.21) b) zf1∗f2 = zf1 zf2 for all f1,f2 ∈ C0(K\G/K), and

c) zf∗f ∗ ≧ 0 for all f ∈ C0(K\G/K).

From this

2.22. Proposition. P is locally compact. Either P is compact and equal to its closure in Cf , or clP = P ∪ 0 is the 1-point compactification of P . Q Proposition 2.22 gives us an analog of the Riemann-Lebesgue Lemma:

2.23. Corollary. If f ∈ L1(K\G/K) then fˆ ∈ C∞(P ), i.e., f is a continuous function on P vanishing at ∞.

C. The Godement– for K\G/K.

Godement’s extension of the classical Pontrjagin–Plancherel Theorem from lo- cally compact abelian groups to Gelfand pairs is JOSEPH A. WOLF

2.24. Theorem. Let (G, K) be commutative. Then there is a unique positive Radon measure ν on P , concentrated on a certain subset M, such that ˆ ˆ if f ∈ C0(K\G/K) then f ∈ L2(P,ν) and ||f||L2(P,ν) = ||f||L2(G) . (2.25)

Moreover f → fˆ extends by continuity to a isomorphism of L2(K\G/K) onto L2(P,ν) which intertwines the (left) convolution representation

ℓ : C0(K\G/K) on L2(K\G/K) by ℓ(f)h = f ∗ h (2.26a) with the (left) multiplication representation

ℓˆ: C0(K\G/K) on L2(P,ν) by ℓˆ(f)q = fqˆ . (2.26b)

ν is called Plancherel measure for (G, K). The uniform closure of ℓ(C0(K\G/K)) ∗ in the algebra of bounded linear operators on L2(K\G/K) is a commutative C algebra, and M is its maximal ideal space. If f ∈ C0(K\G/K) then ℓ(f) has dual ℓ(f) , function on M given by ℓ(f) (m) = ℓ(f) modulo m. If m ∈ M there is a unique zonal spherical function ωm such that b b −1 ℓ(f) (m)= f(x)ωm(x )dmG(x) . (2.27) ZG

Furthermore ωm is positiveb definite, so M is identified with a subset of P , and .{M ֒→ P extends to a homeomorphism of M ∪ {0} onto a closed subset of P ∪ {0 Also,

if f ∈ C0(K\G/K) then ℓ(f) (m)= fˆ(ωm), i.e., ℓ(f) = fˆ|M . (2.28)

With these identifications, the construction of Plancherel measure proceeds along b b the same lines as the standard construction of Haar measure. Along the way one gets the analog of Bochner’s Theorem

ˆ 2.29. Corollary. f(x)= M ωm(x)f(m)dν(m). and the analog of the FourierR inversion formula

ˆ ˆ 2.30. Corollary. f ∈ L1(P,ν) and f(x)= P f(ω)ω(x)dν(ω). for f in the dense subspace R

∗ finite linear combinations aijfi ∗ fj where the fk ∈ C0(K\G/K) X UNCERTAINTY PRINCIPLE FOR GELFAND PAIRS of L2(K\G/K). For comparison it is worth recalling the Pontrjagin–Plancherel Theorem for a locally compact abelian group G. There is a unique normalization of Haar measure b ˆ 1 mG on G such that f → f defines an isometry of L2(G) onto L2(G). We always b use that normalization for mG. The inverse Fourier transform is given by b b φˇ(x)= φ(ω)ω(x)dm b(ω). (2.31) b G ZG This is essentially the same as the Fourier transform on G. So here the Titchmarch Inequality for G (compare [3, §51]), b ||fˆ||p′ ≦ ||f||p whenever 1 ≦ p ≦ 2 and f ∈ Lp(G) (2.32) and the corresponding inequality for G, combine for p = 2 to give the isometry of the Plancherel Theorem. b Section 3. Operator Norm Inequalities.

In this section we prove certain operator norm estimates on Gelfand pairs. Just as K. T. Smith used an extension [11] to locally compact abelian groups of the Donoho–Stark estimate (1.3) to prove his uncertainty principle, the estimates we prove will be used in §4 to extend that uncertainty principle to spaces K\G/K where (G, K) is a Gelfand pair. Fix a Gelfand pair (G, K). To adapt (1.3) we use the spherical transform (2.17) and its inverse (2.30),

−1 ∨ fˆ(ω)= f(x)ω(x )dmG(x) and h (x)= h(ω)ω(x)dν(ω). (3.1) ZG ZP As in the classsical case, straightforward computation gives

∨ ||fˆ||∞ ≦ ||f||1 for f ∈ L1(K\G/K) and ||h ||∞ ≦ ||h||1 for h ∈ L1(P,ν), so Riesz-Thorin interpolation gives us the analog for K\G/K of the Titchmarch Inequality

||fˆ||p′ ≦ ||f||p for f ∈ Lp(K\G/K), 1 ≦ p ≦ 2, ∨ (3.2) ||h ||p′ ≦ ||h||p for h ∈ Lp(P,ν), 1 ≦ p ≦ 2,

1 In fact map f → fˆ is initially defined only for f ∈ L1(G) ∩ L2(G) and then is extended by continuity as in Theorem 2.24. JOSEPH A. WOLF

1 1 with the usual p + p′ = 1. Compare [3, §51].

