Kwapień's property (K) in Lipschitz-free spaces

Introduction

Kwapień’s property (K), in the formulation of Kalton and Pełczyński [15 Section 2], concerns convex blocks of weak*-null sequences in a dual Banach space. A Banach space $X$ has property (K) if every such sequence admits a convex block subsequence converging to zero uniformly on each weakly compact subset of $X$. Equivalently, the blocks converge to zero in the Mackey topology $\mu(X^*, X)$. Avilés and Rodríguez [7 Corollary 2.3] show that every subspace of a strongly weakly compactly generated (SWCG) space has (K). Finite-measure $L_1$-spaces are strongly weakly compactly generated [22], so their subspaces have (K). Kwapień’s argument in Kalton and Pełczyński [15 Section 2] already shows that finite-measure $L_1$-spaces have (K).

The Lipschitz-free space of $I = [0, 1]$ is isometric to $L_1[0, 1]$ [10 Corollary 3.3] and therefore has (K). We show that already a Cantor family of suitable subsets of an interval forces failure of (K). The proof constructs a uniformly null, uniformly Lipschitz sequence whose every convex block subsequence is detected on one fixed fibre. The fibre depends on the whole block sequence. No linear extension operator on the ambient metric space is required.

All Banach spaces are real, $\mathbb{N} = \{1, 2, \ldots\}$, and $\cong$ denotes a Banach space isomorphism. A Cantor metric space is a metric space homeomorphic to $\{0, 1\}^{\mathbb{N}}$. On $C \times K$, where $C \subset\mathbb{R}$ and $(K, d_K)$ is a metric space, we use the metric

Equivalent standard product metrics give the same conclusions. We equip $\mathbb{R}^n \times\mathbb{Z}^m$ with the Euclidean metric inherited from $\mathbb{R}^{n+m}$. Any other norm on $\mathbb{R}^{n+m}$ gives an isomorphic Lipschitz-free space, since the identity map is bi-Lipschitz.

2020 Mathematics Subject Classification. Primary 46B20, 46E15; Secondary 46A20, 28A80.

Key words and phrases. Lipschitz-free space, Kwapień’s property (K), Mackey topology, strongly weakly compactly generated space, Cantor set, Assouad dimension.

RVO: 67985840.

In particular, the theorem applies to $I \times K$ for every uncountable compact metric space $K$, to every cube $I^n$ with $n \ge2$, and to every normed space of dimension at least two. Indeed, every uncountable compact metric space contains a Cantor subspace.

An application to the isomorphic classification of Lipschitz-free spaces follows from Aliaga and Medina’s decomposition [5 Proposition 3.14]: for $n,m \ge1$, the space $\mathcal{F}(\mathbb{R}^n \times\mathbb{Z}^m)$ is isomorphic to the direct sum of $\mathcal{F}(\mathbb{R}^n)$ and $\mathcal{F}(\mathbb{Z}^{n+m})$. Their proof uses the maximum norm; the preceding norm equivalence gives the Euclidean formulation used here. Throughout, $\mathbb{R}^0$ and $\mathbb{Z}^0$ denote singletons.

In particular, $\mathcal{F}(\mathbb{R} \times\mathbb{Z}) \not\cong\mathcal{F}(\mathbb{R}^2)$. This answers [1 Question 5], also recorded in [5 Section 3.5] and [17 Problem 2]. Mason’s recent Dunford–Pettis theorem [17 Theorem 1 and Corollary 2] shows that both spaces have the Dunford–Pettis property, so that property cannot distinguish them. Property (K) does, and the SWCG conclusion yields the stronger nonembeddability result. In the original topological formulation of SWCG, Theorem B also says that the closed unit ball of $\operatorname{Lip}_0(\mathbb{R}^n \times\mathbb{Z}^m)$ is metrisable in its Mackey topology precisely when $n \le1$.

An Eggleston-type product lemma gives a further geometric consequence.

