Papers

In the paper, we prove a recursive version of the weighted Sidelnikov-Welch inequality for real and complex unit vectors. Unlike the classical form, which gives a direct lower bound for a fixed even power sum, our inequality relates two consecutive even power sums. Iteration yields the usual weighted Sidelnikov-Welch bound. We apply this estimate to maximal relative projection constants. If $\mathbb{K}^m$ admits a maximal equiangular tight frame with $M_{\mathbb K}$ vectors, then for every integer $k\geq1$, $$ \lambda_{\mathbb K}(kM_{\mathbb K}-m,kM_{\mathbb K}) = \lambda_{\mathbb K}(m)-\frac{2m}{kM_{\mathbb K}}+1. $$ Moreover, the maximal value is realized by an equiangular tight frame when $k=1$ and by a biangular tight frame when $k\geq2$.

A fundamental result due to Fonf (1981) asserts that the unit sphere of every polyhedral Banach space is covered by true faces of the unit ball. The principal aim of our paper is to study the validity of the same result for polyhedral normed spaces and present some applications. Our first main result is that if $X$ is a polyhedral normed space with property ($\Delta$), then its unit sphere is covered by algebraic true faces. It then follows that if the space is additionally (VI)-polyhedral, then its unit sphere is covered by genuine true faces. We also present several counterexamples showing that these results are optimal in a strong sense. For instance, we show that there exist (V)-polyhedral normed spaces whose unit ball doesn't have any algebraic true face at all, and that there exist polyhedral normed spaces with ($\Delta$) whose unit sphere is not covered by true faces. Further, we give an example of a polyhedral normed space such that the set of strongly exposed points of the unit ball is dense in the sphere, and some results and counterexamples concerning boundaries of polyhedral normed spaces. Our principal application of this material involves locally finite tilings of normed spaces, and we show that a normed space admits a locally finite tiling (by bounded convex bodies) if and only if it admits a (VI)-polyhedral norm with ($\Delta$). In particular, such a tiling exists in every polyhedral Banach space with ($\Delta$), which generalises a result of Fonf.

The Markov kernel based metric $D_1$ was introduced in 2011 in order to construct the scale-invariant dependence measure $\zeta_1$, which assign each bivariate copula $C$ a dependence value in $[0,1]$, with $0$ exclusively for the case of independence, and $1$ exclusively for complete/functional dependence. In the original paper it has been shown that the resulting metric space $(\mathcal{C},D_1)$ is separable and complete, however, no further topological properties were studied. Considering that $D_1$ has proved useful in a variety of contexts, using tools from infinite-dimensional topology, we here close this gap, show that $(\mathcal{C},D_1)$ is homeomorphic to the Hilbert space $(\ell_2,\Vert \cdot \Vert_2)$, and prove that several subfamilies are either homeomorphic to $(\ell_2,\Vert \cdot \Vert_2)$ or to the Hilbert cube $(\mathcal{H},\rho)$. Moreover, allowing for a better assessment of relative sizes, we show that various subfamilies are so-called $Z$-sets in $(\mathcal{C},D_1)$, implying that they are topologically negligible in the full space.

We say that the Poisson operator $P_r$ is Schwarz-contractive from a space $X$ to a space $Y$ if $\|P_r\|_{X\to Y}\le r$ holds for all $0<r<1$. The classical Schwarz lemma is the case $H^\infty_0\to H^\infty$, the subscript indicating zero mean. We are concerned with the non-holomorphic case $L^\infty_0 \to L^p$. For real functions Schwarz-contractivity holds up to the sharp exponent $p_{\mathbb R}=4.109\ldots$. For complex functions we prove it for $p\le 3$ and conjecture that the critical exponent is $4$.

Let $X$ be a Banach space and let $\mathcal{N}(X)$ denote the family of equivalent norms on $X$. We study Hilbert projective metrics on its projectivization $\mathcal{N}'(X)$ induced by ambient cones of nonnegative functions, with particular attention to the intrinsic Hilbert metric induced by the cone $\mathcal{N}(X)\cup\{0\}$. First, we show that the symmetric logarithmic metric on $\mathcal{N}'(X)$ is the Hilbert projective metric induced by the cone of nonnegative real-valued functions, and we characterize the ambient cones that induce the same metric. We then study the intrinsic order on $\mathcal{N}(X)$. For this purpose, we introduce the triangular defect $\Delta_p(x,y)=p(x)+p(y)-p(x+y)$ and prove that two norms $p,q\in\mathcal{N}(X)$ belong to the same intrinsic part if and only if their triangular defects are uniformly comparable, that is, $a\Delta_p\leq\Delta_q\leq b\Delta_p$ for some $a,b>0$. This yields an explicit formula for the intrinsic Hilbert metric in terms of the pointwise comparison of the norms and of their triangular defects. Finally, the map $\Phi(p)=(p,\Delta_p)$ realizes the intrinsic cone order of $\mathcal{N}(X)$ inside a canonical product cone of nonnegative functions and preserves the Hilbert metric on each intrinsic part.

