Papers

We prove that the weak Haagerup constant is multiplicative with respect to direct products, namely $\Lambda_{\mathrm{WH}}(G \times H)=\Lambda_{\mathrm{WH}}(G) \Lambda_{\mathrm{WH}}(H)$ for any locally compact groups $G$ and $H$. Knudby previously proved the submultiplicative inequality and asked whether equality always holds.

We consider an initial value problem for a time-fractional evolution equation in a Hilbert space $X$: $$ \partial_t^{\alpha} (u(t)-a) = Au(t) \qquad \mbox{for $0<t<T$}, $$ where $\partial_t^{\alpha}$ denotes a fractional differential operator of Caputo type with order $0<\alpha<1,$ $u\colon (0,T) \to X,$ $a$ describes an initial value, and $A$, with domain $\mathcal{D}(A),$ is the generator of a contraction C$_0$ semigroup in $X$. We prove a fractional Hille-Yosida theorem characterizing the unique existence of a weak solution for every $a\in X$. We also discuss the unique existence of a strong solution for $a \in \mathcal{D}(A),$ as well as the continuity of weak solutions. Our results are applicable, for example, to time-fractional transport equations and Boltzmann equations. The proofs are mainly based on constructing an approximating sequence of solutions using the Yosida approximation.

In this paper, we study a numeration system with a non–integer base $a>1$ over the binary alphabet $A=\{0,1\}$: $$\left[0;\frac{r}{a-1}\right]\ni x=\sum\limits_{n=1}^{\infty}\frac{\alpha_n}{a^n} \equiv \Delta^{r_a}_{\alpha_1\alpha_2\ldots\alpha_n\ldots}, \quad \alpha_n\in A.$$ We investigate the geometry of $r_a$–representation of numbers, including the geometric interpretation of digits, the structure of cylinder overlaps, and the associated metric properties. The main object of our study is a nowhere monotone function of unbounded variation defined by $$f\left(x=\sum\limits_{n=1}^{\infty}\frac{\alpha_n}{2^n}\right) =\sum\limits_{n=1}^{\infty}\frac{\alpha_n}{a^n},$$ where the classical binary representation of the argument is assumed not to end with the infinite period $(1)$. We derive a system of two functional equations satisfied by $f$. We prove that the function has fractal level sets and a self–affine graph. We also evaluate the integral of $f$ over the interval $[0,1]$.

Let $\mathcal E$ be a separable Hilbert space, $B\in H^\infty_1(L(\mathcal E))$, and $\alpha>-1$. Associated with the Toeplitz operator $T_B$ on the vector-valued weighted Bergman space $A^2_{\alpha,\mathcal E}$ are the sub-Bergman spaces $\mathcal A_\alpha(B)$ and $\mathcal A_\alpha(B^*)$. In this paper, we prove that \[ \mathcal A_\alpha(B)\supset A^2_{\alpha-1,\mathcal H(B(0))} \quad\text{and}\quad \mathcal A_\alpha(B^*)\supset A^2_{\alpha-1,\mathcal H(B(0)^*)}. \] This confirms a conjecture of \cite{GLM26}. We also construct an infinite-dimensional two-sided inner function $B$ such that $\|B(0)\|<1$ and \[ \mathcal A_\alpha(B)\approx \mathcal A_\alpha(B^*)\approx A^2_{\alpha-1,\mathcal E}, \] but $B$ is not a finite Blaschke-Potapov product. This example shows that the finite-dimensional rigidity theorem of \cite{GLM26} does not extend to infinite-dimensional coefficient spaces.

We investigate two natural formulations of equal-length Littlewood–Paley–Rubio de Francia (LPR) inequalities on arbitrary Vilenkin systems \(\Gm\). The canonical formulation uses canonical intervals in \(\mathbb N\) of equal cardinality, while Vilenkin's formulation uses translates of a common initial interval under the group operation on the dual of \(\Gm\). In this paper, our main results show that the definitions of these two types of intervals exhibit dramatically different properties. More precisely, for the canonical formulation, we construct a counterexample showing that, for every \(2<p<\infty\), the Schatten class \(S^p\) fails the equal-length LPR property, in sharp contrast to the corresponding result on the torus; for Vilenkin's formulation, we prove that, for every \(2\le p<\infty\), a Banach space \(X\) has the equal-length LPR property if and only if it is UMD and has type~\(2\). The sufficiency proof combines a transference argument from the torus to finite cyclic groups with an embedding of Vilenkin intervals into digital rectangular envelopes, yielding bounds independent of the generating sequence.

