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Lipschitz free pspaces
Fernando AlbiacPublic University of Navarre
Joint work with José L. Ansorena, Marek Cúth, andMichal Doucha
Banach Spaces and their ApplicationsLVIV, 2629 JUNE 2019

Brief review: pBanach spaces
Consider 0 < p ≤ 1:A pBanach space is a (real) vector spaceX equipped with a map‖ · ‖ : X → [0,∞) such that for all x, y ∈ X and all α ∈ R,(i) ‖x‖ = 0⇔ x = 0,(ii) ‖αx‖ = α‖x‖,(iii) ‖x+ y‖p ≤ ‖x‖p + ‖y‖p.and (X, ‖ · ‖) is complete (i.e. when endowed with the metric(x, y) 7→ ‖x− y‖p, it is a complete metric space).
AokiRolewicz Theorem: (X, ‖ · ‖) is a pBanach space for some0 < p ≤ 1 if and only if it is a quasiBanach space, that is, we can replacecondition (iii) with(iii)′ ∃C ≥ 1 such that ‖x+ y‖ ≤ C(‖x‖+ ‖y‖) ∀x, y ∈ X .

Brief review: pmetric spaces for 0 < p ≤ 1
The corresponding concept for metric spaces is that of a pmetric space.(M, ρ) is a pmetric space if (M, ρp) is a metric space. (Completenessis not assumed now.)
An analogue of the AokiRolewicz theorem holds in this context:Every quasimetric space space (M, ρ), i.e.,
ρ(x, y) ≤ C(ρ(x, z) + ρ(y, z)), x, y, z ∈M,
can be endowed with an equivalent quasimetric which is psubadditivefor some 0 < p ≤ 1.That is, every quasimetric space is Lipschitz isomorphic to a pmetricspace for some choice of 0 < p ≤ 1.

Historical Background
Despite the fact that the first papers on the subject (p < 1) appeared inthe early 1940’, most of the research in Banach space theory has beendone only for p = 1. Why?
It might happen thatX∗ = {0} (e.g. ifX = Lp for p < 1).No HahnBanach theorem. No duality techniques available.Bochner integration is not available in the lack of local convexity.

Motivation to further research in pBanach spaces
Some reasons why it is worth it accepting the challenge to advance thetheory for p < 1:
Some people did it and developed fresh techniques (e.g. N. Kalton).Proving results in pBanach spaces for 0 < p < 1 often provides analternative proof even for p = 1.There are many classical pBanach spaces for p < 1 in the literature:sequence spaces `p, Lorentz sequence spaces d(w, p), functionspaces Lp, Hardy spacesHp, . . . All of them are the nonlocallyconvex counterpart of the corresponding family of Banach spaces.

Lipschitz free Banach spaces over metric spaces
Let (M, d, 0) be a metric space with a distinguished point 0.
We put Lip0(M) = {f : M→ R Lipschitz , f(0) = 0}.
Lip0(M) has a vector space structure and we can define a norm ‖ · ‖Lip by
‖f‖Lip = sup{f(x)− f(y)
d(x, y) : x, y ∈M, x 6= y}.
Then (Lip0(M), ‖ · ‖Lip) is a Banach space.
For any x ∈M, denote by δ(x) ∈ Lip0(M)∗ the evaluation functional
〈δ(x), f〉 = f(x), f ∈ Lip0(M).
DefinitionTheLipschitz free space overM, denotedF(M), is the closureof the linearspan of {δ(x) : x ∈M} in Lip0(M)∗ with respect to the dual norm.

Lipschitz free Banach spaces over metric spaces
In the seminal paper
G. GODEFROY AND N. J. KALTON, Lipschitzfree Banach spaces, StudiaMath. 159 (2003), no.1, 121141.
the authors considered Lipschitz free spaces with the purpose toinvestigate the following general problem in the theory of nonlinearclassification of Banach spaces:
ProblemIfX and Y are Lipschitz isomorphic separable Banach spaces, areX andY linearly isomorphic?

