On the adjoint action of the group of symplectic diffeomorphisms
aa r X i v : . [ m a t h . S G ] S e p On the adjoint action of the group of symplecticdiffeomorphisms
László Lempert ∗ Department of MathematicsPurdue UniversityWest Lafayette, IN 47907-2067, USA
Abstract
We study the action of Hamiltonian diffeomorphisms of a compact symplectic man-ifold (
X, ω ) on C ∞ ( X ) and on functions C ∞ ( X ) → R . We describe various propertiesof invariant convex functions on C ∞ ( X ) . Among other things we show that continuousconvex functions C ∞ ( X ) → R that are invariant under the action are automaticallyinvariant under so called strict rearrangements and they are continuous in the sup normtopology of C ∞ ( X ) ; but this is not generally true if the convexity condition is dropped. Consider a connected, compact, symplectic manifold ( X, ω ) , without boundary, of di-mension n . According to Omori [O], symplectic self–diffeomorphisms of X form aFréchet–Lie group Symp ( ω ) , with Lie algebra the space v ( ω ) of smooth vector fieldson X that are locally Hamiltonian. In this paper we will be interested in the action ofSymp ( ω ) , by pull back, on the Fréchet space C ∞ ( X ) of smooth real functions(1.1) Symp ( ω ) × C ∞ ( X ) ∋ ( g, ξ ) ξ ◦ g − ∈ C ∞ ( X ) , and on functions on C ∞ ( X ) that (1.1) leaves invariant. This action is no adjoint action,but it is close to one. The adjoint action Ad g of g ∈ Symp ( ω ) is, rather, push forwardby g − of vector fields in v ( ω ) . The subspace ham ( ω ) ⊂ v ( ω ) of globally Hamiltonianvector fields, those that are symplectic gradients sgrad ξ of some ξ ∈ C ∞ ( X ) , is invariantunder Ad g , and (1.1) induces via the projection ξ sgrad ξ the restriction of the adjointaction to ham ( ω ) . ∗ Research partially supported by NSF grant DMS 1764167.2020 Mathematics subject classification 53D05, 58D19 ther diffeomorphism groups of X also act on C ∞ ( X ) by pull back. Our focus willbe on the subgroup Ham( ω ) ⊂ Symp ( ω ) of Hamiltonian diffeomorphisms. Hamilto-nian diffeomorphisms are the time 1 maps of time dependent Hamiltonian vector fieldssgrad ξ t , ξ t ∈ C ∞ ( X ) . Continuous norms—and also seminorms—on the Fréchet space C ∞ ( X ) , invariant under Ham( ω ) , are of potential interest in symplectic geometry be-cause they give rise to bi–invariant metrics on Ham( ω ) , and have been investigatedin the past. An obvious norm is k ξ k ∞ = max X | ξ | . That it gives rise to a genuinemetric on Ham( ω ) was proved first by Hofer in R n , and in general by Lalonde andMcDuff; see also Polterovich’s book, [Ho, LM, P]. Work by Ostrover–Wagner, Han, andBuhovsky–Ostrover [BO, Ha, OW] gave the following. Let ( X, µ ) and ( Y, ν ) be measurespaces. We say that measurable functions ξ : X → R , η : Y → R are equidistributed,or strict rearrangements of each other, if µ ( ξ − B ) = ν ( η − B ) for all Borel sets B ⊂ R . When µ ( X ) , ν ( Y ) < ∞ , this is equivalent to µ { x ∈ X : ξ ( x ) > t } = ν { y ∈ Y : η ( y ) > t } for all t ∈ R . We have to use the qualifier ‘strict’, since the notion of rearrangementin harmonic analysis and Banach space theory typically refers to the relation µ { x ∈ X : | ξ ( x ) | > t } = ν { y ∈ Y : | η ( y ) | > t } . Back to our symplectic manifold ( X, ω ) , wewrite µ for the measure on X defined by ω n ; the action (1.1) clearly sends functions on ( X, µ ) to their strict rearrangements. Theorem 1.1 ([BO, H, OW]) . If k k is a Ham( ω ) invariant continuous seminorm onthe Fréchet space C ∞ ( X ) , then k ξ k = k η k whenever ξ, η ∈ C ∞ ( X ) are equidistributed.These seminorms satisfy k k ≤ c k k ∞ with some c ∈ (0 , ∞ ) . Unless k k and k k ∞ are equivalent, the pseudodistance on Ham( ω ) induced by k k is identically 0. One of our goals in this paper is to offer a simpler proof to the first two statements,in fact in a slightly greater generality:
Theorem 1.2.
Suppose p : C ∞ ( X ) → R is a continuous, convex function that isinvariant under the action of Ham( ω ) . Then p is continuous in the topology of C ∞ ( X ) induced by k k ∞ , and is invariant under strict rearrangements: p ( ξ ) = p ( η ) whenever ξ, η are equidistributed. The point is not the modest gain in generality, which can easily be achieved onceTheorem 1.1 is known (for example along the lines of the proof of Theorem 4.1 below).Rather, it is the simplification of the proof. This is how the two proofs compare.[OW] first proved that any
Ham( ω ) invariant seminorm k k ≤ c k k ∞ is invariantunder volume preserving diffeomorphisms. Han in [Ha] subsequently strengthened thisto invariance under strict rearrangements. All this is obtained as a consequence ofa lemma of Katok [K, Section 3]. The final step is in [BO], that takes an arbitrarycontinuous Ham( ω ) invariant seminorm k k on C ∞ ( X ) , and proves by an involvedargument that k k ≤ c k k ∞ .We obtain the simplification by restructuring the proof. First we prove that p in Theorem 1.2 is a limit point of the set of Ham( ω ) invariant functions q that arecontinuous in the L topology on C ∞ ( X ) . This depends on studying linear forms on C ∞ ( X ) , i.e., distributions, and regularizing them using the action of Ham( ω ) . Katok’s emma now gives that the functions q are invariant under strict rearrangements, whenceso must be their limit point p . Another application of Katok’s lemma, combined withreal analysis type arguments then gives the continuity of p with respect to k k ∞ .Continuity of p with respect to k k ∞ in Theorem 1.2 is essentially an upper estimateof p . We will also prove a lower estimate: Theorem 1.3.
