aa r X i v : . [ m a t h . F A ] M a y LIPSCHITZ FREE p -SPACES FOR < p < FERNANDO ALBIAC, JOS´E L. ANSORENA, MAREK C ´UTH,AND MICHAL DOUCHA
Abstract.
This paper initiates the study of the structure of anew class of p -Banach spaces, 0 < p <
1, namely the Lipschitz free p -spaces (alternatively called Arens-Eells p -spaces) F p ( M ) over p -metric spaces. We systematically develop the theory and showthat some results hold as in the case of p = 1, while some newinteresting phenomena appear in the case 0 < p < p -space over a separable ultrametric spaceis isomorphic to ℓ p for all 0 < p ≤
1. On the other hand, solving aproblem by the first author and N. Kalton, there are metric spaces
N ⊂ M such that the natural embedding from F p ( N ) to F p ( M )is not an isometry. Introduction
It is safe to say that most of the research in functional analysis isdone in the framework of Banach spaces. While the theory of thegeometry of these spaces has evolved very rapidly over the past sixtyyears, by contrast, the study of the more general case of quasi-Banachspaces has lagged far behind despite the fact that the first papers in thesubject appeared in the early 1940’s ([5, 8]). The neglect of non-locallyconvex spaces within functional analysis is easily understood. Evenwhen they are complete and metrizable, working with them requiresdoing without one of the most powerful tools in Banach spaces: theHahn-Banach theorem and the duality techniques that rely on it. Thisdifficulty in even making the simplest initial steps has led some toregard quasi-Banach spaces as too challenging and consequently theyhave been assigned a secondary role in the theory. However, these
Mathematics Subject Classification.
Key words and phrases.
Quasimetric space, quasi-Banach space, metric enve-lope, Lipschitz free p -space, Arens-Eells p -space.Accepted in Israel J. Math. In this updated version we corrected some misprintsand moved the result concerning embeddability of ℓ p from the previous version toa new preprint entitled Embeddability of ℓ p and bases in Lipschitz free p -spaces for < p ≤
1, which is available on arXiv. challenges have been accepted by some researchers and the numberof fresh techniques available in this general setting is now increasing(see a summary in [13]). We emphasize that proving new results in p -Banach spaces for 0 < p < p = 1. Hence, quasi-Banach spaces help us appreciatebetter and also shed new light on regular Banach spaces. Taking intoaccount that more analysts find that quasi-Banach spaces have uses intheir research, the task to know more about their structure seems tobe urgent and important.Every family of classical Banach spaces, like the sequence spaces ℓ p ,the function spaces L p , the Hardy spaces H p , and the Lorentz sequencespaces d ( w, p ), have a non-locally convex counterpart corresponding tothe values of 0 < p <
1. In this paper we study Lipschitz free p -spacesover quasimetric spaces. These new class of p -Banach spaces, denotedby F p ( M ), are an analogy of the Lipschitz free spaces F ( M ), whosestudy has become a very active research field within Banach spacetheory since the appearance in 1999 of the important book [23] byWeaver (here we cite the updated second edition) and, more notably,after the seminal paper [11] by Godefroy and Kalton in 2003.Lipschitz free p -spaces were introduced in [3] with the sole instru-mental purpose to build examples for each 0 < p < separable p -Banach spaces which are Lipschitz-isomorphic but fail to be linearlyisomorphic. Whether this is possible or not for p = 1 remains as of to-day the single most important open problem in the theory of non-linearclassification of Banach spaces. However, even though Lipschitz free p -spaces were proved to be of substantial utility in functional analysis,the structure of those spaces has not been investigated ever since. Ourgoal in this paper is to fill this gap in the theory, to encourage furtherresearch in this direction, and help those who want to contribute tothis widely unexplored topic.To that end, after the preliminary Sect. 2 on the basics in quasimetricand quasi-Banach spaces, in Sect. 3 we introduce the notion of metricenvelope of a quasimetric space M and relate it to the existence ofnon-constant Lipschitz maps on M as well as to the Banach envelopewhen M is a quasi-Banach space. In Sect. 4 we recall the definition ofLipschitz free p -space and bring up to light the main differences andsetbacks of this theory with respect to the case p = 1. We also settlea question that was raised in [3] and use molecules and atoms in orderto give an alternative equivalent definition of Lispchitz free p -spaceswhich will be very useful in order to provide examples of Lipschitzfree p -spaces isometrically isomorphic to ℓ p and L p for 0 < p <
1. In
IPSCHITZ FREE p -SPACES 3 Sect. 5 we completely characterise Lipschitz free p -spaces over separableultrametric spaces, showing that for p ≤ ℓ p .The most important results are perhaps the ones in Sect. 6, wherewe study the relation between the subset structure of a quasimetricspace M and the subspace structure of F p ( M ). To be precise, for each p < p < q ≤ N ofa q -metric space M such that F p ( N ) is not naturally a subspace of F p ( M ). This fact evinces a very important dissimilarity with respectto the case p = 1 and solves another problem raised in [3].Throughout this note we use standard terminology and notation inBanach space theory as can be found in [4]. We refer the reader to [23]for basic facts on Lipschitz free spaces and some of their uses, and to[14] for background on quasi-Banach spaces.2. Preliminaries
There are two main goals in this preliminary section. First we reviewthe notion of quasi-metric space along with the related notion of quasi-Banach space and their main topological features. Second, we lay outthe notation and terminology used in this article.2.1.
Quasimetric spaces and Lipschitz maps.
Given an arbitrarynonempty set M , a quasimetric on M is a symmetric map ρ : M ×M → [0 , ∞ ) such that ρ ( x, y ) = 0 if and only if x = y , and for someconstant κ ≥ ρ satisfies the quasi-triangle inequality ρ ( x, z ) ≤ κ ( ρ ( x, y ) + ρ ( y, z )) , x, y, z ∈ M . (2.1)The space ( M , ρ ) is then called a quasimetric space (see [16, p. 109]).A quasimetric ρ on a set M is said to be a p -metric , 0 < p ≤
1, if ρ p is a metric, i.e., ρ p ( x, y ) ≤ ρ p ( x, z ) + ρ p ( z, y ) , x, y, z ∈ M , in which case we call ( M , ρ ) a p -metric space . An analogue of the Aoki-Rolewicz theorem holds in this context (see [16, Proposition 14.5]):every quasimetric space can be endowed with an equivalent p -metric τ for some 0 < p ≤
1, i.e., there is a constant C = C ( κ ) ≥ C − τ ( x, y ) ≤ ρ ( x, y ) ≤ Cτ ( x, y ) , x, y ∈ M . If ( M , ρ ) and ( N , τ ) are quasimetric spaces we shall say that a map f : M → N is Lipschitz if there exists a constant C ≥ τ ( f ( x ) , f ( y )) ≤ Cρ ( x, y ) , x, y ∈ M . (2.2) F. ALBIAC, J. L. ANSORENA, M. C ´UTH, AND M. DOUCHA
We denote by Lip( f ) the smallest constant which can play the role of C in the last inequality (2.2), i.e.,Lip( f ) = sup (cid:26) τ ( f ( x ) f ( y )) ρ ( x, y ) : x, y ∈ M , x = y (cid:27) ∈ [0 , ∞ ) . If f is injective, and both f and f − are Lipschitz, then we say that f is bi-Lipschitz and that M Lipschitz-embeds into N . If there is abi-Lipschitz map from M onto N , the spaces M and N are said to be Lipschitz isomorphic . A map f from a quasimetric space ( M , ρ ) intoa quasimetric space ( N , τ ) is an isometry if τ ( f ( x ) , f ( y )) = ρ ( x, y ) , x, y ∈ M . We shall say that ( M , ρ ) is a pointed quasimetric space (or a pointed p -metric space, or a pointed metric space) , if it has a distinguished pointthat we call the origin and denote 0. The assumption of an origin isconvenient to normalize Lipschitz functions.The Lipschitz dual of a quasimetric space ( M , ρ ), denoted Lip ( M ),is the (possibly trivial) vector space of all real-valued Lipschitz func-tions f defined on M such that f (0) = 0, endowed with the Lipschitznorm k f k Lip = sup (cid:26) | f ( x ) − f ( y ) | ρ ( x, y ) : x, y ∈ M , x = y (cid:27) . It can be readily checked that (Lip ( M ) , k · k Lip ) is a Banach space.2.2.
Quasi-normed spaces and their Banach envelopes.
