Papers

We investigate Campanato and Morrey spaces on metric-measure spaces and establish their relationships with Hölder spaces. In particular, we show that the embedding of a Campanato space into a Hölder space is equivalent to the existence of a lower bound for the underlying measure. Several examples are provided to illustrate the results.

For $0<\theta<\pi$, let $K_\theta=\{e^{is}:|s|\le\theta\}$ and $E_\theta=\operatorname{span}_{\mathbb C}\{1,z\}\subset C(K_\theta)$. We prove that every contractive linear map from $E_\theta$ into $\mathcal B(H)$ admits a factorization through a unitary operator whose spectrum is contained in $K_\theta$, and is therefore completely contractive. Consequently, $E_\theta$ has a unique operator space structure. For $0<\theta,\varphi<\pi$, we also show that $E_\theta$ and $E_\varphi$ are isometrically isomorphic if and only if $\theta=\varphi$, and that none of these spaces is isometrically isomorphic to $\ell_1^2$ or $\ell_\infty^2$. Thus we obtain uncountably many mutually non-isometric two-dimensional complex Banach spaces with a unique operator space structure, giving a negative answer to a question of Pisier.

We prove a simultaneous Williamson's normal form for two symplectically commuting positive operators $A, B$ on a real symplectic Hilbert space $(\mathbf{H}, J)$. Under a natural commutativity condition, there exists a (generally unbounded) operator $\Phi$ that simultaneously diagonalizes $A$ and $B$ in the symplectic setting as $A = \Phi^t M_A \Phi$, $B = \Phi^t M_B \Phi$, and maps $J$ to a canonical symplectic operator. The result generalizes Williamson's normal form to infinite dimensions and yields a complete spectral invariant for the pair $(A,B)$.

For positive matrices $A,B$ and $0<t<1$, Bourin asked whether the crossed-Heinz expression \[ C_t(A,B)=A^tB^{1-t}+B^tA^{1-t} \] satisfies \[ \UIN{C_t(A,B)}\le \UIN{A+B} \] for every unitarily invariant norm. The most direct route to this inequality is to compare $C_t(A,B)$ with the ordinary Heinz expression \[ H_t(A,B)=A^tB^{1-t}+A^{1-t}B^t, \] because the classical Heinz inequality gives $\UIN{H_t(A,B)}\le\UIN{A+B}$. This route, however, encounters a genuine dimensional obstruction: the stronger comparison $\UIN{C_t(A,B)}\le\UIN{H_t(A,B)}$ is known to fail for positive definite $3\times3$ matrices, and hence, by block-diagonal embedding, in every dimension $n\ge3$. We first determine the exact threshold for this obstruction. For every pair of positive $2\times2$ matrices and every $0<t<1$, we prove the weak-majorization relation \[ s(C_t(A,B))\prec_w s(H_t(A,B)). \]

Godbersen's long-standing conjecture states that the mixed volumes of any convex body $K \subset \mathbb R^n$ satisfy $V(K[k], -K[n-k])\le \binom{n}{k} \operatorname{vol}(K)$ for any $k \in \{1, \dots, n - 1\}$, with equality if and only if $K$ is an $n$-simplex. Kotrbatý and Mouamine recently proved the inequality in full generality, and showed that if $K$ is a (full-dimensional) polytope attaining equality, it must be a simplex. We complete the characterization of the equality case: if $K \subset \mathbb R^n$ is a convex body for which $V(K[k], -K[n-k]) = \binom{n}{k} \operatorname{vol}(K)$ for some $k \in \{1, \dots, n -1\}$, then $K$ is an $n$-simplex. By a similar method, we characterize the equality case in the higher-order Godbersen inequality conjectured by Schneider and by Kotrbatý.

In this article, we study a weighted analogue of the $k$-plane transform, where the weight on the $k$-dimensional measure is given by a radial power of distance. Particularly, we first consider the existence of such operators on Lebesgue spaces imbibed with radial power weights. Then, we proceed to see the mapping properties of the said transform when it acts on radial functions. In this regard, we prove certain weighted $L^p$-improving boundedness and end-point Lorentz space estimates. Finally, we look at the weighted $L^p$-$L^p$ boundedness of the operator acting on general functions, and evaluate its operator norm.

