arXiv subjects

Milica Caković

Publications and source records attributed to Milica Caković.

3 recordsLinked to original sources

arXiv2026

Limits of mapping packages and tangents of Sobolev maps

We show the existence of ultralimits of sequences of Hajlasz-Sobolev mappings $f_i:X_i\to Y_i$ 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 mappings pmG-converge to the ultralimit in a suitable sense, and obtain Arzela-Ascoli theorems in the context of pmG-convergence. As an application, we establish a version of Preiss's phenomenon for Hajlasz-Sobolev maps $f:(X,μ)\to V$ 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.

math.MG↗

arXiv2026

Convergence of metric measure spaces via embeddings in the Urysohn universal space

We study different notions of convergence of metric measure spaces by means of isometric embeddings into the Urysohn universal metric space $\mathbb U$. Due to the universality of $\mathbb U$, the collection $\mathbb X_1$ of isomorphism classes of normalised metric measure spaces can be canonically identified with the quotient (set) $\mathscr P_\sim(\mathbb U)=\mathscr P(\mathbb U)/\sim$ of the space $\mathscr P(\mathbb U)$ of Borel probability measures on $\mathbb U$, where $μ\simν$ if $ν$ is the pushforward of $μ$ under an isometry between their respective supports. By making crucial use of the ultrahomogeneity of $\mathbb U$, we show that, under the above identification, Gromov's box topology on $\mathbb X_1$ coincides with the quotient topology induced by the weak topology of $\mathscr P(\mathbb U)$. More quantitatively, the truncated $1$-Wasserstein distance on $\mathscr P(\mathbb U)$ induces a complete and separable distance ${\sf d}_{\rm mG}$ on $\mathbb X_1\cong\mathscr P_\sim(\mathbb U)$, which metrises the quotient topology of $\mathscr P_\sim(\mathbb U)$ and is Hölder equivalent to the box distance $\square$.

math.MG↗

arXiv2025

Tensor products of measurable Banach bundles

We study injective and projective tensor products of measurable Banach bundles. More precisely, given two separable measurable Banach bundles ${\bf E}$, ${\bf F}$ defined over a probability space $({\rm X},Σ,\mathfrak m)$, we construct two measurable Banach bundles ${\bf E}\hat\otimes_\varepsilon{\bf F}$ and ${\bf E}\hat\otimes_π{\bf F}$ over $({\rm X},Σ,\mathfrak m)$ such that $Γ({\bf E}\hat\otimes_\varepsilon{\bf F})\congΓ({\bf E})\hat\otimes_\varepsilonΓ({\bf F})$ and $Γ({\bf E}\hat\otimes_π{\bf F})\congΓ({\bf E})\hat\otimes_πΓ({\bf F})$, where ${\bf G}\mapstoΓ({\bf G})$ is the map assigning to a measurable Banach bundle ${\bf G}$ its space of $L^\infty(\mathfrak m)$-sections, while $Γ({\bf E})\hat\otimes_\varepsilonΓ({\bf F})$ and $Γ({\bf E})\hat\otimes_πΓ({\bf F})$ denote the injective and projective tensor products, respectively, of $Γ({\bf E})$ and $Γ({\bf F})$ in the sense of $L^\infty(\mathfrak m)$-Banach $L^\infty(\mathfrak m)$-modules. In combination with previous results, this provides a fiberwise representation of the injective tensor product $\mathscr M\hat\otimes_\varepsilon\mathscr N$ and the projective tensor product $\mathscr M\hat\otimes_π\mathscr N$ of two countably-generated $L^\infty(\mathfrak m)$-Banach $L^\infty(\mathfrak m)$-modules $\mathscr M$, $\mathscr N$.

math.FA↗