Limits of mapping packages and Preiss's phenomenon

Abstract: We show the existence of ultralimits of sequences of Hajlasz-Sobolev maps $f_i:X_i\to Y_i$ when $X_i$ converges to a limit space in the pointed measured Gromov (pmG) sense, partially extending recent results in [T. Ikonen and S. Wenger, (2026), arXiv:2603.05246]. We moreover demonstrate that the graphs $G(f_i)$ of the mappings pmG-converge to the graph of the ultralimit in a suitable sense. The latter fact stems from a suitable Arzela-Ascoli theorem in the context of pmG-convergence. As an application, we establish a version of Preiss's phenomenon for mapping packages $f:(X,μ)\to V$ into arbitrary Banach spaces. Besides extending it to maps into infinite dimensional targets, our result generalizes existing versions of Preiss's phenomenon [G. C. David, Geom. Funct. Anal., 25 (2015)], [N. Gigli, A. Mondino, and T. Rajala, J. Reine Angew. Math., 705 (2015)] by establishing it for pointed measured Gromov-Hausdorff tangents without a doubling assumption.

• The paper constructs ultralimits of Hajłasz–Sobolev maps on varying metric-measure spaces, preserving local energy bounds and identifying graph limits with ultralimit maps.
• Graphical convergence provides target-independent compactness for equi-Lipschitz mapping packages, including maps into infinite-dimensional Banach spaces where classical Arzelà–Ascoli methods fail.
• The authors prove that graphical tangents are closed under further rescaling almost everywhere, and extend this result to pmGH tangents under finite-dimensional targets and measures that vanish on porous sets.

Overview

This paper, by Caković and Soultanis, develops a limit theory for mappings between varying metric measure spaces and applies it to Preiss's phenomenon — the statement that tangents of Lipschitz maps are themselves tangents at every point of their domain. The two central contributions are: first, an ultralimit construction for sequences of Hajłasz–Sobolev maps $f_i:X_i\to Y_i$ whose domains converge in the pointed measured Gromov (pmG) sense, together with the identification of the pmG-limit of the graphs $G(f_i)$ with the graph of the ultralimit; second, a version of Preiss's phenomenon for "graphical" tangents of Lipschitz mapping packages into arbitrary Banach spaces, which extends prior results to infinite-dimensional targets and to pointed measured Gromov–Hausdorff (pmGH) tangents without a doubling assumption on the source.

The motivation is structural. Classical compactness tools such as Arzelà–Ascoli require target compactness, which fails for infinite-dimensional targets; ultralimits remedy this but are typically too large to be measure-theoretically well behaved or compatible with tangent constructions. The paper bridges these two regimes by showing that graphical limits of equi-Lipschitz packages coincide with ultralimits, thereby importing the compactness of the ultralimit construction into a framework where tangents can be analyzed.

Ultralimits of Sobolev maps and convergence of graphs

The main technical result constructs an ultralimit of Hajłasz–Sobolev maps from varying domains. Suppose $(X_i,\mu_i,\bar x_i)$ is precompact in the complete separable space $(\mathbb X_{pmG},d_{pmG})$ of pointed metric measure spaces, with $\omega$-limit $X_\omega$, and $f_i:X_i\to Y_i$ are Hajłasz–Sobolev maps into complete pointed metric spaces with gradients $g_i$ satisfying uniform local energy bounds

$$\sup_i \int_{B(\bar x_i,R)} g_i^p\,d\mu_i < \infty.$$

Then there exists $f_\omega\in M^{1,p}_{loc}(X_\omega,Y_\omega)$ with a Hajłasz gradient $G(f_i)$0 satisfying the energy inequality

$G(f_i)$1

and, crucially, the graphs converge: $G(f_i)$2 in $G(f_i)$3. Moreover, for $G(f_i)$4-a.e. $G(f_i)$5 in sublevel sets of $G(f_i)$6, the value $G(f_i)$7 is realized as the ultralimit $G(f_i)$8 along any sequence $G(f_i)$9 converging to $(X_i,\mu_i,\bar x_i)$0 within corresponding gradient sublevel sets.