Naor and Schechtman [20] proved that the $L_1$-distortion of planar earthmover metrics on square grids tends to infinity, with a lower bound of order $\sqrt{\log n}$ for grids of side length $n$. Their theorem implies that $\mathcal{F}(I^2)$ and $\mathcal{F}(\mathbb{R}^2)$ do not embed into $L_1$. This also implies $\operatorname{Lip}_0(I^2) \not\cong\ell_\infty$: otherwise $\mathcal{F}(I^2)$ would embed into $\ell_\infty$ and hence, by the measure-domination argument in Lemma 2.4, into a finite-measure $L_1$-space. Theorem A gives a direct qualitative proof of these consequences. Its additional scope includes arbitrary positive-measure sets, products $I \times K$ with null Cantor sets $K$, and the example below, where square grids of uniformly bounded distortion are unavailable.

The Euclidean examples above combine failure of (K) with the Dunford–Pettis property, property ($V^*$) and weak sequential completeness [16, 6, 17]. By contrast, complete purely 1-unrectifiable metric spaces have Schur Lipschitz-free spaces [4 Theorem C] and hence have (K); neither these Schur spaces nor our weakly sequentially complete examples contain $c_0$.

Preliminaries

For a pointed metric space $(M,o)$, let $\operatorname{Lip}_0(M)$ be the Banach space of real Lipschitz functions vanishing at $o$, with norm $\|f\|_{\operatorname{Lip}}$. Its canonical predual is the Lipschitz-free space

Changing the base point gives canonically isometric spaces. McShane’s extension theorem [23 Theorem 1.33] identifies $\mathcal{F}(N)$ isometrically with a closed subspace of $\mathcal{F}(M)$ whenever $N\subset M$ contains the base point. Moreover, $\mathcal{F}(M)$ is canonically isometric to $\mathcal{F}(\overline{M})$, where $\overline{M}$ is the metric completion. A uniformly bounded sequence in $\operatorname{Lip}_0(M)$ is weak$^*$-null if it converges pointwise to zero, by density of the evaluation functionals.

For a Banach space $X$, write $B_X$ for its closed unit ball. The Mackey topology $\mu(X^*,X)$ on $X^*$ is the locally convex topology of uniform convergence on weakly compact subsets of $X$. It is generated by the seminorms

One may equivalently restrict to absolutely convex weakly compact sets, since the closed absolutely convex hull of a weakly compact set is weakly compact. Thus $g_k\to0$ in $\mu(X^*,X)$ means that $p_W(g_k)\to0$ for every such $W$. The weak$^*$ topology $\sigma(X^*,X)$ requires only pointwise convergence on $X$.

A convex block subsequence of $(f_m)$ has the form

where the $E_k\subset\mathbb{N}$ are finite and non-empty. The following standard consequence of Eberlein–Šmulian connects the Mackey formulation of (K) with the pairing formulation in [15]. We recall the proof for completeness; see also the quantitative version in [9 Lemma 3.1].

Proof. One implication follows by applying uniform convergence to the weakly compact set $\{0,x_1,x_2,\ldots\}$. Conversely, failure of uniform convergence on a weakly compact set $W$ gives $\varepsilon>0$, indices $k_j$ and points $x_j\in W$ with $|\langle g_{k_j},x_j\rangle|\ge\varepsilon$. By Eberlein–Šmulian, pass to a subsequence such that $x_j\to x$ weakly. Set $z_{k_j}=x_j-x$ and $z_k=0$ at all other indices. Then $(z_k)$ is weakly null, whereas weak$^*$ convergence gives $|\langle g_{k_j},z_{k_j}\rangle|\ge\varepsilon/2$ eventually. $\square$

The proof of Theorem A uses only the forward implication. The converse reconciles the two formulations of property (K).