Let p : C ∞ ( X ) → R be Ham( ω ) invariant, convex, and continuous.Then either(i) p ( ξ ) = p ( ´ X ξω n ) , where p : R → R is convex; or(ii) there are a ∈ R , b ∈ (0 , ∞ ) such that p ( ξ ) ≥ a + b ˆ X | ξ | ω n if ´ X ξω n = 0 , or ´ X ξω n ≥ and lim R ∋ λ →∞ p ( λ ) = ∞ , or ´ X ξω n ≤ and lim R ∋ λ →−∞ p ( λ ) = ∞ . If p is positively homogeneous ( p ( cξ ) = cp ( ξ ) for positive constants c ), then a = 0 . In particular, if p is a norm, then it dominates L norm, something that [OW] alsofound (cf. Proposition 6.1 there and its proof).Above we have insisted on the difference between rearrangements and strict re-arrangements. Nevertheless, Theorem 1.3 implies that in our setting the differencebetween the two is minimal. The notion of rearrangement invariant Banach spaces inthe next theorem is defined in [BS], see also section 6. Theorem 1.4.
Given a
Ham( ω ) invariant continuous norm p on C ∞ ( X ) , there is arearrangement invariant Banach function space on X whose norm, restricted to C ∞ ( X ) ,is equivalent to p . A natural question is whether Theorem 1.2 holds for all continuous
Ham( ω ) invariantfunctions p , independently of convexity. It does not: Theorem 1.5. If dim X ≥ , there is a smooth Ham( ω ) invariant function p : C ∞ ( X ) → R that is not invariant under volume preserving diffeomorphisms X → X . The last statement of Theorem 1.1 suggests that, after all, the only invariant normon C ∞ ( X ) that is of interest for symplectic geometry, is Hofer’s norm k k ∞ . However,all invariant norms are of interest for Kähler geometry. The groups Symp ( ω ) and Ham( ω ) can be regarded as symmetric spaces. When ( X, ω ) is Kähler, Donaldson,Mabuchi, and Semmes proposed that the infinite dimensional manifold H ω of relativeKähler potentials, endowed with a natural connection on its tangent bundle, should beviewed as the dual symmetric space, at least in a formal sense; see [Do, M, S1, S2]. Ham( ω ) invariant norms on C ∞ ( X ) induce Finsler metrics on H ω that are invariantunder parallel transport, and, perhaps surprisingly, all these Finsler metrics inducegenuine metrics on H ω . Mabuchi was the first to study such a metric, associated with L -norm k ξ k = ( ´ X | ξ | ω n ) / ; more recently, Darvas in [Da] introduced various Orlicznorms on C ∞ ( X ) and the induced metrics on H ω . Generalizing Darvas’s norms andmetrics, in [L] we study general Ham( ω ) invariant Lagrangians and the associated actionon H ω , and most results here are motivated by the needs of that paper. Reduction to linear forms
In this section ( X, ω ) can be any n dimensional symplectic manifold, not necessarilycompact. The space of compactly supported smooth functions on X will be denoted D ( X ) , with its usual locally convex inductive limit topology. Its dual is D ′ ( X ) , thespace of distributions. The group Ham ( ω ) , time 1 maps of compactly supportedHamiltonian flows, acts on D ( X ) by pull back and on D ′ ( X ) by push forward. Wedenote the pairing between D ′ ( X ) and D ( X ) by h , i . The locally convex topologyof D ′ ( X ) is generated by the seminorms k f k ′ ξ = |h f, ξ i| with ξ ∈ D ( X ) . Integrationagainst any smooth n –form defines a distribution. Such distributions will be calledsmooth. If h ∈ D ′ ( X ) , we denote by conv ( h ) the closed convex hull of the Ham ( ω ) orbit of h .The main result of this section is Lemma 2.1.
Suppose p : D ( X ) → R is a Ham ( ω ) invariant, continuous, convexfunction. There is a family A ⊂ R × C ∞ ( X ) such that (2.1) p ( ξ ) = sup n a + ˆ X f ξω n : ( a, f ) ∈ A o , for all ξ ∈ D ( X ) . If p is positively homogeneous as well ( p ( cξ ) = cp ( ξ ) for < c < ∞ ) , then A can bechosen in { } × C ∞ ( X ) . For the proof we need certain regularization maps D ′ ( X ) → D ′ ( X ) . Let U ⊂⊂ X be open, and assume that on a neighborhood of U there are local coordinates x ν inwhich ω takes the form P n dx ν ∧ dx n + ν . Let C ⊂ X \ U be compact. Fix ϕ ν ∈ D ( X ) , ν = 1 , . . . , n , vanishing on a neighborhood of C , such that ϕ ν = x ν in a neighborhoodof U . Let g τν , τ ∈ R , denote the Hamiltonian flow of ϕ ν for ν ≤ n and of − ϕ ν for ν > n ;i.e., the flow of the vector fields ± sgrad ϕ ν . If t = ( t , . . . , t n ) ∈ R n , put g t = g t ◦ g t ◦ · · · ◦ g t n n . Near C we have g t = id; on U , for small t , g t ( x ) = x − t . Let furthermore χ ∈ D ( R n ) benonnegative, ´ R n χ ( t ) dt . . . dt n = 1 . For λ ∈ (0 , ∞ ) define operators R λ : D ′ ( X ) →D ′ ( X ) by R λ h = λ n ˆ R n χ ( λt )( g t ∗ h ) dt . . . dt n ∈ conv ( h ) , h ∈ D ′ ( X ) . Standard properties of convolutions imply
Lemma 2.2. lim λ →∞ R λ h = h for h ∈ D ′ ( X ) . If the support of χ is sufficientlyclose to 0, then R λ h ∈ conv ( h ) is smooth on U and R λ h = h on a neighborhood of C .Furthermore, if V ⊂⊂ W ⊂ X are open, and h is smooth on W , then R λ h is smoothon V for sufficiently large λ . Lemma 2.3.
For any h ∈ D ′ ( X ) , smooth distributions are dense in conv ( h ) . roof. It will suffice to prove that given a finite Ξ ⊂ D ( X ) and ε > , there is a smooth h ′ ∈ conv ( h ) such that |h h ′ − h, ξ i| ≤ ε for all ξ ∈ Ξ . To show this latter, for each z ∈ X construct an open neighborhood V ( z ) ⊂⊂ X so that in a neighborhood of V ( z ) we can write ω = P dx ν ∧ dx n + ν in suitable local coordinates. Select a locally finitecover V ( z ) , V ( z ) , . . . of X . Thus the V ( z j ) form a finite or infinite cover dependingon whether X is compact or not. For each j we can find U j ⊃⊃ V ( z j ) such that { U j } j is still locally finite, and ω = P dx ν ∧ dx n + ν is still valid in some neighborhood of U j .Fix furthermore open sets V ij , i ∈ N , such that U j = V j ⊃⊃ V j ⊃⊃ · · · ⊃ V ( z j ) , and compact sets C j ⊂ X \ S k>j U k , C = ∅ , such that C j − ⊂ int C j and S j C j = X .We let h = h and construct h j ∈ conv ( h ) so that for j ≥ |h h j − h, ξ i| < ε if ξ ∈ Ξ ; h j | V j ∪ · · · ∪ V jj is smooth ; h j = h j − on int C j − . Assuming we already have h j − , we apply Lemma 2.2 with U = U j , C = C j , V = V j ∪ · · · ∪ V jj − , and W = V j − ∪ · · · ∪ V j − j − . If λ is sufficiently large, then h j = R λ h j − will do as the next function. Note that h j is smooth over V ∪ U j ⊃ V j ∪ · · · ∪ V jj − ∪ V jj .Thus h j = h j +1 = . . . on int C j and h j | V ( z ) ∪ · · · ∪ V ( z j ) is smooth. If X iscompact, we take h ′ to be the last h j ; otherwise we take h ′ = lim j →∞ h j . Proof of Lemma 2.1.