Fix subsets T = KTK ⊂ G and U ⊂ P of finite measure. Let 1T and 1U denote their respective indicator functions. Define operators P = PT and Q = QU by

∨ Pf = 1T f and Qf = (1U fˆ) . (3.3)

3.4. Proposition. If 1 ≦ p ≦ 2, q ≧ 1 and f ∈ Lp(K\G/K) then

1/q 1/p ||PQf||q ≦ mG(T ) ν(U) ||f||p .

1/p 1/p Case p = q: the operator norm on Lp(K\G/K) satisfies ||PQ||p ≦ mG(T ) ν(U) .

Proof. We compute

PQf(x) = 1T (x)Qf(x) ∨ = 1T (x)(1U fˆ) (x)

= 1T (x) (1U fˆ)(ω)ω(x)dν(ω) ZP −1 = 1T (x) 1U (ω) f(y)ω(y )dmG(y) ω(x)dν(ω) ZP ZG  −1 = 1T (x) f(y) 1U (ω)ω(y )ω(x)dν(ω) dmG(y). ZG ZP  −1 −1 As ω(y )ω(x)= K ω(y kx)dmK (k) and f is K–bi–invariant, now R −1 PQf(x) = 1T (x) f(y) 1U (ω)ω(y x)dν(ω) dmG(y)= f, kx ZG ZP  −1 ∨ −1 where kx(y) = 1T (x) P 1U (ω)ω(y x)dν(ω) = 1T (x)1U (y x). Using H¨older, ∨ R ′ ′ |PQf(x)| = |f, kx| ≦ ||f||p||kx||p = ||f||p||1U ||p |1T (x)|.

q Integration G |PQf(x)| dmG(x) and the inequality (3.2) give ∨ R ′ 1/q 1/q 1/q 1/p ||PQf||q ≦ ||f||p ||1U ||p mG(T ) ≦ ||f||p ||1U ||p mG(T ) = mG(T ) ν(U) ||f||p . That completes the proof of Proposition 3.4.

Proposition 3.4 is sufficient for the uncertainty principle of Theorem 4.3 below. But there are some other useful operator norm estimates for Gelfand pairs. The estimate analogous to that of Proposition 3.4, but in the other order, is UNCERTAINTY PRINCIPLE FOR GELFAND PAIRS

′ 2 3.5. Proposition. If 1 ≦ p ≦ 2, q ≧ p , and f ∈ Lp(K\G/K), then

1/q 1/p 1/p 1/p ||QPf||q ≦ mG(T ) ν(U) ||f||p and ||QPf||q ≦ mG(T ) ν(U) ||Pf||q .

Proof. As in Proposition 3.4, compute

−1 QPf(x)= 1U (ω) 1T (y)f(y)ω(y )dmG(y) ω(x)dν(ω)= f, jx ZP ZG  ∨ − 1 ′ where jx(y) = 1T (y)1U (y x). So |QPf(x)| ≦ ||f||p||jx||p and

′ 1/q 1/p q ′ ∨ −1 p ||QPf||q ≦ ||f||p 1T (y)|1U (y x)| dmG(y) dmG(x) . ZG (ZG  ) ! Now ′ 1/q ′ q/p ∨ p ||QPf||q ≦ ||f||p (1T ∗|1U | )(x) dmG(x) G Z ′ ′  n ∨ p 1/po = ||f||p(||(1T ∗|1U | )||q/p′ ) ′ ′ 1/p ∨ p ′ ≦ ||f||p ||1T ||q/p′ ||(|1U | )||1 by Young’s Inequality, using q ≧ p ′ ′  1/q ∨ p 1/p = ||f||pmG(T ) (||(|1U | )||1) ∨ 1/q ′ = ||f||pmG(T ) ||1U ||p 1/q ≦ ||f||pmG(T ) ||1U ||p by Titchmarch (3.2), using 1 ≦ p ≦ 2 1/q 1/p = mG(T ) ν(U) ||f||p .