Lipschitz free Banach spaces over metric spaces
To undertake this task, they worked with the following property:
DefinitionA Banach spaceX has the Lipschitzlifting property if there is a continuous linear map T : X → F(X) such that βXT = IdX, where the quotientmap
βX : F(X)→ X,N∑j=1
ajδX(xj) 7→N∑j=1
ajxj ,
is a left inverse of the (nonlinear) isometric embedding δX : X → F(X).
This can be equivalently interpreted by saying that the short exactsequence of Banach spaces
0→ β−1X (0) ↪→ F(X)βX→ X → 0,
which Lipschitz splits (being δX an isometric Lipschitz lifting of βX ), alsolinearly splits.

Lipschitz free Banach spaces over metric spaces
Their construction led to the following criterion:
Proposition (GodefroyKalton; 2003)LetX be a Banach space. Then:
1 F(X) is Lipschitz isomorphic to G(X) = β−1X (0)⊕X .2 F(X) and G(X) are linearly isomorphic if and only ifX has the
Lipschitzlifting property.
Unfortunately, this method fails to produce separable Banach spaceswhich are Lipschitz isomorphic without being linearly isomorphic:
Theorem (GodefroyKalton; 2003)If X is an infinitedimensional separable Banach space, X has theLipschitzlifting property. Hence F(X) is linearly isomorphic to the spaceG(X).

Motivation for introducing the class of Lipschitz freepspaces for 0 < p < 1
However, the method does work for building for every p < 1 twoseparable pBanach spaces which are Lipschitz isomorphic but notlinearly isomorphic:
Theorem (A.Kalton; 2009)Let X be an infinitedimensional separable Banach space which is not aSchur space. For every p < 1, the pBanach spaces Fp(X) and Gp(X) =ker(βX)⊕p X are Lipschitz isomorphic but not linearly isomorphic.
F. ALBIAC AND N. J. KALTON, Lipschitz structure of quasiBanachspaces, Israel J. Math. 170 (2009), 317335.

Motivation for further studying Lipschitz free pspaces
They are the nonlocally convex family members of the Lipschitzfree Banach spaces over metric spacesF(M)Their study got interrupted after 2009They have already proven their usefulness in the general theoryWe know nothing about their linear structure (amongst many otherthings)Since the proofs of the known results in the case p = 1 often rely,more or less explicitly, on duality techiques, obtaining analogueresults for the case of p < 1 requires working out the proofs fromscratch. Hopefully, this could lead to new insights for the case ofp = 1.

Definition of Lipschitz free pspaces over pmetric spaces
Setting: Let (M, ρ, 0) be a pmetric space with a distinguished point 0.
LetRM0 be the space of all maps f : M→ R so that f(0) = 0.
Let P(M) be the linear span in the linear dual (RM0 )# of the evaluationsδ(x), where x runs throughM, defined by
〈δ(x), f〉 = f(x), f ∈ RM0 .
Note that δ(0) = 0.
If µ =∑Nj=1 ajδ(xj) ∈ P(M), put
‖µ‖Fp(M) = sup
∥∥∥∥∥∥N∑j=1
ajf(xj)
∥∥∥∥∥∥Y
,
the supremum being taken over all pnormed spaces (Y, ‖ · ‖Y ) and all1Lipschitz maps f : M→ Y with f(0) = 0.

Definition of Lipschitz free pspaces over pmetric spaces
It is straightforward to check that this defines a pseminorm on P(M).
What is not so straightforward is that, in fact,
‖ · ‖Fp(M) defines a pnorm on P(M)!
That is,{δ(x) : x ∈M} is a linearly independent set in P(M).
DefinitionThe Lipschitz free pspace overM, denotedFp(M), is the pBanach spaceresulting from the completion of the pnormed space (P(M), ‖ · ‖Fp(M)).
In particular, if we apply the above definition to a metric spaceMwithp = 1we obtainF1(M) = F(M).