We study complex rank-two pinor fields on the flat Möbius band and the relation between their deck equivariance and Dirac boundary realizations. With explicit Clifford conventions, the two lifts for each Pin type are represented by $R=c\sigma_1$, where $c=\pm1$ for Pin+ and $c=\pm i$ for Pin-. The relevant longitudinal monodromy is $cJ$, with $J\phi(w)=\sigma_1\phi(-w)$, rather than the fiber matrix alone. Its eigenspaces give shifts $0,1/2$ in the positive case and $1/4,3/4$ in the negative case. We identify the resulting geometric Sobolev spaces and construct self-adjoint elliptic realizations of $D=-i(\sigma_1\partial_x+\sigma_2\partial_w)$ with a local boundary condition defined by the longitudinal tangent field. An exact graph estimate gives compact resolvent and Fredholm mappings on nonnegative Sobolev levels. The spectrum consists of a single unpaired transverse branch and paired branches whose longitudinal shifts alternate with transverse parity. Consequently, the kernel and spectral symmetry depend on the lift and boundary condition; the two negative Pin lifts have eta values $1/2$ and $-1/2$ in the stated convention. We also give a secular equation and an exact kernel criterion for the full translation-invariant family of local self-adjoint elliptic boundary lines.

We prove that the Hilbert matrix operator $H:A^2_{\ln^{\alpha}} \to A^2$ is bounded when $\alpha\geq 2$ and that $H$ does not map $A^2_{\ln^{\alpha}} $ into $A^2$ when $\alpha<2$, solving an open problem posed by B. Karapetrović and M. Jevtić. The boundedness when $\alpha=2$ is shown as a corollary to the more general statement: the Hilbert matrix $H:\ell_v^p \to \ell_w^p$ is bounded, where $v(n) = (\ln n)^p / n$ and $w(n) = 1/n$.

We study the Kadison–Kastler distance and the interior angle between masas in finite-dimensional matrix algebras. In \(\mathbb{M}_2(\mathbb{C})\), we obtain an explicit formula for the Kadison–Kastler distance between two masas and show that every value in \([0,1]\) is attained by \(\mathrm{d}_{\textrm{KK}}(\Delta,u\Delta u^*)\). We further prove that the distance attains its maximal value precisely when the relative unitary is a Hadamard unitary. In addition, for every $v$ in the groupoid normaliser of $\Delta$, the Kadison–Kastler distance between \(\Delta\) and \(v\Delta v^*\) assumes either its minimal or maximal possible value. We also establish the identity \[ \mathrm{d}_{\textrm{KK}}(\mathcal{A},\mathcal{B})=\sin\alpha(\mathcal{A},\mathcal{B}) \] for any two intermediate subalgebras \(\mathbb{C}\subsetneq\mathcal{A},\mathcal{B}\subsetneq\mathbb{M}_2(\mathbb{C})\). Most notably, for any unitaries \(u,v\in\mathbb{M}_2(\mathbb{C})\), we establish the equivalence of maximal Kadison–Kastler distance, maximal interior angle, maximal Connes–Størmer modified entropy, and minimal Popa probabilistic index: \[ \mathrm{d}_{\textrm{KK}}(u\Delta u^*,v\Delta v^*)=1 \iff h(u\Delta u^*,v\Delta v^*)=\log 2 \iff \alpha(u\Delta u^*,v\Delta v^*)=\frac{\pi}{2} \iff \lambda(u\Delta u^*,v\Delta v^*)=\frac{1}{2}. \] For general \(\mathbb{M}_n(\mathbb{C})\), we prove a unitary invariance property for the interior angle and derive an explicit formula for the angle between arbitrary masas of the form \(u\Delta^{(n)}u^*\) and \(v\Delta^{(n)}v^*\). As a consequence, the interior angle is \(\frac{\pi}{2}\) precisely when the relative unitary \(u^*v\) is a Hadamard unitary. Finally, we show that every value in \(\left[0,\frac{\pi}{2}\right]\) is attained as the interior angle between a pair of masas in \(\mathbb{M}_n(\mathbb{C})\).