We introduce a deep kernel hedging framework that combines the flexibility of deep learning with the structural inductive bias of kernel methods. The hedging functional is restricted to a reproducing kernel Hilbert space whose kernel is parameterized through a neural network embedding of the input features. The framework minimizes a regularized empirical risk under convex loss functions and can accommodate path-dependent information through truncated time-augmented signature features. We derive a generalized representer theorem for the joint hedging problem, reducing the empirical optimization to a finite-dimensional problem. To further reduce the computational cost associated with large kernel matrices, we develop a scalable random Fourier feature approximation and establish convergence guarantees. The random Fourier parameters are sampled once and remain fixed throughout training, while the deep kernel adapts to market data through the learned neural representation. We evaluate the performance of the proposed deep kernel approach on both synthetic and real data and compare it with standard kernel methods and classical deep hedging architectures. Numerical results indicate competitive and robust hedging performance, particularly in low-data regimes, which highlights the benefits of combining expressive neural representations with the inductive bias of kernel methods.

In this paper, we introduce and investigate the notion of phi-semi contractibility for Banach algebras. We establish a complete characterization of phi-semi contractibility through the existence of specific elements satisfying particular algebraic conditions. We thoroughly examine the hereditary properties of this concept, including its behavior under closed ideals, continuous homomorphisms with dense range, and projective tensor products. Notably, we prove that the projective tensor product A tensor B is phi tensor psi-semi contractible precisely when both A and B are semi contractible with respect to their respective characters, where phi belongs to the character space of A and psi belongs to the character space of B. Furthermore, we introduce and analyze the approximate version of phi-semi contractibility, providing equivalent characterizations.

For $\alpha\in(0,n)$, $p\in(1,\frac{n}{\alpha})$, $q:=\frac{np}{n-\alpha p}$, and $W\in\mathscr A_{p,q}$, we obtain the following matrix-weighted boundedness of the fractional integral $I_\alpha$: for any $\vec{f}\in L^p(W^p)$, \begin{align*} \left\|I_\alpha\vec f\right\|_{L^q(W^q)} \lesssim[W]_{\mathscr A_{p,q}}^{ (1-\frac{\alpha}{n})\max\{1,\frac{p'}{q}\}} \left\|\vec f\right\|_{L^p(W^p)}, \end{align*} where the implicit positive constant is independent of $W$ and $\vec{f}$. For integer $n\geq2$, $p\in[1,n)$, $q:=\frac{np}{n-p}$, and $W\in\mathscr A_{p,q}$, we establish the following matrix-weighted Sobolev inequality: for any smooth $\mathbb{C}^d$-valued function $\vec{f}$ with compact support, \begin{align*} \left\|\vec{f}\right\|_{L^{q}(W^q)} \lesssim[W]_{\mathscr A_{p,q}}^{\frac{n-1}{n}} \left\|WD\vec{f}\right\|_{L^p(\mathbb R^n,\mathbb C^{d\times n})}, \end{align*} where the implicit positive constant is independent of $W$ and $\vec{f}$ and where $D\vec{f}$ is the Jacobian matrix of $\vec{f}$. Surprisingly, in both bounds above the exponents of $[W]_{\mathscr A_{p,q}}$ coincide with the corresponding known optimal scalar exponents, and hence are optimal, which is different from the Calderón–Zygmund operator case. A key idea to obtain the above matrix-weighted boundedness of the fractional integrals is to establish the off-diagonal chain-packing principle of dyadic cubes, which is inspired by a recent work of A. K. Lerner.

