Functional Analysis Papers

ALT We prove a new functional inequality for uniformly log-concave measures, stronger than the Poincaré inequality. This new inequality bounds the variance of differences of convex functions and is motivated by the stability theory of optimal transport: for a fixed uniformly log-concave source measure, it provides a sharp stability bound for Brenier potentials with respect to variations of the target measure. Applying this inequality to Gaussian measures, we recover the fact that the Bures-Wasserstein distance between two positive semidefinite matrices is equivalent to the Frobenius distance between their square roots. Our proofs hinge upon linearization arguments for the Monge-Ampère equation governing the quadratic optimal transport problem.

ALT Mappings contracting perimeters of triangles are Banach contractions whenever every point of the underlying metric space is an accumulation point. Consequently, applying the perimeter-contraction theorem on a complete metric space without isolated points yields an existence result already covered by Banach's fixed point theorem. This observation applies, in particular, to operators associated with differential and integral equations acting on standard nontrivial function spaces, such as spaces of continuous functions C[a,b], Lebesgue spaces Lₚ, and Sobolev spaces Wᵏᵖ, equipped with their usual norm metrics. Thus, applications extending beyond Banach's theorem for the same operator and metric require a complete invariant domain containing some number of isolated points. Such a domain may be a closed subset of a standard function space, despite the absence of isolated points in the ambient space. The main goal of this paper is to illustrate applications of the fixed point theorem for map

ALT We study noncommutative polynomials and functions invariant under linear actions of infinite (compact Lie) groups, with a focus on SU(2) acting on two noncommuting variables. For SU(2), we give a complete description of the invariant noncommutative polynomials: there are no invariants of odd degree, and in degree 2n the invariants are exactly the span of the S₂ₙ-orbit of (XY-YX)ⁿ. We also construct an explicit basis recursively from the operation (A,B)↦ XAYB-YAXB, and show that the space of degree-2n invariants has dimension equal to the n-th Catalan number. All proofs proceed by direct computation with the Haar integral.

ALT Let ℬα denote the shift operator on the weighted Bergman space A²α(D). We prove that, for every integer k≥1, the Beurling-type theorem holds for ℬαᵏ on A²α(D) if and only if -1<α≤1. We also show that, for every α>-1, there exists a finite Blaschke product B such that M_B admits a cyclic invariant subspace but fails the wandering subspace property.

ALT The aim of this paper is to obtain the characterization of Fourier transform for functions taking values in a separable Hilbert space H, defined on the groups G=ℝⁿ, G=ℤ/nℤ. In the first part of the paper, we consider the Fourier transform as a bijective mapping on S_H, the space of all H-valued smooth rapidly decaying functions on ℝⁿ, which naturally extends bijectively to the dual space S_H'. Within this framework, we characterize the Fourier transform in terms of the convolution theorem, supplemented by additional structural properties of the Fourier transform. In the second part of the paper, we consider the discrete setting, where the Fourier transform is an operator on L¹(ℤ/nℤ, H), and we provide corresponding characterization based on its essential properties. In contrast to the classical setting, a key assumption in our framework involves the interplay between two distinct convolution operations. As a consequence, the structure of our proof diverges notably from the classical ca

ALT We characterize when a de Branges–Rovnyak space coincides with a harmonically weighted Dirichlet space. We obtain a necessary and sufficient condition for this identification, which settles a long-standing open problem. We illustrate the result with several examples, that include measures supported on uncountable sets.

ALT Let R∞(n) be the maximal Banach–Mazur distance from an n-dimensional real normed space to ℓ∞ⁿ. We prove that R∞(n)≥ cn²/³(log(2n))⁻¹⁰. Together with the upper bound of Friedland, Naor, and Youssef, this gives R∞(n) = n²/³⁺ᵒ⁽¹). The examples are polars of symmetric Gaussian polytopes with 2n generators, and the lower bound holds with exponentially high probability. A joint minimization over coefficient representations and positive weights yields an exact second-variation identity and sparse coefficients. A Gaussian projection argument controls all low-rank completions of the recorded matrix. An adaptive choice of exceptional generators then reduces the remaining case to a fixed-shift Gaussian small-ball estimate over two finite nets.

ALT The complex polynomial Bohnenblust–Hille inequality bounds the coefficient ℓ₂ₘ/(ₘ₊₁) norm of a complex m-homogeneous polynomial by its supremum norm, with a constant independent of the number of variables. For each fixed integer N≥3, we characterize the choices of N degree-m monomials for which the inequality holds with optimal constant 1 for every polynomial supported on them, provided m exceeds an explicit threshold depending only on N. This holds precisely when their exponent vectors contain no nontrivial three-term arithmetic progression; equality for a nonzero polynomial then holds only for monomials. We also prove that, in the same degree range, every polynomial with at most N nonzero coefficients that violates the constant-one inequality has a unique coefficient of maximal modulus, whose exponent is the center of such a progression in its support. Exact contractivity also follows from a comparability condition on the two largest coefficients, with a degree threshold that necessa

ALT We construct a bounded closed convex set in c₀(ℂ) on which no nonzero continuous complex-linear functional attains its maximum modulus. Consequently, this answers in the negative the two longstanding questions raised by Deville, Godefroy and Zizler inp. 35]DGZ. The construction uses a small perturbation of the identity on c₀(ℝ).

ALT Let 1≤ p<∞. We prove an Lᵖ-operator analogue of Raeburn's symmetric imprimitivity theorem for countable discrete groups. Let A be an Lᵖ-operator algebra with a contractive two-sided approximate identity, admitting a nondegenerate isometric representation on a σ-finite Lᵖ-space. Given commuting free and proper actions of groups G and H on a second-countable locally compact Hausdorff space Z, together with commuting isometric actions on A, we show that the two corresponding full crossed products, and likewise the two reduced crossed products, are Morita equivalent as Banach algebras. The proof is organized through the linking groupoid. Its main analytic ingredient is a coefficient-valued spatial matrix dilation; in the reduced case we prove that this dilation preserves regularity after a Borel reindexing of the opposite corner. A pairing-valued approximate identity yields fullness and nondegeneracy.