Recallthat a quasi-normed space is a (real) vector space X equipped with amap k · k X : X → [0 , ∞ ) with the properties:(i) k x k X > x = 0,(ii) k αx k X = | α |k x k X for all α ∈ R and all x ∈ X ,(iii) there is a constant κ ≥ x and y ∈ X we have k x + y k X ≤ κ ( k x k X + k y k X ) . (2.3)A quasi-norm k · k X induces a linear metric topology. X is called a quasi-Banach space if X is complete for this metric. Given 0 < p ≤ X is said to be a p -normed space if the quasi-norm k · k X verifies (i),(ii) and it is p -subadditive, i.e.,(iv) k x + y k pX ≤ k x k pX + k y k pX for all x, y ∈ X .Of course, (iv) implies (iii), and, by the Aoki-Rolewicz theorem (see[14]), we also have that (iii) implies (iv). In the case when X is p -normed, a metric inducing the topology can be defined by d ( x, y ) = k x − y k pX . A quasi-Banach space with an associated p -norm is alsocalled a p -Banach space . IPSCHITZ FREE p -SPACES 5 A map k · k X : X → [0 , ∞ ) that verifies properties (ii) and (iv) iscalled a p -seminorm on X . Given a p -seminorm k· k X on a vector space X it is standard to construct a p -Banach space from the pair ( X, k · k X )following the so-called completion method . For that we consider thevector subset N = { x ∈ X : k x k X = 0 } and form the quotient space X/N , which is p -normed when endowed with k · k X . Now we justneed to complete ( X/N, k · k X ). The reader should be acquainted withthe fact that completeness and completion for quasi-metric spaces arecompletely analogous to such notions for metric spaces.Given 0 < p ≤
1, a subset C of a vector space V is said to be absolutely p -convex if for any x and y ∈ C and any scalars λ and µ with | λ | p + | µ | p ≤ λ x + µ y ∈ C . The Minkowski functional k · k C of an absolutely p -convex set C , given by k x k C = inf (cid:8) λ > λ − x ∈ C (cid:9) , defines a p -seminorm on span( C ).Given a nonempty subset Z of a vector space V there is a methodfor building a p -Banach space from it. Let co p ( Z ) denote the p -convexhull of Z , i.e., the smallest absolutely p -convex set containing Z . If N = { x ∈ span( Z ) : k x k co p ( Z ) = 0 } , then the quotient space span( Z ) /N equipped with k · k co p ( Z ) is a p -normed linear space. In the case whenspan( Z ) ∗ separates the points of span( Z ) then k · k co p ( Z ) is a p -norm. Definition . The completion of (span( Z ) /N, k · k co p ( Z ) ) will be calledthe p -Banach space constructed from Z by the p -convexification method and will be denoted by ( X p,Z , k · k p,Z ).Notice that it is possible to give an explicit expression for k · k p,Z .As a matter of fact, for x ∈ X p,Z we have k x k p,Z = inf ∞ X j =1 | a j | p ! /p : x = ∞ X j =1 a i x i , x i ∈ Z . (2.4)When dealing with a quasi-Banach space X it is often convenientto know which is the “smallest” Banach space containing X or, moregenerally, given 0 < q ≤
1, the smallest q -Banach space containing X . Definition . Given a quasi-Banach space X and 0 < q ≤
1, the q -Banach envelope of X (resp. Banach envelope for q = 1), denoted( c X q , k · k c,q ) (resp. ( b X, k · k c ) for q = 1) is the q -Banach space obtainedby applying to the unit ball B X of X the q -convexification method.Obviously k x k c,q ≤ k x k for all x ∈ X , so that the identity map on X induces a (not necessarily one-to-one) bounded linear map i X,q : X → F. ALBIAC, J. L. ANSORENA, M. C ´UTH, AND M. DOUCHA c X q whose range is dense in c X q . This map possesses the followinguniversal property: if T : X → Y is a bounded linear map and Y is anarbitrary q -Banach space then T factors through i X,q , X T / / i X,q ❆❆❆❆❆❆❆❆ Y c X q b T > > ⑥⑥⑥⑥⑥⑥⑥ and the unique “extension” b T : c X q → Y has the same norm as T . Inparticular, X and c X q have the same dual space.For instance, the q -Banach envelope of ℓ p for 0 < p < q ≤ ℓ q .The following formula for the q -Banach envelope quasi-norm will bevery useful. The case q = 1 was shown by Peetre in [19]. Lemma 2.3.
Let X be a quasi-Banach space and < q ≤ . Then for x ∈ X , k x k c,q = inf n X i =1 k x i k q ! /q : n X i =1 x i = x, x i ∈ X, n ∈ N . (2.5) Proof.
Let k · k be the q -seminorm on X defined by the expression in(2.5) and X be the q -Banach space obtained from ( X, k · k ) by thecompletion method. If T : X → Y is a bounded linear map and Y is q -Banach, then k T ( x ) k ≤ k T kk x k . Consequently, X has the sameuniversal property as c X q , thus X and c X q are isometric. (cid:3) p -norming sets in quasi-Banach spaces. Definition . Given a quasi-Banach space X and 0 < p ≤
1, we saythat a subset Z of X is a p -norming set with constants C and D if1 C co p ( Z ) ⊆ B X ⊆ D co p ( Z ) . In the case when C = D = 1 we say that Z is isometrically p -norming .Note that Z is a p -norming set of X if and only if k · k p,Z defines anequivalent quasi-norm on X . Consequently, if X admits a p -normingset then X is isomorphic to a p -Banach space. Conversely, if X is a p -Banach space, then a set Z ⊆ X is p -norming with constants C and D if and only if 1 C Z ⊆ B X ⊆ D co p ( Z ) . (2.6)Adopting the terminology from harmonic analysis it can be said that aset Z is p -norming in X if and only if ( Z, ℓ p ) is an atomic decomposition IPSCHITZ FREE p -SPACES 7 of X . Recall that a pair ( A , S ), where A is a subset of X and S is asymmetric sequence space, is said to be an atomic decomposition of X if there are constants 0 < C, D < ∞ such that(i) Given f = ( a n ) ∞ n =1 ∈ S and ( α n ) ∞ n =1 ⊂ A then P ∞ n =1 a n α n converges in X to a vector x verifying k x k ≤ C k f k S , and(ii) for any x ∈ X there are f = ( a n ) ∞ n =1 ∈ S and ( α n ) ∞ n =1 ⊂ A suchthat x = P ∞ n =1 a n α n and k f k S ≤ D k x k .We conclude this preliminary section enunciating for future referencea few straightforward auxiliary results on p -norming sets. Lemma 2.5.
Suppose Z and Z are subsets of a quasi-Banach X suchthat Z ⊆ Z , Z is dense in Z , and Z is p -norming in X . Then Z is a p -norming set in X with the same constants as Z . Lemma 2.6.
Suppose that Z and Z are p -norming sets for quasi-Banach spaces X and X , respectively. Let T be a one-to-one linearmap from span( Z ) into X such that T ( Z ) = Z . Then T extendsto an onto isomorphism e T : X → X . Moreover, in the case when Z and Z are both isometrically p -norming sets, e T is an isometry. Lemma 2.7.
Suppose that Z is a p -norming set for a quasi-Banachspace X with constants C and C and that Z ⊆ Z . If there is aconstant C such that every x ∈ Z can be written as x = P ∞ n =1 a n x n for some f = ( a n ) ∞ n =1 ∈ ℓ p with k f k p ≤ C and ( x n ) ∞ n =1 in Z , then Z is a p -norming set for X with constants C and CC .Proof. By hypothesis Z ⊆ C co p ( Z ). Therefore co p ( Z ) ⊆ C co p ( Z ),and so 1 C co p ( Z ) ⊆ C co p ( Z ) ⊆ B X ⊆ C co p ( Z ) ⊆ CC co p ( Z ) . (cid:3) The metric envelope of a quasimetric space
Suppose ( M , ρ ) is a pointed quasimetric space. By analogy with theuniversal extension property of the Banach envelope of a quasi-Banachspace, we are interested in the question on how to construct a metricspace ( f M , e ρ ), and a map Q : M → f M with Lip( Q ) ≤ M, d ) is a metric space and f : M → M verifies the Lipschitzcondition d ( f ( x ) , f ( y )) ≤ Cρ ( x, y ) , x, y ∈ M , (3.7)then f induces a Lipschitz map e f : f M → M with f = e f ◦ Q and d ( e f ( x ) , e f ( y )) ≤ C e ρ ( x, y ) for all x, y ∈ f M . F. ALBIAC, J. L. ANSORENA, M. C ´UTH, AND M. DOUCHA
Note that if f verifies (3.7), then we will have d ( f ( x ) , f ( y )) ≤ C n X i =0 ρ ( x i , x i +1 ) , for any finite sequence x = x , x , . . . , x n +1 = y of (possibly repeated)points in M . Therefore, in all fairness we define, for x, y ∈ M , e ρ ( x, y ) = inf n X i =0 ρ ( x i , x i +1 ) , (3.8)where the infimum is taken over all sequences x = x , x , . . . , x n +1 = y of finitely-many points in M . Clearly, e ρ is symmetric, satisfies thetriangle inequality, and does not exceed ρ . Before going on, let uspoint out that e ρ ( x, y ) can be zero for different points x, y in M . Example . A metric space ( M , d ) is metrically convex (see [6]) if forevery x, y ∈ M and any 0 < λ < z λ ∈ M with d ( x, z λ ) = λd ( x, y ) and d ( y, z λ ) = (1 − λ ) d ( x, y ) . Let ( M , d ) be a metrically convex space and, for 0 < p <
1, considerthe p -metric ρ = d /p on M . Then e ρ ( x, y ) = 0 for any x, y ∈ M .Indeed, given x = y in M , by the metric convexity of M for every n ∈ N we can find a chain of points { x , x , . . . , x n } where x = x , x n = y , and d ( x j − , x j ) = d ( x, y ) /n for each j = 1 , , . . . , n . By thedefinition we then have e ρ ( x, y ) ≤ (cid:18) d ( x, y ) n (cid:19) /p n = d ( x, y ) n /p − → . Thus, e ρ ( x, y ) = 0.In view of that, we shall identify points in M that are at a zero e ρ -distance, which leads to the following definition. Definition . Let ( M , ρ ) be a quasimetric space and ˜ ρ as in (3.8).We consider the equivalence relation x ∼ y ⇐⇒ e ρ ( x, y ) = 0 , and define f M to be the quotient space M / ∼ . If e x and e y denote therespective equivalence classes of x and y , we put ˜ ρ ( e x, ˜ y ) = e ρ ( x, y ). Themetric space ( f M , e ρ ), together with the quotient map Q : M → f M willbe called the metric envelope of ( M , ρ ).Our discussion yields that the metric envelope of a quasimetric spaceis characterized by the following universal property. IPSCHITZ FREE p -SPACES 9 Theorem 3.3.