By an affine function we mean a function D ( X ) → R of theform const + linear. Clearly, if an affine function is bounded above on a symmetricneighborhood of ∈ D ( X ) , it is bounded below as well, hence continuous.Let B denote a collection of affine functions β : D ( X ) → R such that β ≤ p . Thus β ∈ B can be written(2.2) β ( ξ ) = a + h h, ξ i , with a ∈ R , h ∈ D ′ ( X ) . The Banach–Hahn separation theorem gives that p = sup β ∈B β with a suitablechoice of B . If p is positively homogeneous, another version of the Banach–Hahn theo-rem, see e.g. [Sc, p.317-319], gives that B can be taken to consists of linear forms, i.e.all a will be 0.By the invariance of p , if β in (2.2) is in B , then for any g ∈ Ham( ω )( g ∗ β )( ξ ) = a + h g ∗ h, ξ i = a + h h, g ∗ ξ i ≤ p ( ξ ) . This means that all g ∗ β can be adjoined to B , and in fact we can arrange that all a + h h ′ , ξ i are in B for any h ∈ B and h ′ ∈ conv ( h ) . Therefore if we take all β ∈ B ofform (2.2) with smooth h and write h as f ω n , the family A of pairs ( a, f ) thus obtainedwill do according to Lemma 2.3. Proof of the second part of Theorem 1.2
The second part was:
Theorem 3.1.
Let ( X, ω ) be a connected, compact, symplectic manifold. Any contin-uous, convex, and Ham( ω ) invariant function p : C ∞ ( X ) → R is strict rearrangementinvariant: p ( ξ ) = p ( η ) if ξ, η are equidistributed. As before, µ denotes the Borel measure on X that the form ω n determines. In ourintegrals below we will often omit dµ and write ´ E f for ´ E f dµ ; and when E = X ,we will even omit X and write ´ f for ´ X f dµ . In the same spirit, we write L q ( X ) for L q ( X, µ ) .We need the following result, an equivalent of Katok’s Basic Lemma, valid for non-compact (but connected) X as well: Lemma 3.2. If ξ, η ∈ L ( X ) are equidistributed, then there is a sequence of g k ∈ Ham ( ω ) such that lim k →∞ ˆ X | ξ − η ◦ g k | dµ = 0 . Proof. (Essentially as in [OW], [Ha, Proposition 1.12].) Given ε > , we will find g ∈ Ham ( ω ) such that ´ | ξ − η ◦ g | < ε . Assume first µ ( X ) < ∞ .The measure | ξ | dµ is absolutely continuous with respect to dµ , hence there is a δ > such that ´ E | ξ | < ε if µ ( E ) < δ . Construct disjoint intervals J , . . . , J N ⊂ R oflength < ε/µ ( X ) so that µ ( X \ S i ξ − J i ) < δ/ , and choose compact sets K i ⊂ ξ − J i so that also(3.1) µ ( X \ [ i K i ) < δ/ . By equidistribution µ ( η − J i ) = µ ( ξ − J i ) , hence there are compact L i ⊂ η − J i suchthat µ ( L i ) = µ ( K i ) . The K i are disjoint among themselves and so are the L i . In thissituation Katok’s Basic Lemma [K, Section 3] provides a g ∈ Ham ( ω ) such that(3.2) µ ( K i \ g − L i ) < δ/ N, i = 1 , . . . , N. If x ∈ K i ∩ g − L i then ξ ( x ) , η ( gx ) ∈ J i and so | ξ ( x ) − η ( gx ) | < ε/µ ( X ) . Conversely, | ξ ( x ) − η ( gx ) | ≥ ε/µ ( X ) can happen only if x ∈ E, where E = (cid:0) X \ [ i K i (cid:1) ∪ [ i (cid:0) K i \ g − L i (cid:1) . By (3.1), (3.2) µ ( E ) < δ , whence µ ( gE ) < δ and ˆ | ξ − η ◦ g | = ˆ X \ E | ξ − η ◦ g | + ˆ E | ξ − η ◦ g | < ε + ˆ E | ξ | + ˆ gE | η | < ε. This takes care of X of finite measure. In general, choose an a > so that thelevel sets Y = {| ξ | ≥ a } and Y = {| η | ≥ a } satisfy ´ X \ Y | ξ | = ´ X \ Y | η | < ε . Then µ ( Y ) = µ ( Y ) < ∞ . The functions ξ ′ = ( ξ on Y on X \ Y and η ′ = ( η on Y on X \ Y re also equidistributed. Construct a connected open X ′ ⊂ X of finite measure con-taining Y ∪ Y . By what we have proved so far, there is a g ∈ Ham ( ω | X ′ ) such that ´ X ′ | ξ ′ − η ′ ◦ g | < ε . Extend g to all of X by identity on X \ X ′ . Denoting this extensionalso by g , we have ˆ | ξ − η ◦ g | ≤ ˆ | ξ ′ − η ′ ◦ g | + ˆ | ξ − ξ ′ | + ˆ | η − η ′ | < ε + ε + ε = 5 ε. To finish the proof, we let ε = 1 /k and g = g k , k ∈ N , and obtain the sequencesought. Proof of Theorem 3.1.
Consider
A ⊂ R × C ∞ ( X ) of Lemma 2.1: p ( ξ ) = sup n a + ˆ f ξ : ( a, f ) ∈ A o . Suppose ξ, η ∈ C ∞ ( X ) are equidistributed, and let g k be as in Lemma 3.2. With any ( a, f ) ∈ A p ( η ) = p ( η ◦ g k ) ≥ a + ˆ ( η ◦ g k ) f → a + ˆ f ξ as k → ∞ . Taking sup over ( a, f ) ∈ A , p ( η ) ≥ p ( ξ ) follows, and in fact p ( ξ ) = p ( η ) by symmetry. This is what the first part says:
Theorem 4.1. If ( X, ω ) is a connected compact symplectic manifold, any continuous,convex, Ham( ω ) invariant function p : C ∞ ( X ) → R is continuous in the sup normtopology on C ∞ ( X ) . We will use the following standard fact:
Lemma 4.2.