We prove sharp Hardy inequalities on $\mathbb{Q}_p^n$ for the quadratic form of the Vladimirov–Taibleson operator, the $L^q$ norm of its fractional derivative, and a nonlinear difference energy. Under the self-dual Fourier normalization, the sharp quadratic-form constant for $0<\alpha<n$ is $[(1-p^{-(n-\alpha)/2})/(1-p^{-(n+\alpha)/2})]^2$, with a remainder given by the ground state representation. For $1<q<\infty$, $q'=q/(q-1)$, and $0<s<n/q$, a weighted Schur estimate determines the exact $L^q$ norm of the weighted Riesz potential $\|\boldsymbol{x}\|_p^{-s}I_s$, giving the sharp linear Hardy constant $[\Gamma_{p}^{(n)}(n/q'+s)/\Gamma_{p}^{(n)}(n/q')]^q$. The two-point Picone inequality and shell decomposition give the sharp constant $2(1-p^{-n})\sum_{m\ge1}p^{-sqm}(1-p^{-(n/q-s)m})^q$ for the Gagliardo-type energy with kernel $\|\boldsymbol{x}-\boldsymbol{y}\|_p^{-n-sq}$. Annular truncations of the critical powers give explicit minimizing sequences with error estimates. Equality is attained only by the zero function in each stated class. For $q=2$, both $L^q$ inequalities reduce to the quadratic-form inequality. Uniform local constancy makes the difference energy finite for every Bruhat–Schwartz function and every $s>0$. The Hardy inequalities on the full test-function space therefore hold precisely when $sq<n$, without the restriction $s<1$.

We establish that the definition of Sobolev mappings from metric measure spaces to metric spaces relying on the post-composition approach of Ambrosio-Reshetnyak admits, under certain transparent assumptions on the spaces, an equivalent reformulation that is substantially weaker than the principal version. In detail, the corresponding definition requires maps of this kind to meet two successive tests in order to be called Sobolev: first a ”qualitative” one and then a ”quantitative” one. At the same time, our result allows the relevant conclusion to be drawn based solely on the ”qualitative” part. Remarkably, the conditions on the spaces sufficient for this to hold fit entirely into the singular flavor of the subject. More explicitly, such a phenomenon emerges if either the source space is of finite Hausdorff dimension and satisfies the Sobolev-to-Luzin-Lipschitz property or the target space is of finite Hausdorff dimension. All this provides a significant generalization of several earlier achievements on the topic.

What becomes of Grothendieck's inequality when its Hilbert-space vectors are replaced by vectors in $\ell_p^r$ and $\ell_{p'}^r$? For an $m\times n$ matrix, we show that the universal dimensional loss is governed not by $r$ alone, but by the effective dimension \[ d=\min\{r,m,n\}. \] More precisely, the optimal constant is bounded above and below, up to dimension-free factors, by \[ d^{|1/p-1/2|}. \] The upper bound rests on a pointwise Hilbertian stabilization: for every matrix $A$, \[ G_2^{(r)}(A)=G_2^{(d)}(A). \] Combining this stabilization with Lewis's Euclidean distortion estimate gives the effective-dimensional Grothendieck–Hölder bound. Beyond the universal power law, we retain the finer rectangular geometry. At $p=\infty$ we prove the exact compression formula \[ \Gamma_\infty^\K(r;m,n) = \rho\!\left( \ell_1^m(\K), \ell_1^{\min\{r,n\}}(\K) \right), \] and derive from it endpoint-sensitive lower bounds throughout the full Hölder scale. Over the complex field, interpolation between the Hilbertian problem and the exact rectangular endpoints yields corresponding upper bounds. Thus the effective dimension determines the universal exponent, whereas the tensor-norm endpoint retains additional aspect-ratio information. We determine the boundedness threshold when $p$ approaches $2$: it is governed by \[ |1/p-1/2|\log d. \] For complex two-row matrices, the endpoint tensor ratios are explicit, and the general rectangular estimates give quantitative two-sided bounds for every $1\le p\le\infty$.

