Papers

Let $I_{\inf}(a):=\lim_{n\to\infty}I_{n}(a)=\inf_{n\ge1}I_{n}(a)$ be the asymptotic isoperimetric profile for product of a weighted Riemannian manifold satisfying $\mathrm{CD}(0,\infty)$ and, more broadly, for product of a nonsmooth subspace with density that can be approximated by a sequence of densities satisfying $\mathrm{CD}(0,\infty)$. We establish the following identity: \[ \lim_{a\downarrow0}\frac{I_{\inf}(a)}{a\sqrt{2\ln(1/a)}}=\sqrt{K_{\mathrm{LS}}}, \] where $K_{\mathrm{LS}}$ is the optimal log-Sobolev constant. The same argument also yields the large-deviation and moderate-deviation asymptotics for isoperimetry.

We study the nonlinear evolution equation $$ \partial_t u+(G+G^\delta)u=I_\alpha(|u|^\rho) \quad\text{on }\mathbb{R}^{N+k}, $$ where $G$ is the nonnegative self adjoint realization of Grushin operator, $0<\delta<1$, and $I_\alpha$ is the potential operator with kernel $|z|^{-\alpha}$, $0<\alpha<N+k$. We develop a well-posedness theory for initial data in Marcinkiewicz spaces, allowing singular profiles outside the corresponding Lebesgue spaces. Using spectral calculus, subordination, and interpolation, we establish Lebesgue and Lorentz smoothing estimates for the mixed semigroup $e^{-t(G+G^\delta)}$, which distinguish the second-order behaviour at short times from the fractional decay at large times. In the subcritical regime, considering $$ \frac1\beta=\frac2Q+1-\frac{\alpha}{d}, \qquad \frac1{\beta_\delta}=\frac{2\delta}{Q}+1-\frac{\alpha}{d}, $$ where $Q$ is homogenous dimension, we prove local existence, uniqueness, Lipschitz dependence on the initial data, and a blow-up alternative in $L^{p,\infty}$ when $$ \rho<1+\frac{p}{\beta}. $$ At the critical relation $\rho=1+p/\beta$, a Yamazaki-type estimate and Lorentz duality yield global mild solutions for sufficiently small data in $L^{p,\infty}$. In the supercritical regime $\rho>1+p/\beta$, well-posedness is recovered in higher-integrability Marcinkiewicz spaces $L^{q,\infty}$ satisfying $$ \beta(\rho-1)<q<\beta_\delta(\rho-1), $$ with global existence for sufficiently small initial data. In all cases, the solutions attain their initial data in the weak-$*$ sense. These results extend the Marcinkiewicz-space theory to mixed Grushin diffusion with a spatially nonlocal potential source and reveal the role of the two competing diffusion scales in determining the admissible integrability regimes.

In this paper, we focus on the boundedness and compactness of the Toeplitz operators and the Hankel operators on the $\mathcal{D}$-weighted Bergman spaces $A_{\omega}^p$ ($1\leq p<\infty$). In particular, the boundedness of Toeplitz operators with $\mathrm{BMO}_\omega^p$-symbols on $A_{\omega}^p$ is characterized in terms of the Berezin-type transform. Several sufficient conditions for the Toeplitz operators $T_f^\omega$ with locally integrable symbols to be bounded (resp. compact) for each $1\leq p<\infty$ are presented. Moreover, sufficient and necessary conditions for the boundedness of the Toeplitz operator $T_{\bar{f}}^\omega:A_\omega^1\rightarrow A_\omega^1$ and the Hankel operator $H_{\bar{f}}^\omega:A_\omega^1\rightarrow L_\omega^1$ with co-analytic symbol are established.

Fofana spaces, introduced on the basis of Wiener amalgam spaces, have attracted some interest recently and have found applications in the field of partial differential equations. In this paper, we study the normed Köthe structure of the predual spaces $\mathcal{H}(p^{\prime}(\cdot),q^{\prime},\alpha^{\prime})$ of Fofana spaces with variable exponent $(L^{p(\cdot)}, \ell^{q})^{\alpha}$, where $1\leq p(\cdot) \leq \alpha \leq q\leq \infty$. As an application of this investigation, we identify the Köthe duals and Köthe biduals of these two spaces, and we obtain a predual of $\mathcal{H}(p^{\prime}(\cdot),q^{\prime},\alpha^{\prime})$. Finally, we prove that neither $(L^{p(\cdot)}, \ell^{q})^{\alpha}$ nor $\mathcal{H}(p^{\prime}(\cdot),q^{\prime},\alpha^{\prime})$ is reflexive.