Characterization of the spacesFp(M)Similarly to Lipschitz free Banach spaces over metric spaces, the spacesFp(M) are characterized by the following universal property:
TheoremGiven (M, ρ) a pointed pmetric space, 0 < p ≤ 1:(i) The map δM : M→ Fp(M), x 7→ δ(x), is an isometric embedding;(ii) The linear span of {δ(x) : x ∈M} is dense inFp(M); and(iii) Fp(M) is the unique (up to isometric isomorphism) pBanach space
such that for every pBanach spaceX and every Lipschitz mapf : M→ X with f(0) = 0 there exists a unique linear mapTf : Fp(M)→ X with Tf ◦ δM = f and ‖Tf‖ = Lip(f).
Mf //
δM ##
X
Fp(M)Tf
;;

Linearization of Lipschitz maps between pmetric spaces
Lipschitz free pspaces provide a canonical linearization process ofLipschitz maps between pmetric spaces
If we identify (through the map δM) a pmetric spaceMwith a subset ofFp(M), then any Lipschitz map f from a pmetric spaceM1 to apmetric spaceM2 which maps 0 to 0 extends to a continuous linearmap fromFp(M1) toFp(M2):
M1f //
δM1��
M2δM2��
Fp(M1)Lf // Fp(M2)
and ‖Lf‖ = Lip(f). In particular, if f is a biLipschitz bijection then Lfis an isomorphism.

The dual ofFp(M) for 0 < p ≤ 1
Proposition
SupposeM is a pointed pmetric space. ThenFp(M)∗ = Lip0(M).
In general, there is no tool for building nontrivial Lipschitz maps from aquasimetric space into the real line.
Lip0(M) could be trivial (in which caseFp(M)∗ = {0}) .For example ifM = (R,  · 1/p), 0 < p < 1, then Lip0(M) = {0}.If (M, d) is a metrically convex metric space and 0 < p < 1, thenLip0(M, d1/p) = {0}.Hence, Lip0(Lp) = {0} for 0 < p < 1.

The metric envelope of a quasimetric spaceM
The existence of (nontrivial) realvalued Lipschitz maps on aquasimetric spaceM is related to the concept ofmetric envelope ofM.
Given a quasimetric space (M, ρ)we look for ametric space (M̃, ρ̃) characterized by the following universal property:Whenever (M,d) is a metric space and f : (M, ρ) → (M,d) is Lipschitz,there is a unique f̃ : (M̃, ρ̃) → (M,d) Lipschitz with Lip(f̃) ≤ Lip(f).Pictorially,
(M, ρ) f //
Q $$
(M,d)
(M̃, ρ̃)f̃
::

An explicit description of the metric envelope
For x, y ∈M, we define
ρ̃(x, y) = infn∑i=0
ρ(xi, xi+1),
where the infimum is taken over all sequences x = x0, x1, . . . , xn+1 = yof finitelymany points inM.ρ̃ is symmetric, satisfies the triangle inequality, and does not exceed ρ.ρ̃(x, y) can be zero for different points x, y inM.We identify points inM that are at a zero ρ̃distance
x ∼ y ⇐⇒ ρ̃(x, y) = 0.
Themetric envelope of (M, ρ) is (M̃, ρ̃) andQ : M→ M̃ =M/ ∼ is aquotient map.

Metric envelope and existence of Lipschitz maps
PropositionGiven a quasimetric space (M, ρ) the following are equivalent.• Its metric envelope (M̃, ρ̃) is trivial.• Lip0(M̃,M) = {0} for any metric space (M,d).• Lip0(M) = {0}.
Wewill see later on that there are conditions that guarantee the existenceof Lipschitz maps on a quasimetric spaceM.

