Papers

Let $d\in [0, n]$ and let $E\subset \mathbb{R}^n$ be a closed $d$-thick set. Given $p\in [1, \infty]$, $q\in (0, \infty]$, and $s\in (\frac{n-d}{p}, 1)$, we provide an intrinsic description of the trace-space of the Besov space $B^s_{p, q}(\mathbb{R}^n)$ to $E$. If, in addition, $p<\infty$, we give an intrinsic description of the trace-space of the Lizorkin–Triebel space $F^s_{p, q}(\mathbb{R}^n)$ to $E$.

We introduce a class of plurisubharmonic weights on \mathbb C^n and develop a theory of the associated weighted Fock spaces F_\varphi^p. The curvature assumptions are imposed on a \mathcal C^2-regularization at bounded distance from the original weight. Thus the original weight need not be smooth or strictly plurisubharmonic, and, even when it is of class \mathcal C^2, its complex Hessian eigenvalues need not be uniformly comparable. This provides a counterpart of Christ's doubling theory in several complex variables. We establish a global upper estimate with decay away from the diagonal and a uniform lower estimate near the diagonal for the weighted Bergman kernel. These estimates yield L^p bounds for the kernel functions, boundedness of the Bergman projection, and duality and complex interpolation for the associated Fock spaces. For a certain class of weights beyond the uniformly controlled curvature setting, we construct an integral solution operator for the \bar\partial equation and establish scale-adapted weighted L^p estimates for 1\leq p\leq\infty. As an application, for all 1\leq p,q<\infty, we characterize the boundedness and compactness of Hankel operators from F_\varphi^p to L_\varphi^q with possibly unbounded symbols in terms of weighted IDA spaces.

We present a simple argument providing a very general Bombieri–Taylor-type result to compute Bragg peaks in diffraction theory as limits. At its abstract core the argument relies on considering the Eberlein convolution as an inner product. This inner product gives an abstract version of a Cauchy–Schwarz type inequality which implies the Bombieri–Taylor-type result as well as generalizations.

Let \(X:M^n\to\mathbb R^{n+p}\) be an \(n\)-dimensional complete self-shrinker in arbitrary codimension \(p\ge 1\). Suppose that the scalar curvature \(R\) is a positive constant and that the squared norm \(S\) of the second fundamental form satisfies \(S\le 1\). We prove that \(S\equiv 1\) and that \(X\) is isometric to either the round sphere \(S^n(\sqrt n)\) or the standard generalized cylinder \(S^k(\sqrt k)\times\mathbb R^{n-k}\), \(2\le k\le n-1\). In particular, no new higher-codimension examples occur under these assumptions. The proof uses an algebraic lemma which yields a uniform positive lower bound for the Bakry–Émery Ricci curvature, together with the comparison theorem of Wei–Wylie, the equivalence of finite Gaussian volume and polynomial volume growth due to Cheng–Zhou, and the gap theorem of Cao–Li.

We use metric functionals to build a weak topology associated with $d$-weak convergence on metric spaces. We also show the existence of a sequence in the space $C[0,1]$ that is unbounded in the sup-norm yet converges $d$-weakly. This points out an error in an earlier work.

Assuming Jensen's diamond principle, we construct an infinite compact zero-dimensional space $K$ such that $C_p(K)$ has no infinite-dimensional Hausdorff separable linear quotient. The space $K$ is separable and crowded, has weight $\aleph_1$ and cardinality $2^{\aleph_1}$, and is an Efimov space. We construct $K$ as an inverse limit of compact metrisable spaces indexed by the countable ordinals. At each nontrivial successor step, the projection has two-point fibres over a chosen closed set and singleton fibres elsewhere; this changes the weak-star limit of a selected sequence of finitely supported measures. We also prove that, for compact $X$, the existence of an infinite-dimensional separable quotient of $C_p(X)$ is equivalent to the existence of an infinite-dimensional metrisable quotient.

We study the rigidity of the orthogonal projections $P_{n}:H^2(\mathbb{D}^2)\rightarrow H_{n}$, where $H^{2}(\mathbb{D}^{2})=\bigoplus_{n=0}^{\infty}H_{n}$ and each $H_{n}=H^{2}(z)w^{n}$ is a reducing subspace for $T_{z}$, and we derive properties of the minimal invariant subspaces of $T_{z}^{*}$. Our main result establishes that the nonzero projections of a minimal invariant subspace onto the one-variable Hardy space share a common $t_z^*$-invariant structure. Furthermore, we relate the results obtained to an approach to the invariant subspace problem.