Suppose ( f M , e ρ, Q ) is the metric envelope of a quasi-metric space ( M , ρ ) . Then: (i) Lip( Q ) = 1 , and (ii) whenever ( M, d ) is a metric space and f : ( M , ρ ) → ( M, d ) is C -Lipschitz, there is a unique map e f : ( f M , e ρ ) → ( M, d ) suchthat f = e f ◦ Q is C -Lipschitz. Pictorially, ( M , ρ ) f / / Q $ $ ■■■■■■■■■ ( M, d )( f M , e ρ ) e f : : ✉✉✉✉✉✉✉✉ Remark . Theorem 3.3 can be rephrased as saying that for everymetric space (
M, d ) the mapping g Q ◦ g defines an isometry fromLip ( f M , M ) onto Lip ( M , M ), and so these two spaces can be natu-rally identified.Note that, in this language, Example 3.1 yields that for 0 < p < R equipped with the p -metric ρ ( x, y ) = | x − y | /p is trivial. On the other hand, by [1, Lemma 2.7], Lip ( R , ρ ) = { } .Next we see that this is not a coincidence. Proposition 3.5.
Given a quasimetric space ( M , ρ ) the following areequivalent. • ( f M , ˜ ρ ) is trivial. • Lip ( M , M ) = { } for any metric space ( M, d ) . • Lip ( M ) = { } .Proof. If ( f M , ˜ ρ ) is trivial it is clear that Lip ( f M , M ) = { } for anymetric space M . Using Remark 3.4 we get Lip ( M , M ) = { } .If Lip ( M , M ) = { } for any metric space M in particular it holdsfor M = R , i.e., Lip ( M ) = { } .Finally, if ( f M , ˜ ρ ) is non-trivial then clearly Lip ( f M ) is non-trivialand so by Remark 3.4 we get Lip ( M ) = { } . (cid:3) Example . Let 0 < p <
1. We know that the p -metric space L p [0 , p -metric induced by the p -norm, given by ρ ( f, g ) = k f − g k p , f, g ∈ L p [0 , ( L p [0 , { } (see [1, Proposition 2.8]). Then, by Proposi-tion 3.5 we infer that its metric envelope is trivial.Let us next show that the fact ^ L p [0 ,
1] = { } is related to thewell-known property that the Banach envelope of the p -Banach space L p [0 ,
1] for 0 < p <
Proposition 3.7.
Let ( X, k · k ) be a p -normed space. Consider on X the p -metric ρ given by ρ ( x, y ) = k x − y k and let be the distinguishedpoint of X . Then e ρ ( x, y ) = k x − y k c for all x, y ∈ X , where k · k c isthe norm introduced in Definition 2.2.Proof. The set of all tuples ( y j ) nj =0 with y = x and y n = y coincideswith the set of all tuples of the form ( x + P jk =0 x k ) nj =0 , where x = 0and P nj =1 x j = y − x . Hence, e ρ ( x, y ) = inf ( n X j =1 ρ x + j X k =1 x k , x + j − X k =1 x k ! : n X j =1 x j = y − x ) = inf ( n X j =1 k x j k : n X j =1 x j = y − x ) = k y − x k c . (cid:3) Remark . Note that it is possible to extend Definition 3.2 and Theo-rem 3.3 to the case when 0 < q <
1. Indeed, given a pointed quasimet-ric space ( M , ρ ) we define its q -metric envelope ( g M q , e ρ q ) following thesame steps as in the construction of its metric envelope. The q -metric e ρ q is given by e ρ q ( x, y ) = inf n X i =0 ρ q ( x i , x i +1 ) ! /q , x, y ∈ M , the infimum being taken over all finite sequences x = x , x , . . . , x n +1 = y of points in M , and g M q is the quotient space M / ∼ q . The equiva-lence relation here is the expected one, i.e., x ∼ q y ⇐⇒ e ρ q ( x, y ) = 0 . Thus ( g M q , e ρ q ) is the pointed q -metric space (having as a distinguishedpoint the equivalence class of 0) characterized by the following universalproperty: whenever ( M, d ) is a q -metric space and f : ( M , ρ ) → ( M, d )is C -Lipschitz then the map e f : ( g M q , e ρ q ) → ( M, d ) such that f = e f ◦ Q is C -Lipschitz, where Q : M → g M q denotes the canonical quotientmap: ( M , ρ ) f / / Q % % ❑❑❑❑❑❑❑❑❑ ( M, d )( g M q , e ρ q ) e f ttttttttt IPSCHITZ FREE p -SPACES 11 Notice also that if we regard a p -Banach space ( X, k·k X ) as a pointed p -metric space in the obvious way (i.e., by taking 0 as the origin of thevector space X equipped with the p -metric ρ ( x, y ) = k x − y k X ), Propo-sition 3.7 can be generalized as well, and we can come to the conclusionthat the q -Banach envelope of X is the completion of its q -metric enve-lope. We leave out for the reader to check the straightforward details.4. Lipschitz free p -spaces over quasimetric spaces Every metric space embeds isometrically into a Banach space. Simi-larly, the natural environment to isometrically embed quasimetric spaceswill be p -Banach spaces. Notice that for 0 < p <
1, every pointed p -metric space M embeds isometrically into a “huge” p -Banach space,namely the space Y = ℓ ∞ ( M ; L p (0 , ∞ )) of bounded functions from M into the real space ( L p (0 , ∞ ) , k · k p ) endowed with the p -norm k f k Y = sup x ∈M k f ( x ) k p . Indeed, with the convention that χ ( a,b ] = − χ ( b,a ] if a > b , the mapΨ : M → ℓ ∞ ( M ; L p (0 , ∞ )) given by Ψ( x ) = (cid:0) χ ( ρ p (0 ,y ) ,ρ p ( x,y )] (cid:1) y ∈M doesthe job (see [3, Proposition 3.3]). Of course, depending on the p -metricspace we can find simpler (isometric) embeddings, like the mapΦ : ( R , | · | /p ) → L p ( R ) , Φ( x ) = χ (0 ,x ] . (4.9)Once we have accomplished the task to embed a p -metric space M into a p -Banach space, it seems natural to look for an “optimal” wayto do it, in the sense that every Lipschitz map from M into a p -Banachspace factors through it. The following construction from [3] attainsthis goal.Let R M be the space of all (not necessarily continuous) maps f : M → R so that f (0) = 0 and let P ( M ) be the linear span in the linear dual( R M ) of the evaluations δ ( x ), where x runs through M , defined by h δ ( x ) , f i = f ( x ) , f ∈ R M . (4.10)Note that δ (0) = 0 . If µ = P Nj =1 a j δ ( x j ) ∈ P ( M ), put k µ k F p ( M ) = sup (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) N X j =1 a j f ( x j ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) Y , (4.11)the supremum being taken over all p -normed spaces ( Y, k · k Y ) and all1-Lipschitz maps f : M → Y with f (0) = 0. It is straightforward tocheck that formula (4.11) defines a p -seminorm on P ( M ). In fact, the following proposition shows that k · k F p ( M ) is a p -norm, thus settling aquestion posed in [3]. Proposition 4.1.