Let V be a locally convex topological vector space over R . If p : V → R is convex and bounded above on some open U ⊂ V , then it is continuous on U .Proof. We can assume U is convex. Say, we want to prove continuity at ∈ U . Let s = sup U p < ∞ . With < λ < and v ∈ ( λU ) ∩ ( − λU ) convexity implies p ( v ) − p (0) ≤ λ ( p ( v/λ ) − p (0)) ≤ λ ( s − p (0)) p (0) − p ( v ) ≤ λ ( p ( − v/λ ) − p (0)) ≤ λ ( s − p (0)) ) → when λ → , as needed.The key to the proof of Theorem 4.1 is the following. Lemma 4.3.
Let
F ⊂ L ( X ) be a Ham( ω ) invariant family of functions. If for every ξ ∈ C ∞ ( X ) (4.1) sup f ∈F ˆ X f ξ dµ < ∞ , then sup f ∈F ´ X | f | dµ < ∞ . his is not hard to show and will suffice to prove Theorem 4.1; but later we will needa more precise statement, whose proof is just a little more involved. Let ξ + = max( ξ, and ξ − = max( − ξ, denote the positive and negative parts of functions ξ : X → R . If E ⊂ X is measurable, write ffl E ξ for the average ´ E ξ/µ ( E ) of an integrable function.If µ ( E ) = 0 , we let ffl E ξ = 0 . Lemma 4.4.
Let f ∈ L ( X ) , ξ ∈ L ∞ ( X ) , and S, T ⊂ X be of equal measure. If ξ ≥ on T and ξ ≤ on X \ T , then (4.2) sup n ˆ X ( f ◦ g ) ξ : g ∈ Ham( ω ) o ≥ S f ˆ ξ + − X \ S f ˆ ξ − . First we show how this implies Lemma 4.3.
Proof of Lemma 4.3.
We can assume µ ( X ) = 1 . Let M ( ξ ) denote the left hand side of(4.1). Fix a nonnegative ξ ∈ C ∞ ( X ) that is not identically , but T ′ = { ξ > } hasmeasure ≤ / . Let f ∈ F . Suppose first that S = { f ≥ } has measure ≥ / , andchoose T ⊃ T ′ so that µ ( S ) = µ ( T ) . By Lemma 4.4 M ( ξ ) ≥ ffl S f ´ ξ , hence ˆ f + ≤ M ( ξ ) . ˆ ξ. If, instead of S , { f ≤ } has measure ≥ / , Lemma 4.4 implies in the same way that ´ f − ≤ M ( − ξ ) / ´ ξ . Since | ´ f + − ´ f − | = | ´ f | ≤ M (1) + M ( − , in both cases weobtain a bound for ´ | f | = ´ f + + ´ f − , as claimed.Given f ∈ L ( X ) , we will write conv ( f ) for the closure, in the L ( X ) topology,of the convex hull of the orbit of f under Ham( ω ) . In light of Lemma 3.2 this is thesame as the closed convex hull of all strict rearrangements of f . To prove Lemma 4.4we need the following. Lemma 4.5. If f ∈ L ( X ) and E ⊂ X has positive measure, then the function f ′ = ( ffl E f on Ef on X \ E is in conv ( f ) .Proof. If two functions f, h ∈ L ( X ) are at L distance ≤ ε , then their Ham( ω ) orbitsare at Hausdorff distance ≤ ε , and so are therefore conv ( f ) and conv ( h ) . Hence,given E , if the lemma holds for a sequence f = f k , k = 1 , , . . . , and f k → f in L ,then the lemma will hold for f as well.Now suppose that E is the disjoint union of E j , j = 1 , . . . , m , of equal measure, and f = c j is constant on each E j . If σ is a permutation of , . . . , m , define f σ ∈ L ( X ) by f σ = c σ ( j ) on E j , f σ = f on X \ E. As a strict rearrangement of f , by Lemma 3.2 f σ is in the closure of the Ham( ω ) orbitof f . Therefore f ′ = X σ f σ /m ! s indeed in conv ( f ) . Since any f ∈ L ( X ) is the limit of functions of the above type,the claim follows. Proof of Lemma 4.4.
Write χ A for the characteristic function of a set A . By Lemma3.2 there is a sequence g k ∈ Ham( ω ) such that χ S ◦ g k → χ T in L . Two applicationsof Lemma 4.5 give that f ′ = ( ffl S f on S ffl X \ S f on X \ S and so f ′′ = lim k f ′ ◦ g k = ( ffl S f on T ffl X \ S f on X \ T are in conv ( f ) . Lemma 4.4. follows, since the left hand side in (4.2) is ≥ ˆ f ′′ ξ = S f ˆ T ξ + X \ S f ˆ X \ T ξ = S f ˆ ξ + − X \ S f ˆ ξ − . Proof of Theorem 4.1.
If a function is continuous in the sup norm topology, we will sayit is k k ∞ –continuous, and use similar terminology for other topological notions. Firstassume that p of the theorem is positively homogeneous as well. By Lemma 2.1 thereis a family F ⊂ L ( X ) such that(4.3) p ( ξ ) = sup n ˆ f ξ : f ∈ F o . If we replace F by its Ham( ω ) orbit, the supremum in (4.3) will not change, for ˆ ( f ◦ g ) ξ = ˆ ( ξ ◦ g − ) f ≤ p ( ξ ◦ g − ) = p ( ξ ) if f ∈ F , g ∈ Ham( ω ) . Therefore we may assume that the family F in (4.3) is already invariant under Ham( ω ) .Hence Lemma 4.3 gives sup F ´ | f | < ∞ . This implies p is bounded on k k ∞ –boundedsubsets of C ∞ ( X ) , and by Lemma 4.2 it is k k ∞ –continuous.For general p , pick a number c > p (0) and consider the Minkowski functional q ofthe convex set { p < c } (see e.g. [Sc, pp. 315-317]), q ( ξ ) = inf { λ ∈ (0 , ∞ ) : p ( ξ/λ ) < c } ∈ [0 , ∞ ) . This is a convex, positively homogeneous, strict rearrangement invariant function, thatis continuous—because locally bounded—in the topology of C ∞ ( X ) . By what we havealready proved, it is k k ∞ –continuous, in particular, the set U c = { q < } ⊃ { p < c } is k k ∞ –open. If ξ ∈ U c then p ( ξ/λ ) < c with some λ < . Also p (0) < c . As ξ is apoint on the segment connecting 0, ξ/λ , convexity implies p ( ξ ) < c . Thus p is boundedabove on the k k ∞ –open set U c , and by Lemma 4.2 it is continuous there. The theoremfollows since S c U c = C ∞ ( X ) . The above ideas can be developed to prove that p can be extended to C ( X ) and, underan additional assumption, to the Banach space B ( X ) of bounded Borel functions, withthe supremum norm. (Thus L ∞ ( X ) is a quotient of B ( X ) , but B ( X ) is more naturalto use in our setting.) efinition 5.1. If V ⊂ B ( X ) is a vector subspace, we say that a function p : V → R is strongly continuous if p ( ξ k ) is convergent whenever ξ k ∈ V is a pointwise convergentsequence of uniformly bounded functions. This is stronger than continuity in the topology inherited from B ( X ) . The limit lim p ( ξ k ) depends only on lim ξ k = ξ , since two such sequences can be combined intoone sequence, converging to ξ . Theorem 5.2.