This extends the construction of Ikonen–Wenger (Ikonen et al., 5 Mar 2026) to varying domains, and it is new even for equi-Lipschitz sequences. The proof rests on two Arzelà–Ascoli-type theorems for maps between precompact families in $(X_i,\mu_i,\bar x_i)$1: one for equi-Lipschitz, coarsely co-uniform maps (where coarse co-uniformity means $(X_i,\mu_i,\bar x_i)$2), and one for Hajłasz–Sobolev maps obtained by decomposing domains into gradient sublevel sets $(X_i,\mu_i,\bar x_i)$3, on each of which the map is $(X_i,\mu_i,\bar x_i)$4-Lipschitz. A key auxiliary device is a functional-analytic lemma producing an $(X_i,\mu_i,\bar x_i)$5 function $(X_i,\mu_i,\bar x_i)$6 representing the ultralimit of the pairings $(X_i,\mu_i,\bar x_i)$7, via duality against $(X_i,\mu_i,\bar x_i)$8 and weak convergence of measures.

The authors note explicitly that the coarse co-uniformity hypothesis cannot be dropped: they exhibit domains $(X_i,\mu_i,\bar x_i)$9 with thin vertical spikes of height $(\mathbb X_{pmG},d_{pmG})$0 attached, with $(\mathbb X_{pmG},d_{pmG})$1, where the pushforward measures $(\mathbb X_{pmG},d_{pmG})$2 converge to $(\mathbb X_{pmG},d_{pmG})$3 while the limit map's pushforward is only $(\mathbb X_{pmG},d_{pmG})$4. This example also illustrates why the graph-level convergence is the correct notion: mass can escape through fibers in ways invisible to pointwise limits but visible in the graphs.

Graphical versus pmG convergence of mapping packages

The paper introduces two pseudometrics on the collection $(\mathbb X_{pmG},d_{pmG})$5 of pointed mapping packages. The graphical distance is

$(\mathbb X_{pmG},d_{pmG})$6

while the pmG distance $(\mathbb X_{pmG},d_{pmG})$7 infimizes a Prokhorov-type distance between joint pushforward measures $(\mathbb X_{pmG},d_{pmG})$8 over all pairs of isometric embeddings of domains and targets. Both yield complete separable metric spaces after passing to equivalence classes.

The relationship between the two notions is asymmetric. pmG-convergence implies graphical convergence, since $(\mathbb X_{pmG},d_{pmG})$9, but not conversely: the authors construct a four-point example in $\omega$0 where two functions have isometric graphs yet no isometry of domains conjugates them, so $\omega$1 while $\omega$2. However, for equi-Lipschitz maps into a fixed proper target, graphical convergence upgrades to pmG-convergence along subsequences, via the Arzelà–Ascoli machinery.

The decisive advantage of graphical convergence is target-independent compactness: if $\omega$3 is precompact and $\omega$4 is any collection of complete pointed spaces, then $\omega$5 is precompact with respect to $\omega$6, and every limit is $\omega$7-Lipschitz. This follows directly from Proposition 3.10-style identification: any graphical limit $\omega$8 satisfies $\omega$9, where $X_\omega$0 is the ultralimit of Theorem 1.1. In other words, graphical limits of equi-Lipschitz packages are exactly ultralimits, which is the conceptual link the title advertises.

Compactness and measurability of tangent families

For a Lipschitz map $X_\omega$1 into a Banach space, rescaled maps are defined by $X_\omega$2, where $X_\omega$3 uses a fixed normalizing kernel $X_\omega$4. The set of graphical tangents is

$X_\omega$5

in $X_\omega$6, and analogously for pmG-tangents when $X_\omega$7 is finite dimensional.