Let ( M , ρ ) be a pointed p -metric space, < p ≤ .Then ( P ( M ) , k · k F p ( M ) ) is a p -normed space.Proof. Suppose that k P Nj =1 a j δ ( x j ) k F p ( M ) = 0 for some ( a j ) Nj =1 scalarsand some ( x j ) Nj =1 in M \ { } . Then P Nj =1 a j f ( x j ) = 0 for every p -Banach space X and every Lipschitz map f : M → X with f (0) = 0.Pick i ∈ { , . . . , N } and for the sake of convenience denote the dis-tinguished point of M by x . Since the set N = { x j : 0 ≤ j ≤ N } is finite, the map from the metric space ( N , ρ p ) into ( R , | · | ) given by x i x j j = i is Lipschitz. By McShane’s theorem,it extends to a Lipschitz map g from ( M , ρ p ) into ( R , | · | ). In otherwords, the map g is Lipschitz from ( M , ρ ) into ( R , | · | /p ). If Φ is as in(4.9), then f := Φ ◦ g : M → L p ( R ) is Lipschitz as well. Since f (0) = 0,we infer that a i χ (0 , = P Nj =1 a j f ( x j ) = 0. Hence, a i = 0. (cid:3) Definition . Given a p -metric space M , the Lipschitz free p -space over M , denoted by F p ( M ), is the p -Banach space resultingfrom the completion of the p -normed space ( P ( M ) , k · k F p ( M ) ). We willrefer to the map δ M : M → F p ( M ) given by δ M ( x ) = δ ( x ) as thenatural embedding of M into F p ( M ).In [15], Kalton uses the symbol F ω ( M ) to denote Lipschitz-free Ba-nach spaces associated with metric spaces equipped with distances ω ◦ d that arise after snowflaking, where ω is a gauge. This is of course dif-ferent from what is considered in the present work, but we want thereader to be warned to avoid possible confusions. Note that our con-siderations are also independent of the work of Petitjean in [20], wherehe studies Lipschitz-free spaces over metric spaces induced by p -norms. Remark . The choice of a base point in M is not relevant in the def-inition of F p ( M ). Indeed, if we change the origin in M and apply theconstruction, we have a natural linear isometry between the resultingLipschitz free p -spaces.For expositional ease and further reference, let us point out the fol-lowing easy consequence of the proof of Proposition 4.1. Lemma 4.4.
Let ( M , ρ ) be an infinite p -metric space, < p ≤ .Then F p ( M ) is infinite dimensional. Similarly to Lipschitz free Banach spaces over metric spaces, thespaces F p ( M ) for 0 < p < IPSCHITZ FREE p -SPACES 13 Theorem 4.5.
Let ( M , ρ ) be a pointed p -metric space. Then: (a) δ M is an isometric embedding. (b) The linear span of { δ M ( x ) : x ∈ M} is dense in F p ( M ) . (c) F p ( M ) is the unique (up to isometric isomorphism) p -Banachspace such that for every p -Banach space X and every Lipschitzmap f : M → X with f (0) = 0 there exists a unique linear map T f : F p ( M ) → X with T f ◦ δ M = f . Moreover k T f k = Lip( f ) .Pictorially, M f / / δ M ❍❍❍❍❍❍❍❍❍ X F p ( M ) T f ; ; ✇✇✇✇✇✇✇✇✇ Corollary 4.6.
The space F p ( M ) is separable whenever M is.Proof. Note that the map δ : M → F p ( M ) is an isometric embeddingand that F p ( M ) is the closed linear span of δ ( M ). (cid:3) Remark . If p = 1 (so that ρ is a metric) then it follows from theHahn-Banach theorem that F ( M ) is the space denoted by F ( M ) in[11, 15] and the norm of µ = P Nj =1 a j x j ∈ P ( M ) can be computed as k µ k F M ) = sup (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) N X j =1 a j f ( x j ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) , the supremum being taken over all 1-Lipschitz maps f : M → R with f (0) = 0. Moreover, it is known (see, e.g., [23]) that F ( M ) ∗ =Lip ( M ). We advance that the corresponding result also holds for p <
1, i.e., F p ( M ) ∗ = Lip ( M ). We will prove this in Corollary 4.23.Lipschitz free p -spaces provide a canonical linearization process ofLipschitz maps between p -metric spaces: if we identify (through themap δ M ) a p -metric space M with a subset of F p ( M ), then any Lips-chitz map from a p -metric space M to a p -metric space M whichmaps 0 to 0 extends to a continuous linear map from F p ( M ) to F p ( M ). That is: Lemma 4.8 (cf. [11, Lemma 2.2]) . Let M and M be pointed p -metric spaces (0 < p ≤ and suppose f : M → M is a Lipschitzmap such that f (0) = 0 . Then there exists a unique linear operator L f : F p ( M ) → F p ( M ) such that L f δ M = δ M f , i.e., the following diagram commutes M f / / δ M (cid:15) (cid:15) M δ M (cid:15) (cid:15) F p ( M ) L f / / F p ( M ) and k L f k = Lip( f ) . In particular, if f is a bi-Lipschitz bijection then L f is an isomorphism.Proof. Since δ M is an isometric embedding, the map g := δ M ◦ f isLipschitz with g (0) = 0 and Lip( g ) =Lip( f ). Now the result followsfrom Theorem 4.5. (cid:3) Molecules and atomic decompositions.
Given a set M and x ∈ M , let χ x denote the indicator function of the singleton set { x } .Now, for x and y ∈ M we put m x,y := χ x − χ y . Let ( M , ρ ) be a p -metric space for some 0 < p ≤
1. A molecule of M is a function m : M → R that is supported on a finite subset of M andthat satisfies P x ∈M m ( x ) = 0. The vector space of all molecules of ametric space M will be denoted by Mol( M ).A simple induction argument shows that every molecule has at leastone expression as a linear combination of molecules of the form m x,y , sothat Mol( M ) coincides with the linear span of the family of molecules A ′ ( M ) = (cid:26) m x,y ρ ( x, y ) : x, y ∈ M , x = y (cid:27) ⊆ R M . Definition . We define the
Arens-Eells p -space over M , denotedÆ p ( M ), as the p -Banach space constructed from the set A ′ ( M ) usingthe p -convexification method (see Definition 2.1).This way, if we give Mol( M ) the p -seminorm k m k Æ p = inf N X i =1 | a i | p ! /p : m = N X i =1 a i m x i ,y i ρ ( x i , y i ) , N ∈ N , (4.12)we have that Æ p ( M ) is the completion of Mol( M ) (a priori, modulothe set of molecules with zero p -seminorm) with respect to k · k Æ p .However, as we will see below, formula (4.12) defines in fact a p -normon Mol( M ).The following result establishes that the Arens-Eells p -space over M can be identified with the Lipschitz free p -space over M . IPSCHITZ FREE p -SPACES 15 Theorem 4.10.
Let < p ≤ and ( M , ρ ) be a pointed p -metric space.Then F p ( M ) and Æ p ( M ) are isometrically isomorphic. In fact, thereis a linear onto isometry T : F p ( M ) → Æ p ( M ) such that T ( δ ( x )) = χ x − χ = m x, for all x ∈ M .Proof. Consider the map f : M → Æ p ( M ) given by f ( x ) = m x, for x ∈ M . Clearly, f (0) = 0 and f ( x ) − f ( y ) = m x,y for all x , y ∈M . Since f is 1-Lipschitz, Theorem 4.5 yields a norm-one linear map T f : F p ( M ) → Æ p ( M ) such that T f ( δ ( x )) = m x, .Since ( χ x ) x ∈M is a linearly independent family in R M , there is alinear map from span { χ x : x ∈ M} into P ( M ) that takes χ x to δ ( x )for every x ∈ M . Let S be its restriction to Mol( M ). For x, y ∈ M with x = y we have S (cid:18) m x,y ρ ( x, y ) (cid:19) = δ ( x ) − δ ( y ) ρ ( x, y ) , and (cid:13)(cid:13)(cid:13)(cid:13) δ ( x ) − δ ( y ) ρ ( x, y ) (cid:13)(cid:13)(cid:13)(cid:13) F p ( M ) ≤ , so by density S extends to a norm-one operator S from Æ p ( M ) into F p ( M ). Since T ( S ( m )) = m for every molecule m , and S ( T ( µ )) = µ for every µ ∈ P ( M ), by continuity and density it follows that T ◦ S =Id Æ p ( M ) and S ◦ T = Id F p ( M ) . (cid:3) The following two results are re-formulations of Theorem 4.10. Whilethe expression of the norm in (4.11) relies on extraneous ingredients,Corollary 4.12 provides an intrinsic formula for the p -norm on F p ( M ),i.e., an expression that relies only on the quasimetric on the space M . Corollary 4.11.
Let ( M , ρ ) be a pointed p -metric space, < p ≤ .The subset of P ( M ) given by A ( M ) = (cid:26) δ ( y ) − δ ( x ) ρ ( x, y ) : x, y ∈ M , x = y (cid:27) is isometrically p -norming for F p ( M ) . Corollary 4.12.
