Papers

We study the $p$-Bures-Wasserstein distance $D_p(A,B)=(\mathrm{tr}\,A^p+\mathrm{tr}\,B^p-2\|A^{1/2}B^{1/2}\|_p^p)^{1/2}$ on positive semidefinite matrices; it equals the Bures-Wasserstein distance for $p=1$, and the Hilbert-Schmidt distance for $p=2$. We prove that $D_p$ is a metric for every $1\le p\le2$, by proving a stronger 2014 conjecture of Lin and van den Driessche on $3\times3$ block matrices. Our main tool is a partition inequality for Schatten norms, proved by complex interpolation. We also characterize Fréchet means for $D_p$ by a fixed-point equation and prove geometric convergence of the associated iteration.

Given $1 < p < \infty$ and a function $m \colon \mathbf{N}_0 \to \mathbf{C}$, we provide a characterization of radial Schur multipliers $M(x,y) = m(d(x,y))$ on the Schatten $p$-class of a tree $(X,d)$ equipped with its graph metric. That is, the operator norm of the radial Schur multiplier $S_M$ is controlled by the norm of the Fourier multiplier $T_m$ on the torus: \[ \big\| S_M \colon S_p(X) \to S_p(X) \big\|_{\mathrm{cb}} \lesssim \frac{p^4}{(p-1)^2} \, \big\| T_m \colon L_p(\mathbf{T}) \to L_p(\mathbf{T}) \big\|_{\mathrm{cb}}, \] with both quantities being comparable when the tree $X$ has infinite diameter. This is a rather unexpected inequality, which necessarily fails at the endpoints $p=1,\infty$ due to the work of Haagerup, Steenstrup and Szwarc. As an application, we construct the first examples of completely $S_p$-bounded Herz-Schur multipliers in the free group that are unbounded as Fourier multipliers. This result disproves Fourier-Schur transference—a longstanding open problem since the work of Neuwirth and Ricard in 2011. In line with this, our main result also implies the surprising fact that free balls Schur multipliers are uniformly $S_p$-bounded, in sharp contrast to their Fourier behavior. In addition, we disprove Fourier-Schur transference over the algebraic groups $\mathrm{PGL}_2(\mathbf{Q}_q)$ and $\mathrm{PSL}_2(\mathbf{R})$ with $\mathrm{K}$-biinvariant counterexamples, the latter involving additional geometric and probabilistic techniques.

We consider symplectic analogues of the fractional Fourier transforms and obtain convenient representations in terms of Bargmann transforms and Wigner distributions. We show that they act as homeomorphisms on Schwartz spaces, the space of square integrable functions, the space of tempered distributions, and on Fourier-invariant Gelfand-Shilov spaces and their corresponding distribution spaces. More generally, we establish their homeomorphism properties on a broad class of spaces defined through (Wigner-)Hermite expansions. In addition, we prove that these transforms preserve norms on Schatten-von Neumann symbol classes associated with the Weyl calculus.

Let $\mathbf{T}=(T_1,\ldots,T_n)$ be a commuting tuple of bounded linear operators on a separable Hilbert space $\mathcal{K}$, and let $\{a_\alpha\}_{\alpha\in\mathbb{Z}_+^n}$ be strictly positive weights. Under mild assumptions on the weights, we characterize the weighted frame vectors and weighted Parseval frame vectors, that is, those vectors $\xi$ for which $\{a_\alpha \mathbf{T}^\alpha\xi\}_{\alpha\in\mathbb{Z}_+^n}$ is a frame or a Parseval frame. We reduce the problem to characterizing surjective module maps from the reproducing analytic Hilbert module associated with the weights onto the module $\mathcal{K}$, equipped with the module structure defined by $\mathbf{T}$. It is shown that weighted Parseval frame vectors differ by a unimodular scalar whenever they exist. If the kernel defined by the weights has the complete Nevanlinna–Pick property, then any two associated weighted frame vectors are similar whenever they exist. We also focus on the case of the Hardy kernel on the polydisc, that is, the case $a_\alpha=1$ for all $\alpha\in\mathbb{Z}^n_+$. This is related to the famous Douglas–Foias problem: whether two similar quotient modules of the Hardy spaces $H^2(\mathbb{D}^n)~(n>1)$ are equal. Our result shows that the answer is negative. This leads to the conclusion that frame vectors for representations of $\mathbb{Z}_+^n$ $(n>1)$ may fail to be similar. It means that, for the case $a_\alpha=1$, there exists a tuple $\mathbf{T}=(T_1,\dots,T_n)$ with $n>1$ on $\mathcal{K}$ and two distinct frame vectors for $\mathbf{T}$ which are not similar.