It is well known that every hyponormal operator on a complex Hilbert space whose spectrum has planar Lebesgue measure zero is normal. In particular, every compact hyponormal operator on a complex Hilbert space is normal. In this paper, we investigate analogous structural and spectral phenomena for so-called sum-hyponormal $d$-tuples on a complex Hilbert space $\mathcal H$, namely, commuting $d$-tuples ${\bf T}=(T_1, \ldots, T_d)$ satisfying $\sum_{j=1}^d [T^*_j, T_j] \geqslant 0$. We show that every sum-hyponormal $d$-tuple of compact operators decomposes as the direct sum of a normal $d$-tuple and a quasinilpotent sum-hyponormal $d$-tuple. As a consequence, every sum-hyponormal $d$-tuple on a finite-dimensional Hilbert space is normal. In contrast to the single-operator case, a sum-hyponormal $d$-tuple need not be normaloid. Nevertheless, we establish a multivariable analogue of Putnam's inequality for sum-hyponormal $d$-tuples with commuting imaginary parts. As a consequence, we prove that ${\bf T}$ is normal whenever the planar Lebesgue measure of $s(\sigma({\bf T}))$ is zero, where $\sigma({\bf T})$ denotes the Taylor spectrum of ${\bf T}$ and ${s}$ is the complex-linear polynomial with alternating coefficients $1$ and $i$.

We study the Gabor frame properties of the rational window $$ g(x)=\frac{x^2-1}{(x^2+1)(x^2+4)(x^2+9)}, $$ whose poles occur in symmetric pairs. We prove that every lattice satisfying $0<\alpha\beta<1/3$ generates a frame, and we establish an additional frame region for $\beta\ge1$ and $1/3\le \alpha\beta<0.47373$. Furthermore, we show that the rational hyperbolas $\alpha\beta=\frac{p}{3p-1}$, for $p\ge2,$ are entirely contained within the frame set. In contrast, we construct explicit non-frame lattice points on the hyperbolas $$\alpha\beta\in\left\{\frac{1}{3},\frac{1}{2},\frac{2}{3},\frac{3}{4}\right\}.$$ Finally, for the density family $\alpha\beta=p/(p+1)$, we derive a symmetry reduction of the associated Zibulski-Zeevi matrix, providing numerical evidence for a richer structure of non-frame obstructions.

Given a family $\mathcal T = (T_\lambda)_{\lambda\in\Lambda}$ of bounded linear operators acting on the same $F$-space $X$ and indexed by a set of parameters $\Lambda \subset \mathbb{R}^d$, $d\geq1$, we explore the existence and non-existence of vectors that simultaneously satisfy, for all elements of $\mathcal T$, one of the following three dynamical properties: hypercyclicity, upper frequent hypercyclicity, and frequent hypercyclicity. Our approach relies on associating to $\mathcal T$ a "Lipschitz constant function” $F$, whose growth encodes the interaction between the operators and the parameter set $\Lambda$. We show that the asymptotic behavior of $F$ determines sharp thresholds between existence and non-existence of common vectors. In the hypercyclicity setting, we identify the divergence of $\sum 1/F(n)^s$, where $s$ is the Hausdorff dimension of $\Lambda$, as a natural condition for existence, and prove complementary non-existence results, which are optimal for weighted shifts. For upper frequent hypercyclicity, we obtain criteria of non-existence in terms of series of the form $\sum 1/F(\gamma^n)$ for parameter sets in $d$-dimensions. Considered in the one-dimensional case, these criteria are optimal and allow us to contrast known existence results of Mestiri for logarithmic-type growth with non-existence beyond this scale. For frequent hypercyclicity, we establish a dichotomy showing that common vectors exist essentially only when $F$ is bounded, while any unbounded growth prevents their existence. Our results apply to a broad class of parameter sets, including self-similar fractals, providing a unified perspective on how growth conditions and geometric features of $\Lambda$ determine common dynamical behavior. Finally, in the specific context of weighted shifts, we prove that two operators do not necessarily share the same frequently hypercyclic vectors if the sequence of their weight products ratios admits two distinct non-zero cluster points, establishing the optimality of a result of Grivaux, Matheron and Menet.