That proves the first inequality of Proposition 3.5. For the second inequality replace f by Pf in the first inequality and use P 2 = P :

1/q 1/p ||QPf||q ≦ mG(T ) ν(U) ||Pf||p . (i) q 1 1 If r = p and r + r′ = 1 then

p p ||Pf||p = |Pf| , 1T p ≦ ||(|Pf|) ||r ||1T ||r′ bytheH¨olderinequality (ii)

p 1 p =(||Pf|| ) ||1 || ′ since rp = q and = . q T r r q

2There is a typographical error (a misplaced prime) in the published statement of [11, Theorem 3.2]. The correction is implicit in the argument below. The argument for the second inequality of Proposition 3.5 follows [12]. JOSEPH A. WOLF

1 1 1 Use pr′ = p − q to combine (i) and (ii) as

′ 1/q 1/p 1/pr 1/p 1/p ||QPf||q ≦ mG(T ) ν(U) ||Pf||qmG(T ) = mG(T ) ν(U) ||Pf||q .

That completes the proof.

If K\G/K has finite measure, then G is compact, and one has a stronger form of part of (3.2), analog of the Hausdorff–Young Inequality (again compare [3, §51]):

1 1 1 ||fˆ|| ≦ ||f|| for f ∈ L (K\G/K) with 1 ≦ p ≦ ∞ and ≦ min( , ) . (3.6) q p p q p′ 2

Similarly, if P has finite measure, for example if P is compact, then

1 1 1 ||h∨|| ≦ ||h|| for h ∈ L (P,ν) with 1 ≦ p ≦ ∞ and ≦ min( , ) . (3.7) q p p q p′ 2

These lead to slightly stronger forms of Propositions 3.4 and 3.5.

Section 4. The Uncertainty Principle for K\G/K.

As in §3 fix a Gelfand pair (G, K) and subsets T = KTK ⊂ G and U ⊂ P of finite measure. Define operators P = PT and Q = QU as in (3.3). Given ǫ ≧ 0 we say that

f ∈ Lp(K\G/K) is Lp ǫ − concentrated on T if ||f − 1T f||p ≦ ǫ||f||p (4.1a) and similarly, given δ ≧ 0,

h ∈ Lp′ (P,ν) is Lp′ δ − concentrated on U if ||h − 1U h||p′ ≦ δ||h||p′ (4.1b)

Somewhat analogously, we say (compare [4] and [11]) that

f ∈Lp(K\G/K) is Lp δ − bandlimited to U if there exists (4.2) fU ∈ Lp(K\G/K) with fU supported in U and ||f − fU ||p ≦ δ||f||p .

c 4.3. Theorem. Suppose that 0 = f ∈ Lp(K\G/K) with 1 ≦ p ≦ 2 and ǫ,δ ≧ 0 such that f is ǫ–concentrated on T and δ–bandlimited to U. Then

1 − ǫ − δ m (T )1/pν(U)1/p ≧ ||PQ|| ≧ . G p 1+ δ UNCERTAINTY PRINCIPLE FOR GELFAND PAIRS

And if p = 2 then, further, ||PQ||2 ≧ 1 − ǫ − δ.

Proof. The first inequality is the case p = q of Proposition 3.4. Note fU = QfU because fU is supported in U, and of course ||P || ≦ 1, to compute

||cf||p − ||PQf||p ≦ ||f − PQf||p

≦ ||f − Pf||p + ||Pf − PfU ||p + ||PQfU − PQf||p

≦ ǫ||f||p + δ||f||p + ||PQ||pδ||f||p

=(ǫ + δ + δ||PQ||p)||f||p , so ||PQf||p ≧ (1 − ǫ − δ − δ||PQ||p)||f||p. Now ||PQ||p ≧ 1 − ǫ − δ − δ||PQ||p so

(1 + δ)||PQ||p ≧ 1 − ǫ − δ .

That proves the general assertion. If p = 2 we can take fU = Qf so that Qf = QfU and the ||PQfU − PQf||p term does not occur. c

Section 5. Noncommutative Function Theory of Gelfand Pairs.