We establish the weak type \((1,1)\) endpoint estimate for operator-valued rough homogeneous singular integrals with kernels satisfying the classical \(L\log L\) integrability and cancellation conditions. This extends Seeger's endpoint theorem [{J. Amer. Math. Soc.} 9 (1996)] to the noncommutative setting. A key ingredient is an off-diagonal analysis that controls interactions between different kernel scales and avoids an additional logarithmic loss.

We prove an explicit lower spectral bound for independent random vectors with prescribed group multiplicities. Let $v_{ij}\in\mathbb C^d$, $1\le i\le l$, $1\le j\le k_i$, be independent random vectors with finite support and $\mathbb E v_{ij}v_{ij}^*=A_i$, where $\sum_{i=1}^l A_i=I_d$ and $\operatorname{Tr}(A_i)\ge\epsilon$. Set $k=\max_i k_i$, $n=\max_i\operatorname{rank}(A_i)$, and $K=\sum_{i=1}^l k_i$. If $n>(6+4\sqrt2)k$ and $2\sqrt{nk}-k<\epsilon\le n/2$, then, with positive probability, \[ \lambda_K\left(\sum_{i=1}^l\sum_{j=1}^{k_i}v_{ij}v_{ij}^*\right) \ge n-(\sqrt{k}+\sqrt{n-\epsilon})^2. \] As a consequence, we obtain a restricted invertibility result selecting exactly $k_i$ vectors from the $i$th group of a Parseval frame. The proof combines interlacing families, a reciprocal transformation of mixed characteristic polynomials, and a multivariate barrier estimate that accommodates repeated differentiation in the same variable.

We study isomorphic embeddings into Musielak-Orlicz sequence spaces \(\ell_\Phi\), where the normalized generating sequence \(\Phi=\{\phi_n\}\) satisfies the \(\delta_2\) condition. Using admissible normalized blocks, we construct an associated limiting family \(\mathcal C_{\Phi,\tau}\) of Orlicz functions. If a Banach space \(X\) with a symmetric basis embeds into \(\ell_\Phi\), then there is a function \(\psi\in\mathcal C_{\Phi,\tau}\) such that the basis of \(X\) is equivalent to the canonical basis of \(\ell_\psi\). An Orlicz sequence space \(\ell_\psi\) embeds if and only if \(\psi\) is equivalent near zero to a member of \(\mathcal C_{\Phi,\tau}\); in particular, \(\ell^p\) embeds exactly when such a member is equivalent to \(t^p\). Assuming differentiability of the generators on a common interval near zero, we also derive the necessary bound \(p\leq b_{\Phi,(c_n),\delta}\), where the upper growth index incorporates the summable errors in the \(\delta_2\) condition. The same index yields cotype \(q\) for every \(q\geq2\) with \(q>b_{\Phi,(c_n),\delta}\). Examples with nonuniform growth and lacunary weights illustrate this bound. For alternating Nakano exponents, the explicit limiting family gives a sharper restriction than the index alone. For general Nakano exponents, we recover the known exact cotype criterion, including attainment of the optimal exponent. For weighted Orlicz spaces, we express the cotype criterion through summable scalar errors and recover the formula using the global upper growth index when \(w\in\mathcal A\).

In this paper, we investigate the Schatten class membership of the product of a composition operator and the adjoint of another composition operator on the Hardy and Bergman spaces. We provide several sufficient conditions and necessary conditions for these operators to belong to the Schatten classes. Moreover, for some specific symbols including linear fractional maps, finite Blaschke products, and inner functions, we obtain necessary and sufficient conditions.