Any continuous, convex,
Ham( ω ) invariant p : C ∞ ( X ) → R has aunique continuous extension to C ( X ) ; this extension is convex and Ham( ω ) (hencestrict rearrangement) invariant. If p is strongly continuous, then it has a unique stronglycontinuous, strict rearrangement invariant extension q : B ( X ) → R . This extension isconvex, and satisfies lim k q ( ξ k ) = q ( ξ ) whenever uniformly bounded ξ k ∈ B ( X ) convergealmost everywhere to ξ . Since C ∞ ( X ) is dense in C ( X ) , and p is known to be continuous in supremum norm,for the first part of Theorem 5.2 one only needs to prove that a continuous extensionexists. This is a special case of the following: Lemma 5.3.
Let W be a locally convex topological vector space over R , V ⊂ W adense subspace. Any continuous, convex p : V → R can be extended to a continuous q : W → R .Proof. First we show that any w ∈ W has a convex neighborhood U such that p isbounded on V ∩ U . By continuity, there certainly is a symmetric, convex neighborhood U ⊂ W of such that p is bounded on V ∩ U . Now w + 2 U is a neighborhood of w ,and if v ∈ V is sufficiently close to w , then U = v + 2 U is also. For any v ∈ V ∩ U convexity implies p ( v ) ≤ p (2 v ) + p (cid:0) v − v ) (cid:1) . Since v − v ∈ U , the right hand side is bounded as v varies in V ∩ U . Thus p isbounded above on V ∩ U . But then p ( v ) + p (2 v − v ) ≥ p ( v ) gives that p is alsobounded below. Set s = sup U | p | .We let U ′ = v + U and show that p is uniformly continuous on V ∩ U ′ . For suppose λ ∈ (0 , ∞ ) . If u, v ∈ V ∩ U ′ and v − u ∈ U /λ , then v + λ ( v − u ) ∈ v + U + U = U ,hence by convexity p ( v ) − p ( u ) ≤ p (cid:0) v + λ ( v − u ) (cid:1) − p ( u )1 + λ ≤ s λ . Since the roles of u, v are symmetric, this indeed proves locally uniform continuity;which in turn implies continuous extension.The proof of the second part of Theorem 5.2 requires some preparation.
Lemma 5.4.
There is a continuous θ : X → [0 , µ ( X )] that is smooth away fromthe preimage of finitely many t ∈ [0 , µ ( X )] , and that preserves measure (the target isendowed with Lebesgue measure). roof. If ζ ∈ C ∞ ( X ) is a Morse function, its reverse distribution function λ ( t ) = µ ( ζ < t ) , t ∈ [min ζ, max ζ ] , is continuous, strictly increasing, and smooth away from the set C of critical values of ζ . It is a homeomorphism [min ζ, max ζ ] → [0 , µ ( X )] , and a diffeomorphism away from C . The function θ = λ ◦ ζ will therefore do, as µ ( θ < s ) = µ ( ζ < λ − ( s )) = λ ( λ − ( s )) = s, s ∈ [0 , µ ( X )] . We will need the notion of decreasing rearrangement of a measurable ξ : X → R .It is the decreasing, say, upper semicontinuous function ξ ⋆ : [0 , µ ( X )] → R that isequidistributed with ξ . Thus µ ( s ≤ ξ ≤ t ) is equal to the length of the maximalinterval on which s ≤ ξ ⋆ ≤ t . In particular,(5.1) µ ( ξ ≥ ξ ⋆ ( s )) = s. The upper semicontinuity requirement translates to left continuity of the decreasingfunction ξ ⋆ , which differs from the more usual convention of right continuity, but thedifference is inconsequential. Obviously, with θ of Lemma 5.3 ξ and ξ ⋆ ◦ θ are equidis-tributed. Lemma 5.5. If ξ ∈ C ( X ) , then ξ ⋆ is continuous.Proof. Since ξ ⋆ is always u.s.c., i.e., left continuous, all we need to show is that if s j ∈ [0 , µ ( X )] decreases to s , then lim j ξ ⋆ ( s j ) cannot be > ξ ⋆ ( s ) . Suppose it were,and let ξ ⋆ ( s ) < α < β < lim j ξ ⋆ ( s j ) . Then ξ − ( α, β ) ⊂ X would be a nonempty opensubset, of positive measure, contradicting (cf.(5.1)) µ ( ξ ≥ ξ ⋆ ( s j )) = s j → s = µ ( ξ ≥ ξ ⋆ ( s )) . The continuity property in Definition 5.1 implies a stronger property:
Lemma 5.6.