We construct a one-parameter family $(X_c)_{0<c<1}$ of real reflexive Banach spaces, isomorphic to $\ell_2$ and satisfying $d_{\mathrm{BM}}(X_c,\ell_2)\le c^{-1}$, such that $(X_c,X_c)$ fails the weak maximizing property while $(X_c^*,X_c^*)$ has it. The failure of the WMP for $X_c$ is witnessed by a diagonal operator which does not attain its norm but admits a non-weakly null maximizing sequence. On the dual side, a separated-block estimate yields, after dualization, a reverse Pythagorean inequality and a quantitative form of the Opial property; together with property $(M)$, this gives the weak maximizing property for $X_c^*$. We also show that the compact perturbation property is self-dual for reflexive pairs. Since the weak maximizing property implies the compact perturbation property, each $(X_c,X_c)$ has the compact perturbation property. Consequently, the compact perturbation property does not imply the weak maximizing property even arbitrarily close to Hilbert space, and the weak maximizing property is not stable under duality in the reflexive setting.

The Asplund property plays an important role in Banach space theory due to its connections with differentiability properties of continuous convex functions, optimization problems, and the weak topology of Banach spaces. Motivated by the variety of nonequivalent definitions proposed in the literature for locally convex spaces, in this survey we provide a unified framework for studying the Asplund property beyond the Banach space setting. The main object of our study is the locally convex space of continuous functions $C_k(X)$ endowed with the compact-open topology, where $X$ is an arbitrary Tychonoff space. We completely characterize the Asplund property for $C_k(X)$ in terms of topological properties of the underlying space $X$. As an essential step, we revisit the proof of several classical results, including the Namioka–Phelps theorem. Our approach is independent of differentiability techniques and relies solely on topological methods. The exposition is self-contained, and all major results are provided with complete proofs, making the paper accessible to both specialists and newcomers.

Comparing probability measures in machine learning trades transport geometry against computational cost: Wasserstein distances encode the geometry of $\mathbb R^d$ but require solving a transport problem, while kernel discrepancies are cheap to evaluate yet depend delicately on their test class. We study the Toscani–Fourier family $\mathrm T_{s,p}$, the weighted $L^p$ norm of the difference of two characteristic functions, as a continuous Fourier-side discrepancy on $\mathbb R^d$. For $1\le p<\infty$ we show that $d/p<s<1+d/p$ is exactly the window in which $\mathrm T_{s,p}$ is finite on $\mathcal P_p(\mathbb R^d)$, both endpoints already failing for a pair of Dirac measures, and we establish the metric, embedding, and compactness structure of the resulting space, which we prove to be complete. Duality identifies $\mathrm T_{s,p}$ as an integral probability metric over a homogeneous Fourier–Lebesgue ball, with an explicit extremizer when $1<p<\infty$. We prove the global bound $\mathrm T_{s,p}\lesssim W_p^{\,s-d/p}$, whose exponent is sharp, show that no global converse of any form can hold, and recover topological equivalence with $W_p$ on bounded-support and uniform-tail classes, together with explicit reverse moduli on bounded-support classes that improve the imported energy-kernel exponent at $p=2$. There, $\mathrm T_{s,2}$ is a constant multiple of the classical energy distance, which yields an exact finite-sample identity for the mean of the empirical discrepancy; the numerical experiments are otherwise diagnostic.

We investigate the operator distance between the Identity and the Copson operator on the cone of nonnegative, nonincreasing sequences within the Lebesgue sequence spaces $\ell^p(\mathbb{N})$. By proving the exact value of this distance, we confirm a conjecture originally formulated by A. Ben Said, S. Boza, and J. Soria in “The norm of Hardy-type oscillation operators in the discrete and continuous settings” (Analysis and Applications, 2025).