Let ( M , ρ ) be a pointed p -metric space, < p ≤ .For µ ∈ F p ( M ) we have k µ k F p ( M ) = inf ∞ X k =1 | a k | p ! /p : µ = ∞ X k =1 a k δ ( x k ) − δ ( y k ) ρ ( x k , y k ) . Applications: early examples and results.
Next we use Corol-lary 4.11 to identify the first examples of Lipschitz-free p -spaces overquasimetric spaces for 0 < p <
1. The informed reader will see a re-lation between the map considered in (4.13) and the (quite forgotten)theory of flat spaces, developed by J.J. Sch¨affer and others in the 1970’(see e.g. [12, 21]).
Theorem 4.13.
Let < p ≤ . Let I be an interval of R equippedwith the p -metric ρ ( x, y ) = | x − y | /p for x, y ∈ I . Then F p ( I ) ≈ L p ( I ) isometrically. To be precise, if a is the base point of I , the map F p ( I ) → L p ( I ) , δ I ( x ) χ ( a,x ] (4.13) extends to a linear isometry.Proof. Choose an arbitrary a ∈ I as the base point of ( I, | · | /p ). Set A p,I = (cid:26) χ ( x,y ] | y − x | /p : x, y ∈ I, x < y (cid:27) , and let T : P ( I ) → R I be the linear map determined by δ ( x ) χ ( a,x ] for x ∈ I \ { a } . Using the notation of Corollary 4.11, we put A ( I ) = (cid:26) δ ( x ) − δ ( y ) | x − y | /p : x, y ∈ I, x = y (cid:27) . Since T ( A ( I )) = {± f : f ∈ A p,I } , taking into account Corollary 4.11and Lemma 2.6, it suffices to show that A p,I is an isometric p -normingset for L p ( I ). To that end we need to verify that f ∈ co p ( A p,I ) forevery f ∈ L p ( I ) with k f k p ≤
1. By density it is sufficient to prove itfor step functions. Let f : I → R be a step function, i.e., f = N X j =1 a j χ ( x j − ,x j ] , for some x < x < · · · < x j − < x j < · · · < x N in I and some scalars( a j ) Nj =1 . Then, if b j = ( x j − x j − ) /p a j , we have f = P Nj =1 b j f j with f j ∈ A p,I and N X n =1 | b j | p = n X j =1 | a j | p ( x j − x j − ) = k f k pp . (cid:3) Recall that a quasimetric space M is uniformly separated ifinf { ρ ( x, y ) : x, y ∈ M , x = y } > . Let us note that each bounded and uniformly separated quasimetricspace is Lipschitz isomorphic to the { , } -metric space , i.e., the metricspace whose distance attains only the values 0 and 1. IPSCHITZ FREE p -SPACES 17 Theorem 4.14.
Let M be a bounded and uniformly separated quasi-metric space. For < p ≤ we have F p ( M ) ≈ ℓ p ( M \ { } ) . To beprecise, the map F p ( M ) → ℓ p ( M \ { } ) , δ M ( x ) e x where e x denotes the indicator function of the singleton { x } , extendsto a linear isomorphism.Proof. Without loss of generality we can assume that ( M , ρ ) is the { , } -metric space. If x and y are two different points in M \ { } wecan write δ ( y ) − δ ( x ) ρ ( x, y ) = a xy δ ( x ) ρ (0 , x ) + b x,y δ ( y ) ρ (0 , y ) , where a x,y = − b x,y = 1. Since | a x,y | p + | b x,y | p = 2, by Corol-lary 4.11 and Lemma 2.7, the set A = (cid:26) δ ( x ) ρ (0 , x ) : x ∈ M \ { } (cid:27) = { δ ( x ) : x ∈ M \ { }} , is p -norming for F p ( M ) with constants 1 and 2 /p . Consider the linearmap T : P ( M ) → R M\{ } given by δ ( x ) e x . We have that T ( A ) = A ( M\{ } ) := { e x : x ∈ M\{ }} . Since A ( M\{ } ) is an isometrically p -norming set for ℓ p ( M \ { } ), Lemma 2.6 finishes the proof. (cid:3) Notice that, quantitatively, the proof of Theorem 4.14 gives that if M is equipped with the { , } -metric, then2 − /p X x ∈M\{ } | a x | p /p ≤ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X x ∈M\{ } a x δ ( x ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) F p ( M ) ≤ X x ∈M\{ } | a x | p /p for all scalars ( a x ) x ∈M\{ } eventually null. Going further we are goingto be able to compute the quasi-norms (cid:13)(cid:13)(cid:13)P x ∈M\{ } a x δ ( x ) (cid:13)(cid:13)(cid:13) F p ( M ) in thecase when a x ≥
0. Our argument relies on the construction of a suitable d -dimensional absolutely p -convex body for every d ∈ N . Proposition 4.15.
For every d ∈ N and every < p ≤ , there is a p -norm k · k ( p ) on R d such that: (a) k ( x j ) nj =1 k ( p ) = ( P dj =1 x pj ) /p if x j ≥ for j ∈ { , . . . , d } , and (b) k e i − e j k ( p ) ≤ for all i , j ∈ { , . . . , d } .Proof. Given a vector space V and Z ⊆ V , setco + p ( Z ) = ( k X j =1 λ j v j : k ∈ N , λ j ≥ , k X j =1 λ pj ≤ , v j ∈ Z ) . For d ∈ N , put N [ d ] = { , . . . , d } . Given s = ( s j ) dj =1 ∈ R d , we let M s be the endomorphism of R d given by M s (( x j ) dj =1 ) = ( s j x j ) dj =1 . Given A ⊆ N [ d ], we put M A = M s where s = ( s j ) nj =1 is defined by s j =1 for j ∈ N [ d ] \ A , and s j = − j ∈ A ; that is, M A is the symmetrywith respect to the subspace { ( x j ) dj =1 ∈ R d : x j = 0 for all j ∈ A } .Denote R d + = (cid:8) ( x j ) dj =1 ∈ R d : x j ≥ j ∈ N [ d ] (cid:9) ,B p = ( ( x j ) dj =1 ∈ R d : d X j =1 | x j | p ≤ ) ,B ∞ = (cid:8) ( x j ) dj =1 ∈ R d : | x j | ≤ j ∈ N [ d ] (cid:9) . Given i, j ∈ N [ d ] with i = j , we define Z i,j ⊆ R d + by Z i,j = { a e i + b e j : 0 ≤ a, b ≤ } . Given disjoint sets A , B ⊆ N [ d ] we define T A,B ⊆ R d + by T A,B = { } if A = B = ∅ co + p ( { e i : i ∈ A } )) if A = ∅ and B = ∅ , co + p ( { e j : j ∈ B } ) if A = ∅ and B = ∅ , co + p ( ∪ ( i,j ) ∈ A × B Z i,j ) otherwise.It is routine to prove that the the family of bodies ( T A,B ) enjoys thefollowing properties.(a1) If λ , µ ≥ λ p + µ p ≤
1, then λ T A,B + µ T A,B ⊆ T
A,B for all disjoint
A, B ⊆ N [ d ].(a2) B p ∩ R d + ⊆ T N [ d ] \ A,A ⊆ B ∞ , for all A ⊆ N [ d ].(a3) B p ∩ R d + = T N [ d ] , ∅ .(a4) If A ⊆ A and B ⊆ B , then T A,B ⊆ T A ,B .(a5) If x = ( x j ) dj =1 ∈ T A,B and x j = 0 for every j / ∈ D , where D ⊆ N [ d ], then x ∈ T A ∩ D,B ∩ D .(a6) If 0 ≤ x j ≤ y j for every j ∈ N [ d ] and ( y j ) dj =1 ∈ T A,B , for
A, B ⊆ N [ d ] disjoint, then ( x j ) dj =1 ∈ T A,B .Given A ⊆ N [ d ], put C A = M A ( T N [ d ] \ A,A ). By definition,(b1) −C A = C N [ d ] \ A .We infer from (a1), (a2), (a3) and (a6), respectively, that(b2) if λ , µ ≥ λ p + µ p ≤
1, then λ C A + µ C A ⊆ C A ,(b3) { ( x j ) dj =1 ∈ B p : { j : x j < } = A } ⊆ C A ⊆ B ∞ ,(b4) C ∅ = B p ∩ R d + , and IPSCHITZ FREE p -SPACES 19 (b5) if s ∈ [0 , d and x ∈ C A then M s ( x ) ∈ C A .Consider the d -dimensional body C ( p ) = ∪ A ⊆ N [ d ] C A . Properties (b2),(b3) and (b5) give, respectively,(c1) −C ( p ) = C ( p ) ,(c2) B p ⊆ C ( p ) ⊆ B ∞ , and that(c3) if x ∈ C ( p ) and s ∈ [0 , d , then M s ( x ) ∈ C ( p ) .We infer from (a4) and (a5) that(c4) if x = ( x j ) dj =1 ∈ C ( p ) and { j ∈ N [ d ] : x j < } ⊆ B ⊆ { j ∈ N [ d ] : x j ≤ } , then x ∈ C B .Combining (c4) with (b4) we obtain(c5) C ( p ) ∩ R d + = B p ∩ R d + .Let us prove that C ( p ) is absolutely p -convex. Let x = ( x j ) dj =1 , y =( y j ) dj =1 ∈ C ( p ) and λ, µ ∈ R with | λ | p + | µ | p ≤
1. By (c1) we can assumethat λ, µ ≥
0. Let A = { j ∈ N [ d ] : sgn( x j ) sgn( y j ) = − } ,D = { j ∈ N [ d ] \ A : sgn( λx j + µy j ) = 0 } ,E = { j ∈ N [ d ] \ A : sgn( λx j + µy j ) = sgn( x j ) } ,F = { j ∈ N [ d ] \ A : sgn( λx j + µy j ) = sgn( y j ) } . By construction (
