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Lipschitz free p-spaces Fernando Albiac Public University of Navarre Joint work with José L. Ansorena, Marek Cúth, and Michal Doucha Banach Spaces and their Applications L, 2629 J 2019
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• Lipschitz free p-spaces

Fernando AlbiacPublic University of Navarre

Joint work with José L. Ansorena, Marek Cúth, andMichal Doucha

Banach Spaces and their ApplicationsLVIV, 26-29 JUNE 2019

• Brief review: p-Banach spaces

Consider 0 < p ≤ 1:A p-Banach 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).

Aoki-Rolewicz Theorem: (X, ‖ · ‖) is a p-Banach space for some0 < p ≤ 1 if and only if it is a quasi-Banach space, that is, we can replacecondition (iii) with(iii)′ ∃C ≥ 1 such that ‖x+ y‖ ≤ C(‖x‖+ ‖y‖) ∀x, y ∈ X .

• Brief review: p-metric spaces for 0 < p ≤ 1

The corresponding concept for metric spaces is that of a p-metric space.(M, ρ) is a p-metric space if (M, ρp) is a metric space. (Completenessis not assumed now.)

An analogue of the Aoki-Rolewicz 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 p-subadditivefor some 0 < p ≤ 1.That is, every quasimetric space is Lipschitz isomorphic to a p-metricspace 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 Hahn-Banach theorem. No duality techniques available.Bochner integration is not available in the lack of local convexity.

• Motivation to further research in p-Banach 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 p-Banach spaces for 0 < p < 1 often provides analternative proof even for p = 1.There are many classical p-Banach 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 non-locallyconvex 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, Lipschitz-free Banach spaces, StudiaMath. 159 (2003), no.1, 121-141.

the authors considered Lipschitz free spaces with the purpose toinvestigate the following general problem in the theory of non-linearclassification 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 Lipschitz-lifting property if there is a continu-ous 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 (Godefroy-Kalton; 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

Lipschitz-lifting property.

Unfortunately, this method fails to produce separable Banach spaceswhich are Lipschitz isomorphic without being linearly isomorphic:

Theorem (Godefroy-Kalton; 2003)If X is an infinite-dimensional separable Banach space, X has theLipschitz-lifting property. Hence F(X) is linearly isomorphic to the spaceG(X).

• Motivation for introducing the class of Lipschitz freep-spaces for 0 < p < 1

However, the method does work for building for every p < 1 twoseparable p-Banach spaces which are Lipschitz isomorphic but notlinearly isomorphic:

Theorem (A.-Kalton; 2009)Let X be an infinite-dimensional separable Banach space which is not aSchur space. For every p < 1, the p-Banach 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 quasi-Banachspaces, Israel J. Math. 170 (2009), 317-335.

• Motivation for further studying Lipschitz free p-spaces

They are the non-locally 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 p-spaces over p-metric spaces

Setting: Let (M, ρ, 0) be a p-metric 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 p-normed spaces (Y, ‖ · ‖Y ) and all1-Lipschitz maps f : M→ Y with f(0) = 0.

• Definition of Lipschitz free p-spaces over p-metric spaces

It is straightforward to check that this defines a p-seminorm on P(M).

What is not so straightforward is that, in fact,

‖ · ‖Fp(M) defines a p-norm on P(M)!

That is,{δ(x) : x ∈M} is a linearly independent set in P(M).

DefinitionThe Lipschitz free p-space overM, denotedFp(M), is the p-Banach spaceresulting from the completion of the p-normed 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 p-metric 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) p-Banach space

such that for every p-Banach 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 p-metric spaces

Lipschitz free p-spaces provide a canonical linearization process ofLipschitz maps between p-metric spaces

If we identify (through the map δM) a p-metric spaceMwith a subset ofFp(M), then any Lipschitz map f from a p-metric spaceM1 to ap-metric 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 bi-Lipschitz bijection then Lfis an isomorphism.