We study the Riesz transform $\mathcal{R}=\partial(-\Delta)^{-\frac{1}{2}}$ on uniform local trees, metric measure spaces that are locally real trees and whose canonical Dirichlet form is built from weak derivatives along the skeleton. The reference measure $m$ may be singular with respect to the length measure $\nu$, so boundedness of $\mathcal R$ is understood from $L^{p}(m)$ to $L^{p}(\nu)$. In contrast with fractal-like manifolds and cable systems, the diffusion is sub-Gaussian at small scales and Gaussian at large scales. Under uniform volume growth, two-sided heat kernel estimates and a pointwise gradient estimate for the heat kernel, we prove that a local Dini condition on the scale function implies boundedness of $\mathcal{R}$ on $L^{p}$ for every $p\in[2,\infty)$, and hence the reverse Riesz inequality for every $p\in(1,2]$. Conversely, boundedness of $\mathcal{R}$ for some $p<2$, or a reverse Riesz inequality for some $p>2$, forces the space to be one-dimensional at small scales. We show that a reverse Hölder inequality for harmonic functions yields the gradient estimate, and verify all hypotheses for spaces carrying a geometric group action whose generators have bounded displacement. As an application, for the alternating Vicsek fractafold in $\mathbb Z^{d}$ we determine the exact ranges of $p$ for which the Riesz and reverse Riesz inequalities hold.

We study $\mathcal F$-transitivity of backward shifts on finite- and infinite-width directed lattice graphs. For the unilateral and bilateral finite-width lattices on weighted $\ell^p$- or $c_0$-spaces, we obtain exact weight characterizations of $\mathcal F$- and $\widetilde{\mathcal F}$-transitivity for an arbitrary Furstenberg family $\mathcal F$. If $\mathcal F$ is finitely invariant, these conditions also characterize topological $\mathcal F$-recurrence. For the infinite quadrant lattice with radial weights on weighted $\ell^2$-spaces, a finite-dimensional reduction and sharp smallest-singular-value estimates for Pascal-type transfer matrices yield equivalent characterizations of $\mathcal F$-transitivity, hypercyclicity and mixing. This provides a partial answer, in the radial $\ell^2$-setting, to the open problem posed by Baranov, Lishanskii and Papathanasiou concerning hypercyclicity on the infinite lattice graph. Examples at the critical exponential growth rate illustrate the finer dynamical information provided by our criteria.

We study the weighted Karcher mean on the Lie group $\operatorname{UT}_n$ of upper triangular matrices with positive diagonal entries. Using a divided-difference description of analytic functions on upper triangular matrices and the Parlett recurrence, we prove the existence, uniqueness, and analyticity of the weighted Karcher mean and derive recursive formulas for its computation. The theory yields explicit expressions in low dimensions and formulas for weighted geometric means on $\operatorname{UT}_n$. More generally, the results reveal a close relationship between divided differences, analytic matrix functions, and nonlinear matrix means on triangular Lie groups.

We study the weighted bilinear Hardy inequality for the same-direction product $H_I f H_I g$ on an arbitrary real interval $I$, with $0<q<\infty$, $1\le p_1,p_2\le\infty$, and measurable weights allowed to take the values $0$ and $+\infty$. The characterisation is formulated through directed compact restrictions $J\Subset I$. On each compact interval, endpoint-safe input profiles retain the $p_i=1$ boundary contribution as a Stieltjes atom, while the lower-triangle reduction separates a closed Hardy part from a strict Copson part so that common atoms are counted exactly once. Upper and mixed regimes follow from exact freezing and compact linear Hardy estimates. The lower regimes are obtained by power lifting and a measure-valued Hardy-Copson theorem. After simultaneous regularisation of the weights, the compact optimal constant is equivalent, with exponent-only constants, to an operational local characteristic, and the global constant satisfies $C_I\asymp\sup_{J\Subset I} A_J^{\mathrm{op}}$. In the regular interior range, the resulting conditions recover the classical $A_1,\ldots,A_7$ characterisations.