The Banach envelope ofFp(M)
Roughly speaking, the Banach envelope X̂ of a quasiBanach spaceX isthe “closest” Banach space toX .
PropositionSupposeM is a pointed pmetric space. Then:
1 The Banach envelope F̂p(M) ofFp(M) is the Lipschitz free spaceF(M̃) over the metric envelope M̃ ofM.
2 In particular, whenX is a quasiBanach space, F̂p(X) = F(X̂), theLipschitz free space over the Banach envelope ofX .
CorollaryLetM1 andM2 be pointedmetric spaces. If Fp(M1) ≈ Fp(M2) forsome 0 < p < 1 thenF(M1) ≈ Fp(M2).

The Banach envelope ofFp(M)
We use the last Corollary to tell apart some Lipschitz free pspaces(without knowing what they are).
Example1 Fp(N,  · ) 6≈ Fp(R,  · ) sinceF̂p(N) = F(N) ≈ `1 6≈ L1 = F(R) = F̂p(R).
2 Fp(R) 6≈ Fp(R2) for 0 < p ≤ 1 since (by NaorSchechtman)F̂p(R) = F(R) 6≈ F(R2) = F̂p(R2) .

Computation of the norm in Lipschitz free pspaces
Lipschitz free pspaces over quasimetric spaces can be definedequivalently viamolecules, considering the vector space spanned by theevaluation functionals of the form δ(x)− δ(y), x, y ∈M.
This approach has the advantage to allow us to work with the followingformula of the pnorm in Lipschitzfree pspaces:
For µ ∈ Fp(M)we have
‖µ‖Fp(M) = inf
( ∞∑k=1akp
)1/p: µ =
∞∑k=1
akδ(xk)− δ(yk)ρ(xk, yk)
, xk 6= yk
,which relies only on the pmetric on the spaceM.

Early examples
That alternative expression of the pnorm inFp(M) is very useful andpermits to provide the first examples of Lipschitz free pspaces overpmetric spaces:
Example (Fp(M)spaces isomorphic to `p andLp, 0 < p ≤ 1)1 ConsiderN equipped with the pmetric ρ(m,n) = m− n1/p form,n ∈ N. ThenFp(N) ≈ `p isometrically.
2 Let I be an interval of the real line equipped with the pmetricρ(x, y) = x− y1/p for x, y ∈ I . ThenFp(I) ≈ Lp(I) isometrically.
3 LetM be a bounded and uniformly separated pmetric space. Then
Fp(M) ≈ `p(M\ {0}).

First results
The spacesFp(M) over quasimetric spaces are a new class of pBanachspaces that are difficult to identify.By imposing a stronger condition onM, like being ultrametric, we willbe able to recognize the Lipschitz free pspace overM.Recall that a metric d on a setM is an ultrametric if the triangle law isreplaced with the stronger condition d(x, z) ≤ max{d(x, y), d(y, z)}.
TheoremIf (M, d) is a separable infinite ultrametric space and 0 < p ≤ 1, thenFp(M) ≈ `p.
CorollaryLetM be an infinite subset of R and 0 < p ≤ 1. If the closure ofM hasmeasure zero, thenFp(M,  · 1/p) ≈ `p isometrically. In particular, theresult holds ifM is the range of a monotone sequence of real numbers.

Linearization of Lipschitz embeddings: case p = 1
In the case when p = 1, the study of Lipschitz free spaces over subsets ofM is a powerful tool to understand the linear structure ofF(M).
Indeed, ifM is a pointedmetric space andN is a subset ofM containing0, the linearization process applies to the inclusion : N →M.Then, MacShane’s theorem ensures that L : F(N )→ F(M) is a linearisometric embedding.ThusF(N ) can be naturally identified with a subspace ofF(M).
`1 embeds isomorphically inF(M) for every infiniteMThis can be proved directly with an argument that shows that everyinfinite metric spaceM contains a sequenceN := {xn}∞n=1 such thatF(N ) is isomorphic to `1. Now we linearly embed `1 intoF(M) via themap L.