This is the original definition of Schlüchtermann and Wheeler. By [22 Theorem 2.1], it is equivalent to the existence of a weakly compact absolutely convex set $H\subset Y$ such that

Here $nH=\{nh:h\in H\}$, and $H$ is said to strongly generate $Y$. We use this characterisation below. Equivalently, there is a sequence $(A_j)$ of weakly compact absolutely convex subsets of $Y$ such that, for every weakly compact $W\subset Y$ and every $\varepsilon>0$, some $A_j$ satisfies $W \subset A_j + \varepsilon B_Y$. This expresses the weakly compact density condition: a countable family approximates every weakly compact set to arbitrary norm accuracy. The three equivalent formulations are also recorded in [13 Theorem 10]; for the density-condition viewpoint, see [12 Section 6.7, pp. 177–183].

The result below for separable Schur spaces is due to Schlüchtermann and Wheeler [22]; see also Mercourakis and Stamati [18 Example 1.3(ii)]. We include its short proof and the elementary permanence properties needed for Theorem B.

SWCG itself does not pass to closed subspaces: Mercourakis and Stamati [18 Corollary 3.10] construct a non-SWCG closed subspace of $L_1[0,1]$. Thus the nonembedding conclusion in Theorem A uses the stronger fact that every subspace of an SWCG space has (K), due to Avilés and Rodríguez [7 Corollary 2.3]. Together with the finite-measure $L_1$ case of [22], this also gives the following observation for arbitrary measures and dual isomorphisms.

The derivative map identifies $\operatorname{Lip}_0(I)$ isometrically with $L_\infty(I)$; Pełczyński’s theorem [21] gives $L_\infty(I)\cong\ell_\infty$. We shall also use that Schur spaces have (K): their weakly compact sets are norm compact, and bounded weak$^*$-null sequences converge uniformly on norm-compact sets. Recall also that the Dunford–Pettis property is equivalent to $\langle x_j^*,x_j\rangle\to0$ whenever both $(x_j)\subset X$ and $(x_j^*)\subset X^*$ are weakly null. In contrast, the dual sequences in the definition of (K) are only assumed to be weak$^*$-null.

The product construction

Passing to a compact subset and rescaling, it suffices to consider compact $C\subset I$ with $\min C=0$, $\max C=1$ and $c=|C|>0$. When normalising $C$, rescale the metric on $K$ by the same factor; this rescaling leaves the distortion of the embedding unchanged. Let

The Rademacher functions are orthonormal in $L_2(I)$. By Bessel’s inequality, applied to $\mathbf{1}_{C\cap[0,x]}$, we have $H_j(x)\to0$ for every $x$, and their common Lipschitz bound 1 implies

Proof. We first describe the affine extension used in [10 Theorem 3.2]. Since $C$ is compact and contains 0 and 1, each component of $I\setminus C$ is an open interval $(a,b)$ with $a,b\in C$. For $h\in\operatorname{Lip}_0(C)$, define $\Lambda h=h$ on $C$ and

This defines a linear operator $\Lambda:\operatorname{Lip}_0(C)\to\operatorname{Lip}_0(I)$. On $[a,b]$, the slope of $\Lambda h$ is $(h(b)-h(a))/(b-a)$, whose modulus is at most $\lVert h\rVert_{\mathrm{Lip}}$. If $x<y$ do not lie in the closure of one gap, let $a$ and $b$ be the first and last points of $C\cap[x,y]$. Applying the preceding estimate on $[x,a]$ and $[b,y]$, and the Lipschitz bound for $h$ on $[a,b]$, gives

Since restriction to $C$ recovers $h$, $\Lambda$ is an isometry. Its adjoint sends an evaluation at $x\in C$ to the same evaluation on $C$, and for $x\in(a,b)\subset I\setminus C$ it satisfies

where the evaluations on the right belong to $\mathcal{F}(C)$. Thus $\Lambda^*$ maps the linear span of the evaluations on $I$ into $\mathcal{F}(C)$ and, by continuity, restricts to a contraction $P:\mathcal{F}(I)\to\mathcal{F}(C)$. The identity $\langle h,P\mu\rangle=\langle\Lambda h,\mu\rangle$ for $h\in\operatorname{Lip}_0(C)$ and $\mu\in\mathcal{F}(I)$ gives $P^*=\Lambda$.