We show that for every $n\in \mathbb{N}$, the Lipschitz-free space $\mathcal{F}(\mathbb{R}^n)$ has the Dunford-Pettis property. As a consequence, for every $n\in \mathbb{N}$ and every nonempty $M\subset \mathbb{R}^n$, the space $\mathcal{F}(M)$ has the Dunford-Pettis property. This implies that $\mathcal{F}(M)$ is not isomorphic to $\mathcal{F}(X)$ for any separable Banach space $X$ that fails the Dunford-Pettis property. In particular, this applies to $X=\ell_2$, and more generally to every infinite-dimensional separable reflexive space. We also deduce that if $1<p<\infty$, then $\mathcal{F}(\ell_p)$ is not isomorphic to $\left(\bigoplus\limits_{n=1}^\infty \mathcal{F}(\ell_p^n) \right)_{\ell_1}$.

This paper presents a comprehensive formulation and a systematic investigation of g-dual Banach frames in a Banach space X and its dual space X?. We study the fundamental structural properties of these frames and establish sufficient conditions for their existence. Furthermore, we explore their relationships with other generalized notions of duality in Banach frame theory. Finally, we investigate the stability of g- dual Banach frames under bounded perturbations, demonstrating their robustness and potential applications in functional analysis.

We prove that every character on a commutative complex complete Hausdorff locally multiplicatively convex algebra, not necessarily unital, is bounded on bounded sets. Consequently, every character on a commutative complex Fréchet–Arens–Michael algebra is continuous, giving an affirmative answer to Michael's problem. In the Fréchet case, each character is bounded by one seminorm from any increasing defining sequence of submultiplicative seminorms.

Let $(\mathcal{H}\_t)_{t \geq 0}$ denote the standard Ornstein-Uhlenbeck semigroup over ambient space $\mathbb R^d$ equipped with Gaussian measure $d\gamma(x) = e^{-|x|^2} dx$. Further let $\mathcal{H}^*(f)(x) = \sup_{t > 0} \mathcal{H}_t(|f|)(x)$ denote the associated maximal operator. Since $(\mathcal{H}_t)_{t \geq 0}$ is a Markovian semigroup over $(\mathbb R^d,d\gamma(x))$, it is well known that $\mathcal{H}^*$ is bounded on $L^p(\mathbb R^d,d\gamma(x))$ for all $1 < p \leq \infty$. We show that $\mathcal{H}^*$ is also bounded on the weighted space $L^p(\mathbb{R}^d,\omega(x)e^{-\frac{p}{2}|x|^2}dx)$ for $1 < p <\infty$, for weights $\omega\,: \mathbb R^d \to (0,\infty)$ belonging to a certain Muckenhoupt type class $A^\alpha_p$, where $\alpha \in [0,1)$. Essentially, a weight $\omega$ belongs to $A^\alpha_p$ if and only if $\omega(Q) \omega^{-\frac{1}{p-1}}(Q)^{p-1} \leq C |Q|^p$ for all cubes $Q$ contained in any reference cube $N_\alpha(R_x) \subseteq \mathbb R^d$, where the well chosen family $(N_\alpha(R_x))_{x \in \mathbb R^d}$ covers $\mathbb R^d$, $N_\alpha(R_x)$ contains $x$ and the side length of $N_\alpha(R_x)$ is comparable to $1/\max(1,|x|)^\alpha$, with constants depending only on $d$. The class $A^\alpha_p$ contains the usual Muckenhoupt class. We also use the weight class $A_p^{loc}$ from \cite[Definition 2.2]{B} defined in a similar way as $A^\alpha_p$ but with dyadic subcubes contained in some $N(R)$ built over the Gaussian dyadic grid (see below), and we show that the local part $\sup_{t >0}\mathcal{H}_t(|f|\chi_{N(R_x)})(x)$ is bounded on $L^p(\mathbb{R}^d,\omega(x)e^{-\frac{p}{2}|x|^2}dx)$ if and only if $\omega \in A^{loc}_p$.