Linearization of Lipschitz embeddings: case p < 1
For 0 < p < 1, not only do we not know how to extend real Lipschitzfunctions on quasimetric spaces but, even in the case when we considerFp(M) over a metric space (or a Banach space), the Lipschitz functionsin the definition of ‖ · ‖Fp(M) map into a pBanach space!
We start by exhibiting that the previous arguments break down whenp < 1 (solving a problem raised 10 years ago):
TheoremFor each 0 < p < 1 and p ≤ q ≤ 1 there is a qmetric space (M, ρ) and asubsetN ⊆M such that the inclusion map : N →M induces anonisometric isomorphic embedding L : Fp(N )→ Fp(M) with‖L−1 ‖ ≥ 21/q .

Linearization of Lipschitz embeddings: case p < 1
Remark1 We have not found examples of quasimetric spacesN ⊂MwhereL : Fp(N )→ Fp(M) is not an isomorphic embedding.
2 There are cases whenN ⊂M always implies thatL : Fp(N )→ Fp(M) is an isometric embedding. For example:
WhenM is a metric space endowed with the {0, 1}metric.WhenM is a subset ofR equipped with the quasimetricρ(x, y) = x− y1/p such thatM has measure 0.

Linear structure ofFp(M), 0 < p < 1
Here is where we encounter the main difficulties in making headwaysince the lack of those tools that we take for granted when p = 1 is animportant stumbling block in the development of the theory.
However, not everything is lost and we still can developmethods, specificof the case p < 1, that permit to shed light onto the structure ofFp(M).
Theorem (Embeddability of `p inFp(M) for p ≤ 1)IfM is an infinite pmetric space, 0 < p ≤ 1, thenFp(M) contains asubspace isomorphic to `p.

Linear structure ofFp(M), 0 < p < 1
The proof of this theorem is very technical.
Proof.Wemust find a “suitable” sequence of molecules (µn)∞n=1 insideFp(M),
µn =δ(xn)− δ(yn)ρ(xn, yn)
, n = 1, 2, . . .
so that (µn)∞n=1 is equivalent to the canonical `pbasis.
The pthtriangle law gives us one inequality for free:∥∥∥∥∥∞∑n=1
anµn
∥∥∥∥∥p
Fp(M)
≤∞∑n=1anp‖µn‖pFp(M) =
∞∑n=1anp,
for any sequence of scalars (an).

Linear structure ofFp(M), 0 < p < 1
Proof.To prove the converse inequality, we need to choose 1Lipschitz mapsgn : M→ Y and a target pBanach space Y (where we know how toexplicitly compute pnorms), in such a way that∥∥∥∥∥
∞∑n=1
anµn
∥∥∥∥∥p
Fp(M)
≥
∥∥∥∥∥∞∑n=1
angn(xn)− gn(yn)
ρ(xn, yn)
∥∥∥∥∥p
Y
≥ C∞∑n=1anp,
for some constantC ≥ 1.
We accomplish this by taking Y = Lp(R) and defining mapsgn : M→ Lp(R) on sequences (xn), (yn) insideM that can beuniformly separated in a certain sense.

Projections on spacesFp(M)
Cúth, Doucha, andWojtaszczyk proved in 2016 that `1 embedscomplementably inF(M) for every infinite metric spaceM.
Proof.First they showed that `∞ ↪→ Lip0(M).IfX is Banach, `∞ ↪→ X∗ if and only if `1 ↪→c X .Apply this toX = F(M).
None of this can be transferred to Lipschitz free pspaces overquasimetric (even metric) spaces when p < 1. In fact, it does not holdthat `p is isomorphic to a complemented subspace ofFp(M) when0 < p < 1 for every quasimetric spaceM.
ExampleFor 0 < p < 1, the space `p is not complemented inLp = Fp([0, 1],  · 1/p).