For $H\in\operatorname{Lip}_0(I)$, write $H'$ for its almost-everywhere derivative. Such a function is absolutely continuous, $H(x)=\int_0^x H'(s)\,\mathrm{d}s$, and $\lVert H'\rVert_\infty=\lVert H\rVert_{\mathrm{Lip}}$. The interval representation in [10 Corollary 3.3] can therefore be described directly by

Indeed, $J\mathbf{1}_{(a,b]}=\delta_b-\delta_a$ whenever $0\le a<b\le1$. Interval step functions are dense in $L_1(I)$, so the estimate $\lVert Ju\rVert\le\lVert u\rVert_1$ shows that $Ju\in\mathcal{F}(I)$ for every $u\in L_1(I)$. Taking $H(x)=\int_0^x\operatorname{sgn}u(s)\,\mathrm{d}s$ gives the reverse inequality, hence $J:L_1(I)\to\mathcal{F}(I)$ is an isometry. Its range is closed and contains every evaluation $\delta_x=J\mathbf{1}_{(0,x]}$, so it is onto.

Put $\nu_j=c^{-1}PJ(\mathbf{1}_C r_j)$. Since $\lVert\mathbf{1}_C r_j\rVert_1=c$, we have $\lVert\nu_j\rVert\le1$. For every $h\in\operatorname{Lip}_0(C)$,

by Bessel’s inequality, since $\mathbf{1}_C(\Lambda h)'\in L_2(I)$. Thus $(\nu_j)$ is weakly null. The function $H_j$ is constant on every gap of $C$, so $\Lambda(H_j|_C)=H_j$. Consequently

As $\lVert H_j|C\rVert_{\mathrm{Lip}} \le1$, this also gives $\lVert\nu_j\rVert= 1$.

Let $K$ be a Cantor metric space. Fix a homeomorphism $S \mapsto t_S$ from the subsets of $\mathbb{N}$, identified with $\{0,1\}^{\mathbb{N}}$, onto $K$. For a finite binary word $\sigma$, define the compact cylinder

For $b \ge1$, put

The sequence $(\varepsilon_b)$ decreases to zero. By (3), choose strictly increasing indices $j_b$ with $\lVert H_{j_b}\rVert_\infty\le\varepsilon_b$. For a fixed $m$ and a subset $S\subset\mathbb{N}$, call the elements of $S\cap\{m,m+1,\ldots\}$ the selected indices. Define on $C\times K$

These functions vanish on $\{0\}\times K$, satisfy $\lVert f_m\rVert_\infty\le\varepsilon_m$, and are 1-Lipschitz in the first coordinate. To check the second coordinate, suppose that $S,U$ first differ at $b$. If $b\ge m$ and a selected index lies in $[m,b)$, the first selected indices agree and the values in (4) coincide. Otherwise all selected indices are at least $b$; this also holds when $b<m$, since they are then at least $m>b$. The difference is therefore at most $2\varepsilon_b\le d_b\le d_K(t_S,t_U)$. Going through $(x',t)$ now gives the explicit estimate

Proof of Theorem A. Reduce $C$ to the compact normalised case above and $K$ to a Cantor subspace. Let $\Phi:C\times K\to M$ be a bi-Lipschitz embedding, and choose the base point $o=\Phi(0,t_{\varnothing})$. Write $d_M$ for the metric on $M$, put $Q=\Phi(C\times K)$, and define $g_m=f_m\circ\Phi^{-1}$ on $Q$. By (5), these functions have a common Lipschitz bound $L=\sqrt{2}\lVert\Phi^{-1}\rVert_{\mathrm{Lip}}$. McShane’s extension theorem [23 Theorem 1.33] gives the extensions

Here $L d_M(p,q)$ is the distance between $p$ and $q$ multiplied by the constant $L$. The triangle inequality gives $\widetilde{F}_m(p)\le\widetilde{F}_m(p')+L d_M(p,p')$; interchanging $p$ and $p'$ proves the Lipschitz bound $L$. For $p\in Q$, the Lipschitz bound on $g_m$ gives $g_m(p)\le g_m(q)+L d_M(p,q)$ for every $q\in Q$, and taking $q=p$ shows that $\widetilde{F}_m(p)=g_m(p)$.