A, D, E, F ) is a partition of N [ d ]. Note that λ > E = ∅ and µ > F = ∅ . We define ˜ x = ( x j ) dj =1 , ˜ y = ( y j ) dj =1 and s = ( s j ) dj =1 by(˜ x j , ˜ y j , s j ) = ( x j , y j ,
1) if j ∈ A, (0 , ,
0) if j ∈ D, ( x j , , ( λx j ) − ( λx j + µy j )) if j ∈ E, (0 , y j , ( µy j ) − ( λx j + µy j )) if j ∈ F. By construction, s ∈ [0 , d and λ x + µ y = M s ( λ ˜ x + µ ˜ y ). Hence, takinginto account (c3), it suffices to prove that λ ˜ x + µ ˜ y ∈ C ( p ) . Note that,by construction, sgn(˜ x j ) sgn(˜ y j ) = − j ∈ N [ d ]. Therefore,the set { j ∈ N [ d ] : ˜ x j < } ∪ { j ∈ N [ d ] : ˜ y j < } is contained in B := { j ∈ N [ d ] : ˜ x j ≤ } ∩ { j ∈ N [ d ] : ˜ y j ≤ } . Since, by (c3), ˜ x , ˜ y ∈ C ( p ) , we infer from (c4) that ˜ x , ˜ y ∈ C B . Then,by (b2), λ ˜ x + µ ˜ y ∈ C B ⊆ C ( p ) . Let k · k ( p ) be the Minkowski functional associated to C ( p ) . Takinginto account (c2) we infer that k · k ( p ) is a p -norm on R d . By (c5), k x k ( p ) = k x k p for every x ∈ R d + . (cid:3) Proposition 4.16.
Let M be the { , } -metric space and ( a x ) x ∈M\{ } be an eventually null family of scalars. Then (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X x ∈M\{ } a x δ ( x ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) F p ( M ) ≥ X x ∈M\{ } a x ≥ a px /p . Proof.
Let ( a x ) x ∈M\{ } ∈ [0 , ∞ ) M\{ } be eventually null. Pick d ∈ N and a one-to-one map φ : N [ d ] → M \ { } such that { x ∈ M \ { } : a x > } ⊆ φ ( N [ d ]) ⊆ { x ∈ M \ { } : a x ≥ } . Let k · k ( p ) be as in Proposition 4.15 and consider the mapping f : M → ( R d , k · k ( p ) ) given by φ ( k ) e k for all k ∈ N [ d ] and x x / ∈ φ ( N [ d ]). Since k e i k ( p ) , k e i − e j k ( p ) ≤ i, j ∈ N [ d ], f is1-Lipschitz. Therefore (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X x ∈M\{ } a x δ ( x ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) F p ( M ) ≥ (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) X x ∈M\{ } a x f ( x ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ( p ) = (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) d X k =1 a φ ( k ) e k (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) ( p ) = d X k =1 a pφ ( k ) ! /p = X x ∈M\{ } a x ≥ a px /p . (cid:3) On occasion it will be convenient to know that the Lipschitz free p -space over a quasimetric space and the Lipschitz free p -space over itscompletion are the same. Let us state this basic fact for reference andprovide a proof using the tools that we introduced before. Proposition 4.17.
Let M be a p -metric space for some < p ≤ and let N be a dense subset of M equipped with the same quasimetric.Then F p ( N ) ≈ F p ( M ) isometrically. In fact, the canonical linear mapis an isometry.Proof. The canonical linear map L : P ( N ) → P ( M ) induced by theinclusion from N into M is one-to-one on P ( N ). By density, the setof molecules of M of the form L ( A ( N )) = (cid:26) δ M ( y ) − δ M ( x ) ρ ( x, y ) : x, y ∈ N , x = y (cid:27) IPSCHITZ FREE p -SPACES 21 is an isometrically p -norming set in F p ( M ). Lemma 2.6 and Corol-lary 4.11 yield that L extends to a linear isometry from F p ( N ) onto F p ( M ). (cid:3) We will study in detail some properties of the canonical map L j in Section 6. For the time being, to finish this section we provide asufficient condition for L to be an isomorphic embedding. Definition . Let M be a p -metric space, 0 < p ≤
1, and let N be a subset of M . A Lipschitz map r : M → N is called a
Lipschitzretraction if it is the identity on N . When such a Lipschitz retractionexists we say that N is a Lipschitz retract of M . Lemma 4.19 (cf. [11, Lemma 2.2]) . Let M be a pointed p -metric space (0 < p ≤ and N be a Lipschitz retract of M . Then the inclusion map : N → M induces an isomorphic embedding L : F p ( N ) → F p ( M ) onto a complemented subspace.Proof. Without loss of generality we may and do assume that 0 ∈ N .Let : N → M be the inclusion map and let r : M → N be a Lipschitzretraction. Lemma 4.8 yields L r ◦ L = Id F p ( N ) , i.e., L ◦ L r is a linearprojection from F p ( M ) onto the linear subspace L ( F p ( N )) of F p ( M )and L is an isomorphism. (cid:3) Envelopes and duality.Proposition 4.20.
Suppose M is a pointed p -metric space with q -metric envelope g M q , where < p < q ≤ . Then: (a) The q -Banach envelope of F p ( M ) is F q ( g M q ) . (b) In the particular case that M is a p -Banach space X with q -Banach envelope c X q , the q -Banach envelope of F p ( X ) is F q ( c X q ) .Proof. The universal properties of q -metric envelopes and q -Banachenvelopes yield the commutative diagram M Q / / δ (cid:15) (cid:15) g M qδ (cid:15) (cid:15) F p ( M ) L Q / / i (cid:15) (cid:15) F q ( g M q ) \ F p ( M ) q f L Q ttttttttt Since co p ( A ( M )) is dense in the unit ball B of F p ( M ) and i (co q ( B )) isdense in the unit ball of X := \ F p ( M ) q , we infer that co q ( i ( A ( M ))) is dense in the unit ball of X . Therefore, by Lemma 2.5, A := i ( A ( M )) isan isometrically q -norming set for X . Moreover, f L Q is a bijection from i ( P ( M )) onto P ( g M q ) = P ( Q ( M )) and f L Q ( A ) = A ( g M q ) = A ( Q ( M )).We deduce from Lemma 2.6 that f L Q is an isometric isomorphism. (cid:3) Remark . The previous proposition implies, for example, that F p ( R )is not isomorphic to L p for p <
1. Indeed, its Banach envelope is L ,while the Banach envelope of L p is trivial, and if two p -spaces are iso-morphic, their envelopes are also isomorphic. Remark . Roughly speaking, it could be argued that given a r -metric space M , 0 < r ≤
1, the family ( F p ( g M p ))
1, Proposition 4.20provides a canonical range-dense linear map L q,p : F p ( g M p ) → F q ( g M q )with k L q,p k ≤ q < s ≤
1, we have L s,p = L s,q ◦ L q,p .Let us restrict our attention to the case when p < r . Then L r,p : F p ( M ) → F r ( M )is the identity map on P ( M ) and, hence, it is one-to-one on a densesubspace. However we do not know if this map is always injective.In the case when r = 1 we would like to point out that the map L ,p : F p ( M ) → F ( M ) is one-to-one if and only if F p ( M ) ∗ separatesthe points of F p ( M ). Corollary 4.23.
Let M be a pointed p -metric space, < p ≤ .Then F p ( M ) ∗ = Lip ( M ) , i.e., given φ ∈ F p ( M ) ∗ there is a unique f ∈ Lip ( M ) so that φ (cid:0) P a i δ ( x i ) (cid:1) = P a i f ( x i ) for every P a i δ ( x i ) ∈F p ( M ) , and the map φ f is a linear isometry of F p ( M ) ∗ onto Lip ( M ) . In particular, F p ( M ) ∗ = { } if Lip ( M ) = { } .Proof. By identifying M with δ ( M ) ⊆ F p ( M ) we get that the restric-tion of any φ ∈ F p ( M ) ∗ to M belongs to Lip ( M ). And conversely,any f ∈ Lip ( M ) uniquely extends by the universal property to anelement of F p ( M ) ∗ . This correspondence is a linear isometry. (cid:3) Corollary 4.24.