The complex polynomial Bohnenblust–Hille inequality states that, for a $d$-homogeneous polynomial $P:\mathbb C^N\to\mathbb C$, $P(z)=\sum_{|\alpha|=d}c_\alpha z^\alpha$, the coefficient norm $(\sum_{|\alpha|=d}|c_\alpha|^{p_d})^{1/p_d}$ is bounded by $B_d^{\mathrm{pol}}\|P\|_\infty$, where $p_d=2d/(d+1)$ and $B_d^{\mathrm{pol}}$ is independent of $N$. Here $\|P\|_\infty$ is the supremum norm on the unit polydisc. This inequality has a Boolean analogue for the Fourier–Walsh coefficients of functions of degree at most $d$, with the supremum taken over $\{-1,1\}^N$. In this setting, Defant, Mastyło and Pérez obtained subexponential bounds, and Ivanisvili established polynomial growth. Further polynomial estimates were obtained by Slote, Tseng and Volberg in the Boolean setting and by Slote and Volberg for cyclic groups. We study these coefficient inequalities in a broader framework of finite systems of functions that encompasses and extends the Boolean and cyclic settings. This framework also admits a natural $\ell_p$ formulation, leading to Hardy–Littlewood-type inequalities. For orthonormal systems, we obtain polynomial bounds for the Bohnenblust–Hille constants and, for sufficiently large $p$, for the constants in these Hardy–Littlewood-type inequalities. In the Boolean case, the polynomial power is $5/2$, with a cubic logarithmic factor.

We study the numerical radius \(\omega(T)\) of a bounded linear operator \(T\) on a complex Hilbert space. Write \(T=A+iB\), where \(A\) and \(B\) are the real and imaginary parts of \(T\). A central result is the exact formula \[ \omega(T)=\frac{\pi}{2}\sup_{\|x\|=1}\int_0^1 \left|\left\langle (A\cos \pi t+B\sin \pi t)x,x\right\rangle\right|\,dt . \] For every \(n\in\mathbb N\), we also prove the power form \[ \omega^{2n}(T)=\frac{4^n}{\binom{2n}{n}} \sup_{\|x\|=1}\int_0^1 \left\langle (A\cos \pi t+B\sin \pi t)x,x\right\rangle^{2n}\,dt . \] This yields sharp upper bounds; for \(n=1\), together with the matching integral lower bound, it recovers Kittaneh's classical two-sided inequality \[ \frac14\|T^*T+TT^*\|\le \omega^2(T)\le \frac12\|T^*T+TT^*\|. \] Furthermore, we obtain exact numerical radius formulas for some block operator matrices, including a \(3\times3\) nilpotent matrix and longer alternating paths. The factor \(\pi/2\) is best possible.

For a $C^*$-algebra $A$ let $\mathcal I(A)$ be the set of all elements $a+b$ with $a,b\in A_+$, $ab=0$ and $\|a\|=\|b\|=1$. We study the algebraic, metric and topological properties of $\mathcal I(A)$. We describe $\mathcal I(A\oplus B)$ and show that $\mathcal I(A/J)\subseteq\pi(\mathcal I(A))$ for the quotient map $\pi:A\to A/J$. For finite-dimensional $A$ we prove $\mathcal I(A)=\operatorname{Sph}^+_A(E(A))$, where $E(A)$ is the set of minimal projections of $A$ and $\operatorname{Sph}^+_A(E)$ is the set of positive norm-one elements at distance one from every element of $E$; we also show that the analogous statement with the minimal projections of the bidual fails in general. When $\mathcal I(A)\neq\varnothing$ the distance from $1$ to $\mathcal I(A)$ is $0$ or $1$, according to whether $A$ has a nontrivial projection. Moreover, $\mathcal I(A)$ is norm compact if and only if $A$ is finite-dimensional, closedness of $\mathcal I(A)$ passes to hereditary subalgebras, $\mathcal I(A)$ is norm closed for every von Neumann algebra, and the extreme points of $\mathcal I(A)$ are exactly its projections. For a compact, locally path connected space $X$ the set $\mathcal I(C(X))$ is closed, whereas $\mathcal I(\ell_c(A))$ fails to be closed when $A$ is unital without nontrivial projections.