To retain the uniform bound $\varepsilon_m$, define

This replaces values above $\varepsilon_m$ by $\varepsilon_m$ and values below $-\varepsilon_m$ by $-\varepsilon_m$. The scalar map $s\mapsto\max\{-\varepsilon_m,\min\{\varepsilon_m,s\}\}$ is 1-Lipschitz and fixes every point of $[-\varepsilon_m,\varepsilon_m]$. Since $|g_m|\le\varepsilon_m$ on $Q$, it follows that $F_m$ still extends $g_m$, vanishes at $o$, and satisfies $\lVert F_m\rVert_{\mathrm{Lip}}\le L$ and $\lVert F_m\rVert_\infty\le\varepsilon_m$. In particular, $(F_m)$ is weak*-null in $\operatorname{Lip}_0(M)$.

For each fixed $t\in K$, the bounded linear map

has $\lVert T_t\rVert\le\lVert\Phi\rVert_{\mathrm{Lip}}$ by the universal linearisation property of Lipschitz-free spaces [23 Theorem 3.6], and sends $(\nu_j)$ to a weakly null sequence. Let $(G_k)$ be any convex block subsequence of $(F_m)$, with notation as in (1). Put $b_k=\max E_k$ and $t=t_{\{b_k:k\geq1\}}$. For every $m\in E_k$, the least selected index at least $m$ is $b_k$. Consequently

Lemma 2.1 proves that $\mathcal{F}(M)$ fails (K). Notice that the same fibre is used for the whole block sequence.

The subspace theorem from [7] excludes an embedding into an SWCG space. To exclude an embedding into an arbitrary $L_1$-space, restrict it to the separable subspace $\mathcal{F}(Q)$. Applying the construction with ambient space $Q$ shows that $\mathcal{F}(Q)$ fails (K), contradicting Lemma 2.4. Finally, if $M$ is separable then so is $\mathcal{F}(M)$, and the same lemma gives $\operatorname{Lip}_0(M)\not\cong\ell_\infty$.

The construction also gives a quantitative estimate. Following [9 Section 3], for a bounded sequence $(g_k)\subset X^*$ put

The quantity $K_2(X)$ is the supremum, over weak*-null sequences in $B_{X^*}$, of the infimum of $\beta$ over their convex block subsequences. If $\Phi$ is as in the proof above and $D=\|\Phi\|_{\operatorname{Lip}}\|\Phi^{-1}\|_{\operatorname{Lip}}$, then

Indeed, for any $\eta>0$, choose the indices $j_b$ so that $\|H_{j_b}\|_\infty\leq\eta\varepsilon_b$. The functions in (4) are then $\eta$-Lipschitz in the second coordinate and have Lipschitz norm at most $\sqrt{1+\eta^2}$. Run the extension construction with $L_\eta=\sqrt{1+\eta^2}\|\Phi^{-1}\|_{\operatorname{Lip}}$ and truncate the extensions to $[-\eta\varepsilon_m,\eta\varepsilon_m]$. For every convex block sequence $(G_k)$ of these extensions, the weakly null sequence

satisfies

Letting $\eta\downarrow0$ gives (6). Since $K_2(X)\leq1$, we obtain $K_2(\mathcal{F}(M))=1$ whenever the product embeds isometrically.

This proof also explains the significance of positive horizontal measure. If a non-empty compact set $C\subset I$ has Lebesgue measure zero, then $\mathbf{1}_C r_j=0$ in $L_1(I)$ and $H_j$ is identically zero. Moreover, $\mathcal{F}(C)$ has the Schur property: if $C$ is infinite, it is linearly isometric to $\ell_1$ by [10 Corollary 3.4], whereas for finite $C$ it is finite-dimensional. Thus $\mathcal{F}(C)$ contains no normalised weakly null sequence. This explains why the fibre argument requires $C$ to have positive measure; it does not show that this condition is necessary for $\mathcal{F}(C\times K)$ to fail (K).