In Proposition 3.5 of our paper [J. Langemets and K. Pirk, Stability of diametral diameter two properties, RACSAM (2021)], we claimed that the convex DLD2P lifts from an $M$-ideal to its superspace. The proof contains a quantifier error. We show that its valid part yields the following replacement: for every $\varepsilon>0$, every slice of the unit ball of the superspace contains a $(2-\varepsilon)$ $\Delta$-point of norm arbitrarily close to one. In particular, the unit ball is the closed convex hull of these points at each fixed scale, and the superspace has the local diameter two property. Finally, we prove that the original claim was false, we construct Banach spaces $Y$ and $X$ such that $Y$ has the convex DLD2P, $Y$ is an $M$-ideal in $X$, yet $X$ fails the convex DLD2P.

We extend the Besov rough path framework of Friz and Seeger to paths lying in Besov spaces that are Muckenhoupt-weighted in time. We establish a weighted Besov sewing lemma, construct Young and rough integrals, and prove local Lipschitz continuity of the Itô-Lyons map in weighted Besov norms. The resulting estimates deliver "location-sensitive" control of errors in RDE solution increments. We illustrate the framework with a diffusion whose coefficient is singular at a specified time.

The Baringhaus-Henze-Epps-Pulley (BHEP) tests for multivariate normality are affine-invariant goodness-of-fit tests based on a Gaussian-weighted $L^2$ distance between empirical and Gaussian characteristic functions. In 1990, Henze and Zirkler expressed the limiting null distribution through the eigenvalues of an integral operator on the standard Gaussian space. In 1997, Henze and Wagner obtained a simpler covariance kernel and raised the problem of calculating the eigenvalues of the resulting operator on a Gaussian-weighted space. Although subsequent work treated the univariate case and numerical approximations in a few low dimensions, the complete all-dimensional spectral problem remained open. This paper determines both complete spectra for every dimension $d \in \mathbb{N}$ and every smoothing parameter $\beta > 0$. The two operators are shown to have the forms $\mathcal{X}_{\beta,d}^*\mathcal{X}_{\beta,d}$ and $\mathcal{X}_{\beta,d}\mathcal{X}_{\beta,d}^*$ for the same Hilbert-Schmidt operator $\mathcal{X}_{\beta,d}$. Consequently, their nonzero eigenvalues agree, including multiplicities, while the null space of the Henze-Zirkler operator is identified exactly. The Gaussian integral operator in the Henze-Wagner decomposition is diagonalized by Mehler's formula, and rotational symmetry confines the finite-rank correction to the sectors associated with spherical harmonics of degrees $0$, $1$, and $2$. The degree-$1$ and degree-$2$ eigenvalues are characterized by scalar transcendental equations, and the radial eigenvalues by an explicit pole-safe Fredholm determinant. The paper establishes nonnegativity, multiplicities, eigenfunction reconstruction, completeness, the trace identity, and a complete characterization of all exceptional pole cases.

The objective of this paper is to identify a finite group based on the orbit of a vector under its action. When a finite group acts unitarily on a Hilbert space, the associated Gram matrix lies in the group algebra and satisfies identities that encode the algebraic group structure. We ask about the converse: Given a Gram matrix obtained from the orbit of a vector under the group action, can one recover the underlying abstract group structure? Our main result identifies a short list of conditions on a family of orthogonal matrices that force the family to be the right regular representation of a finite group. Building on this characterization, we develop a staged optimization framework that enforces these conditions progressively. Numerical experiments on the dihedral group $D_4$ and the tetrahedral rotation group $A_4$ recover $D_4$ exactly and all twelve $A_4$ matrices up to modest precision.

We study supinf problems for perturbations of a given function of two variables by continuous bounded functions in the underlying spaces and the properties of the associated solution mapping. We provide conditions on the initial data such that most (in the Baire category sense) of the functions from the perturbation space secure existence and uniqueness of the solution of the supinf problem for the corresponding perturbed functions. In this way we obtain generic variational principles for such problems.

We prove results about stability under mixtures for transport–entropy inequalities and restricted functional inequalities of Poincaré and log-Sobolev type. We treat Talagrand inequalities of type $\mathbf{T}_2$ and $\mathbf{T}_1$ together with their generalizations, as well as weak transportation inequalities introduced by Gozlan et al. Our main results provide mild sufficient conditions under which a given inequality may be transferred from mixture components to the mixture itself.