Three lemmas drive the analysis. First, the correspondence $X_\omega$8 is lower hemicontinuous with respect to both $X_\omega$9 and (for finite-dimensional $f_i:X_i\to Y_i$0) $f_i:X_i\to Y_i$1; this relies on continuity of the rescaling operator under graphical and pmG convergence, using Lemma 2.4 identifying $f_i:X_i\to Y_i$2 with the rescaled graph $f_i:X_i\to Y_i$3. Second, $f_i:X_i\to Y_i$4 is upper semicontinuous, giving weak measurability of the correspondence $f_i:X_i\to Y_i$5. Third, precompactness: for $f_i:X_i\to Y_i$6-a.e. $f_i:X_i\to Y_i$7, the family $f_i:X_i\to Y_i$8 is precompact in $f_i:X_i\to Y_i$9 (a result of Bate and Pasqualetto–Schultz, transferred to the $g_i$0-normalization via the comparability estimate $g_i$1 at pointwise doubling points), and Corollary on graphical precompactness then gives precompactness of $g_i$2 in $g_i$3.

Combining these yields the structural theorem: for a pointwise doubling space and Lipschitz $g_i$4, outside a $g_i$5-null Borel set the correspondence $g_i$6 is non-empty, compact-valued, and measurable; when $g_i$7 is finite dimensional the same holds for $g_i$8 in $g_i$9. Note that when $$\sup_i \int_{B(\bar x_i,R)} g_i^p\,d\mu_i < \infty.$$0 is infinite dimensional, requiring tangent targets to equal $$\sup_i \int_{B(\bar x_i,R)} g_i^p\,d\mu_i < \infty.$$1 would be too restrictive — the authors state they do not know whether any such tangents exist — but the ultralimit construction always permits taking the target to be the ultrapower $$\sup_i \int_{B(\bar x_i,R)} g_i^p\,d\mu_i < \infty.$$2, so tangent maps may always be assumed linearly targeted.

Preiss's phenomenon

Preiss's phenomenon asserts that tangents are closed under further blow-up: if $$\sup_i \int_{B(\bar x_i,R)} g_i^p\,d\mu_i < \infty.$$3, $$\sup_i \int_{B(\bar x_i,R)} g_i^p\,d\mu_i < \infty.$$4, and $$\sup_i \int_{B(\bar x_i,R)} g_i^p\,d\mu_i < \infty.$$5, then $$\sup_i \int_{B(\bar x_i,R)} g_i^p\,d\mu_i < \infty.$$6 again belongs to $$\sup_i \int_{B(\bar x_i,R)} g_i^p\,d\mu_i < \infty.$$7. The paper proves this in three progressively stronger settings.

Graphical tangents, arbitrary Banach targets. For pointwise doubling $$\sup_i \int_{B(\bar x_i,R)} g_i^p\,d\mu_i < \infty.$$8 and Lipschitz $$\sup_i \int_{B(\bar x_i,R)} g_i^p\,d\mu_i < \infty.$$9 into any Banach space, $f_\omega\in M^{1,p}_{loc}(X_\omega,Y_\omega)$0 is non-empty and compact for $f_\omega\in M^{1,p}_{loc}(X_\omega,Y_\omega)$1-a.e. $f_\omega\in M^{1,p}_{loc}(X_\omega,Y_\omega)$2, and closed under iterated rescaling $f_\omega\in M^{1,p}_{loc}(X_\omega,Y_\omega)$3. This extends David's result [DG2015] and Gigli–Mondino–Rajala [GMR15] to infinite-dimensional targets and to pmG-based tangents without doubling.

The proof follows the scheme of [GMR15, Theorem 3.2]. Two auxiliary facts are essential. The first is a density-point reduction: if $f_\omega\in M^{1,p}_{loc}(X_\omega,Y_\omega)$4 is closed and $f_\omega\in M^{1,p}_{loc}(X_\omega,Y_\omega)$5 is a Lebesgue density point with finite pointwise doubling constant, then $f_\omega\in M^{1,p}_{loc}(X_\omega,Y_\omega)$6 converges to $f_\omega\in M^{1,p}_{loc}(X_\omega,Y_\omega)$7 if and only if the restricted rescalings $f_\omega\in M^{1,p}_{loc}(X_\omega,Y_\omega)$8 do — this holds because $f_\omega\in M^{1,p}_{loc}(X_\omega,Y_\omega)$9 at density points, with matching normalizations. The second is Borel measurability of the graph of $G(f_i)$00, obtained by composing the lower hemicontinuous $G(f_i)$01 with the measurable $G(f_i)$02. After covering the separable space $G(f_i)$03 by countably many sets of small diameter, the bad set is shown to be contained in projections of Borel sets, hence analytic and $G(f_i)$04-measurable; a covering argument on a compact positive-measure subset, iterating the density-point lemma along sequences $G(f_i)$05 with $G(f_i)$06, produces a chain of inequalities forcing $G(f_i)$07, a contradiction.