In this Section we describe the vector–valued transform that leads to an analysis of L2(G/K) where (G, K) is a Gelfand pair. As before, we defer to [2] and [5] for reference to original sources. Recall Theorem 2.15. Every positive definite zonal spherical function ω ∈ P determines an essentially unique irreducible unitary representation πω and unit vector u ∈ H = H such that ω(x) = u , π (x)u for all x ∈ G. It is ω ω πω ω ω ω Hω immediate that

if f ∈ L (G/K) and x ∈ G then (f ∗ ω)(x)= π (f)u , π (x)u . (5.1) 1 ω ω ω ω Hω

On the other hand, a calculation shows that

if f ∈ C(G/K) ∩ L1(G/K) and x ∈ G then f(x)= (f ∗ ω)(x)dν(ω) . (5.2) ZP

So f ∗ ω ∈ C(G/K) ∩ L1(G/K) corresponds to πω(f)uω ∈ Hω . In fact (5.1) and (5.2) combine to prove JOSEPH A. WOLF

5.3. Lemma. If f ∈ C0(G/K) and x ∈ G then f(x)= π (f)u , π (x)u dν(ω) and ||f||2 = ||π (f)u ||2 dν(ω) . ω ω ω ω Hω L2(G/K) ω ω Hω ZP ZP

We need the details of the notion of direct integral. Let (Y,τ) be a measure space. For each y ∈ Y let Hy be a Hilbert space. Let {sα} be a countable family of maps Y → y∈Y Hy such that

sα(y) ∈Hy Sand {sα(y)} spans Hy for all y ∈ Y , and the functions y → s (y),s (y) belong to L (Y,τ). α α Hy 1

Then the direct integral H = Y Hydτ(y) modeled on the {sα} is the linear span of all the maps s : Y → H such that y∈Y R y s(y) ∈H a.e. (Y,τ) and the functions y → s(y),s (y) belong to L (Y,τ). y S α Hy 1 H is a separable Hilbert space with inner product s,s′ = s(y),s′(y) dτ(y). H Y Hy

More generally, fix 1 ≦ p ≦ ∞. Then the Lp direct integralR Hp is the linear span of the maps s : Y → y∈Y Hy such that S 1/2 s(y) ∈Hy a.e. (Y,τ) and the functions y → s(y),s(y)Hy belong to Lp(Y,τ). 1/p H is a with norm ||s|| = s(y),s(y)p/2 for 1 ≦ p < ∞, p p P Hy 1/2 ∞   norm ||s|| = ess sup(P,ν)s(y),s(y)Hy . OfR course H2 is the Banach structure underlying the Hilbert space structure of H.

Let BL(Hy) denote the algebra of bounded linear operators on the Hilbert space Hy. Let T : Y → y∈Y BL(Hy) such that ′ ′ T (y) ∈ BL(HyS) a.e. (Y,τ), and if s,s ∈ H then y → T (y)s(y),s (y) belongs to L1(Y,τ). Then s → Ts ∈H, Ts(y)= T (y)s(y), defines an element T ∈ BL(H). This element is denoted T = Y T (y)dν(y) and is called the direct integral of the T (y). In our case,R (P,ν) is the measure space; for ω ∈ P we have the Hilbert space

Hω; for f ∈ C0(G/K) we have sf (ω)= πω(f)uω , and {sα} = {sfα } where {fα} is a countable dense subset of C0(G/K).

5.4. Definition. The Fourier transform on G/K is the map

F : L1(G/K) → Hωdν(ω) ZP UNCERTAINTY PRINCIPLE FOR GELFAND PAIRS given by F(f)(ω)= sf (ω)= πω(f)uω .

Now Lemma 5.3 gives us the Godement–Plancherel Theorem for G/K:

5.5. Theorem. Let (G, K) be commutative and let ν be its Plancherel measure. Then the Fourier transform

F : C0(G/K) →H , whereH = Hωdν(ω) , (5.7) ZP satisfies

f(x)= F(f), π (x)u dν(ω) and ||F(f)||H = ||f|| . (5.8) ω ω Hω L2(G/K) ZP

Moreover f → F(f) extends by continuity to a Hilbert space isomorphism of L2(G/K) onto H which intertwines the (left) regular representation

λ : L1(G) on L2(G/K) by λ(ψ)f = ψ ∗ f (5.9a) with the direct integral representation

π = πω dν(ω): ZP (5.9b) L1(G) on H = Hωdν(ω) by π(ψ)= πω(ψ)dν(ω) . ZP ZP

The standard Plancherel-type formula is of the form

f(x)= trace π(ℓ(x−1)f) d(π), (5.10) b ZG where G is the unitary dual of G and ℓ(y)f(g) = f(y−1g). The connection with Theorem 5.5 is given by b −1 trace πω(ℓ(x )f)= πω(f)uω, πω(x)uω. (5.11)

Formula (5.11) depends on the fact that if vω ⊥ uω in Hω and f ∈ L1(G/K) then πω(f)vω = 0. The same calculation shows that if π ∈ G has no K–fixed vector and f ∈ L1(G/K) then π(f) = 0. So for functions on G/K, only the πω occur in the expression (5.10). b JOSEPH A. WOLF