Proof of Theorem B. Each $\mathbb{Z}^k$ is countable and uniformly discrete, so $\mathcal{F}(\mathbb{Z}^k)$ is a separable Schur space by [14 Proposition 4.4]; it is therefore SWCG by Lemma 2.3. This settles the case $n=0$, including the zero space if $m=0$. For $n=1$ and $m\geq1$, [5 Proposition 3.14] gives

where the second isomorphism uses [10 Corollary 3.3]. The first summand is isomorphic to $L_1[0,1]$ and hence is SWCG; Lemma 2.3 applies again. The case $n=1,m=0$ is the same $L_1$ observation. Every SWCG space has (K) by [7]. If $n\geq2$, then the underlying space contains an isometric copy of $I^2$, so Theorem A gives failure of (K). Finally, $\mathcal{F}(\mathbb{R}^d)$ embeds into no SWCG space when $d\geq2$, which proves the final part of the theorem.

Positive-measure Euclidean sets

We include the following Eggleston-type argument for completeness; see [19] for related product theorems and their history.

Proof. By inner regularity, replace $E$ by a compact subset of positive measure. For $y \in\mathbb{R}^{d}$, set $E_y = \{x : (x,y) \in E\}$ and $\Psi(y) = 1_{E_y} \in L_1(\mathbb{R})$. The map $\Psi$ is measurable: its distances to a countable dense family in $L_1(\mathbb{R})$ are measurable by Fubini. Fubini and Lusin give a compact parameter set $P_0$ of positive measure on which $\Psi$ is continuous and every $E_y$ has positive measure. Replace $P_0$ by the support of Lebesgue measure restricted to $P_0$. The resulting compact set $P$ has no isolated points and retains both properties.

Choose $y_{\varnothing} \in P$ and let $a = |E_{y_{\varnothing}}| > 0$. By continuity and the absence of isolated points, choose inductively $y_s \in P$ and relatively open neighbourhoods $U_s$ for finite binary words $s$, with $y_s \in U_s$, nested closures, disjoint sibling closures, diameters tending uniformly to zero, and

Here and below closures are taken in $P$. Then

are compact, and the nested construction makes $K$ a Cantor set. Every point lost from the root fibre is lost at some edge of a finite branch, whence

For $y \in K$, the points $y_s$ along its branch converge to $y$. Closedness of $E$ then gives $C \times\{y\} \subset E$. □

Proof of Theorem C. Apply Lemma 4.1 to $\overline{E}$, with $d=n-1$, and then Theorem A. Since $\mathcal{F}(E)=\mathcal{F}(\overline{E})$ canonically, all the conclusions follow. □

Aliaga [1 Theorem 3.7 and Corollary 3.8] proves $\operatorname{Lip}_0(E) \cong\operatorname{Lip}_0(\mathbb{R}^{n})$ whenever $E \subset\mathbb{R}^{n}$ is non-porous, in particular whenever it has positive measure. That dual isomorphism alone does not identify the weak* topologies determined by the canonical preduals. Theorem C establishes failure of (K) directly for those preduals. Aliaga’s subsequent theorem [2 Theorem 1] gives a complemented copy of $\mathcal{F}(\mathbb{Z}^{n})$ in $\mathcal{F}(E)$ whenever $E$ is non-porous. It does not yield failure of (K), since $\mathcal{F}(\mathbb{Z}^{n})$ is Schur. Indeed, the positive-measure hypothesis in Theorem C cannot be replaced by non-porosity: $\mathbb{Z}^{n}$ is non-porous in the all-scales sense of [1 Definition 2.1], but its Lipschitz-free space is SWCG. That definition requires a fixed $\lambda>0$ such that every ambient ball $B(z,R)$, at every scale $R>0$, contains a ball of radius $\lambda R$ disjoint from the set. Every point of $\mathbb{R}^{n}$ is within $\sqrt{n}/2$ of $\mathbb{Z}^{n}$, so every ball of radius greater than $\sqrt{n}/2$ meets $\mathbb{Z}^{n}$. Taking $\lambda R>\sqrt{n}/2$ rules out the required hole. This is compatible with the porosity of $\mathbb{Z}^{n}$ at small scales.