We prove Chvátal's conjecture, posed in 1972: every hereditary family of subsets of a finite set has a largest intersecting subfamily that is a star. More generally, we prove a sharp correlation inequality for increasing Boolean functions $f,g:\{0,1\}^n\to\{0,1\}$. Writing $g^*(x)=1-g(1-x)$, we show that $$ \sum_{\varnothing\ne S\subseteq[n]}\hat{g}(S)^2\max_{i\in S}\mathrm{Inf}_i[f]\le\frac{2\mathrm{Cov}(f,g)\mathrm{Cov}(f,g^*)}{\mathrm{Cov}(f,g)+\mathrm{Cov}(f,g^*)}. $$ When $g$ is antipodal, that is, $g=g^*$, this yields $\mathrm{Cov}(f,g)\ge\frac{1}{4}\min_{i\in[n]}\mathrm{Inf}_i[f]$, the correlation formulation of Chvátal's conjecture due to Friedgut, Kahn, Kalai and Keller.

This paper solves the Wehrl-type entropy problem for arbitrary compact connected semisimple Lie groups. Let $G$ be a compact connected semisimple Lie group, and let $\pi:G\to U(V_\lambda)$ be a finite-dimensional irreducible unitary representation associated with the highest weight $\lambda$. We prove that coherent projectors are the unique minimizers of the Wehrl entropy over all density matrices on $V_\lambda$. They also uniquely maximize every Husimi power moment of order $p>1$. The proof uses a second-variation at the extremizer by perturbation in directions associated with Killing fields, similar to the strategy of Frank and Lieb used in \cite{FrankLieb}. Then the problem reduces to the extreme-moment property of highest-weight vectors.

\cite{mss2015} proved Weaver's discrepancy result existentially, resolving the Kadison–Singer conjecture . Finding such signs efficiently for general inputs remained an open algorithmic question. In the real-arithmetic model, we give a deterministic algorithm running in polynomial time with discrepancy at most $35\sqrt\varepsilon$. The algorithm walks from the origin of the hypercube to a vertex, fixing coordinates as they hit a face. Its potential measures a soft spectral edge of the discrepancy matrix perturbed by an operator-valued free semicircular element. The perturbation's covariance vanishes as the coefficients reach their endpoints. Inspired by the free interpolation approach of Bandeira, Boedihardjo, and van Handel \cite{bbvh2023}, we combine Lehner's variational formula \cite{lehner1999} with spectral Tsallis–$1/2$ regularization used in \cite{allenZhuLiaoOrecchia2015} and \cite{pesentivladu2026}. The resulting potential has a finite-dimensional SDP formulation, allowing the discrepancy and remaining covariance to be analyzed together. We analyze the optimizer's stability through the linearized Karush–Kuhn–Tucker (KKT) system of a regularized min–max problem, whose stationarity equations are related to the matrix Dyson equation \cite{erdos2019}. This gives the movement rule: either a coordinate can move toward its nearer endpoint at small spectral cost, or a low-curvature direction orthogonal to the current coefficient vector allows further progress. Choosing the better sign of this direction controls discrepancy while increasing the squared distance from the origin. Upcoming work \cite{kathuria2026higherRank} will address higher-rank Kadison-Singer and spectrally thin trees. Lean formalizations of our main discrepancy theorems have been completed and will be released shortly.

A compactification of the discrete space $\omega$ of all integers is called symmetric if it is the quotient space obtained by gluing together the remainders of two copies of some other compactification of $\omega$. This is a generalization of both a convergent sequence, which is in a way the minimal symmetric compactification of $\omega$, and the Arkhangel'skiĭ–Bereznitskiĭ–Schachermayer space studied in $C_p$-theory, which is in a sense the maximal symmetric compactification of $\omega$. We investigate symmetric compactifications of $\omega$ and their relations to the Separable Quotient Problem for spaces $C_p(X)$ and to the existence of Josefson–Nissenzweig sequences of finitely supported Borel measures on spaces $X$, in particular with supports of bounded size. Further, we reduce the Separable Quotient Problem for spaces $C_p(K)$, $K$ compact, to the case when $K$ is a totally asymmetric compactification of $\omega$. Our results shed some new light on the Grothendieck property of Banach spaces $C(K)$.