We investigate the von Neumann–Jordan constant of Banach tensor products equipped with the injective and projective tensor norms. We first establish lower estimates in terms of the geometric properties of the factor spaces and show that, whenever both factors are infinite-dimensional, the von Neumann–Jordan constants of both canonical tensor products attain the maximal value \(2\). For finite-dimensional factors, we obtain explicit results for classical sequence spaces and characterize the extremal case through operator-space and orthogonal-contact conditions. We further derive dimension-sensitive estimates based on Euclidean sections. Finally, we study the constant under equivalent renormings and determine its exact renorming envelope for injective and projective tensor products. As consequences, we characterize when these tensor products are superreflexive and when they are weak Hilbert spaces. These results clarify the influence of tensor norms, dimension, and renorming on the von Neumann–Jordan geometry of Banach tensor products.

For a fixed modulus h and residue class c, we study the logarithmic mass of primes p in a fixed power band for which the exponent of p in n! is congruent to c modulo h. We establish the asymptotic ((b-a)/h + o(1)) log(n!) for primes satisfying n^a < p <= n^b, where 0 <= a < b <= 1. The argument combines a fixed-order derivative estimate, Vaughan's identity, and finite Fourier approximation of a discontinuous valuation function. We also derive a necessary capacity condition for infinite families of solutions to polynomial-factorial equations P(x) = n!.

We show that the natural tensor product multiplication of two Archimedean almost \(f\)-algebras does not, in general, extend to their Fremlin tensor product. The two algebras in the counterexample have the same underlying vector lattice \(C[0,1]\). Their multiplications are explicit, positive and nilpotent of index three. The obstruction is the fact that the function \((s,t)\mapsto e^{st}\) does not belong to \(C[0,1]\ftensor C[0,1]\). The same construction gives Banach almost \(f\)-algebras and has consequences for previously published assertions on Riesz and Fremlin projective tensor products of almost \(f\)-algebras.

We present an expository excursus on modulation spaces, from Feichtinger's original construction on locally compact abelian groups to recent symplectic formulations. We trace the same local-global principle through bounded uniform partitions of unity, the short-time Fourier transform, coorbit methods, and Gabor coefficients, while distinguishing equivalent descriptions from the weighted, quasi-Banach, and Gelfand-Shilov extensions of the underlying functional setting. We also discuss Gabor matrices, convolution and embedding estimates, and an application to a nonlinear dispersive equation. The final part is devoted to metaplectic Wigner distributions $W_{\mathcal A}$. In the shift-invertible case these representations are, up to chirps and a linear change of phase-space variables, rescaled short-time Fourier transforms. This structural fact explains both their characterization of modulation and Wiener amalgam spaces and their discretization by metaplectic Gabor frames, and places the recent symplectic theory naturally within Feichtinger's modulation-space framework.

We establish Bourgain–Brezis selection maps for Hodge systems in new ranges of Besov and Triebel–Lizorkin spaces. Our approach combines the trace-free Beurling–Ahlfors transform with a nonlinear duality method. To control concentrations, we use tools developed by Stolyarov in the study of Maz'ya's $\Phi$-inequalities. These methods yield selection maps in Besov spaces $\dot B^{n/p}_{p,q}(\mathbb{R}^n)$ for every $1\leq q\leq p\leq 2$, and in Triebel–Lizorkin spaces $\dot F^{n/p}_{p,q}(\mathbb{R}^n)$ for every $1<p\leq q\leq 2$. The theorems hold for every dimension $n\geq 2$ and every form degree $1\leq l\leq n-1$. Although our results overlap with some earlier work, they considerably extend the known range of solutions to the Bourgain–Brezis problem and underscore the importance of the trace-free Beurling–Ahlfors transform in its resolution.