Recent years have seen significant interest in finding equivalent norms on Cesàro function spaces together with their optimal equivalence constants. In this work, we continue this line of research by deriving sharp two-sided inequalities relating the $L^p$-norms of $\mathcal{C}_m f$ and $\mathcal{C}_n f$, where $\mathcal{C}_k$ is the generalized Cesàro operator of order $k \in \mathbb{N}$ acting on non-negative measurable functions $f$ on $(0,\infty)$. A parallel study is also conducted for the corresponding adjoint operators.

In this paper, we study the characterizations of Schatten \(p\)-class Toeplitz operators on smoothly bounded convex domains of finite type in \(\C^n\). On one hand, by using discrete Kobayashi lattice, we characterize the Schatten \(p\)-class Toeplitz operators by employing a probabilistic selection method for \(0<p<1\). At the same time, we show that the characterizations obtained by Kobayashi lattice are equivalent to the Berezin transform for \(p > \frac{n}{n+1}\). On the other hand, as an application, the criterion gives a geometric characterization for Schatten \(p\)-class weighted composition operators for \(0<p<2\) and also gives an analytic characterization for \(p>\frac{2n}{n+1}\). Our main results give an answer to the question that find a geometric or analytic characterization of the Schatten \(p\)-class weighted composition operators whenever \(0<p<2\) raised by Xiao-Yang-Yuan \cite{Xiao2026}.

In this paper, we continue to investigate Campanato spaces via semigroups on von Neumann algebras. One of the main results is their coincidence with Lipschitz spaces for all regularity indices $\alpha > 0$ and for all analytic semigroups on von Neumann algebras not necessarily being finite; the desired self-improving property of Campanato spaces has also been verified. As a consequence, we demonstrate that the column Campanato spaces are isomorphic to the row ones for all $\alpha > 0$. This not only removes several restrictions in the previous work \cite{HJ24} by the first two authors, but also extends the previous results to any analytic semigroup, and thus resolves several problems left open in \cite{HJ24}. Both the results and the proof are new even in the commutative setting.

For all $0< s \le 1$, we provide a new deterministic construction of Cantor sets whose Fourier dimension and Hausdorff dimension are both equal to $s$. The construction is based on a straightforward Cantor-Moran construction with contraction ratios given by reciprocals of integers. The key tool to obtain the fast Fourier decay is due to the Weil bound in analytic number theory. Furthermore, we show that the natural equal-weighted Cantor-Moran measure is the desired measure admitting the near optimal Fourier decay and the measure admits an exponential orthonormal basis $\{e^{2\pi i \lambda x}: \lambda\in \Lambda\}$ for its $L^2$ space. This gives the first examples of singular Salem spectral measures in ${\mathbb R}^1$.

Let \(\Gm=\prod_{k\ge0}\mathbb Z_{m_k}\) be a Vilenkin group that is not necessarily bounded, i.e., \(\sup_k m_k=\infty\). We prove that, for every UMD Banach space \(X\) and every \(1<p<\infty\), the Vilenkin partial-sum operators are uniformly bounded on \(L^p(\Gm;X)\), with a bound depending only on \(p\) and the UMD constant of \(X\), and not on \(\mathbf m\). This resolves an open problem arising from the work of Clément et al.~\cite{ClementDePagterSukochevWitvliet2000} and later recorded explicitly in the book of Hytönen et al.~\cite[p.~362]{HNVWI}. The proof reduces the partial-sum estimate, via a Paley conjugation identity and a tangent-sequence decoupling argument, to a decoupling inequality for Fourier projections on finite cyclic groups, which appears to be new. The same approach also yields \(\mathcal R\)-boundedness for the family of partial-sum operators associated with the finer block decomposition, thereby resolving another related problem communicated to us by Fedor Sukochev.

For a positive integer $m$ and a real parameter $\alpha>-1$, consider the generalized Cesàro matrix on $\ell^2(\mathbb N_0)$ with entries $m(n-j+1)_{m-1}/(n+\alpha+1)_m$ for $j\le n$. We give a computer-assisted proof that this operator is hyponormal if and only if $\alpha\ge0$. The main algebraic ingredient is an explicit decomposition of an auxiliary defect operator into a positive diagonal operator and at most $m$ rank-one operators. Discrete Gram polynomials determine the signs and coefficients of these terms. For nonnegative parameters, weighted estimates reduce positivity to scalar inequalities. A uniform analytic estimate covers every $m\ge32768$, while finite rational certificates cover the remaining orders and entire parameter intervals, without parameter sampling or truncating an infinite operator. The proof also gives a uniform weighted lower bound for the self-commutator. For $-1<\alpha<0$, finite-support vectors give negative quadratic forms for every order.