Given $p \in (1,\infty)$, let $\operatorname{X}=(\operatorname{X},\rho,\mu)$ be a metric measure space such that the measure $\mu$ is uniformly locally doubling and $\operatorname{X}$ supports a weak local $(1,p)$-Poincaré inequality. Let $(\operatorname{Y},\operatorname{d},\underline{y})$ be a complete pointed metric space. We prove that the equivalence class of a Borel map $ u:\operatorname{X} \to \operatorname{Y}$ (modulo coincidence $\mu$-a.e.) belongs to the Ambrosio–Reshetnyak–Sobolev class $W_{p}^{1}(\operatorname{X},\operatorname{Y})$ if and only if $h \circ u$ belongs to the Sobolev space $W_{p}^{1}(\operatorname{X})$ for every $1$-Lipschitz function $h:\operatorname{Y} \to \mathbb{R}$.

Let \(X\in M_n(\mathbb C)\), \(n\geq2\), and let \[ \delta_2(X) = \sup_{\dim S=2}\delta(X_S), \] where \(X_S\) denotes the compression of \(X\) to \(S\), and \(\delta(X_S)\) is the diameter of the largest disk contained in the numerical range \(W(X_S)\). Bourin and Lee proved that every positive block matrix \[ M= \begin{pmatrix} A&X X^*&B \end{pmatrix} \] satisfies \[ \norm{M}\leq\norm{A+B}+\delta_2(X). \] In the same work, they conjectured that the inequality \[ \norm{M}\leq\norm{A+B} \] for every positive block matrix with prescribed off-diagonal block \(X\) characterizes essentially Hermitian matrices. We resolve this conjecture by proving the stronger exact formula \[ \sup_{\left(\begin{smallmatrix}A&X\\X^*&B\end{smallmatrix}\right)\geq0} \left\{ \norm{\begin{pmatrix}A&X\\X^*&B\end{pmatrix}} -\norm{A+B} \right\} = \delta_2(X). \] Thus the elliptical width is precisely the optimal norm defect associated with a fixed off-diagonal block. The reverse inequality is obtained by lifting an arbitrary two-dimensional compression of \(X\) to an explicit family of positive block matrices, yielding quantitative two-sided estimates with error of order \(t^{-1}\). As a consequence, the norm inequality above holds universally if and only if \(X\) is essentially Hermitian, proving Conjecture 3.3 of Bourin and Lee. We also construct an explicit \(3\times3\) normal example showing that the radius constant in their normal-off-diagonal eigenvalue estimate is optimal at the leading eigenvalue level. By direct-sum amplification of this construction, we further prove that the same radius constant is sharp at every eigenvalue level \(j\geq0\), thereby resolving their corresponding sharpness question.

Let $N\in\mathbb N$, $p\in[1,\infty)$, and $Q_0:=(0,1)^N$. In this article, we establish sharp Poincaré inequalities for the nonlocal nonconvex functionals $$ \Phi_{\lambda,p}^\gamma(g;Q_0) :=\lambda^p\iint_{\{(x,y)\in Q_0^2:\,x\ne y,\,|g(x)-g(y)|>\lambda|x-y|^{1+\frac{\gamma}{p}}\}} |x-y|^{\gamma-N}\,dx\,dy, $$ where $g:Q_0\to\mathbb R$ is measurable, $\lambda\in(0,\infty)$, and $\gamma\in\mathbb R$. Specifically, we prove that the estimate $$ \int_{Q_0}\int_{Q_0}|g(x)-g(y)|^p\,dx\,dy \lesssim\lambda^p+\Phi_{\lambda,p}^\gamma(g;Q_0) $$ holds for any such $g$ and $\lambda$ if and only if $\gamma\in(-\infty,-1]$, where the implicit positive constant depends only on $N$, $p$, and $\gamma$. This extends Nguyen's Poincaré inequality for $\gamma=-p$ and provides a useful tool for studying regularity and compactness through these functionals. As applications, we obtain a lower bound for the $\Gamma$-convergence of $\Phi^\gamma_{\lambda,p}$ and related asymptotic characterizations for Sobolev and BV spaces. Moreover, we identify the optimal range of $\gamma$ for which both the Rellich–Kondrachov theorem and a criterion for VMO regularity involving $\Phi^\gamma_{\lambda,p}$ at a fixed threshold hold. These results also answer the question on nonlocal Poincaré inequalities posed by H.-M. Nguyen in [C. R. Math. Acad. Sci. Paris 363 (2025)]. For $\gamma\in(-\infty,-1]$, they further answer questions on Sobolev and BV characterizations posed in the same article and by H. Brezis et al. in [Anal. PDE 17 (2024)].

