Differentiability of Lipschitz curves in $p$-Banach spaces

Fernando Albiac; José L. Ansorena; Pietro Wald

Home/Archive/arXiv

arXiv · 2609.26992

Differentiability of Lipschitz curves in $p$-Banach spaces

Fernando Albiac·José L. Ansorena·Pietro Wald

View original source ↗Read source document ↗Export metadata ↓

Abstract

The problem of differentiating Lipschitz curves in Banach spaces goes back to Tamarkin in the early 1930s and is one of the origins of the Radon--Nikodým property in classical Banach space theory. In this article we characterize those quasi-Banach spaces $X$ for which every Lipschitz curve $F\to\mathbb{R}\to X$ admits a point of differentiability and prove that this property holds if and only if $X$ is isomorphic to a Banach space with the Radon--Nikodým property, thus substantiating a conjecture of Kalton, communicated personally to the first-named author in 2008. Equivalently, every nonlocally convex quasi-Banach space admits a nowhere differentiable Lipschitz curve. Our result is obtained from a quantitative characterization of local convexity in terms of the infinitesimal oscillation of Lipschitz curves. Motivated by this obstruction, we then investigate positive differentiability phenomena in $\ell_p$, $0<p<1$, and identify natural decoupled-coordinate constructions for which Lipschitz regularity nevertheless implies almost-everywhere differentiability. We also contrast this behavior with metric differentiability for the natural metric and the quasi-metric on $\ell_p$.

Explore related subjects

math.FA

Keep this discovery

Explore connections, maps & timelines

BibTeX RIS

Fernando Albiac, José L. Ansorena, Pietro Wald. 2026-09-22. Differentiability of Lipschitz curves in $p$-Banach spaces. https://arxiv.org/abs/2609.26992

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Norm attaining dual truncated Toeplitz operators

In this paper, we investigate norm attainment for dual truncated Toeplitz operators $D_\vp$ acting on $\clk_u^\perp=uH^2\oplus H^2_{-}$, where $u$ is a nonconstant inner function and $\vp\in L^\infty(\T)$. Our main focus is the structure of extremal vectors and the distinction between global and componentwise norm attainment. For arbitrary $\vp\in L^\infty(\T)$, we establish an exact norm-defect identity and characterize the extremal space in terms of the essential maximum set $E_\vp=\{ζ\in\T:|\vp(ζ)|=\|\vp\|_\infty\}$. As a consequence, when $u$ is a finite Blaschke product, \[ D_\vp\in\mathcal{NA}\quad\Longleftrightarrow\quad m(E_\vp)>0. \] In this case, whenever $D_\vp$ is norm attaining, its extremal space is infinite-dimensional. We further show that the sets of symbols generating norm attaining and non-norm attaining DTTOs are both norm dense in $L^\infty(\T)$. Consequently, both the norm attaining and the non-norm attaining DTTOs are operator-norm dense in the class of all DTTOs associated with $u$. For unimodular symbols, we characterize extremality by the condition $M_\vp f\in\clk_u^\perp$ and equivalently by a truncated Hankel kernel condition. For mixed extremal vectors $f=x\oplus y$, we derive the identity \[ \|D_\vp x\|^2-\|x\|^2=\|D_\vp y\|^2-\|y\|^2=-\langle D_\vp x,D_\vp y\rangle, \] which yields a phase-rotation criterion and coupled Toeplitz--Hankel relations. We also show that global norm attainment may occur even when neither the analytic nor the coanalytic component contains a nonzero extremal vector. Under additional Hardy-space hypotheses, we obtain factorization criteria for componentwise extremals, construct explicit extremal families for quotient-inner symbols, and relate norm attainment of Toeplitz operators to that of dual truncated Toeplitz operators.

math.FA↗

arXiv2026

A Constructive Framework for Generalized Fourier Transforms via Truncate-and-Generalized Limits

This paper introduces a constructive definition of generalized Fourier transforms based entirely on ordinary truncated Fourier integrals and ordered dual-domain limits, within the framework of improper Riemann integration and classical analysis. The proposed truncate-and-generalized-limit (t.g.l.) formulation does not require test-function spaces, Lebesgue measure theory, or duality pairings in its proofs: the forward and inverse transforms are defined directly through finite-domain truncation of the target function, followed by successive ordered limits in the time and frequency domains. As consequences of this constructive definition, the formulation provides a unified treatment of non-decaying, oscillatory, and locally singular functions beyond the classical L1(R) setting; reveals an inherent asymmetry between the forward transform, interpreted as a first-order generalized-limit family, and the inverse transform, which requires frequency-domain truncation to generate pointwise reconstruction through Dirichlet-type oscillatory localisation; and clarifies the distinction between the t.g.l. approach and distribution theory, where generalized Fourier transforms are introduced through duality pairings rather than constructed from ordinary integrals. The inversion formula is established rigorously for two concrete admissible classes using only the classical Dirichlet convergence theorem. Several examples confirm that the framework covers constants, polynomials, periodic functions, singular kernels, and chirp signals within a single constructive scheme.

math.FA↗

arXiv2026

Every compact operator is a commutator of compact operators

We prove that every compact operator $T$ on a separable infinite-dimensional complex Hilbert space is a commutator of two compact operators, thereby answering a question of Pearcy and Topping. Moreover, the compact factors $A$ and $B$ can be chosen such that $[A, B] = T$ and $\max\{\Vert{}A\Vert{},\Vert{}B\Vert{}\}\leq c\Vert{}T\Vert{}^{1/2}$ for a universal constant $c$.

math.FA↗