Let $\mathcal R$ be a normalized root system in $\mathbb R^N$ with a nonnegative multiplicity function $k$, and let $\mathcal F$ be the associated Dunkl transform. We prove a Hörmander multiplier theorem on the Hardy spaces $H^p_{\mathrm{Dunkl}}$, $0<p\le1$, defined by conical Littlewood–Paley square functions. Let $W_2^s$ denote the classical Sobolev space in $\mathbb{R}^N$. More precisely, if a multiplier $m$ satisfies \[ \sup_{t>0} \|\psi(\cdot)m(t\cdot)\|_{W_2^s}<\infty \] for a nonzero radial cutoff $\psi\in C_c^\infty(\mathbb R^N\setminus\{0\})$ and \[ s>\mathbf N\left(\frac1p-\frac12\right), \] where $\mathbf N$ is the homogeneous dimension, then the Dunkl multiplier \[ \mathcal T_mf=\mathcal F^{-1}(m\mathcal Ff) \] extends to a bounded operator on $H^p_{\mathrm{Dunkl}}$.

We present a uniform method for proving meromorphic function identities by comparing complete zero and pole multisets. A fixed annular Weierstrass normalization represents every nonzero meromorphic function uniquely as \[ q_t=e^{h_t}\frac{p_{Z_t}}{p_{P_t}},\qquad t=([h_t],Z_t,P_t),\qquad [h_t]\in\mathcal O(\mathbb C)/(2\pi i\mathbb Z). \] Thirteen principal comparisons, together with their specializations and consequences, use the same four steps: represent the two sides, compare their divisors, apply a divisor-comparison theorem (DCT), and normalize $h_s-h_t$. A small library packages classical finite-order, parity, recurrence, and lattice-rigidity arguments for repeated use. The examples span sine, Gamma, Barnes $G$, Bessel, completed zeta, theta, and elliptic functions. Their breadth demonstrates reuse of one proof structure across these families; the family-specific hypotheses and scalar calculations remain explicit. The contribution is a methodological synthesis: shorter proofs where available, and otherwise a modular organization of classical arguments. The fixed factors supply common coordinates, rather than a new uniqueness hypothesis. Historically, the viewpoint is motivated by the nineteenth-century contrast between Riemann's global geometric approach to complex function theory and Weierstrass's analytic construction of functions; the present method deliberately combines global divisor data with fixed Weierstrass factors. An appendix develops the associated ring and field presentations, separately from the ordinary operations used in the proofs.

We give negative answers to Kolmogorov's rearrangement problem and Garsia's conjecture. We construct a complete uniformly bounded orthonormal system for which every rearrangement admits a square-summable series divergent almost everywhere. The finite construction uses two copies of the trigonometric system in different orderings. The main ingredient is a combinatorial lemma which guarantees a prescribed ordering along an arithmetic progression in at least one of two related permutations. Its proof uses Szemerédi's theorem and a counting argument. A Walsh variant gives $N$-term $\{\pm1\}$-valued systems with $L^2$ maximal norm at least $c\log\log N$ in every ordering, matching Bourgain's upper bound.

We prove that $L^2(\Omega)$ admits no Riesz basis of exponentials when $\Omega$ is a ball in $\R^n$, $n\geq2$, or a nondegenerate triangle in $\R^2$. The ball result extends to every nonempty bounded convex open set with $C^2$ boundary in $\R^n$, $n\geq2$. Starting from a hypothetical Riesz basis, we use Beurling weak limits of translates of its frequency set to construct a stationary system. On the associated Hilbert space, we define spectral cutoff projections and a comparison family of orthogonal projections obtained from the synthesis operator. The distance between corresponding projections is uniformly bounded by a constant strictly smaller than one. The comparison family is continuous in operator norm, whereas the boundary geometry of $\Omega$ produces two strong limits of the cutoff projections with strictly nested ranges. Both limiting projections would then be at distance less than one from the same comparison projection, which is impossible.

We show that the norm-closed linear span of products of two Dunford-Pettis operators on $L^1(0,1)$ lies strictly between the representable and the Dunford-Pettis operators. This gives an additional proper large closed ideal in $\mathcal{B}(L^1(0,1))$, answering the existence question of Johnson, Pisier and Schechtman. The quotient of the Dunford-Pettis ideal by this square ideal contains a contractively complemented isometric copy of $L^1(0,1)$. We also prove that the Dunford-Pettis ideal has no right approximate identity, answering a question of Johnson and Schechtman. More precisely, a positive norm-one convolution operator $R$ satisfies $\|R-RT\|\geq1$ and $\|R-TR\|\geq1$ for every Dunford-Pettis operator $T$. The proofs combine translation averaging, Rajchman measures supported on a strongly independent set, and two atomless disintegrations of positive operators. The approximate-identity obstruction extends to every finite measure space with a nonzero atomless part. Finally, we prove that quantitative multiplication estimates for an ideal pass to all its closed power ideals.