We study diffeomorphisms of Banach spaces satisfying generalized hyperbolicity: the stable and unstable subspaces are invariant only up to inclusion, need not depend continuously on the point, and expansivity may fail. On closed manifolds, this condition is equivalent to Axiom~A with strong transversality. We introduce a continuous-selection method that converts uniform orbitwise estimates into bounded continuous solutions of cohomological equations for generalized-hyperbolic cocycles. For diffeomorphisms in the uniform global \(C^1\) category, this yields continuous semiconjugacies in both directions under sufficiently small bounded Lipschitz perturbations, and topological stability in the sense of Walters. For the infinite product of one-dimensional Morse-Smale systems on \(\ell^p(\mathbb Z)\), \(1\leq p\leq\infty\), we construct compatible forward and reverse cohomological solution operators and prove structural stability, including for perturbations that couple coordinates, although the product is not expansive. For \(p=\infty\) this gives a structurally stable diffeomorphism with uncountably many hyperbolic fixed points, in contrast with the finitely many basic sets of Axiom~A diffeomorphisms. The cocycle analysis yields Lipschitz shadowing for the induced operator on bounded continuous vector fields.

For $n\ge2$, we establish second differentiability of horizontal perimeter for compact $C^2$ boundaries in $\mathbb H^n$ and the continuous extension of their affine second variation to $C^1$ vector fields along the boundary. For the unit Pansu sphere, the Hessian agrees with the closed Jacobi form across both characteristic poles, including variations which move them. We compute the full Friedrichs spectrum of its Jacobi operator. The operator has compact resolvent, Morse index one and nullity $2n+1$; its kernel consists exactly of normalized translation speeds. Consequently, the sphere is stable under volume-preserving ambient variations.

A Hardy space approach to the Nyman-Beurling and Báez-Duarte criterion for the Riemann Hypothesis (RH) was introduced in [7]. It states that the RH holds if and only if the span of a particular sequence of functions $(h_k)_{k\geq2}$, denoted $\mathcal{N}$, is dense in the Hardy space $H^2$. In this note we construct a bounded operator $T$ on $H^{2}$ such that if the closure of $\mathcal{N}$ is $T$-invariant, then either the RH holds or there exists an infinite progression of $\zeta$-zeros. In particular, this yields an operator-theoretic criterion for the Riemann hypothesis as an invariant subspace question.

We improve an earlier characterization of the Sobolev space $\dot W^{1,p}(\mathbb R^d)$, $1<p<\infty$, in terms of the oscillations of a function on balls of varying centers and radii. In contrast to earlier work, finiteness of the supremum is replaced by the formally weaker assumption of finiteness of the liminf, thereby answering a question by Brezis.

We study the QWEP property of twisted crossed products and its relation to Fourier multiplier transference on noncommutative $L_p$-spaces. We show that QWEP is not preserved by crossed products of QWEP von Neumann algebras by hyperlinear groups. Our examples include trace-preserving actions of a torsion-free residually finite Kazhdan group on a diffuse abelian algebra and on the hyperfinite type II$_1$ factor. We also obtain non-QWEP crossed products with nontrivial scalar cocycles. For scalar twists, we prove that a finite trace-preserving crossed product is QWEP if and only if its ordinary crossed product and its twisted group von Neumann algebra are QWEP. In the positive direction, we establish a restriction theorem for scalar twists of regular wreath products. These results provide QWEP inputs for Fourier multiplier transference and identify a completely bounded multiplier norm obstruction to QWEP. Finally, operator-valued cocycles give finite QWEP crossed products whose acting groups are not hyperlinear.