We study two invariants of $C^*$-algebras built from positive zero divisors. First, we associate with a $C^*$-algebra $A$ the graph $\Gamma_+(A)$ whose vertices are the nonzero positive zero divisors of $A$, two vertices being adjacent when they are orthogonal. We show that $\Gamma_+$ of the Calkin algebra is connected of diameter $3$, that $\Gamma_+(B(H))$ is connected of diameter $4$ (for $\dim H\ge3$), that $\Gamma_+(A)$ is connected of diameter at most $4$ for every von Neumann factor $A\neq\mathbb C,M_2(\mathbb C)$, and of diameter at most $3$ if $A$ is not of type~I, whereas $\Gamma_+(M_2(\mathbb C))$ is disconnected. Second, we study the zero-divisor real rank $ZDRR(A)$, obtained by replacing invertibility in the definition of real rank with the weaker condition of being a non-zero-divisor, so that $ZDRR(A)\le RR(A)$. For commutative algebras $C(X)$ we prove $ZDRR(C(X))=RR(C(X))=\dim X$ when $X$ is a compact almost-$P$-space, while $ZDRR(C(X))=0$ for separable $X$. Using Levy's connected, totally ordered almost-$P$-space we construct unital commutative $C^*$-algebras $A_n$ with $ZDRR(A_n)=n$ for every $n\in\mathbb N_0$. On the other hand $ZDRR(C(X\times[0,1]))=0$ for every compact Hausdorff space $X$, so $ZDRR$ and $RR$ can differ arbitrarily.

In this paper, sufficient conditions are established for diagonal perturbations of weighted shifts to be supercyclic, hypercyclic, and frequently hypercyclic, extending several classical results for weighted shifts to this more general setting. In addition, a complete analytic characterization of the shadowing property for these operators is obtained. Consequently, examples of frequently hypercyclic perturbed weighted shifts with the shadowing property are constructed.

Let $\mathcal{H}(b)$ be the de Branges-Rovnyak space associated to a non-extreme point $b$ of the unit ball of $H^\infty$, and let $\phi=b/a$, where $a$ is the Pythagorean mate of $b$. In an earlier article, the author showed that, if $\phi\in H^2$, then \[ f\in\mathcal{H}(b) \iff \Bigl(f\in H^2 \text{ and }\sum_{m\ge0}\Bigl|\sum_{n\ge0}\overline{\hat{\phi}(n)}\hat{f}(m+n)\Bigr|^2<\infty\Bigr), \] in which case we have \[ \|f\|_{\mathcal{H}(b)}^2=\sum_{m\ge0}|\hat{f}(m)|^2 +\sum_{m\ge0}\Bigl|\sum_{n\ge0}\overline{\hat{\phi}(n)}\hat{f}(m+n)\Bigr|^2. \] It was also shown that there exists a space $\mathcal{H}(b)$ for which $\phi\in\cap_{p<2}H^p$, yet \[ f\in\mathcal{H}(b)\not\Rightarrow \sum_{m\ge0}\Bigl|\sum_{n\ge0}\overline{\hat{\phi}(n)}\hat{f}(m+n)\Bigr|^2<\infty. \] Here we complete this circle of ideas by showing that there exists a space $\mathcal{H}(b)$ for which $\phi\in\cap_{p<1}H^p$, yet \[ \Bigl(f\in H^2\text{ and } \sum_{m\ge0}\Bigl|\sum_{n\ge0}\overline{\hat{\phi}(n)}\hat{f}(m+n)\Bigr|^2<\infty\Bigr) \not\Rightarrow f\in\mathcal{H}(b). \]

We prove Kahane's conjecture: if a continuous unimodular function $F$ on the circle satisfies $\|F^n\|_{A(\mathbb{T})}=o(\log|n|)$ as $|n|\to\infty$, $n\in\mathbb{Z}$, then $F(x)=e^{i(mx+b)}$ for some $m\in\mathbb{Z}$ and $b\in\mathbb{R}$. The main step is a logarithmic lower bound for averages: for every nonconstant real-valued $\varphi\in A(\mathbb{T})$, $\frac{1}{T}\int_a^{a+T}\|e^{it\varphi}\|_{A(\mathbb{T})}dt \ge c_\varphi\log T \quad(a\in\mathbb{R},\ T\ge T_\varphi)$, where $c_\varphi>0$ and $T_\varphi>1$ are independent of $a$. The proof uses exact sampling on finite subgroups of the circle, uniformly bounded localized modulations, and the weighted non-harmonic Littlewood inequality of Jaming–Kellay–Saba. A suitable cutoff and a small linear perturbation of the phase ensure that a fixed positive proportion of the coefficients at the beginning of the frequency ordering have modulus one. Passing to integer powers and removing the degree of the circle map yield Kahane's conjecture without any additional smoothness assumptions. We also give equivalent phase, convolution, and operator formulations, and derive consequences for homomorphisms of weighted Wiener algebras and for higher-dimensional tori.