This 30 years old book is posted for free download by the generosity of Oxford University Press that granted permission to allow access to future generations of young scientists. A printed copy can be purchased from the publisher. The authors received for this book "The 1999 INFORMS Computing Society Prize for Research Excellence in the Interface Between Operations Research and Computer Science" from the Institute for Operations Research and the Management Sciences (INFORMS). For a book review by Janos D. Pinter go to: Journal of Global Optimization, 16: 107-108, 2000.

We establish several sufficient conditions ensuring that a perturbed family of operators in a given Hilbert-Schmidt frame for a separable Hilbert space remains a Hilbert-Schmidt frame. The perturbation criteria are formulated through Nagy-type and Pollard-Hilding-type inequalities, yielding explicit estimates for the resulting frame bounds. Several examples are presented to illustrate the applicability of the results.

We study frames of unilateral iterations generated by finitely many vectors under an invertible bounded operator $T$. Suppose that the spectrum of $T$ lies in $\{\rho e^{it}:0<\rho\le1,\ |t|\le c\}$, where $c<\pi$, and that its essential spectrum lies in $\{e^{it}:|t|\le c_{\mathrm e}\}$, where $0\le c_{\mathrm e}\le c$. Real powers are defined through the principal logarithm. Our structural result identifies fractional-orbit synthesis, up to an invertible map and a compact perturbation, with synthesis of projected vector-valued exponentials on an arc containing the essential spectrum. Semi-Fredholm properties and Fredholm index therefore transfer. For a uniformly separated temporal set $\Lambda$, this reduction gives a frame when its one-sided lower uniform density satisfies $D_+^-(\Lambda)>c_{\mathrm e}/\pi$ and its logarithmic block density satisfies $L(\Lambda)>c/\pi$. Completeness at the latter threshold requires only an invertible operator with spectrum in the corresponding angular sector and a complete integer multiorbit. At $L(\Lambda)>2c/\pi$, the complement property is equivalent to a finite condition in channel-cyclic subspaces, with real phase-retrieval consequences. Generalized divided differences admit the same compact reduction and the analogous frame theorem with multiplicities counted. For arbitrary one-point-per-cell sampling with cell length $r$, the sufficient raw-sampling conditions are $rc<\pi$ and $2rc_{\mathrm e}<\pi$. Thus, the full range $rc<\pi$ holds for raw samples when the essential spectrum is contained in $\{1\}$; over the unrestricted class, the sharp universal range is $rc<\pi/2$. Divided-difference preconditioning restores $rc<\pi$ for all operators considered and converts exact collisions into logarithmic-derivative data.

We show that if a sequence of expander graphs equi-coarsely embeds into a Banach space, then this Banach space fails Kasparov and Yu's Property (H). Consequently, no Banach space with Property (H) can be coarsely universal for all countable groups. We provide a Lean verification of our results.

We derive explicit lower bounds for relative Fisher information by combining a variational principle with suitably orthogonalized Hermite-polynomial test functions. The resulting cumulant bounds are asymptotically sharp and yield lower bounds for Gaussian entropy deficits. We also establish quantitative versions of sharp min-entropy power inequalities in all dimensions. En route, we develop a stability result for Brzezinski's sharp bound for block sections of products of Euclidean balls, which may be of independent interest.

We develop a representation-theoretic framework for computing projection constants of finite-dimensional subspaces of $C(K)$, where $K$ is a compact homogeneous $G$-space. Within this framework, we characterize the finite-dimensional $G$-invariant subspaces for which the $G$-equivariant projection is unique. These are precisely the finite orthogonal sums of full isotypic components in the Peter–Weyl decomposition of $L_2(K)$. For such subspaces, averaging shows that this unique $G$-equivariant projection is minimal; it is the restriction to $C(K)$ of the $L_2(K)$-orthogonal projection. Consequently, their projection constants are given by the $L_1$-norm of an explicit reproducing-kernel slice. We apply this method to spaces of $d$-homogeneous polynomials in high dimension. The examples range from Fourier analysis on the torus, through spherical harmonic analysis on real and complex Euclidean spheres, to genuinely noncommutative harmonic analysis on the unitary group, where representation theory becomes indispensable. For all these families, we obtain precise high-dimensional asymptotics at the square-root-of-dimension scale.