Let $(\mathcal{M},\tau)$ be a semi-finite von Neumann algebra. We prove that $L_1(\mathcal{M})$ has Talagrand type $(1,\psi_{1,1})$, where $\psi_{1,1}(t)=t/\log(e+t)$. As an consequence, the Schatten trace class $S_1$ has Talagrand type $(1,\psi_{1,1})$, thus settled an open problem of Question 2 in Cordero-Erausquin and Eskenazis (2023). Our proof is based on the noncommutative Mazur maps initiated by Ricard.

We develop a differential-geometric approach to contractive projections on complex Banach spaces based on Levi forms of plurisubharmonic functions. We show that if a projection $P$ contracts a $\mathrm{C}^2$ plurisubharmonic function $\Phi$, then, at any point of its range, the Levi form of $\Phi$ defines a tangent Hilbertian geometry for which $P$ becomes an orthogonal projection. When the range is finite-dimensional and the relevant Levi forms are nondegenerate, comparing the Levi geometries associated with different points leads naturally to positive relative operators and associated one-parameter groups. We apply this framework to tracial noncommutative $\mathrm{L}^4$-spaces. Exploiting the cubic form of the duality mapping, we characterize contractive projections by a Jordan triple orthogonality property and prove that a projection on $\mathrm{L}^4(\mathcal{M})$ is $2$-contractive if and only if it is completely contractive. When the ambient von Neumann algebra is finite-dimensional, we identify the relative operator associated with the left and right tangent forms on the range with a restriction of an inverse square root of a modular operator on a linking von Neumann algebra. This entails that the range of any $2$-contractive projection is completely isometric to a rectangular noncommutative $\mathrm{L}^4$-space associated with a $\mathrm{W}^*$-ternary ring of operators. As a consequence, every such projection is contractively decomposable. Finally, for every $1 < p < \infty$ with $p \neq 2$, we prove that every finite-rank $2$-contractive projection on a tracial noncommutative $\mathrm{L}^p$-space is completely contractive. For $p > 2$, we obtain a weighted ternary description of its range in terms of a finite-dimensional $\mathrm{W}^*$-$\mathrm{TRO}$.

We present a simple proof of the $\varepsilon$-Dvoretzky conjecture, which asserts that the dependence on the approximation parameter $\varepsilon$ in Dvoretzky's theorem is polynomial in $1/\varepsilon$. In particular, if $n \geq (C/\varepsilon)^{\ell/2+1}$, then any $n$-dimensional convex body has, through any given interior point, an $\ell$-dimensional section that is $\varepsilon$-close to a Euclidean ball. Here, $C > 0$ is a universal constant. We in fact obtain a sharper dependence on $\varepsilon$. The proof is probabilistic, but uses a different probabilistic model from those employed previously. We also prove a simultaneous version of our theorem for a finite family of convex bodies with the origin in their interior, yielding a common linear subspace on which all of the bodies have nearly-Euclidean sections. Finally, we compare the radius of the approximating Euclidean ball with familiar geometric parameters, such as the mean widths of the body and its dual.

We prove strict necessary density conditions for smooth coherent frames and Riesz sequences in the orbit of a connected, simply connected, nilpotent Lie group. This extends previous results for homogeneous groups to general nilpotent Lie groups. The main novelty in our approach is that we prove a stability result for coherent frames and Riesz sequences based on deforming their representation rather than their index set.

For a Banach space $X$, we study the completed direct limit of its even iterated duals along the canonical embeddings and the bounded inverse limit of the corresponding odd duals. We give explicit constructions, establish their duality relation, and describe the comparison maps associated with iterated dualization. We then examine several assertions in three preprints of Uglešić, giving counterexamples concerning dimension stability and the commutation of bidualization with direct limits. The finite quotient shape (the category whose objects are the finite-dimensional quotients, with all continuous linear maps between them) is classified by the algebraic dimension of the continuous dual. We also compute the dimensions of the finite iterated duals and the completed tower of $\ell^{\infty}(\mathbb{N})$, and prove that, inside the tower space, the canonical copy of a higher layer need not be the range of an $M$-projection. Throughout, uncompleted limits, Banach completions, and the different meanings of isomorphism are distinguished explicitly.