Discretization can change a sharp stability threshold even when the continuum threshold is recovered at infinity. We study this phenomenon for the Hardy inequality on the integer lattice in dimensions at least three: its best coefficient agrees with the continuum value through dimension eight and is smaller from dimension nine onwards. We characterize this transition by the total critical defect, the summable error obtained by testing the lattice operator against the critical continuum power. A theta-function representation proves that the defect is negative and strictly decreasing from dimension nine; explicit positive Hardy remainders establish its positivity in dimensions three through eight. An exact energy identity relates this defect to the energy saved by optimizing the profile outside the neighbours of the origin. This exterior remains stable in dimensions eight and nine, locating the first loss of positivity in its coupling to those neighbours. We give self-contained proofs of both dimensional ranges. We then describe how local attraction creates a negative eigenvalue in a general class of lattice operators with critical inverse-square tails. Its magnitude is exponentially small near the threshold, with both the rate and the multiplicative prefactor determined by zero-energy solutions. The result allows connected finite modifications of the lattice and nonradial corrections to the tail. The same analysis controls the spread of the emerging state and gives a two-dimensional return scale for the associated Markov chain in every ambient dimension.

We present a unified treatment of the data-fitting problems with quadratic (Hilbertian) regularization that arise in approximation theory, machine learning, engineering, and statistics. The generic problem is an optimization in the continuum: given $M$ linear measurements of an unknown function, one minimizes a convex data-fidelity term that is penalized by $\|\mathrm{L} f\|^2_{L_2}$, where $\mathrm{L} $ is a differential-like (spline-admissible) operator. The natural domain of the minimization is the native space $\mathcal{H}_\mathrm{L}$ of $\mathrm{L} $ - the largest Hilbert space over which the regularizer remains meaningful - for which we provide a concise and directly usable characterization. Central to our formulation is a measurement-adapted (conditioned) basis that decouples the null-space and spline components of the solution. This basis serves in three places. First, it tightens the formulation of our representer theorem, which confines the unique solution to an $M$-dimensional space without any orthogonality constraint on the coefficients. Second, it streamlines the implementation, as the exact discretization of the problem becomes an unconstrained convex program with closed-form solutions in the least-squares case, which covers ridge regression, Tikhonov regularization, and smoothing splines. Third, it makes the bridge with the minimum-mean-square-error estimation of generalized Gaussian processes, which is the topic of a companion paper.

The variational theory of splines requires a native space: a Hilbert space over which the regularization energy $\|\mathrm{L} f\|_{L_2}^2$ is well defined and the minimum-energy interpolation problem admits a spline solution. In this paper, we characterize the operators $\mathrm{L}$ that enable the construction of such spaces. Our framework is based on the completion of a space $\mathcal{S}$ of test functions in a controlled Gelfand-triple configuration $\mathcal{S}\subset\mathcal{H}\subset\mathcal{S}'$, which we call a Hilbert-Gelfand space. This departs from the classical reproducing-kernel techniques. Our construction is substantially more general, as it supports arbitrary linear operators and singular behaviors, and it recovers the reproducing-kernel setting as a particular case. It is also complete and canonical: the admissibility conditions on $\mathrm{L}$ are necessary and sufficient, and $\mathrm{L}$ determines its native space as the unique Hilbert-Gelfand space on which it extends to a bounded surjection onto $L_2$. We first show that Hilbert-Gelfand spaces are in one-to-one correspondence with compatible Riesz pairs $(\mathrm{A},\mathrm{A}^{-1})$ of positive-definite operators $\mathcal{S}\to\mathcal{S}'$. We then extend this correspondence to conditional Riesz pairs, where $\mathrm{A}$ is only conditionally positive-definite, which is the scenario that is relevant to splines. The theory identifies the attainable null space of the energy as the unique admissible conditioning, realizes the abstract completions as concrete spaces of distributions, and provides a verifiable admissibility condition in terms of a dual energy functional. The framework is illustrated with the fractional Laplacian, which covers polyharmonic and thin-plate splines, and with ordinary differential operators, which cover exponential splines.

We construct an equivalent norm on $\mathbb{R}\oplus\ell_2$ and a nonempty closed convex unbounded subset having the fixed point property for nonexpansive self-mappings. The ambient renormed space has the fixed point property on bounded closed convex subsets. The main point is a geometric localization estimate at a distinguished point of the unbounded set; the fixed point conclusion then follows from the Kirk–Ray localization theorem.