Let $X$ and $Y$ be real Banach spaces, and let $\cC(X)$ and $\cC(Y)$ denote the families of all nonempty bounded closed convex subsets of $X$ and $Y$, respectively, equipped with the Hausdorff metric. We prove that every surjective isometry $F:\cC(X)\to\cC(Y)$ is induced by a surjective affine isometry of the underlying spaces. More precisely, $F$ maps singleton sets onto singleton sets, and the map $T:X\to Y$ defined by $F(\{x\})=\{T(x)\}$ is a surjective affine isometry satisfying \[ F(A)=T[A]:=\{T(x):\;x\in A\},\qquad(A\in\cC(X)). \] In particular, our theorem extends the result of Gruber and Lettl for finite-dimensional Euclidean spaces to arbitrary real Banach spaces, without imposing any additional assumptions on the underlying spaces.

Let $A$ and $B$ be nonzero unital $C^*$-algebras with units $1_A$ and $1_B$, respectively, and let $T\colon A\to B$ be a bijection satisfying \[ \|T(a+b)\|=\|T(a)+T(b)\|, \qquad \|T(ab)\|=\|T(a)T(b)\|, \qquad T(a^*)=T(a)^* \] for all $a,b\in A$. We prove that there exist a central symmetry $u$ in $B$ and a real $*$-isomorphism $\Phi\colon A\to B$ such that \[ T(a)=u\Phi(a) \qquad(a\in A). \] Moreover, $u$ and $\Phi$ are uniquely determined by $T$. Conversely, if $u\in B$ is a central symmetry and $\Phi\colon A\to B$ is a real $ * $-isomorphism, then $T=u\Phi$ is a bijection satisfying the three identities above. In particular, the two norm identities, together with involution preservation, force the normalized map $uT$ to preserve the full product, not merely the Jordan product.

Let $\mathcal M$ be a diffuse semifinite von Neumann algebra endowed with a faithful normal semifinite trace $\tau$. We prove that, for every nonzero Banach space $Y$, the projective tensor product $L_1(\mathcal M,\tau)\widehat{\otimes}_{\pi}Y$ has the operator Daugavet property. Moreover, the witnessing operators may always be chosen contractive. This extends the operator Daugavet phenomenon from atomless vector-valued $L_1$-spaces to the semifinite noncommutative setting and yields further Daugavet-type consequences for projective symmetric tensor products, all without approximation assumptions.

We investigate structural properties of the ideal-null sequence spaces c_{0,I} and c_{0,I}(X), and of the quotient l_infty/c_{0,I}, emphasizing the interaction between Banach space theory and combinatorial, topological, and measure-theoretic properties of the underlying ideal. We study classical problems related to c_0, including the dual space, compactness criteria, bounded and compact c_{0,I}-valued operators, Sobczyk-type extension phenomena, and complemented copies of c_0. We identify c_{0,I}^* isometrically with a space of bounded finitely additive measures on I and obtain a canonical atomic-singular decomposition, with l_1 as an isometrically complemented summand. A Dini principle for ideal convergence yields characterizations of relatively compact subsets of c_{0,I} and of compact operators with range in c_{0,I}. We prove a Sobczyk theorem for c_{0,I} and show that its separable-injectivity constant is exactly 2 for every proper ideal. We also construct, in ZFC, a class of statistical ideals for which c_{0,I} is not complemented in l_infty, and characterize complemented copies of c_0 in c_{0,I}(X) via the corresponding properties of c_{0,I} and X. Finally, we describe quotients associated with direct and Frolik sums of ideals, characterize Banach-lattice copies of c_0(kappa) in l_infty/c_{0,I} through I-almost disjoint families, identify ad(I) with the cellularity of the associated Stone space, and determine its behavior under Fubini products.

In this paper, we investigate the Fourier uncertainty principle through infinite banded matrices by way of a variational approach to minimizing the uncertainty product. Traditional methods for analyzing the Fourier uncertainty principle rely heavily on the Heisenberg–Weyl structure that underlie the Fourier transform and therefore do not generalize well beyond this setting. We provide three distinct approaches to developing the uncertainty principle based on the spectrum of a shifted quantum harmonic oscillator Hamiltonian, $H_{a,b}$. In the Hermite–Gauss basis, $H_{a,b}$ has an infinite, tridiagonal form which is central to the analysis. One approach exactly solves for the spectrum through analytic methods, another decomposes $H_{a,b}$ into an infinite sum of $2\times 2$ matrices, and the third approach is an algebraic approach to establishing the spectrum via a unitary equivalence with the quantum harmonic oscillator. The approaches taken herein provide alternate avenues that are more broadly applicable to other canonically conjugate operators for other integral transforms which maintain a tridiagonal form.