We introduce the skew Schäffer type constant as an infimal counterpart of the recently developed skew James type constant \cite{RaoWuLiuNiWang} and establish its basic geometric theory in Banach spaces. The new constant admits sharp universal bounds and characterizes uniform nonsquareness through a strict gap condition. Exact values are obtained in Hilbert and Lebesgue spaces, where an exact relation with the skew James type constant is established. We further derive a James-modulus estimate leading to sufficient conditions for uniform normal structure.

We construct hypocoercive Kolmogorov-type operators on the Euclidean motion group $\mathrm{SE}(3)=\mathbb{R}^3\rtimes\mathrm{SO}(3)$ using ZXZ Euler angles and study their long-time behaviour using the abstract hypocoercivity method. The operator $L=S-A$ decomposes into a symmetric part $S$, corresponding to a degenerate diffusion on the rotation component $\mathrm{SO}(3)$, and an antisymmetric part $A$, which captures transport and torque effects induced by the Lie group structure. The associated invariant probability measure is $\mu_\Phi=\gamma_\Phi\otimes\nu_3$, where $\gamma_\Phi$ is a probability measure on $\mathbb{R}^3$ with density $Z(\Phi)^{-1}e^{-\Phi}$ with respect to the Lebesgue measure, and $\nu_3$ is the normalized Haar measure on $\mathrm{SO}(3)$. As a consequence, if $\Phi\in C^\infty(\mathbb{R}^3)$ is bounded from below, $\gamma_\Phi$ satisfies a Poincaré inequality with constant $\Lambda>0$, and there exists $c<\infty$ such that \begin{equation*} |\nabla_\xi^2\Phi(\xi)| \leq c\bigl(1+|\nabla_\xi\Phi(\xi)|\bigr), \quad \xi\in\mathbb{R}^3, \end{equation*} then the semigroup generated by the closure of $L$ converges exponentially fast to equilibrium in $L^2(\mathrm{SE}(3),\mu_\Phi)$.

We study weighted bilinear Hardy-Steklov operators on the half-line for $1<p_1,p_2<\infty$ and $0<q<\infty$. The operator-theoretic core is an exact output-localisation principle. For a bounded bilinear map into $L^q$, compact middle truncations identify compactness with vanishing of a directed output-tail invariant and, with constant one, identify that invariant with the essential distance and a radius-based Hausdorff measure of noncompactness. Under the additional hypothesis of local finite-rank approximability, the same invariant also equals the finite-rank distance and governs the rank-approximation conclusions. We verify these hypotheses for the Hardy-Steklov map and then translate the tail invariant through the five weighted boundedness regions. In Regions $I,\mathrm{II}_1,\mathrm{II}_2,\mathrm{III}_1$, boundedness must be supplemented by lower- and upper-endpoint vanishing, and the essential distance is comparable to the larger endpoint defect. In the strict subcritical Region $\mathrm{III}_2$, boundedness alone implies compactness. This yields a sharp structural transition between the equality boundary $P=Q$ and the summable regime $P>Q$. The quasi-Banach range $0<q<1$ is treated throughout with the powered metric $d_q(f,g)=\|f-g\|_q^q$.

This paper provides a complete characterization of the spectrum, point spectrum, continuous spectrum, and residual spectrum of bounded convolution operators acting on the non-separable space $\sigma_\infty$, the dual of the Hahn sequence space. Because standard duality and separability-based techniques fail in this setting, a linear functional approach is employed with the fine spectral decomposition of the unilateral shift operator. Furthermore, as a major application of this methodology, we establish a complete description of the fine spectrum for operator polynomials acting on general sequence spaces without requiring separability. This result provides a unified framework that subsumes classical spectral determinations for general banded triangular matrices across various sequence spaces and enables the explicit computation of fine spectrum on non-separable spaces.

In their seminal work, Baum-Douglas-Taylor showed that under a certain compactness assumption, local boundary conditions for first-order elliptic operators over compact manifolds with boundary define cycles for relative K-homology. Using a systematic approach to regular boundary conditions, we generalize their result to elliptic operators of arbitrary order and to non-compact manifolds with potentially non-compact boundary. In addition, we show that under stronger regularity conditions, local boundary conditions even define cycles for absolute K-homology. In contrast to the relative classes, the resulting absolute K-homology classes are in general not independent of the choice of boundary condition.