pmGH tangents under porosity assumptions. If $G(f_i)$08 vanishes on porous sets and $G(f_i)$09 is Lipschitz into a finite-dimensional Banach space, then for $G(f_i)$10-a.e. $G(f_i)$11, $G(f_i)$12 is non-empty and compact, and closed under rescaling. Applying this to constant maps yields the analogous statements for space tangents $G(f_i)$13 and $G(f_i)$14.

The bridge from pmG to pmGH is Proposition 5.5: if $G(f_i)$15 is $G(f_i)$16-doubling along a closed set $G(f_i)$17, and $G(f_i)$18 is a point where $G(f_i)$19 is not porous, then pmG and pmGH convergence of the rescalings $G(f_i)$20 coincide. The proof works with approximate-isometry characterizations of the two convergences: weak $G(f_i)$21-approximations characterizing pmG are upgraded to genuine $G(f_i)$22-approximations by extending them off the good set $G(f_i)$23 using countable dense subsets, with non-porosity supplying the density needed to control distortion and surjectivity errors, and the doubling bound along $G(f_i)$24 ruling out holes in the image via quantitative volume comparisons. Since measures vanishing on porous sets admit a countable decomposition into closed sets along which $G(f_i)$25 is uniformly doubling (via Bate–Li), and $G(f_i)$26-a.e. point of each piece is a non-porosity point, the corollary identifies the two tangent families almost everywhere, and Preiss's phenomenon transfers from pmG to pmGH tangents.

Two caveats deserve emphasis. The equivalence of pmG and pmGH tangents genuinely requires more than pointwise doubling: as the authors remark, Preiss's phenomenon for pmGH tangents fails for merely pointwise doubling measures, citing [LeDonne2011, Remark 3.4]; vanishing on porous sets is strictly stronger than pointwise doubling and is exactly what licenses the passage. Also, the pmGH theorem is stated for finite-dimensional $G(f_i)$27; the infinite-dimensional case remains available only in the graphical formulation, where the target may need to be enlarged to $G(f_i)$28.

Limitations and open questions

Several restrictions are intrinsic to the methods. The ultralimit theorem assumes uniform local energy bounds and basepoints lying in a common gradient sublevel set closure; without some such control neither existence nor the energy inequality can hold. The identification of graphical limits with ultralimits is up to $G(f_i)$29-equivalence, and the four-point example shows this equivalence cannot be strengthened to $G(f_i)$30-equivalence, nor can graphical convergence of the full sequence imply pmG-convergence without passing to subsequences even for proper targets. The pmGH version of Preiss's phenomenon depends on the assumption that $G(f_i)$31 vanishes on porous sets, and the authors' own citation of the pointwise-doubling counterexample shows this hypothesis cannot simply be weakened. Finally, whether tangent maps of Lipschitz maps into an infinite-dimensional $G(f_i)$32 exist with target exactly $G(f_i)$33 (rather than the ultrapower $G(f_i)$34) is left open.

Conclusion

The paper supplies a coherent limit framework for mappings between varying metric measure spaces: Arzelà–Ascoli theorems valid without target compactness, an ultralimit construction for Hajłasz–Sobolev maps preserving energy and graph structure, and a metric theory (graphical convergence) that renders ultralimits usable for tangent analysis. Its main applications — Preiss's phenomenon for graphical tangents into arbitrary Banach spaces, and for pmGH tangents of Lipschitz differentiability spaces — generalize David's and Gigli–Mondino–Rajala's results in both target generality and hypotheses on the source measure. The framework is already positioned for use in forthcoming work on embeddability and rectifiability of Lipschitz differentiability spaces.