Suppose p : C ∞ ( X ) → R is Ham( ω ) invariant, continuous, and convex.If ξ k ∈ C ∞ ( X ) is a uniformly bounded sequence that converges almost everywhere, then p ( ξ k ) is also convergent.Proof. By the first part of Theorem 5.2, already proved, p has a continuous invariantextension to C ( X ) , still denoted p . Suppose uniformly bounded ξ k ∈ C ∞ ( X ) convergea.e. to ξ ∈ B ( X ) . This implies that the rearrangements ξ ⋆k converge everywhere to ξ ⋆ , see [BS, Proposition 1.7, p.41]. Immediately that Proposition only gives a ξ ⋆ ≤ lim inf k ξ ⋆k , when ξ k ≥ ; but applying it with c + ξ k , c − ξ k and a suitable constant c we do obtain what we need. If θ : X → [0 , µ ( X )] is as in Lemma 5.4, ξ k and η k = ξ ⋆k ◦ θ are equidistributed, and η k → η = ξ ⋆ ◦ θ . For each k we can uniformly approximate ξ ⋆k by smooth functions on [0 , µ ( X )] , and η k = ξ ⋆k ◦ θ by their pullbacks along θ . Since p (extended to C ( X ) ) is continuous, there are u k ∈ C ∞ [0 , µ ( X )] such that max | u k − ξ ⋆k | < /k and | p ( u k ◦ θ ) − p ( η k ) | < /k. We can arrange that u k is constant in a neighborhood of the critical values of θ . Thisimplies that u k ◦ θ ∈ C ∞ ( X ) , and lim k u k ◦ θ = lim k η k = η pointwise. Hence p ( u k ◦ θ ) converges, and so does p ( ξ k ) = p ( η k ) . roof of Theorem 5.2. We have already seen that the first half of the theorem followsfrom Lemma 5.3. As to the extension to B ( X ) , note that with θ of Lemma 5.4 forany ξ ∈ B ( X ) its strict rearrangement ξ ⋆ ◦ θ is u.s.c. Thus it is the pointwise limitof a uniformly bounded sequence of continuous, hence also of smooth functions ξ k .Therefore at ξ the extension q of p must take the value q ( ξ ⋆ ◦ θ ) = lim k p ( ξ k ) , so it isunique. What remains is to construct the required extension q .If ξ ∈ B ( X ) , predictably we let q ( ξ ) = lim k p ( ξ k ) , where the uniformly boundedsequence ξ k ∈ C ∞ ( X ) converges to ξ a.e. By Lemma 5.6 the limit exists and, as wesaw, it is independent of the choice of the sequence ξ k . Clearly p = q on C ∞ ( X ) . Ifuniformly bounded η k ∈ C ∞ ( X ) converge to η ∈ B ( X ) a.e., and λ ∈ [0 , , then q (cid:0) λξ + (1 − λ ) η (cid:1) = lim k p (cid:0) λξ k + (1 − λ ) η k (cid:1) ≤ lim k λp ( ξ k ) + (1 − λ ) p ( η k ) = λq ( ξ ) + (1 − λ ) q ( η ) , i.e., q is convex. It is also strongly continuous, and in fact if uniformly bounded ξ k ∈ B ( X ) a.e. converge to ξ ∈ B ( X ) , then q ( ξ k ) → q ( ξ ) . It suffices to show that this latterconvergence holds along some subsequence.By dominated convergence,(5.2) lim k ˆ | ξ k − ξ | = 0 . Let each ξ k be the a.e. limit of a uniformly bounded sequence ξ ik ∈ C ∞ ( X ) , as i →∞ . We can arrange that the double sequence ξ ik is also uniformly bounded. Thus lim i →∞ p ( ξ ik ) = q ( ξ k ) . For each k choose i = i k so that η k = ξ ik satisfies(5.3) | p ( η k ) − q ( ξ k ) | < /k, ˆ | η k − ξ k | < /k. In view of (5.2) lim k ´ | η k − ξ | = 0 , so a subsequence η k j converges to ξ a.e. Hence, by(5.3) q ( ξ ) = lim j p ( η k j ) = lim j q ( ξ k j ) , as needed.Finally, to show that q is invariant under strict rearrangements, consider equidis-tributed ξ, η ∈ B ( X ) . By Lemma 3.2 there are g k ∈ Ham( ω ) such that ´ | η − ξ ◦ g k | → as k → ∞ . Choose uniformly bounded ξ k ∈ C ∞ ( X ) converging to ξ a.e. In particular, lim k ´ | ξ k − ξ | = 0 . Then lim k ˆ | ξ k ◦ g k − η | ≤ lim sup k ˆ | ( ξ k − ξ ) ◦ g k | + lim sup k ˆ | ξ ◦ g k − η | = 0 . Again, this means that a subsequence of ξ k ◦ g k converges a.e. to η , whence q ( ξ ) = lim k p ( ξ k ) = lim k p ( ξ k ◦ g k ) = q ( η ) , which proves that q is indeed invariant under strict rerrangements. ere is the last result in this section. Theorem 5.7.
If a strict rearrangement invariant convex p : B ( X ) → R is stronglycontinuous, then it is Lipschitz continuous on bounded sets.Proof. Let θ be as in Lemma 5.4. We start by showing that p is bounded on boundedsets. Otherwise there would be a bounded sequence ξ k ∈ B ( X ) such that | p ( ξ k ) | → ∞ .The decreasing rearrangements ξ ⋆k are uniformly bounded, hence by Helly’s theoremcontain a pointwise convergent subsequence. But along that subsequence ξ ⋆k ◦ θ convergespointwise and therefore by strong continuity p ( ξ k ) = p ( ξ ⋆k ◦ θ ) also converges, a contradiction.Now boundedness on bounded sets implies Lipschitz continuity on bounded sets.For suppose ξ = η have norm ≤ R , and let ρ be the unit vector in the direction of ξ − η .With M = sup || ζ || ∞ ≤ R +1 | p ( ζ ) | , by convexity p ( ξ ) − p ( η ) || ξ − η || ∞ ≤ p ( ξ + ρ ) − p ( η ) || ξ + ρ − η || ∞ ≤ M. The roles of ξ, η being symmetric, we obtain Lipschitz continuity.
To simplify notation, we will assume µ ( X ) = 1 . By Lemma 2.1 a Ham( ω ) invariantconvex, continuous, p : C ∞ ( X ) → R can be written(6.1) p ( ξ ) = sup n a + ˆ f ξ : ( a, f ) ∈ A o with a family A ⊂ R × C ∞ ( X ) , that can be chosen convex and invariant under Ham( ω ) .The possible behaviors of p described in Theorem 1.3 are determined by whether allfunctions f that occur in A are constant or not.If in A only constant functions occur, then (6.1) gives p ( ξ ) = p ( ´ ξ ) . Henceforwardwe will assume A contains a pair ( a, f ) with a nonconstant function f . According to(ii) of Theorem 1.3, we must estimate p ( ξ ) from below with the L norm of ξ ∈ C ∞ ( X ) .We do this do in a somewhat greater generality, that we will need in the next section. Lemma 6.1.