We study Riesz and reverse Riesz inequalities for the fractional powers of the Laplacian $\Delta$ on the unbounded Vicsek set. The space carries its Hausdorff measure $m$. The gradient $\partial$ is a weak gradient defined on the skeleton, and it is measured with respect to a length measure $\nu$ that is singular with respect to $m$. The relevant fractional order for the Riesz inequalities is not $1/2$ but $\gamma_{p}=\frac{d_f+p-1}{pd_w}$, where $d_f=\log_{3}5$ and $d_w=d_f+1$. For $p\in[1,2]$, we prove the weak-type reverse Riesz inequality $\|(-\Delta)^{\gamma_p}f\|_{L^{p,\infty}(m)}\lesssim\|\partial f\|_{L^p(\nu)}$. The corresponding strong estimate fails for $1\le p<2$, whereas the weak estimate fails for $2<p<\infty$, even after heat regularization. For the Riesz transform $\mathcal{R}_p=\partial(-\Delta)^{-\gamma_p}$, we show that $\mathcal{R}_1$ is not of weak type $(1,1)$ but that $\mathcal{R}_p$ is bounded from $L^p(m)$ to $L^{p,\infty}(\nu)$ for $p\in(1,2)$, and from $L^{p,1}(m)$ to $L^p(\nu)$ for $p\in(2,\infty)$. However, if $p \ge 1$, $\mathcal{R}_p$ is bounded from $L^p(m)$ to $L^p(\nu)$ only for $p=2$.

In this paper, we study the Fisher–KPP equation \[ \partial_t u = \alpha\Delta_{\TT_{q+1}}u+\beta u(1-u), \] on the homogeneous \((q+1)\)-regular tree \(\TT_{q+1}\), with \(q\geq2\), for nontrivial, compactly supported radial initial data. We first obtain an explicit representation of the fundamental solution of the corresponding linearized problem, together with sharp two-sided pointwise estimates. These estimates identify the propagation threshold \[ \beta>\alpha(\sqrt q-1)^2 \] and determine the associated critical speed \(c_\ast\) and decay rate \(\lambda_\ast\). In this regime, we prove that, for every fixed \(\theta\in(0,1)\), the outermost \(\theta\)-level set satisfies \[ \kappa_\theta(t) = c_\ast t-\frac{3}{2\lambda_\ast}\log t+O(1), \qquad t\to\infty. \] Therefore, while the branching geometry affects the threshold for propagation and the values of the critical parameters, the classical Bramson factor \(3/2\) persists. Our results therefore extend classical propagation and logarithmic-delay phenomena, previously known for one-dimensional continuous and discrete Fisher–KPP models, to homogeneous trees.

Given a reproducing kernel Hilbert space $\mathcal H$ and a bounded operator $T$ on $\mathcal H$, we say that $T$ obeys the strong reproducing kernel thesis (SRKT) if $\|T\|=\sup_k\|Tk\|$, where $k$ ranges over the normalized reproducing kernels in $\mathcal H$. Two problems are posed. One of these is to determine which Hankel operators on the Hardy space $H^2$ obey the SRKT. The other concerns the geometry of the unit ball of $BMO$ with respect to the Garsia norm.

We study Schatten embeddings through circular means of trace powers. In finite dimensions, a decomposition by singular-value vanishing orders shows that every term of order at most two has a nonnegative coefficient. This yields the classification $\ell_q^2(\C)\hookrightarrow\Sp_p^n$ if and only if $q=p$ or $q=2$, for $n\ge2$ and $p<\infty$. In particular, it gives negative answers to parts (ii), (iii), and (iv) of Problem 3.1 in Chattopadhyay–Pradhan–Skripka, arXiv:2603.07359v2. A regularized curvature estimate also excludes $\ell_q^2$ from infinite-dimensional $\Sp_p$ for $0<p\le2<q\le\infty$ and gives finite-rank distortion bounds. The other parts of that problem are not claimed to be solved.Our results extend previous studies on Schatten embeddings, particularly those in \cite{CHPPR, CHPR, CPS}, by developing a complex-curvature framework based on complex convexity theories \cite{BR,DGT}, which resolves several remaining quasi-Banach cases and yields new geometric obstructions for isometric embeddings.