We show that an autoregressive time series on Banach space can be canonically identified with a finite direct product of self-contained and self-determined natural components defined on separate subspaces by the spectral projections for each spectral point of the characteristic linear pencil. We assume the resolvent of the pencil is analytic everywhere except for the spectral points which may be poles or isolated essential singularities. Each natural component is represented as a forward, backward or outward flow fixed by an upstream boundary condition in the far distant past, the far distant future, or at a nominated initial time—depending on whether the corresponding spectral point is outside, inside, or on the unit circle.

This paper presents a study of dispersion for the two-dimensional Friedlander model with Neumann and Robin boundary conditions, thereby extending previous works treating only Dirichlet boundary conditions. By considering an Airy-type special function, we obtain a parametrix construction similar to the Dirichlet case but with additional terms requiring careful control. By studying the dispersion of this parametrix, we obtain the same dispersive estimates as in previous results for the Dirichlet case, and therefore the same Strichartz estimates.

Let $X$ be a three-dimensional real Banach space and let $f:S_X\to S_Y$ be a surjective isometry. We study the propagation of an affine formula for $f$ on a relatively open part of $S_X$. A finite family of antipodal distance coordinates propagates a linear anchor except at three explicitly described degeneracies: a facet, an open face-star, or a family of chord cones with a common cylindrical kernel. We prove that, if the norm has no fixed-direction cylindrical open cone, every affine open anchor is linear and global. This yields a source-geometric criterion for the Mazur–Ulam property and covers, among other non-strictly-convex examples, the Euclidean double cone. We also give a segment-saturation criterion which proves the property for every prism $Z\oplus_\infty\mathbb R$, where $Z$ is an arbitrary real Banach plane. Finally, in the remaining cylindrical regime, we prove that a chord-saturated same-kernel network cannot be confined to one proper projective quotient arc. We then show that this multi-arc difficulty and the transverse-ruled part of producing an initial affine anchor reach the same final obstruction: upgrading norm calibration on an ambient open cone, together with calibration on a few spherical segments, to pointwise calibration of the sphere map on a two-dimensional open patch.

Let $1 < \rho < n$ and let Omega be in $L^\rho(S^{(n-1)})$ with vanishing mean. We prove that the maximal truncation $T^*_\Omega$ of the rough singular integral $T_\Omega$ is pointwise dominated by finitely many sparse potentials of the form: $\sum_{Q \in S} l(Q) * ( (1/|Q|) * \int_Q |\nabla f|^p )^{1/p}$, where $1/\rho~ = 1/\rho' + 1/n$ and $\rho~ \leq p < n$. This estimate is uniform in the truncation parameter and extends the subcritical bound of Hoang, Moen, and Perez for $T_\Omega$ to $T^*_\Omega$. Since the argument does not require the boundedness of $T^*_\Omega$ on the target space, it yields two-weight Sobolev inequalities: $\parallel T^*_\Omega f \parallel L^q(u) \leq C * \parallel \nabla f \parallel L^p(v)$, under joint two-weight conditions, while the target weight u itself is only required to belong to $A_\infty$. Additionally, we prove a Hedberg-type estimate involving a Morrey norm of the gradient and apply it to weighted grand Lebesgue spaces. Further consequences are obtained in weighted Lebesgue, Orlicz, and variable Lebesgue spaces.

We study linearizations of dynamical systems and some of its topological properties. Special attention is paid to the case of Lipschitz-free operators, and they are shown to model the dynamics of very general linearizations. We provide a new criterion, called the targeting property, for the (weakly) mixing property in a linear dynamical system through a (possibly) non-linear restriction of it. We then apply this criterion to the linearization $T_f$ of a Lipschitz map $f\colon M\to M$, where $M$ is a metric space and the operator $T_f$ is defined on the corresponding Lipschitz-free space $\mathcal F (M)$—the so-called Lipschitz-free operators. We show that, under some natural assumptions on the distance considered in $M$, the operator $T_f$ is hypercyclic if and only if the map $f$ has the targeting property.

We provide a universal characterisation of the Banach spaces of equivalence classes of $L^p$ Bochner integrable functions, under equality almost everywhere, as initial objects in appropriate categories of bifunctors, for all $p \in [1,\infty)$. As a by-product of our construction, we also obtain a unique characterisation of the Bochner integral.