Suppose
A ⊂ R × L ( X ) is convex and invariant under Ham( ω ) . For ξ ∈ L ∞ ( X ) let q ( ξ ) = sup ( a,f ) ∈A a + ´ f ξ . If A contains a pair ( a, f ) with f nonconstant,then there are a ∈ R and b ∈ (0 , ∞ ) such that q ( ξ ) ≥ a + b ˆ | ξ | if ´ ξ = 0 , or ´ ξ ≥ and lim R ∋ λ →∞ q ( λ ) > q (0) , or ´ ξ ≤ and lim R ∋ λ →−∞ q ( λ ) > q (0) . If A ⊂ { } × L ( X ) , then a can be chosen . roof. Fix ( a, f ) ∈ A with f nonconstant. If α ∈ (0 , let(6.2) s α = s α ( f ) = sup µ ( E )= α E f, i α = i α ( f ) = inf µ ( E )= α E f, and let s = ess sup f , i = ess inf f . For every α > there is an S = S α ⊂ X ofmeasure α for which ffl S f = s α . Indeed, consider u = inf (cid:8) t ∈ R : µ { f > t } ≤ α (cid:9) . Since µ { f > u } ≤ α ≤ µ { f ≥ u } , any set S of measure α sandwiched between { f > u } and { f ≥ u } will provide the sup in (6.2). Similarly, S ′ = X \ S , of measure − α ,satisfies i − α = ffl S ′ f . This implies that s α > i − α . From the absolute continuity of f dµ with respect to dµ we deduce that s α , i α are continuous functions of α > ; continuitytrivially holds at α = 0 as well. Hence(6.3) c = 2 c ( f ) = min ≤ α ≤ ( s α − i − α ) > , m = 2 m ( f ) = max ≤ α ≤ | s α | + | i − α | < ∞ . Consider a ξ ∈ L ∞ ( X ) and let T = { ξ ≥ } . With α = µ ( T ) and S = S α as above,Lemma 4.4 implies(6.4) q ( ξ ) ≥ a + s α ˆ ξ + − i − α ˆ ξ − = a + s α − i − α ˆ | ξ | + s α + i − α ˆ ξ (even if α = 0 ). When ´ ξ = 0 , by (6.3) we obtain q ( ξ ) ≥ a + c ´ | ξ | .Next suppose that lim λ →∞ q ( λ ) > q (0) . There are λ > and ( a , f ) ∈ A with a + ´ f λ > q (0) ≥ a ; hence ´ f > . Because A is convex, we can arrange that ourfixed ( a, f ) ∈ A already satisfies ´ f > . Let b = s c/ ( s + m ) . We will show that if ´ ξ ≥ , then q ( ξ ) ≥ a + b ´ | ξ | . Note that the constant function f ′ = ´ f is in conv ( f ) according to Lemma 4.5, and ( a, f ′ ) is in A . Hence q ( ξ ) ≥ a + ´ f ′ ξ = a + s ´ ξ . By(6.4) q ( ξ ) ≥ a + c ´ | ξ | − m ´ ξ . Combining these two we can eliminate ´ ξ and obtain mq ( ξ ) + s q ( ξ ) ≥ ( m + s ) a + s c ˆ | ξ | , as needed. Finally, if lim λ →−∞ q ( λ ) > q (0) , we choose ( a, f ) ∈ A such that f isnonconstant and ´ f < . Letting b = c ( f ) | s ( f ) | / ( | s ( f ) | + m ( f )) we can similarlyprove q ( ξ ) ≥ a + b ´ | ξ | whenever ´ ξ ≤ . This completes the proof of the lemma, andalso of the theorem. This was the theorem:
Theorem 7.1.
Given a
Ham( ω ) invariant continuous norm p on C ∞ ( X ) , there is arearrangement invariant Banach function space on X whose norm, restricted to C ∞ ( X ) ,is equivalent to p . e will get to the notion of rearrangement invariant Banach spaces shortly, but firstwe formulate a few auxiliary results that we will need. Let us say that two functions φ, ψ : X → R are similarly ordered if (cid:0) φ ( x ) − φ ( y ) (cid:1)(cid:0) ψ ( x ) − ψ ( y ) (cid:1) ≥ for all x, y ∈ X .Put it differently, φ ( x ) > φ ( y ) should imply ψ ( x ) ≥ ψ ( y ) . In spite of what the languagemay suggest, this is not an equivalence relation (all functions are similarly ordered asa constant). However, it is true that if φ and ψ are similarly ordered, and U : R → R is increasing, then φ and U ◦ ψ are also similarly ordered.We will write φ ∼ ψ for measurable functions X → R if they are equidistributed.The following lemma in one form or another is known and, like Lemmas 7.3, 7.4, 7.5,holds in any finite measure space ( X, µ ) without atoms. Lemma 7.2.
Let φ ∈ L ( X ) be bounded below and ψ ∈ L ∞ ( X ) .(a) sup φ ∼ φ ´ φψ = sup ψ ∼ ψ ´ φ ψ .(b) The suprema in (a) are attained, by φ and ψ that are similarly ordered as ψ and φ .(c) ´ φψ is independent of the choice of φ ∼ φ , ψ ∼ ψ , as long as φ, ψ are similarlyordered.Proof. (b) That the suprema are attained, at least when φ , ψ ≥ , is proved in [BS,Chapter 2, Theorems 2.2 and 2.6]. The general result follows upon adding a constant tothe functions. The proof in [BS, pp. 49-50], say, for the first supremum in (a), proceedsby first considering simple φ and representing the maximizing φ by an explicit formula,then passing to a limit. The formula shows that φ and ψ are similarly ordered when φ is simple; but similar ordering is preserved under pointwise limits, and must hold ingeneral.