We investigate the hypercyclicity of Toeplitz operators on the Bergman space $L_{A}^{2}(\mathbb{D})$ with symbols of the form $\Psi(z) = \gamma\bar{z}+\psi(z)$, where $\gamma \in \mathbb{C} \setminus \{0\}$ and $\psi$ is analytic on an open neighborhood of the closed unit disc $\overline{\mathbb{D}}$. Our approach bypasses the classical Hardy space techniques (reproducing kernel linear combinations, Nevanlinna factorization) by directly solving the integro-differential resolvent equation arising from the Bergman projection. A key winding number argument shows that for sense-reversing symbols ($|\gamma| > \sup_{z \in \overline{\mathbb{D}}} |\psi'(z)|$), the symbol's image $\Psi(\mathbb{D})$ is contained in the point spectrum of $T_\Psi$. In the tridiagonal case $\Psi(z) = a\bar{z}+b+cz$, we fully resolve the longstanding eigenvector completeness problem by linking the recurrence coefficients to a rotated Favard spectral measure on the major axis of the symbol's ellipse. Combined with self-commutator positivity, this establishes an unconditional, exact necessary and sufficient characterization of hypercyclicity: $T_\Psi$ is hypercyclic if and only if $|a| > |c|$ and $\Psi(\mathbb{D})$ intersects both the unit disc and its exterior, completely eliminating the $(3+\sqrt{2})$ restriction of previous literature.

In this paper, we prove a boundedness result for parabolic Riesz transforms in the range $p \le 2$ associated with degenerate parabolic operators whose elliptic part is in divergence form with complex coefficients depending measurably on space and time. The degeneracy is determined by a spatial weight in the Muckenhoupt class $A_2(\mathbb{R}^n)$. The argument relies on off-diagonal estimates for the parabolic gradient of the resolvent family, together with weighted embeddings. We obtain boundedness in a range of exponents strictly below $2$, with explicit ranges quantified by the reverse Hölder class of the weight. For real coefficients, this quantification is sharp and extends down to $1$.

It is shown that if one knows "sufficiently many" snapshots (samples) from an individual solution of a Schrödinger-type equation-which is bandlimited in an appropriate sense-then the solution can be reconstructed by a simple formula similar to the Shannon formula for bandlimited signals. Furthermore, results demonstrate that by using snapshots of bandlimited solutions of a Schrödinger-type equation, one can reconstruct bandlimited solutions of some relevant diffusion- and wave-type equations. Interestingly, the reconstruction formulas are essentially independent of the operators defining the equations.

In this paper, we resolve the quantum KKL conjecture of Montanaro and Osborne \cite{MO2010} for quantum Boolean functions on the $n$-qubit hypercube. More precisely, for every self-adjoint unitary $T$, we bound the largest $L_2$-influence from below by a constant multiple of $\mathrm{Var}(T)\log(n)/n$. The proof heavily depends on upper and lower commutator estimates on the Hilbert space.

We study the dynamics of weighted backward shifts on Ces`aro spaces associated with leafless locally finite rooted trees. We first characterize their boundedness in terms of adjacent level cardinalities and edge weights. We then characterize their $\mathcal{F}$-transitivity by a growth condition involving level cardinalities, products of weights along paths, and a level-dependent Ces`aro factor. As consequences, we obtain criteria for hypercyclicity, weak mixing, topological ergodicity, and topological mixing. We also characterize the existence of nonzero orbit limit points and chaotic weighted shifts, the latter in terms of normalized fixed points and unit flows satisfying an explicit summability condition. Examples show that $\mathcal{F}_{\underline{d}>0}$-transitivity need not imply frequent hypercyclicity, that a nonhypercyclic weighted shift may nevertheless have a nonzero orbit limit point, and that topological mixing need not imply chaos.