Candido, Cúth and Doucha [8 Theorem 1.16] proved $\operatorname{Lip}_0(\mathbb{Z}^{d}) \cong\operatorname{Lip}_0(\mathbb{R}^{d})$ for every $d\ge1$. The non-isomorphism of the canonical preduals was already known: $\mathcal{F}(\mathbb{Z}^{d})$ is Schur, whereas $\mathcal{F}(\mathbb{R}^{d})$ contains a copy of $L_{1}$ coming from a line. For $d \ge2$, the present argument also distinguishes these preduals by property (K).

An example of Assouad dimension one

For a metric space $M$, its Assouad dimension is the infimum of the exponents $s$ for which there is $A_s < \infty$ such that

where $\operatorname{diam} M$ may be infinite and $N_r$ is the least number of balls of radius $r$ needed to cover the indicated set.

Proof of Theorem D. Starting with $[0,1]$, at stage $j \ge1$ retain the two endpoint subintervals of relative length $a_j = 2^{-j-2}$ in each remaining interval. Let $K_\ast$ be the resulting Cantor set. The intervals at level $j$ have length $L_j = \prod_{i=1}^{j} a_i = 2^{-j(j+5)/2}$, with $L_0 = 1$.

We verify that $\dim_A K_\ast= 0$. Fix $\varepsilon> 0$ and choose $J$ so that $a_j^\varepsilon\le1/2$ for $j \ge J$. For $0 < r < R < 1$, choose $k,n$ with $L_k \le R < L_{k-1}$ and $L_n \le r < L_{n-1}$. Then $n \ge k$. Distinct level-$(k-1)$ intervals are separated by gaps of length at least $6L_{k-1} > 2R$; for $k=1$ there is just one such interval. Thus a ball of radius $R$ meets at most one, and $N_r(B(p,R) \cap K_\ast) \le2^{n-k+1}$. If $n \le k+1$, this is at most 4. Otherwise,

The same global estimate treats $R=1$, so $\dim_A K_\ast= 0$. Product covers give $\dim_A(I \times K_\ast) \le1$, and the contained interval gives the reverse inequality. Set $M=I \times K_\ast$ and apply Theorem A.

Since $M$ is a compact subset of the plane, [16] gives $(V^\ast)$ and weak sequential completeness, while [17 Corollary 2] gives the Dunford–Pettis property.

Finally, suppose that $f:\{0,\ldots,N\}^2 \to M$ satisfies

for some $\lambda> 0$ and fixed $D$. Its $(N+1)^2$ image points are $\lambda$-separated and lie in a ball of radius $\sqrt{2}D\lambda N$. The Assouad estimate with exponent $3/2$ and $r=\lambda/3$ gives

If the displayed outer radius exceeds $\operatorname{diam} M$, use $\operatorname{diam} M$ instead. The constant $A$ depends only on $M$, so $N$ is bounded for each fixed $D$. □

The dimension in Theorem D is minimal even among complete metric spaces. If $\dim_A M < 1$, every bounded ball in $M$ has zero $\mathcal{H}^1$-measure, and a countable covering by such balls gives $\mathcal{H}^1(M)=0$. Thus $M$ is purely 1-unrectifiable, and [4 Theorem C] gives the Schur property and (K).

For context, Kochanek and Pernecká [16] establish $(V^\ast)$ and weak sequential completeness for Lipschitz-free spaces over compact subsets of superreflexive spaces. Aliaga, Pernecká and Quero [6 Theorem C] prove $(V^\ast)$ over Hilbert spaces; as this property passes to closed subspaces, it holds for all our Euclidean examples. More generally, Jung, Petitjean, Procházka and Quilis [11 Proposition 6.1(ii)] obtain $(V^\ast)$ for metric spaces of finite Nagata dimension, including doubling spaces. Mason’s theorem [17 Corollary 2] gives the Dunford–Pettis property for every subset of a finite-dimensional Euclidean space. The example above therefore has all three properties despite failing (K).