Let M and N be pointed metric spaces and suppose < p < . If F p ( M ) ≈ F p ( N ) then F ( M ) ≈ F ( N ) .Proof. Just take Banach envelopes in Proposition 4.20 (a). (cid:3)
The last theorem of this section extends to the case when 0 < p <
1a result of Naor and Schechtman [18].
Theorem 4.25.
For any < p ≤ , the p -Banach spaces F p ( R ) and F p ( R ) are not isomorphic. IPSCHITZ FREE p -SPACES 23 Proof.
The case p = 1 was proved in [18]. The case 0 < p < (cid:3) Lipschitz free p -spaces over ultrametric spaces The spaces F p ( M ) over quasimetric (or even metric) spaces provide anew class of quasi-Banach spaces that in general are difficult to identify.The point of this section is to see that by imposing a stronger conditionon M , namely being ultrametric, we can recognize the Lipschitz free p -space over M .Recall that a distance d on a set M is called an ultrametric providedthat in place of the triangle inequality, d satisfies the stronger condition d ( x, z ) ≤ max { d ( x, y ) , d ( y, z ) } , x, y, z ∈ M . Note that ultrametrics can be characterized as metrics d such that d p is a metric for every p >
0. Indeed, if ( M , d p ) is a metric space for p ∈ A , and the set A ⊆ R is unbounded, then d ( x, z ) ≤ ( d p ( x, y ) + d p ( y, z )) /p , x, y, z ∈ M , p ∈ A. Letting p tend to infinity we get d ( x, z ) ≤ max { d ( x, y ) , d ( y, z ) } . Theconverse implication is clear.Before proceeding, let us digress a bit with the help of an exam-ple. Let ( M , ≤ ) be a totally ordered set and λ = ( λ x ) x ∈M a non-decreasing family of positive numbers (it could also be non-increasing,in which case we would consider the reverse order on M ). Equippedwith d λ : M × M → [0 , ∞ ) defined by d λ ( x, y ) = ( λ max { x,y } if x = y x = y, M is an ultrametric space. Since every set can be equipped with a totalorder, if we put λ x = 1 for all x ∈ M , we infer that the { , } -metric on M and the ultrametric d λ coincide. Thus the following theorem, whichextends to the case p < Theorem 5.1.
Let ( M , d ) be an infinite separable pointed ultrametricspace. Then F p ( M , d ) ≈ ℓ p for every < p ≤ . The techniques we use to prove this theorem rely on the conceptsof R -tree and length measure. For the convenience of the reader weinclude these definitions, which we borrow from [10, Sect. 2]. For moredetails concerning R -trees see for instance [9, Chapter 3]. Definition . An R -tree is a metric space ( T , d ) satisfying: (i) For any points a and b in T , there exists a unique isometry φ from the closed interval [0 , d ( a, b )] into T such that φ (0) = a and φ ( d ( a, b )) = b .(ii) Any one-to-one continuous mapping ϕ : [0 , → T has the samerange as the isometry φ associated to the points a = ϕ (0) and b = ϕ (1).If T is an R -tree, given any x and y in T we denote by φ xy theunique isometry associated to x and y as in Definition 5.2, and write[ x, y ] for the range of φ x,y . Such subsets of T are called segments .Moreover, we say that v ∈ T is a branching point of T if there are threepoints x , x , x ∈ T \ { v } such that [ x i , v ] ∩ [ x j , v ] = { v } whenever i, j ∈ { , , } , i = j . We say that a subset A of T is measurablewhenever φ − xy ( A ) is Lebesgue-measurable for any x and y in T . If A is measurable and S is a segment [ x, y ], we write λ S ( A ) for λ ( φ − xy ( A )),where λ is the Lebesgue measure on R . We denote by R the set of allsubsets of T that can be written as a finite union of disjoint segments.For R = ∪ nk =1 S k (with disjoint S k ) in R , we put λ R ( A ) = n X k =1 λ S k ( A ) . Now, λ T ( A ) = sup R ∈R λ R ( A )defines a measure on the σ -algebra of T -measurable sets called the length measure . Note that this is nothing but the 1-dimensional Haus-dorff measure (multiplied by the constant 2).Suppose ( S , d ) is a closed subset of an R -tree T with a base point0 ∈ S . For s ∈ S we put L S ( s ) := inf x ∈ [0 ,s ) ∩S d ( s, x ) . If L S ( s ) >
0, we denote by σ S ( s ) the unique point from [0 , s ) ∩ S with d ( s, σ S ( s )) = L ( s ). Finally, we put S + := { s ∈ S : L S ( s ) > } . Lemma 5.3.
Let ( S , d ) be a closed subset of an R -tree T with a point ∈ S . Let S have length measure zero. Then for all y ∈ S and all x ∈ [0 , y ] ∩ S , d ( x, y ) = X z ∈ ( x,y ] ∩S + L S ( z ) . IPSCHITZ FREE p -SPACES 25 Proof.
Using the transformation φ ,y , we can assume without loss ofgenerality that x, y ∈ R , 0 ≤ x ≤ y and S ⊆ [0 , y ]. Then, the subset( x, y ) \ S of R is open and, so, it can be expressed as ∪ i ∈ I ( a i , b i ),where the intervals are disjoint. Since S has measure zero, we have y − x = P i ∈ I ( b i − a i ). It is clear that b i ∈ S + and σ S ( b i ) = a i forevery i ∈ I . Thus, it suffices to see that the map b : I → S + , i b i isonto. Given s ∈ S + , since ( σ S ( s ) , s ) ∩ S = ∅ , there exists i ∈ I suchthat ( σ S ( s ) , s ) ⊆ ( a i , b i ). Taking into account that neither σ S ( s ) nor s belong to ( a i , b i ), we infer that a i = σ S ( s ) and b i = s . (cid:3) In the case when p = 1 the following result was proved by Godard(see [10, Proposition 2.3]). Below we give an alternative proof whichworks for every 0 < p ≤ R -trees T . Proposition 5.4.
