We introduce the $G$-Daugavet property ($G$-DPr, for short) for Banach spaces endowed with an action of a group $G$ by surjective linear isometries. This notion provides a common framework for the classical Daugavet property and the alternative Daugavet property, which correspond respectively to the trivial action and to the scalar action of $S_{\mathbb{K}}$. We establish several characterizations of the $G$-DPr in terms of $G$-slices and closed convex $G$-invariant hulls, recovering the usual slice descriptions of the DPr and the aDPr as particular cases. We show that the presence of a group action leads to new behavior in Daugavet theory. In particular, the $G$-DPr may hold on classical reflexive spaces in sharp contrast with the classical Daugavet property. We relate this phenomenon to convex transitivity, almost transitivity and finite-dimensional rotation problems. We also prove group-action versions of the classical characterizations for $L^1(μ, X)$- and $C(K,X)$-spaces. The paper also studies group separable determination, $G$-versions of numerical radius and numerical index, and connections between the $G$-DPr and strong Radon-Nikodým and SCD operators. Finally, we introduce a parameter which measures how far the $G$-DPr is from the classical DPr in a quantitative manner. As a consequence of these results, we obtain conditions under which the $G$-DPr recovers several classical implications, including the failure of the RNP for both $X$ and $X^*$, the presence of copies of $\ell_1$ and the failure of the unit ball to be an SCD set.

We give a Bellman-function proof of the dimension-free estimate \[ \Big\| \vec{R} f \Big\|_{L^p(Ω;\,\ell^2)} \lesssim (p-1) \,\|f\|_{L^p(Ω)}, \qquad 2\le p<\infty, \] for the vector of Riesz transforms associated with the Walsh number operator on the Hamming cube $Ω=\{-1,1\}^n$, as well as for locally compact abelian groups, in particular $Ω=\mathbb{Z}^n$. The argument is based on a Poisson semigroup representation, symmetrized estimates along edges of $Ω$, and a two-point inequality. This is the first non noncommutative proof of this result, after the seminal papers of Lust-Piquard and later Junge-Mei-Parcet. According to an example of Lamberton, for $1<p<2$ such a dimension-free bound is known to be false.

We characterize the pointwise notion of almost equicontinuity for families of real-valued measurable functions on subsets of $\mathbb R^n$ of finite measure. The characterization is given by means of an integral truncated translation condition. We also provide examples showing that the finite measure assumption and the truncation are essential.

This paper investigates the one-dimensional Fourier phase retrieval problem for indicator functions of finite unions of intervals. Specifically, we study the recovery of a set $Ω= \bigcup_{j=1}^m I_j \subset\mathbb{R}$ from the magnitude of its Fourier transform $|\widehat{\mathbf{1}_Ω}|$, where each $I_j \subset \mathbb{R}$ is a bounded interval. For $m\le 2$, we prove that $Ω$ is uniquely determined by $ |\widehat{\mathbf{1}_Ω}|$ up to the natural ambiguities of translation and reflection, and we further establish a stability result for this reconstruction. In contrast, for $m\ge 3$, uniqueness fails in general. More precisely, for every $m\ge 3$, we explicitly construct functions $f_m,g_m\in\mathcal{I}_m$ such that $|\widehat{f_m}|=|\widehat{g_m}|,$ while $f_m$ cannot be obtained from $g_m$ by any translation or reflection, where $\mathcal{I}_m$ denotes the class of indicator functions of unions of exactly $m$ intervals. Furthermore, building on the theory of the turnpike problem, in which a finite integer set is uniquely determined by its multiset of pairwise differences under a collision-free condition, we establish an analogous result for finite subsets of $\mathbb{R}$. This, in turn, yields a sufficient condition for recovering indicator functions of finite unions of intervals. These results provide a complete characterization of the Fourier phase retrieval problem for indicator functions of finite unions of intervals and offer new insights into Fourier phase retrieval for indicator functions of more general domains in higher dimensions.

We provide a partial answer to Problems 1 and 2 raised in the recent article by Blaschke et al., concerning the analytic continuation of weighted $H$-harmonic Bergman spaces. These are spaces of functions annihilated by the Möbius-invariant Laplacian on the unit ball. More precisely, we identify some of the discrete Wallach sets and show, among others, that structure depends on the parity of the dimension.