Further questions

Let $\Delta$ be the middle-third Cantor set. Both $I \times\Delta$ and the product in Theorem D are porous in the plane. Indeed, all retention ratios are at most $1/3$. For a point of either Cantor set and $0 < r < 4$, the first level interval of length at most $r/4$ has a parent with a middle gap longer than $r/12$, at distance at most $r/4$ from the point. This gives a complementary planar ball of radius $r/48$ inside the ball of radius $r$ centred at the corresponding product point. Larger scales follow from boundedness. For an arbitrary ambient ball $B(z, R)$, either $B(z, R/2)$ misses the product or it contains a product point to which this construction applies with radius $R/2$. Thus one may take porosity constant $1/96$.

Porosity, rather than measure zero alone, places these products outside Aliaga’s non-porosity theorem. We know that $\operatorname{Lip}_0(I \times\Delta) \not\cong\ell_\infty$, but the argument does not decide whether $\operatorname{Lip}_0(I \times\Delta) \cong\operatorname{Lip}_0(I^2)$. A negative answer would disprove the line-or-plane dichotomy in [1 Question 3]; a positive answer would supply a porous example for [1 Question 4]. The recent theorem of Aliaga, Cúth and Vico [3 Theorem 1.1] makes every Lipschitz-free space the strongly unique isometric predual of its Lipschitz space. This does not require arbitrary Banach space isomorphisms of Lipschitz spaces to preserve their canonical weak* topologies, and therefore does not settle the question above. The example $\mathbb{Z}^2$ versus $\mathbb{R}^2$ already shows that such an inference would fail.

Question 6.1. If $M \subset\mathbb{R}^2$ is compact and $\mathcal{F}(M)$ fails (K), must $M$ contain a bi-Lipschitz copy of $C \times K$ as in Theorem A?

Compactness is a substantive restriction here: a countable dense subset of the square has the same Lipschitz-free space as the square, but cannot contain such an uncountable product.

Question 6.2. Can the construction give explicit lower bounds for the $L_1$-distortion of finite-dimensional subspaces of $\mathcal{F}(I \times K)$ in terms of the separation scales of $K$?

The estimate (6) quantifies failure of (K), but does not by itself give finite-dimensional $L_1$-distortion bounds. In particular, it does not recover the grid estimates of [20].

Use of artificial intelligence

This project originated in 2019. ChatGPT (OpenAI) was subsequently used to assess whether our preliminary results could address more recent problems, notably Aliaga’s question concerning the isomorphism of $\mathcal{F}(\mathbb{R} \times\mathbb{Z})$ and $\mathcal{F}(\mathbb{R}^2)$ [1 Question 5], also recorded in [17 Problem 2], and his questions about the isomorphism types of Lipschitz spaces over planar sets and the rôle of porosity [1 Questions 3 and 4]. It also assisted with literature searches and the revision of the exposition. We retain responsibility for the mathematical content and references. (T. Kania) MATHEMATICAL INSTITUTE, CZECH ACADEMY OF SCIENCES, ŽITNÁ 25, 115 67 PRAHA 1, CZECH REPUBLIC AND INSTITUTE OF MATHEMATICS AND COMPUTER SCIENCE, JAGIELLONIAN UNIVERSITY, ŁOJASIEWICZA 6, 30-348 KRAKÓW, POLAND

Email address: [email protected], [email protected]

(J. Kąkol) DEPARTMENT OF FUNCTIONAL ANALYSIS, FACULTY OF MATHEMATICS AND COMPUTER SCIENCE, ADAM MICKIEWICZ UNIVERSITY, UNIWERSYTETU POZNAŃSKIEGO 4, 61-614 POZNAŃ, POLAND

Email address: [email protected]