Projections on spacesFp(M)
Note that the existence of a complemented subspaceX ofFp(M)withX∗ 6= {0} guarantees the existence of nonzero Lipschitz maps onM.
Indeed,Lip0(M) = Fp(M)∗ = (X ⊕ Y )∗ = X∗ ⊕ Y ∗.
So it is natural to try to find bounded linear projections on spacesFp(M)where Lip0(M) 6= {0}.
Example1 Of course, ifM is metric (hence pmetric for all p < 1) it has plentyof realvalued Lipschitz maps.
2 Uniformly separated pmetric spaces have nontivial Lipschitzfunctions. In fact, if (M, ρ) is rseparated, for every y ∈M the mapx 7→ ρp(x, y) is Lipschitz with constant rp−1.

Projections on spacesFp(M)
Using specific techniques (that also apply to the case p = 1!) we haveobtained:
TheoremLet p ∈ (0, 1]. Suppose that (M, ρ) is either(a) an infinite metric space, or(b) a uniformly separated uncountable pmetric space.Then there existsN ⊂M such that(i) Fp(N ) ' `p,(ii) L : Fp(N )→ Fp(M) is an isomorphic embedding, and(iii) L(Fp(N )) is complemented inFp(M).
That is, in the two cases of the Theorem we are able to identify the copyof `p insideFp(M) as a Lipschitz free pspace on a subset ofM.

Open questions
Our work leaves many open problems. Let us single out a few:We know there are cases, such as whenM is a subset ofR equipped withthe quasimetric ρ(x, y) = x− y1/p for which L : Fp(N )→ Fp(M) isan isometric embedding wheneverN ⊂M .
Question 1Can we identify the pmetric spacesM for which the canonical linearmap L : Fp(N )→ Fp(M) is an isometrywheneverN ⊂M?

Open questions
Related to the previous question:
Question 2Suppose 0 < p < 1. Is the canonical linearization L : Fp(N )→ Fp(M)always an isomorphismwheneverN ⊂M andM is a pmetric (evenmetric) space.
The following partial result establishes a dichotomy in that respect:
LemmaLet 0 < p ≤ q ≤ 1. Either there is a qmetric spaceM and a subsetN ofM for whichL is not an isomorphism or there is a universal constantC(depending on p and q) such that for every qmetric spaceM and everysubset N ofM we have L : Fp(N ) → Fp(M) is an isomorphism with‖L−1 ‖ ≤ C.

Open questions
ConsiderN equipped with the standard metric d(n,m) = m− n.
Question 3What is the spaceFp(N) for 0 < p < 1?
Facts we know about it:1 Its Banach envelope isF(N) = `1;2 It contains `p complemented;3 It has a monotone Schauder basis.
It seems only natural to guess thatFp(N) ≈ `p . . . but we don’t know howto substantiate this conjecture.

Open questions
Question 4What is the spaceFp([0, 1]) for 0 < p < 1?
We know that:1 Its Banach envelope isF([0, 1]) = L1 (henceFp([0, 1]) cannot be `por Lp);
2 It contains `p complemented;3 It has a monotone Schauder basis which cannot be unconditional.Fp([0, 1]) for 0 < p < 1 is a pBanach space with a lot of theoreticalusefulness (which reinforces the importance of understanding Lipschitzfree pspaces better):−On one hand, it provides a concrete example of a quasiBanach spacewhose Banach envelope is L1. Nigel had proved that such pspacesexisted inside `p for every p < 1. IsFp([0, 1]) ⊂ `p? DoesFp([0, 1])contain `1 complemented? (We know `1 6= Fp(K) for anyK ⊂ [0, 1]!)−On the other hand, it is a nontrivial example of a quasiBanach spacewith a basis, which does not have an unconditional basis.

References
F. ALBIAC, J.L. ANSORENA, M. CÚTH, ANDM. DOUCHA, Lipschitz freepspaces for 0 < p < 1. To appear in Israel J. Math.
F. ALBIAC, J.L. ANSORENA, M. CÚTH, ANDM. DOUCHA, Embeddabilityof `p and bases in Lipschitz free pspaces for 0 < p < 1. Submitted.

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