Conjecture~2 of Hinrichs and Vybíral asserts that every continuous, nonnegative, positive definite function $f$ on $\R^d$, normalized by $f(0)=1$, satisfies $[f(x_j-x_k)]_{j,k=1}^n\succeq \one\one^T/n$. We give an explicit trigonometric polynomial on $\R^7$ and eight points for which this inequality fails. All hypotheses are verified directly: positive definiteness follows from nonnegative Fourier coefficients, and pointwise nonnegativity follows from interpolation on a cube. For an explicit vector of signs, the quadratic form is $22/5$, whereas the proposed lower bound is $9/2$. The resulting gap is exactly $-1/10$. The example is compatible with the established bound for functions of the form $|g|^2$ with $g$ positive definite.\\ The solution was found in a single prompt try by a colleague using his private licence of ChatGPT 6. Our aim is to make the solution public, as well as to collect some ideas that emerged during the analysis of the solution.

Let $\mu$ be a positive Borel measure on $[0,1)$ and $\omega$ a radial weight. In this paper we consider the Cesàro-type operator $C_{\mu,\omega}$ induced by the reproducing kernel $B^{\omega}$ of the weighted Bergman space $A^2_{\omega}$, given by $$ C_{\mu,\omega}(f)(z)=\int_{0}^{1}f(tz)B^{\omega}_t(z)\,d\mu(t), \quad z \in \mathbb{D}, $$ for functions $f$ analytic in $\mathbb{D}$. Under the assumption that $\omega$ satisfies a natural doubling property, we study the boundedness of $C_{\mu,\omega}$ acting on several spaces of analytic functions, including Hardy spaces $H^p$ and weighted Bergman spaces $A^p_{\nu}$. For $0<p,q<\infty$ and a two-sided doubling weight $\nu$, we completely characterize when $C_{\mu,\omega}: H^p \to H^q$ and $C_{\mu,\omega}: A^p_{\nu} \to A^q_{\nu}$ are bounded in terms of the interplay of tail integrals or moments of the inducing weights and the measure $\mu$. Many of the results obtained are new even in the setting of standard weights or when the Bergman reproducing kernel is replaced by the Cauchy kernel. In addition, we consider $C_{\mu,\omega}$ acting on $H^{\infty}$, Korenblum spaces and weighted Hardy spaces.

In this paper, we present a refinement of the symplectic Lidskii theorem for directional derivatives of symplectic eigenvalues, which is inspired by the work of Sendov [Electron. J. Linear Algebra 41(2025), 338-341]. We show that for a $2n \times 2n$ real positive definite matrix $A$ and symmetric matrices $H, K$ \begin{align*} d'(A;H+K)-d'(A;H) \prec_{\pi_A} d'(A;K). \end{align*} Here $\prec_{\pi_A}$ represents block-majorization corresponding to the partition $\pi_A$ of $\{1,\ldots, n\}$ given by the equality blocks of symplectic eigenvalues of $A$. We know that Hermitianization of a symmetric matrix occurs in the directional derivative expression of symplectic eigenvalues. We establish a componentwise inequality comparing the symplectic eigenvalues of a positive definite matrix with the ordinary eigenvalues of its associated Hermitianization, and also characterize the equality case. By using the aforementioned inequality, we then show that the symplectic Lidskii theorem follows from the directional derivative Lidskii theorem.

We construct Riesz sequences whose Beurling density is arbitrary close to the critical density in the setting of reproducing kernel Hilbert spaces on metric measure spaces. In particular, our results apply to exponential systems on unbounded sets, nonlocalized Gabor systems and general coherent systems arising from nilpotent Lie groups. The methods used in this paper are based on our previous work on the redundancy of frames and a selector form of Weaver's conjecture.

For finite measures with positive densities on convex Euclidean supports, we prove that $MCP(\kappa,N)$ passes with unchanged parameters to almost every isometric leaf of an arbitrary nonexpansive map. The proof rests on a sharp contraction inequality for geometric conditional densities, with exponent equal to the leaf codimension. The inherited dimension parameter is optimal. A total-variation limit on resolvent graphs extends the result to inverse fibres of maximal monotone relations, including convex gradient fibres. We also disprove Klartag's curvature-dimension inheritance conjecture by a firmly nonexpansive example in dimension three and a gradient example in dimension four. In codimension one, affinity of the geometric density yields curvature-dimension inheritance. The first example also gives failure on monotone fibres. Both constructions admit arbitrarily large curvature loss, including for a fixed Gaussian ambient measure on families of leaves of positive quotient measure.