We proved recently that there are no unconditional Schauder frames of exponentials in the space $L^p(\Omega)$, $1 \le p < 2$, for any set $\Omega \subset \mathbb{R}^d$ of finite measure. For $p > 2$, we proved that the same holds provided that $\Omega$ has nonempty interior, but the general case remained open. In this note we close this gap, i.e., we establish that exponentials cannot form an unconditional Schauder frame in $L^p(\Omega)$, $p \ne 2$, without any additional assumptions on the set $\Omega$ of finite measure other than measurability.

We study weighted Bergman spaces of quasiregular mappings on the unit ball in $\mathbb{R}^n$, with weight $(1-|x|)^\alpha$, $\alpha>-1$. For every fixed distortion bound $K\ge1$ and every $p>0$, we characterize Bergman integrability by averaged derivatives. We also give geometric conditions under which integration over a subset, or against another measure, controls the full Bergman integral. When $K$ is sufficiently large, the dominating sets are precisely the relatively dense sets. For comparison, we determine the dominating sets for the Möbius class $K=1$ in dimensions $n\ge3$. At the critical exponent $p=n+\alpha$, its domination criterion differs from that for every fixed $K>1$. Our results extend aspects of the Bergman theory developed by Nolder~\cite{Nolder90} and by Oleinik and Vuorinen~\cite{OV}, and complement the recent work of Adamowicz and González~\cite{AG21,AG25} and Adamowicz and Caamaño~\cite{AC} on quasiregular Hardy spaces.

We prove that, on a non-compact Ahlfors-regular PI space, every function of bounded variation $f$ can be decomposed as a countable sum $\sum_{n=1}^\infty f_n$ of monotone functions of bounded variation $f_n$, in such a way that the total variation of $f$ is the sum of the total variations of the functions $f_n$. Our result extends a Euclidean result of Bianchini and Tonon to the metric setting.

We study products $B_{\alpha,m}^{(d)}B_{\beta,m}^{(d)}$ of Berezin transforms on generalized Fock spaces over $\mathbb C^d$. We establish a result for $m\neq2$: if $B_{\alpha,m}^{(d)}B_{\beta,m}^{(d)}$ coincides with another such product, then the weight parameters of the second product must be $(\alpha,\beta)$. As consequences, we characterize when two Berezin transforms commute and when the product of two Berezin transforms is again a Berezin transform.

We develop a finite-tree formalism for sharp inequalities on real-valued, mean-centered $\operatorname{L}^1$-BMO over homogeneous $q$-ary filtrations. A reduction theorem identifies local and global sharp constants with the supremum of finite-depth constants. Finite anchoring gives compact optimization problems and rational extremizers under rational piecewise-linear hypotheses, while stable centering shows that imposing zero root average preserves the supremum over all depths. The principal application is the absolute-value map: we construct centered examples in which taking absolute values increases the BMO seminorm and obtain bounds strictly below the universal factor $2$. Jensen-defect Carleson functionals and martingale-difference envelopes provide further applications, illustrating distinct anchoring mechanisms and the differences between finite-depth rationality, stabilization, and global attainment.

In this work, we show that if $K$ is a nonempty, closed, bounded, and convex subset of $X$ and $1\leq \gamma<\sqrt{N(X)},$ where $N(X)$ denotes the Bynum normal structure coefficient, then every uniformly asymptotically Hölder $\gamma$-Lipschitz mapping $T\colon K\to K$ has a fixed point. To achieve this, it was necessary to guarantee the existence of a subsequence of a bounded sequence whose asymptotic diameter is controlled by the quantity $D(\{x_n\}),$ introduced by Jiménez-Melado, while simultaneously preserving the asymptotic radii at every point of the closure of the linear span generated by the original sequence. This allowed us to remove the Hölder exponent from the calculations and obtain a Hölder-type extension of the Casini–Maluta, Lim–Xu, and, consequently, Kim–Xu theorems.