Let {\Psi} be an Orlicz function, and let T denote any finite linear combination of composition operators. Motivated by the work of Choe, Koo, and Wang on the compactness of linear combinations of composition operators on weighted Bergman spaces, we completely characterize the compactness of T on the Bergman-Orlicz space A{\Psi}(D). Unlike the classical weighted Bergman space setting, the precise analysis on A{\Psi}(D) depends on the growth behavior of {\Psi} and the associated Luxemburg norm. Hence, the power-type estimates available in weighted Bergman spaces cannot be applied directly. To overcome this difficulty, under suitable growth assumptions on {\Psi} and using the Julia-Caratheodory type function theory and Carleson type measure technique, respectively, we establish two necessary and sufficient compactness characterizations for T on A{\Psi}(D).

We show functional inequalities with logarithmic weights as a critical case of Caffarelli-Kohn-Nirenberg inequalities. Original CKN inequalities have polynomial weights and are shown via scaling argument. In this paper, we develop this argument into it for nonlinear scaling. Our inequalities are generalizations of critical Hardy type inequalities.

We solve Nikolski's norm-controlled inversion problem for measure algebras of locally compact Abelian groups. For every $\delta>1/2$ there is a constant $C_M(\delta)$, independent of the group $G$, such that $\|\mu\|_{M(G)}\le1$ and $\inf_{\gamma\in\widehat G}|\widehat\mu(\gamma)|\ge\delta$ imply that $\mu$ is invertible in $M(G)$ and $\|\mu^{-1}\|_{M(G)}\le C_M(\delta)$. Together with Nikolski's negative results, this shows that $1/2$ is the sharp universal threshold. The main step is a uniform inversion theorem for Fourier algebras $A(K)$ of compact Abelian groups; its proof is based on a profile decomposition and a resolvent argument. We also obtain norm-controlled inversion for Toeplitz operators with non-vanishing Wiener symbols of winding number zero and give direct endpoint constructions showing failure of norm control at $\delta=1/2$ in $A(K)$ for every infinite compact Abelian group $K$, and in the unitization of $L^1(G)$ for every nondiscrete locally compact Abelian group $G$.

In this paper, we introduce the class of weakly localized operators on the weighted Bergman spaces induced by two-sided doubling weights. We first establish the characterization of the compactness of the operator belonging to the norm closure of weakly localized operators in terms of the boundary vanishing of the Berezin transform. Our approach here is different from the methods in the literature since the “translation" operators are not served us a direct applicable tool in our setting. We further characterize the compactness of Toeplitz operators with bounded symbols in the two-weight setting, a boundedness criterion of a two-weight Bergman projection is also presented.

We prove that, for real Banach spaces of dimension greater than one, each of the identities $\|Id+T^2\|=1+\|T^2\|$ and $\|Id-T^2\|=1+\|T^2\|$, required for every rank-one operator $T$, characterizes the Daugavet property, thereby answering a question posed by Kadets, Martín, and Merí (2007). Consequently, every extremely non-complex Banach space of dimension greater than one has the Daugavet property, answering a question of Martín and Merí (2011). We also compare pointwise versions of the square Daugavet properties with Daugavet points, $\Delta$-points, and $\nabla$-points.

We study generalized Volterra companion operators on Bergman spaces over smoothly bounded convex domains of finite type. A derivative Carleson embedding gives boundedness and compactness criteria between reflexive Bergman spaces. When the target exponent is smaller than the source exponent, boundedness already implies compactness. In the other range, we also obtain an essential-norm formula. Local masses on McNeal polydiscs describe these criteria. Their norm estimates require an extra weight because radial derivatives vanish at the origin. On the Hilbert Bergman space, we characterize Schatten-class membership at and above the Hilbert–Schmidt threshold and prove a sufficient condition below it. We also give a Hilbert–Schmidt kernel test. For bounded symbols and self-maps with relatively compact image, singular values decay exponentially in a power of their index. An ellipsoid example gives a sharp power law governed by boundary type and dimension. The results extend to positive radial shifts.

In this paper, we introduce fractional Cesàro–Morrey function spaces over $p$-adic fields and investigate their fundamental properties. We first study the behavior of dilation operators on these spaces and establish a Minkowski-type integral inequality. We then prove the boundedness of $p$-adic Hardy–Hilbert–type integral operators on fractional Cesàro–Morrey function spaces. As an applications, we derive the $p$-adic Hardy inequality, the Hilbert inequality, and the Hardy–Littlewood–Pólya inequality. Finally, we establish the boundedness of the Erdélyi–Kober fractional integral operator and the Hadamard fractional integral operator on these spaces.