We introduce two James-type constants associated with approximate isosceles and approximate Birkhoff–James orthogonality in real normed spaces. The isosceles family coincides with the classical James constant of Gao and Lau [J. Austral. Math. Soc. Ser. A 48 (1990), 101–112], while the Birkhoff family extends the orthogonal James constant of Baronti and Papini [Constr. Math. Anal. 5 (2022), no.~1, 37–45]. We show that approximate orthogonality constraints can, in certain settings, recover classical geometric constants exactly. In particular, the approximate isosceles family collapses to the classical James constant, while the approximate Birkhoff–James family forms a genuine parameter-dependent interpolation between the orthogonal and classical James constants.We also compare the Birkhoff profile with several classical geometric constants and study how it changes with the parameter.Several examples are also given to illustrate the relation between geometric constants and approximate orthogonality.

In 2002 Nielsen and Zhou conjectured that the shifted wavelet packet system associated with a Daubechies filter of length at least four fails to be a Schauder basis of $L^p(\mathbb{R})$ for $1\le p\le\infty$, $p\ne2$, and that the packets are not uniformly bounded in $p$-mean across scales for any $p>2$. The basis obstruction throughout the range $1<p<\infty$, $p\ne2$, was previously established for length four, via an explicit computation; the full $p$-mean assertion remained open. We prove both for every Daubechies filter of length at least four: the high-pass transition matrices satisfy $\rho_p\rho_{p'}>2$ for $1<p<\infty$, $p\ne2$, and for $p>2$ the full family of transition matrices has $p$-norm joint spectral radius exceeding the Haar value $4^{1/p}$. The key step is structural: an affine piece of the matrix pressure forces constant spectral radius, which by a theorem of Protasov and Voynov forces simultaneous orthogonality of the transition matrices, incompatible with the double zero of the high-pass symbol. The results extend to every real spectral factor with the Daubechies magnitude response, including the least-asymmetric filters.

We characterize the continuous mappings of the unit disk that preserve every Carleson interpolating sequence. Preservation in one direction is equivalent to being a disk homeomorphism whose inverse is uniformly continuous in the pseudohyperbolic metric and whose associated weighted pushforward is bounded on positive Carleson measures. Preservation in both directions is equivalent to being a disk homeomorphism that, together with its inverse, is uniformly continuous in that metric and has a strongly quasisymmetric boundary extension. Within this class, comparability of the boundary defects is equivalent to a bi-Lipschitz boundary map. Building on the canonical factorization established in our earlier work, we give explicit criteria for membership in the continuous boundary kernel. For a homeomorphism of the closed disk with identity boundary values and a uniform boundary expansion of first order, kernel membership is equivalent to positivity of the determinant of the real boundary differential, or equivalently of the inward normal derivative of the boundary defect. No differentiability in the interior is required. The $C^1$ criterion on the closed disk follows as a corollary, while a counterexample shows that pointwise boundary differentiability does not suffice. We also characterize factorizations with a quasiconformal kernel and record the exact maximal dilatation of its radial factor.

We prove two dimension-free estimates in Gaussian space. The first is an optimal Meyer-type inequality for the Gaussian divergence: for $p\geq2$, \[ \|\delta V\|_{L^p(\gamma_d)} \leq \sqrt p\,|\mathbb E V| +Cp\,\|DV\|_{L^p(\gamma_d;\mathrm{HS}_d)}\, . \] Here $\mathrm{HS}_d$ is the space of $d\times d$ matrices with its Hilbert-Schmidt norm. Its main ingredient is a dimension-free weak-type $(1,1)$ bound for the second-order transform $D^2\mathcal N^{-1}$, where $\mathcal N := -\Delta+x\cdot D$ is the Ornstein-Uhlenbeck operator. We also include a direct change-of-variables proof of the weaker bounded derivative divergence inequality in the appendix. The second result is a Gaussian Bernstein-Markov inequality. If $P$ is a polynomial, $p\geq2$, and $m_p := (\mathbb E|g|^p)^{1/p}$ for a standard Gaussian $g$, then \[ \|DP\|_{L^p(\gamma_d;\mathbb R^d)} \leq\frac{2\sqrt{2e\, \mathrm{deg}(P)}}{m_p}\,\|P\|_{L^p(\gamma_d)}\, . \] The factor $2$ is unnecessary when $p$ is even, or $P$ is even or odd. Thus, the constant is of order $\sqrt{\mathrm{deg}(P)/p}$ throughout the range $p\geq2$.