(c) Again, first assume that ψ is simple, and takes values a < a < · · · < a k . Let A j = { x : ψ ( x ) = a j } . If necessary, we can change the values of φ, ψ on a set of zeromeasure to arrange that each µ ( A j ) > . Let m j = inf A j φ, M j = sup A j φ. If x ∈ A j and y ∈ A j +1 , then ψ ( x ) < ψ ( y ) and φ ( x ) ≤ φ ( y ) , hence(7.1) . . . ≤ m j ≤ M j ≤ m j +1 ≤ . . . It follows that the set B j = { x : m j < φ ( x ) < M j } is included in A j . With C j = { x ∈ A j : φ ( x ) = m j } and D j = { x ∈ A j : φ ( x ) = M j } therefore ˆ φψ = X j a j ˆ A j φ = X j a j ( ´ B j φ + m j µ ( C j ) + M j µ ( D j ) if m j < M j m j µ ( A j ) if m j = M j . We will show that each term on the right is determined by φ , ψ .To start,(7.2) m j = sup n m : µ ( φ ≥ m ) ≥ µ (cid:16) k [ i = j A i (cid:17)o , ecause by (7.1) φ ≤ m j on j − [ i =1 A i , φ ≥ m j on k [ i = j A i . Since µ ( φ ≥ m ) = µ ( φ ≥ m ) and µ (cid:0) S kj A i (cid:1) = µ ( ψ ≥ a j ) , (7.2) shows that the m j are determined by φ , ψ ; and so are the M j . It follows that µ ( B j ) = µ ( m j < φ < M j ) and ˆ B j φ = ˆ { m j <φ Lemma 7.4. If φ , ψ ∈ L ∞ ( X ) , then (7.3) sup φ ∼ φ ˆ | φ | ψ ≤ sup φ ∼ φ ˆ φψ + sup φ ∼ φ ˆ ( − φ ) ψ + | φ | ˆ ψ. Proof. First we estimate ´ φ + ψ . By Lemma 7.2 we can choose φ ∼ φ , similarlyordered as ψ , that realizes sup φ ∼ φ ´ φψ . It follows that φ +1 , a composition of φ withan increasing function, is also similarly ordered as ψ . Using Lemma 7.2 once more weobtain sup φ ∼ φ ˆ φ + ψ = ˆ φ +1 ψ = ˆ φ ψ + ˆ φ − ψ. As − φ − and ψ are similarly ordered, Lemma 7.3 gives − ´ φ − ψ ≥ − ffl φ − ´ ψ , and so(7.4) sup φ ∼ φ ˆ φ + ψ ≤ ˆ φ ψ + φ − ˆ ψ = sup φ ∼ φ ˆ φψ + φ − ˆ ψ. eplacing φ with − φ ,(7.5) sup φ ∼ φ ˆ φ − ψ ≤ sup φ ∼ φ ˆ ( − φ ) ψ + φ +0 ˆ ψ, and (7.3) follows by adding (7.4) and (7.5). Lemma 7.5. If f , ξ ∈ L ∞ ( X ) then sup f ∼ f ´ | f ξ | ≤ f ∼ f | ´ f ξ | + 3 ffl | f | ´ | ξ | .Proof. Let us start with a simple ξ . Lemma 7.4, with φ = f , ψ = | ξ | gives(7.6) sup f ∼ f ˆ | f ξ | ≤ f ∼ f (cid:12)(cid:12)(cid:12) ˆ f | ξ | (cid:12)(cid:12)(cid:12) + | f | ˆ | ξ | . By Lemma 7.2(7.7) sup f ∼ f ˆ f | ξ | = sup ζ ∼| ξ | ˆ f ζ. Any ζ ∼ | ξ | can be written as ζ = | η | with η ∼ ξ . Indeed, suppose ξ takes distinctvalues a , . . . , a k . If for some i there is no j with a i = − a j , we let η ≡ a i on the set ( ζ = | a i | ) . If for some i there is a (necessarily unique) j with a i = − a j , for each suchpair we divide the set ( ζ = | a i | = | a j | ) in two parts, of measures µ ( ξ = a i ) , µ ( ξ = a j ) ,and define η ≡ a i on the former, η ≡ a j on the latter.Hence, applying Lemma 7.4 again, this time with φ = ξ , ψ = f , we obtain sup ζ ∼| ξ | ˆ f ζ = sup η ∼ ξ ˆ f | η | ≤ η ∼ ξ (cid:12)(cid:12)(cid:12) ˆ f η (cid:12)(cid:12)(cid:12) + f ˆ | ξ | . In light of (7.7) and Lemma 7.2 therefore sup f ∼ f ˆ f | ξ | ≤ f ∼ f (cid:12)(cid:12)(cid:12) ˆ f ξ (cid:12)(cid:12)(cid:12) + | f | ˆ | ξ | . Substituting this, and its counterpart with f replaced by − f , into (7.6) gives thelemma, when ξ is simple. A general ξ can be uniformly approximated by simple func-tions ξ m , and knowing the estimate for each ξ m gives the estimate for ξ in the limit. Proof of Theorem 7.1. By Lemma 2.1 p ( ξ ) = sup { ´ f ξ : f ∈ F } with a family F ⊂ L ∞ ( X ) , that we can choose to be invariant under Ham( ω ) . Because of Lemma 3.2 wecan even choose it to be invariant under strict rearrangements. For any measurable ζ : X → [ −∞ , ∞ ] define q ( ζ ) = sup n ˆ | f ζ | : f ∈ F o ∈ [0 , ∞ ] , and let B = { ζ : q ( ζ ) < ∞} , k k = q | B . Some obvious properties of q are: it is positivelyhomogeneous, q ( η + ζ ) ≤ q ( η ) + q ( ζ ) , and | η | ≤ | ζ | a.e. implies q ( η ) ≤ q ( ζ ) . If q ( ζ ) = 0 then ζ = 0 a.e. on any set where some f ∈ F is nonzero; since F is invariant understrict rearrangements, this simply means ζ = 0 a.e. By Lemma 4.3 sup f ∈F ´ | f | < ∞ , ence L ∞ ( X ) ⊂ B . Furthermore, q is invariant under all rearrangements, strict or not;this also implies by Lemma 6.1, with a suitable b > ,(7.8) q ( ζ ) ≥ b ˆ | ζ | if ζ ∈ L ∞ ( X ) .Following Bennett–Sharpley’s definition [BS, pp. 2, 59], ( B, k k ) is a rearrange-ment invariant Banach space if, in addition to the properties above, (7.8) holds for allmeasurable ζ , and(7.9) lim k →∞ q ( ζ k ) = q ( ζ ) for every increasing sequence ζ k ≥ converging to ζ . We start with the latter. On theone hand, since q is monotone, the limit in (7.9) exists, and is ≤ q ( ζ ) . On the other,the monotone convergence theorem implies that with any f ∈ F ˆ | f ζ | = lim k →∞ ˆ | f ζ k | ≤ lim k →∞ q ( ζ k ) . Taking the sup over all f ∈ F we obtain q ( ζ ) ≤ lim k q ( ζ k ) , which proves (7.9). That(7.8) holds for all measurable ζ now follows because | ζ | is the limit of an increasingsequence of functions in L ∞ ( X ) .It remains to verify that p and k k are equivalent on C ∞ ( X ) . Clearly p ≤ k k . ByLemma 7.5 k ξ k = sup f ∈F ˆ | f ξ | ≤ f ∈F (cid:12)(cid:12)(cid:12) ˆ f ξ (cid:12)(cid:12)(cid:12) + 3 sup f ∈F | f | ˆ | ξ | , ξ ∈ C ∞ ( X ) . Equivalence follows, because the first supremum on the right is p ( ξ ) and the last termis ≤ Cp ( ξ ) by Lemma 4.3 and Theorem 1.3. The construction of a smooth,