Bhatt-Morrow-Scholze motivic filtration on topological Hochschild homology ($THH$) is extended to Azumaya algebras over quasisyntomic rings. We prove the Azumaya descent theorem: $THH$ of such an algebra is equivalent, as an underlying spectrum, to $THH$ of its center, normalized on $\pi_0$ by the reduced trace instead of the unit map. This comparison $\Phi_A$ is not shown to be étale-locally cyclotomic. But what is cyclotomic étale-locally is a separate Morita trace, agreeing with $\Phi_A$ only on $\pi_0$. The resulting transported filtration is exactly the center BMS filtration, well defined up to the automorphism ambiguity of $\Phi_A$. It is insensitive to the Brauer class, reflected instead by $K$-theoretic and module-categorical pairings $THH$ does not see. We establish the analogous statement for derived Azumaya algebras over connective $\mathbb{E}_\infty$-rings, under a connectivity hypothesis.

Let $C^1(I)$ denote the space of all continuously differentiable real-valued functions on the closed unit interval $I=[0,1]$. For $1\le p\le\infty$, we equip $C^1(I)$ with the norm \[ \Vert{f}\Vert_{(p)} := \begin{cases} \left(|f(0)|^p+\Vert{f'}\Vert_{I}^p\right)^{1/p}, & 1\le p<\infty, \\ \max\{|f(0)|,\Vert{f'}\Vert_{I}\}, & p=\infty, \end{cases} \qquad (f\in C^1(I)), \] where $\Vert{\cdot}\Vert_I$ denotes the supremum norm on $I$. Let $\mathcal{S}_p^+:=\{f\in C^1(I):f(0)\ge0,\,f'\ge0,\,\Vert{f}\Vert_{(p)}=1\}$ be the positive unit sphere of $C^1(I)$. A map $T:\mathcal{S}_p^+\to \mathcal{S}_p^+$ is called norm-additive if \[ \|T(f)+T(g)\|_{(p)}=\|f+g\|_{(p)} \qquad (f,g\in \mathcal{S}_p^+). \] For $1<p\le\infty$, we prove that every surjective norm-additive map $T\colon\mathcal{S}_p^+\to\mathcal{S}_p^+$ is of the form \[ T(f)(t) = f(0)+\int_0^t f'(\tau(s))\,ds \] for all $f\in\mathcal{S}_p^+$ and $t\in I$, where $\tau\colon I\to I$ is a homeomorphism. Conversely, every map of this form is a surjective norm-additive map. Consequently, every such map extends to a surjective linear isometry from $(C^1(I),\Vert{\cdot}\Vert_{(p)})$ onto itself. For $p=1$, however, we construct a surjective norm-additive map on $\mathcal{S}_1^+$ which is not of the above form and admits no linear extension to $C^1(I)$, showing that the assumption $p>1$ is essential.

Let $\eta_{1},\cdots,\eta_{d}$ be nonconstant one-variable inner functions satisfying $\eta_{i}(0)=0$ for all $1\leq i\leq d$. Denote by $M_{\eta}$ the submodule of the Hardy space $H^{2}(\mathbb{D}^{d})$ generated by the function family $\{\eta_{i}(z_{i})-\eta_{i+1}(z_{i+1})\mid 1\leq i\leq d-1\}$. For the corresponding quotient module $N_{\eta}=H^{2}(\mathbb{D}^{d})\ominus M_{\eta}$, we define the compressed multiplication operators $S_{z_{i}}=P_{N_{\eta}}M_{z_{i}}\vert_{N_{\eta}}$, where $P_{N_{\eta}}$ is the orthogonal projection from $H^{2}(\mathbb{D}^{d})$ onto $N_{\eta}$. In this paper, we characterize the Hilbert-Schmidt property of the self-commutators $[S_{z_{i}}^{\ast},S_{z_{i}}]$ and cross-commutators $[S_{z_{i}}^{\ast},S_{z_{j}}]$ with $i\neq j$. Moreover, explicit formulas for their Hilbert-Schmidt norms are established whenever these commutators are Hilbert-Schmidt.