Let ( S , d ) be a closed subset of an R -tree T suchthat S contains all the branching points of T and has length measurezero. Then F p ( S , d /p ) ≈ ℓ p ( S + ) isometrically. To be precise, the map T ( δ ( s )) := X x ∈ (0 ,s ] ∩S + ( L S ( x )) /p e x , s ∈ S , extends to a linear isometry between F p ( S , d /p ) and ℓ p ( S + ) .Proof. Without loss of generality we assume that 0 ∈ S , and for sim-plicity, for each s ∈ S + we denote σ S ( s ) by s . For every y ∈ S and x ∈ [0 , y ] ∩ S we have δ ( y ) − δ ( x ) = X z ∈ ( x,y ] ∩S + (cid:0) δ ( z ) − δ ( z ) (cid:1) . (5.14)To see this, if we consider in [ x, y ] the total order induced by the isom-etry φ x,y , by Lemma 5.3 for any ε > z < z < · · · If neither Case 1 nor Case 2 occurs, there exists a branchingpoint c ∈ [ x, y ] with [0 , x ] ∩ [0 , y ] = [0 , c ]. Then we can write a x,y = λa y,c + µa c,x , λ = d /p ( y, c ) d /p ( x, y ) , µ = d /p ( c, x ) d /p ( x, y ) . From the two previous cases we have a c,y , a c,x ∈ A , and since λ p + µ p = d ( x, c ) + d ( c, y ) d ( x, y ) = 1 , we conclude that a x,y ∈ A . Hence, (5.15) is fulfilled. (cid:3) Since the real line is a trivial example of an R -tree we obtain: Corollary 5.5. Let M be an infinite subset of R and < p ≤ . If theclosure of M has measure zero then F p ( M , | · | /p ) ≈ ℓ p isometrically.In particular, the result holds if M is the range of a monotone sequenceof real numbers.Proof of Theorem 5.1. Since d p is also an ultrametric whenever d is,we need only show that F p ( M , d /p ) ≈ ℓ p . By [7, Proposition 12],there exists a closed subset S of a separable R -tree T containing allits branching points in such a way that S has length measure zeroand ( M , d ) is bi-Lipschitz isomorphic to a Lipschitz retract of S . De-noting the metric on S by η , we have that ( M , d /p ) is bi-Lipschitzisomorphic to a Lipschitz retract of ( S , η /p ). By Proposition 5.4, IPSCHITZ FREE p -SPACES 27 Lemma 4.4, and Corollary 4.6, F p ( S , η /p ) ≈ ℓ p hence, by Lemma 4.4and Lemma 4.19, F p ( M , d /p ) is isomorphic to an infinite-dimensionalcomplemented subspace of ℓ p . Since every infinite-dimensional comple-mented subspace of ℓ p is isomorphic to ℓ p by a classical result of Stiles[22], we infer that F p (( M , d /p )) ≈ ℓ p , and the proof is over. (cid:3) Theorem 4.13 and Corollary 5.5 allow us to identify the free p -spacesover some subsets of the real line equipped with the “anti-snowflaking”quasimetric | · | /p . However, identifying the free p -space over subsets of R equipped with the Euclidean distance seems to be a more challengingtask. For the time being, let us just mention that since the Banachenvelope of F p ( I ) is L ( I ) for any interval I with the Euclidean distance(to see this, apply Proposition 4.20(b) and Theorem 4.13 for p = 1),the spaces F p ( I ) constitute a new class of p -Banach spaces.6. Linearizations of Lipschitz embeddings Lipschitz free p -spaces over quasimetric spaces constitute a nice familyof new p -Banach spaces which are easy to define but whose geometryseems to be difficult to understand. To carry out this task sucessfulyone hopes to be able to count on “natural” structural results involv-ing free p -spaces over subsets of M . In this section we analyse thispremise and confirm an unfortunate recurrent pattern in quasi-Banachspaces: the lack of tools can be an important stumbling block in thedevelopment of the nonlinear theory. However, as we will also see, noteverything is lost and we still can develop specific methods that permitto shed light onto the structure of F p ( M ).If M is a pointed p -metric space and N is a subset of M containing0, the linearization process of Lemma 4.8 applies in particular to thecanonical injection : N → M . If p = 1, McShane’s theorem [17]ensures that L : F ( N ) → F ( M ) is an isometric embedding. Thus F ( N ) can be naturally identified with a subspace of F ( M ) . However,if p < p = 1, F ( N ) isisometric to a subspace of F ( M ) and so the study of Lipschitz freespaces over subsets is a powerful tool. We start by exhibiting that thisargument breaks down when p < 1, settling a problem raised in [3]. Theorem 6.1. For each < p < and p ≤ q ≤ there is a q -metric space ( M , ρ ) and a subset N ⊆ M such that the inclusionmap : N → M induces a non-isometric isomorphic embedding L : F p ( N ) → F p ( M ) with k L − k ≥ /q .Proof. Let N = { } ∪ N and M = { , z } ∪ N , where z / ∈ N . De-fine ρ : M × M → [0 , ∞ ) by ρ ( x, x ) = 0 for all x ∈ M , ρ ( j, z ) = ρ ( z, j ) = 2 − /q for all j ∈ N , and ρ ( x, y ) = 1 otherwise. It is clear that( M , ρ ) is a bounded and uniformly separated q -metric space. Then,by Theorem 4.14, L is an isomorphism.Given ǫ > 0, there is k ∈ N and a k -tuple ( a j ) kj =1 such that a j ≥ j = 1, . . . , k , P kj =1 a pj = 1, and P kj =1 a j ≤ ǫ . Indeed, itsuffices to choose k ≥ ε p/ ( p − and put a j = k − /p for 1 ≤ j ≤ k . Onthe one hand, since ρ is the { , } -metric on N , by Proposition 4.16, k P kj =1 a j δ ( j ) k F p ( N ) = 1 . On the other hand, considering the decom-position k X j =1 a j δ ( j ) = k X j =1 a j ! δ ( z ) + k X j =1 a j /q δ ( j ) − δ ( z )2 − /q and using Corollary 4.12, we have (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) k X j =1 a j δ ( j ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) p F p ( M ) ≤ k X j =1 a j ! p + k X j =1 (2 − /q a j ) p ≤ ǫ p + 2 − p/q . Hence, k L − k ≥ sup ǫ> ǫ p + 2 − p/q ) /p = 2 /q . (cid:3) The following problem seems to be open. Question . Let 0 < p < N ⊆ M be two p -metric (or metric)spaces in inclusion. Is the canonical linear map of F p ( N ) into F p ( M )an isomorphic embedding?The answer to Question 6.2 is positive in some special cases. Proposition 6.3. If M is a { , } -metric space, then the canonicalmap of F p ( N ) into F p ( M ) is isometric for every N ⊆ M .Proof. Notice that a map f from a { , } -metric space M into a quasi-Banach space X is 1-Lipschitz if and only if k f ( x ) − f ( y ) k ≤ x , y ∈ M . Then, a 1-Lipschitz map f : N → X with f (0) = 0extends by f ( x ) = 0 for x ∈ M \ N to a 1-Lipschitz map from M into X . This gives k L ( µ ) k F p ( M ) ≥ k µ k F p ( N ) for every µ ∈ P ( N ). (cid:3) Proposition 6.4. Let M be a subset of R equipped with the quasimetric ρ ( x, y ) = | x − y | /p for < p ≤ . If the closure of M has measurezero and N ⊆ M then the canonical mapping L : F p ( N ) → F p ( M ) isan isometry.Proof. By Proposition 4.17 we can assume that both N and M areclosed. Let T M : F p ( M ) → ℓ p ( M + ) and T N : F p ( N ) → ℓ p ( N + ) be the IPSCHITZ FREE p -SPACES 29 isomotries provided by Proposition 5.4. If U = T M ◦ L ◦ T − N , where : N → M is the inclusion map, we have U ( e s ) = 1 | s − σ N ( s ) | /p X z ∈ ( σ N ( s ) ,s ] ∩M + L /p M ( z ) e z , s ∈ N + . By Lemma 5.3, k U ( e s ) k p = 1 for every s ∈ N + . Pick s , t ∈ N + with s < t . Since ( σ N ( s ) , s ) ∩ N = ∅ , we have s ≤ σ N ( t ). Then,( σ N ( s ) , s ] ∩ ( σ N ( t ) , t ] = ∅ and so ( U ( e s )) s ∈N + is a disjointly supportedfamily in ℓ p ( M + ). Hence ( U ( e s )) s ∈N + is isometrically equivalent to theunit vector basis of ℓ p ( M + ), i.e., U is an isometric embedding. (cid:3) Question . Let 0 < p < 1. Can we identify the p -metric spaces M for which the canonical linear map from F p ( N ) into F p ( M ) is anisometry for every N ⊆ M ? Acknowledgment. F. Albiac acknowledges the support of the Span-ish Ministry for Economy and Competitivity Grants MTM2014-53009-P for An´alisis Vectorial, Multilineal y Aplicaciones , and MTM2016-76808-P for Operators, lattices, and structure of Banach spaces . J.L.Ansorena acknowledges the support of the Spanish Ministry for Econ-omy and Competitivity Grant MTM2014-53009-P for An´alisis Vecto-rial, Multilineal y Aplicaciones M. C´uth has been supported by CharlesUniversity Research program No. UNCE/SCI/023 and by the Researchgrant GA ˇCR 17-04197Y. M. Doucha was supported by the GA ˇCRproject 16-34860L and RVO: 67985840.F. Albiac and J.L. Ansorena would like to thank the Faculty of Math-ematics and Physics at Charles University in Prague for their hospi-tality and generosity during their visit in September 2018, when mostwork on this paper was undertaken. References [1] F. Albiac, Nonlinear structure of some classical quasi-Banach spaces and F -spaces , J. Math. Anal. Appl. (2008), no. 2, 1312–1325.[2] , The role of local convexity in Lipschitz maps , J. Convex Anal. (2011), no. 4, 983–997.[3] F. Albiac and N. J. Kalton, Lipschitz structure of quasi-Banach spaces , IsraelJ. Math. (2009), 317–335.[4] , Topics in Banach space theory , 2nd ed., Graduate Texts in Mathe-matics, vol. 233, Springer, [Cham], 2016. With a foreword by Gilles Godefroy.[5] T. Aoki, Locally bounded linear topological spaces , Proc. Imp. 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Sch¨affer, Inner diameter, perimeter, and girth of spheres , Math. Ann. 173(1967), 59–79; addendum, ibid. (1967), 79–82.[22] W. J. Stiles, Some properties of l p , < p < 1, Studia Math. (1972), 109–119.[23] N. Weaver, Lipschitz algebras , 2nd ed., World Scientific Publishing Co. Pte.Ltd., Hackensack, NJ, 2018. IPSCHITZ FREE p -SPACES 31 Mathematics Department–InaMat, Universidad P´ublica de Navarra,Campus de Arrosad´ıa, Pamplona, 31006 Spain E-mail address : [email protected] Department of Mathematics and Computer Sciences, Universidad deLa Rioja, Logro˜no, 26004 Spain E-mail address : [email protected] Faculty of Mathematics and Physics, Department of MathematicalAnalysis, Charles University, 186 75 Praha 8, Czech Republic E-mail address : [email protected] Institute of Mathematics, Czech Academy of Sciences, 115 67 Praha1, Czech Republic E-mail address ::