Section 6. The Uncertainty Principle for G/K

We now modify the considerations of §§3 and 4 to carry them from K\G/K to G/K. We start with the formulae for the Fourier transform (5.4) and its inverse (5.8),

F(f)(ω)= π (f)u and F −1(v)(x)= v , π (x)u dν(ω). (6.1) ω ω ω ω ω Hω ZP Again as in the classical case,

||F(f)||∞ = ess sup(P,ω) ||F(f)(ω)||Hω = ess sup(P,ω) ||πω(f)uω||Hω

≦ ess sup(P,ω) ||f||1 = ||f||1 for f ∈ L1(G/K), and

−1 ||F (v)||∞ = ess sup v(ω), π (x)u dν(ω) ≦ ||v(ω)||H dν(ω)= ||v|| ω ω Hω ω 1 ZP ZP for v ∈H1. So as before, Riesz-Thorin interpolation results in a G/K analog of the Titchmarch Inequality

||Ff||p′ ≦ ||f||p for f ∈ Lp(G/K), 1 ≦ p ≦ 2, −1 (6.2) ||F (v)||p′ ≦ ||v||p for v ∈Hp , 1 ≦ p ≦ 2, 1 1 with the usual p + p′ = 1. Fix sets T = TK ⊂ G and U ⊂ P of finite measure. As in (3.3) define −1 Pf = 1T f and Qf = F (1U F(f)) . (6.3)

6.4. Proposition. If 1 ≦ p ≦ 2,q ≧ 1 and f ∈ Lp(G/K) then 1/q 1/p ||PQf||q ≦ mG(T ) ν(U) ||f||p . 1/p 1/p Case p = q: the operator norm on Lp(G/K) satisfies ||PQ||p ≦ mG(T ) ν(U) .

Proof. We compute essentially as in the proof of Proposition 3.4: −1 PQf(x) = 1T (x)F (1U F(f))(x)

= 1 (x) 1 (ω)π (f)u , π (x)u T U ω ω ω ω Hω ZP = 1 (x) 1 (ω) f(y)π (y)u , π (x)u dm (y) dν(ω) T U ω ω ω ω Hω G ZP ZG  = 1 (x) f(y) 1 (ω)u , π (y−1x)u dν(ω) dm (y) T U ω ω ω Hω G ZG ZP  = f, kxL2(G/K) UNCERTAINTY PRINCIPLE FOR GELFAND PAIRS where k (y) = 1 (x) 1 (ω)u , π (y−1x)u dν(ω) x T U ω ω ω Hω P Z −1 −1 = 1T (x)F (ω → 1U (ω)uω)(y x) . q Using H¨older, integration G |PQf(x)| dmG(x), and the inequality (6.2), we see, as before, that R −1 1/q ||PQf||q ≦ ||f||p ||F (ω → 1U (ω)uω)||p′ mG(T ) 1/q 1/q 1/p ≦ ||f||p ||1U ||p mG(T ) = mG(T ) ν(U) ||f||p .

That completes the proof of Proposition 6.4. One can also prove operator norm inequalities corresponding to (3.5), (3.6) and (3.7), but we leave their formulation and proof to the reader. We now proceed as for K\G/K in §4. Given ǫ ≧ 0 we say that

f ∈ Lp(G/K) is Lp ǫ − concentrated on T if ||f − 1T f||p ≦ ǫ||f||p (6.5a) and similarly, given δ ≧ 0,

v ∈Hp′ is Lp′ δ − concentrated on U if ||v − 1U v||p′ ≦ δ||v||p′ (6.5b)

Analogously,

f ∈Lp(G/K) is Lp δ − bandlimited to U if there exists (6.6) fU ∈ Lp(G/K) with F(fU ) supported in U and ||f − fU ||p ≦ δ||f||p .

We proved Theorem 4.3, the uncertainty principle for K\G/K, as a formal conequence of Proposition 3.4. Now exactly the same argument proves the un- certainty principle for G/K as a formal consequence of Proposition 6.4:

6.7. Theorem. Suppose that 0 = f ∈ Lp(G/K) with 1 ≦ p ≦ 2 and ǫ,δ ≧ 0 such that f is ǫ–concentrated on T and δ–bandlimited to U. Then 1 − ǫ − δ m (T )1/pν(U)1/p ≧ ||PQ|| ≧ . G p 1+ δ

And if p = 2 then, further, ||PQ||2 ≧ 1 − ǫ − δ.

There is an equivalent uncertainty principle for the operator–valued transform f → F(f) where F(f)(ω)= πω(f) for f ∈ Lp(G/K) and ω ∈ P . See the discussion surrounding (5.10) and (5.11). JOSEPH A. WOLF

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