On a totally bounded metric space every non-contractive self-map is an isometry. For quasi-metric spaces, taken $T_1$ throughout ($d(x,y) = 0$ implies $x = y$), this fails: Zavarzina gave a hereditarily precompact counterexample and asked whether bijectivity restores rigidity; a recent example, the toll spiral, shows that it does not. We prove that compactness does: every compact quasi-metric space is expand-contract plastic, that is, each of its bijective non-contractive self-maps is an isometry. Countable compactness suffices, as does compactness of the conjugate topology. Bijectivity cannot be dropped: a compact, hereditarily precompact quasi-metric on the natural numbers makes the shift non-contractive and not an isometry. For arbitrary non-contractive self-maps we prove rigidity when the space is hereditarily precompact and its conjugate topology is sequentially compact, in particular when it is hereditarily precompact and Yoneda complete; neither hypothesis can be dropped. In both theorems the $T_1$ axiom can be weakened to $R_0$ ($d(x,y) = 0$ implies $d(y,x) = 0$), with hereditary precompactness read sequentially (every sequence has a left K-Cauchy subsequence), but not to $T_0$. We construct a $T_0$ space with all the remaining hypotheses of both theorems, Yoneda complete and compact both in its own topology and in the conjugate one, carrying a bijective non-contractive self-map that is not an isometry. Under $R_0$ the two-sided case reduces to the metric theorem, both topologies being sequentially compact exactly when the symmetrized distance is compact; the one-sided theorems do not reduce to it, and plasticity does not pass to the symmetrization even for compact spaces. The two theorems are machine-checked in Lean, and so are the counterexamples, apart from a few statements listed at the end.

Motivated by the role of piecewise linear functions in ReLU neural networks, we study systems of dilated functions in the Lebesgue spaces $L_q([0,1]^d)$, $q\in(1,\infty)$. For a sequence $\tau\in\ell_1$ we consider the operator $T_\tau g(x)=\sum_{k=1}^\infty \tau_k g(kx)$ and the function system $\mathcal Z^\tau$ obtained by applying it to the trigonometric system. We show that $\tau\mapsto T_\tau$ is an isomorphism of algebras from $\ell_1$ with the Dirichlet convolution as multiplication onto an algebra of operators and connect the inverse of $T_\tau$ to the inverse of $\tau$ with respect to the Dirichlet convolution. We give explicit conditions on $\tau$, under which the resulting system $\mathcal Z^\tau$ is a Schauder basis of $L_q([0,1]^d)$ and, for $q=2$, a Riesz basis for which we estimate the Riesz constants in terms of $\tau$. We characterise the Riesz bases of this type in terms of a sequence space of Dirichlet convolution multipliers, and we treat mixed systems of cosine and sine functions with different sequences. Finally, we apply these results to the system generated by two piecewise linear functions inspired by ReLU networks and improve the known lower Riesz constant to $2/3$, independently of the dimension.

For the bilinear spherical averages in dimension $d\geq2$, we show that the lacunary maximal operator is not bounded from $L^1(\mathbb R^d)\times L^q(\mathbb R^d)$ to $L^r(\mathbb R^d)$ whenever $1\leq q\leq\infty$ and $1/r=1+1/q$. The obstruction is realized by fixed nonnegative inputs; a second construction proves quantitative failure on smooth compactly supported functions and finitely many dyadic scales. The same conclusions hold for the full maximal operator. We give a sharp Lorentz-index criterion in the range $q>d/(d-1)$ and extend the negative result to all multilinear spherical maximal operators with an $L^1$ input. We indicate the precise strong-type boundary statements to which these conclusions apply, without drawing conclusions about separate weak-type or interior estimates.

For either real or complex scalars, we improve the known upper bound for the Banach–Mazur distance between $L_\infty[0,1]$ and $\ell_\infty$, proving $d_{\operatorname{BM}}(L_\infty[0,1],\ell_\infty)\leqslant (2+\sqrt5)^2=9+4\sqrt5=17.944\ldots$. The additional input is a stabilised kernel estimate: the copy of $\ell_\infty$ in $L_\infty$ is chosen so that its kernel is an $\ell_\infty$-sum of singular two-point hyperplanes, and Cohen's bound-two construction is then applied coordinatewise. The quantitative part of the proof is an asymmetric Pełczyński-type factorisation, with the elementary optimisation included below.

We propose a new class of cyclic (noncyclic) operator type condensing mappings and establish conditions that guarantee the existence of best proximity points (pairs) using measure of noncompactness. The results obtained are further applied to investigate the optimum solution of an infinite system of differential equations in $c_0$ and $\ell_1$.