Consider the space of $C^\infty$-functions on the two-dimensional sphere $S^2$ and its decomposition $\oplus\mathcal H_n$ into a direct sum of minimal rotation-invariant spaces. We consider elements of $\oplus\mathcal H_n$ as functions of two variables, a nonnegative integer variable $n$ and a complex variable $u$ (a restriction of such function to the set $n=k$ is a polynomial in $u$ of degree $\le 2k$). For operators of multiplication by $x_1$, $x_2$, $x_3$ in $C^\infty(S^2)$ we obtain the corresponding operators in $\oplus\mathcal H_n$, they are differential-difference operators in the variables $u$, $n$ (including second derivatives in $u$ and shifts $n\mapsto n\pm1$). We obtain the similar correspondence for operators of differentiation along conformal vector fields on $S^2$.

We study ball-covering properties in Lipschitz-free spaces. We establish an extension criterion for proving that $\mathcal F(M)$ fails the ball-covering property and apply it to several classes of nonseparable metric spaces. In contrast, we construct a nonseparable uniformly discrete metric space $M$ such that $\mathcal F(M)$ has the uniform ball-covering property and is isomorphic to $\ell_1(2^\omega)$. More precisely, $\mathcal F(M)$ has the $\alpha$-ball-covering property for every $\alpha\in[-1,1)$. This example also shows that these quantitative ball-covering properties are not hereditary within the class of Lipschitz-free spaces. Finally, we prove some stability results under sufficiently small bi-Lipschitz perturbations of the metric.

We study strong measurability and adapted approximations for maps taking values in possibly nonseparable metric spaces. After recalling the metric-space Pettis criterion, we give conditions under which measurability implies strong measurability. Under the continuum hypothesis (CH), every measurable map from a countably generated measurable space into a metric space is strongly measurable for every measure on the domain. For finite or $\sigma$-finite measures, the same conclusion holds when the target density is strictly smaller than every real-valued measurable cardinal. Under CH this includes targets of density at most $\mathfrak c$; if no real-valued measurable cardinal exists, it holds for all metric targets. For function-valued maps, we show that pointwise measurability and continuity of the sections need not ensure measurability in the function-space topology. We provide a verification criterion using countably many evaluations and essential separability of the range. Further, we obtain elementary $L^p$-approximations of progressive or jointly measurable adapted processes under suitable filtration assumptions, Lipschitz functionals with path and Wasserstein variables, and smooth approximations based on finitely many observations of a process with left- or right-continuous paths. Finally, under CH and suitable assumptions, we apply these results to controlled stochastic differential equations in separable Hilbert spaces, allowing for jumps, random coefficients, path dependence, and state- and control-law dependence; we establish convergence of states and costs uniformly over admissible controls, convergence of optimal values, and transfer of near-optimal controls.

The exponential decay of weak solutions towards the constant steady state of general cross-diffusion systems in bounded domains with no-flux boundary conditions is investigated. The proof of quantitative decay rates is based on the relative entropy method and involves two parameters: the exponent determining the entropy density and the exponent in the entropy production integral. The key ingredient are novel convex Sobolev inequalities, which extend existing results in the literature to a broad range of values of the two parameters. These inequalities are established using convexity arguments and the Gagliardo-Nirenberg inequality.

We study bounded operators whose prescribed powers remain uniformly close to the identity, and resolvent conditions sampled along sequences of powers. The displacement bound $\sup_{n\ge1}\|I-T^n\|\le q<2$ forces a finite peripheral spectrum consisting of odd-order roots of unity and makes an explicitly determined odd power of $T$ a Ritt operator. The sharp unconditional threshold for $T$ itself is $\sqrt3$. If $\Phi_*(q)$ denotes the optimal universal Ritt resolvent bound below this threshold, then $\Phi_*(q)=2/(\sqrt3-q)+O(1)$ as $q\uparrow\sqrt3$. At every higher peripheral threshold the optimal finite-peripheral resolvent bound has reciprocal order of growth. We also obtain constructive bounds for general weighted Wiener symbols. For each $1<q<2$, an angular escape invariant characterises exactly the sampling sequences for which a displacement bound by $q$, together with the peripheral spectral condition, forces the Ritt property. We compute this invariant for asymptotically geometric sequences. At the endpoint $q=1$, suitable phase conditions imply the full displacement bound without assuming power boundedness; normal contractions admit an exact scalar criterion. Finally, we establish quantitative Hilbert-space and $L^p$ resolvent estimates, a characterisation of operators with odd Ritt powers on uniformly convex spaces, and strict norm gaps for operator-valued disc-algebra functions, with explicit matrix-polynomial bounds from Fejér–Riesz factorisation.