• The dual ofFp(M) for 0 < p ≤ 1

Proposition

SupposeM is a pointed p-metric 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 (non-trivial) real-valued 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) char-acterized 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 finitely-many 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 ρ̃-distance

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 quasi-Banach spaceX isthe “closest” Banach space toX .

PropositionSupposeM is a pointed p-metric 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 quasi-Banach 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 p-spaces(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 Naor-Schechtman)F̂p(R) = F(R) 6≈ F(R2) = F̂p(R2) .

• Computation of the norm in Lipschitz free p-spaces

Lipschitz free p-spaces 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 p-norm in Lipschitz-free p-spaces:

For µ ∈ Fp(M)we have

‖µ‖Fp(M) = inf

( ∞∑k=1|ak|p

)1/p: µ =

∞∑k=1

akδ(xk)− δ(yk)ρ(xk, yk)

, xk 6= yk

,which relies only on the p-metric on the spaceM.

• Early examples

That alternative expression of the p-norm inFp(M) is very useful andpermits to provide the first examples of Lipschitz free p-spaces overp-metric spaces:

Example (Fp(M)-spaces isomorphic to `p andLp, 0 < p ≤ 1)1 ConsiderN equipped with the p-metric ρ(m,n) = |m− n|1/p form,n ∈ N. ThenFp(N) ≈ `p isometrically.

2 Let I be an interval of the real line equipped with the p-metricρ(x, y) = |x− y|1/p for x, y ∈ I . ThenFp(I) ≈ Lp(I) isometrically.

3 LetM be a bounded and uniformly separated p-metric space. Then

Fp(M) ≈ `p(M\ {0}).

• First results

The spacesFp(M) over quasimetric spaces are a new class of p-Banachspaces that are difficult to identify.By imposing a stronger condition onM, like being ultrametric, we willbe able to recognize the Lipschitz free p-space 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 p-Banach 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 q-metric space (M, ρ) and asubsetN ⊆M such that the inclusion map : N →M induces anon-isometric 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− y|1/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 p-metric 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 `p-basis.

anµn

∥∥∥∥∥p

Fp(M)

≤∞∑n=1|an|p‖µn‖pFp(M) =

∞∑n=1|an|p,

for any sequence of scalars (an).

• Linear structure ofFp(M), 0 < p < 1

Proof.To prove the converse inequality, we need to choose 1-Lipschitz mapsgn : M→ Y and a target p-Banach space Y (where we know how toexplicitly compute p-norms), in such a way that∥∥∥∥∥

∞∑n=1

anµn

∥∥∥∥∥p

Fp(M)

∥∥∥∥∥∞∑n=1

angn(xn)− gn(yn)

ρ(xn, yn)

∥∥∥∥∥p

Y

≥ C∞∑n=1|an|p,

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 p-spaces 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 p-metric for all p < 1) it has plentyof real-valued Lipschitz maps.

2 Uniformly separated p-metric spaces have nontivial Lipschitzfunctions. In fact, if (M, ρ) is r-separated, 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 p-metric 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 p-space 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− y|1/p for which L : Fp(N )→ Fp(M) isan isometric embedding wheneverN ⊂M .

Question 1Can we identify the p-metric 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 p-metric (evenmetric) space.

The following partial result establishes a dichotomy in that respect:

LemmaLet 0 < p ≤ q ≤ 1. Either there is a q-metric 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 q-metric 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 p-Banach space with a lot of theoreticalusefulness (which reinforces the importance of understanding Lipschitzfree p-spaces better):−On one hand, it provides a concrete example of a quasi-Banach spacewhose Banach envelope is L1. Nigel had proved that such p-spacesexisted 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 non-trivial example of a quasi-Banach spacewith a basis, which does not have an unconditional basis.

• References

F. ALBIAC, J.L. ANSORENA, M. CÚTH, ANDM. DOUCHA, Lipschitz freep-spaces 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 p-spaces for 0 < p < 1. Submitted.

• THANK YOU FOR YOUR ATTENTION!!