We establish a dimension-free vector-valued Poincaré-type inequality on the discrete cube for Banach spaces with Rademacher type $p$, $1\le p\le2$. As a consequence, we obtain $$ T_p^R(X)\le T_p^E(X)\le p\,T_p^R(X). $$ This gives an alternative proof that Rademacher type and Enflo type coincide, while improving the constant $\pi/\sqrt{2}$ in the quantitative estimate of Ivanisvili, van Handel, and Volberg \cite{IVV} to $p$.

Little is known about the dynamics of Toeplitz operators on the Bergman space, and even in the case of harmonic symbols, few results are available. In this paper, we study the dynamical properties of Toeplitz operators with symbols of the form $a \bar{z} + p (z)$ on the Bergman space, where $a\neq 0$ and $p$ is analytic on the closed unit disk. We obtain some necessary conditions and some sufficient conditions for such operators to be hypercyclic, weakly mixing, mixing, chaotic, or frequently hypercyclic. As an application, we obtain a complete characterization of hypercyclicity for Toeplitz operators with harmonic linear polynomial symbols. We also show that Toeplitz operators with the same symbol may exhibit different hypercyclic behavior on the Hardy and Bergman spaces.

We establish a quaternionic Hansen–Pedersen inequality for real operator-convex functions on right quaternionic Hilbert spaces. The proof is based on the continuous real functional calculus for selfadjoint quaternionic operators, a direct-sum functional-calculus identity, and a block-unitary construction. As an application, let \(T = U|T|\) be an injective \(p\)-hyponormal operator. We prove that, for \(0 < r \leq 1/2\), the transform \(S_{r}(T) = U|T|^{r}U\) is \(2p\)-hyponormal when \(0 < p \leq 1/2\) and hyponormal when \(1/2 < p \leq 1\). We also show that \(S_{r}(T)\) preserves log-hyponormality for \(r > 0\). These results illustrate the use of quaternionic Jensen-type operator inequalities in the study of transforms associated with the polar decomposition.

The least regular norm of a factorization of antipodal differences through sign differences on \((\mathbb Z/m\mathbb Z)^n\) is \(m/2\) when \(4\mid m\). On these tori, when \(2\sqrt n\le m\le4\sqrt n\), normalized divergence equations admit selections with pointwise norm bound \(3/2\), while the least linear lifting norm and global Lipschitz selection constant equal \(\sqrt n/2\). Higher-moment lifting bounds follow from a quoted sharp \(L_1\) inequality.

Let $m\geq1$ be fixed. We consider the space $\mathcal{P}_m(\mathbb{R}^n)$ of real $m$-homogeneous polynomials on $\mathbb{R}^n$, endowed with the standard Gaussian measure $\gamma_{m,n}$ associated with the Bombieri norm. We study how well the norm of a typical polynomial on the unit ball of $\ell_\infty^n$, namely the cube $[-1,1]^n$, can be recovered from its values at the vertices. For $P\in\mathcal{P}_m(\mathbb{R}^n)$, set \[ M(P)=\max_{x\in[-1,1]^n}|P(x)|, \qquad V(P)=\max_{\varepsilon\in\{-1,1\}^n}|P(\varepsilon)|. \] If $P_n$ is chosen according to $\gamma_{m,n}$, we prove that the relative loss \[ 1-\frac{V(P_n)}{M(P_n)} \] is of order at most $n^{-1/2}$ in probability. Consequently, for every $0<\beta<1/2$, \[ \gamma_{m,n}\left\{ P\in\mathcal{P}_m(\mathbb{R}^n): V(P)\geq (1-n^{-\beta})M(P) \right\} \longrightarrow1 \] as $n\to\infty$. Thus, with respect to the Bombieri Gaussian measure, the vertices of the cube are asymptotically norming for homogeneous polynomials of fixed degree.