For a broad class of Delone sets in $\mathbb{R}^n$ that are of significance in both mathematics and physics, we prove a non-uniform Balian-Low type theorem and settle the converse problem on the existence of Gabor frames, for arbitrary dimension $n$. To this end, we introduce a class of Gabor frames, termed full Gabor frames, and prove that the existence of such a frame on the Delone set with Schwartz window functions is equivalent to the condition that the lower Beurling density be strictly greater than one. In fact, the usual Balian-Low direction using window functions from the Feichtinger's algebra can be proven for arbitrary point sets, thereby improving an earlier density theorem by Christensen, Deng, and Heil. The corresponding dual result for Riesz sequences is also obtained. The main technical tools employed in this paper are tiling groupoid constructions and $C^*$-algebraic methods. As a byproduct, we resolve an open question from Ito's thesis concerning the bounded dynamical asymptotic dimension of tiling groupoids. Furthermore, this result allows us to extend the classification theorem of Ito, Whittaker, and Zacharias to the twisted case.

The paper extends previous results on the nth root problem to a large class of Hilbert-space operators, namely, the class of all normaloid operators with normaloid parts, which includes the paranormal operators, and also the $k$-paranormal operators. It is shown that if a normaloid operator with normaloid parts has a normal nth power, then it is normal.

This paper studies a Grothendieck-type finite-dimensionality problem for closed linear subspaces embedded in functional Banach spaces. Let $S_p^{(q)} \subset L_p(M,dμ)$ be a closed linear subspace of the Banach space $L_p(M,dμ)$ defined with respect to a probability measure $dμ$ on $M$. We prove that if $S_p^{(q)}$ is continuously (identically) embedded into $L_q(M,dμ)$ for $q>p$, then its dimension $\dim S_p^{(q)} = N \in \mathbb{N}$ satisfies the estimate \[ \frac{1}{N}\left(\frac{\sqrtπ,Γ!\left(\frac{N+\tilde q}{2}\right)}{Γ!\left(\frac{\tilde q+1}{2}\right)Γ!\left(\frac{N}{2}\right)}\right)^{2/\tilde q}\le K_{p,q(m)}^2, \] where $1/\tilde q + 1/q = 1$, $q = 2 + (p-2)2^m > p$ with $p \neq 2$ and $m \in \mathbb{N}$, and $K_{p,q(m)}>0$ is a bounded constant. We also prove that certain closed linear subspaces of $L_p(M,dμ)$ consisting of continuous functions on $M$ must be finite-dimensional.

In this paper we prove ``$H=W$" in the context of a Banach function space $X(Ω)$. Let $Ω$ be a subset of ${\mathbb R}^n$ and denote by $W^1_X(Ω)$ the collection of all those $f\in X(Ω)$ whose distributional derivatives $\partial_jf$ are contained in $X(Ω)$. Our main result provides a small collection of ``universal" hypotheses on $X(Ω)$ that ensure $W^1_X(Ω)$ is equal to $H^1_X(Ω)$, the formal closure of ${Lip}(Ω)\cap W^1_X(Ω)$ with respect to the norm \[\|f\|_{W^1_X(Ω)} = \|f\|_{X(Ω)} + \|\nabla f\|_{X(Ω)}.\] The main theorem has two corollaries. The first gives a slightly stronger set of hypotheses for ``$H=W$", and the second gives density of $C^\infty_c({\mathbb R}^n)$ in $W^1_X({\mathbb R}^n)$.

Given $(A_0, A_1)$ a compatible couple of Banach spaces, we describe the dual of the limiting real interpolation space $(A_0, A_1)^{K}_{ 1,q,b}$ for $0 < q < 1$ and $b$ a slowly varying function. In the process, we use the $J$-spaces $(A_0, A_1)^{J}_{ 1,q,b}$ and we establish an equivalence theorem for $K$ and $J$ spaces of independent interest. We also give examples that recover known results on this topic.

In this paper we deal with geometrical versions of Hardy type inequalities with additional positive terms in convex domains. The constant $λ(Ω)$ multiplying the additional term depends on the geometry of the multidimensional domain $Ω$ and the numerical parameters of the problem. The constant (functional) $λ(Ω)$ is called Brezis-Marcus constant. In 2010, F.G. Avkhadiev and K.-J. Wirths proposed the hypothesis that among all $n$-dimensional domains with given inradius the maximum of the best Brezis-Marcus constant is achieved for the $n$-dimensional ball of radius. Using one dimensional Hardy type inequalities we proved the Avkhadiev-Wirths conjecture on Brezis-Marcus constants in the cases $n=2$ and $n\geq 4$. The sharp constants are solutions of the equation in terms of special functions and fixed eigenvalues of the Sturm-Liouville differential operators. The corresponding eigenfunctions in the $2$-d case are spheroidal wave functions and for dimensions greater than or equal to $4$ are confluent Heun functions. New properties of the Heun functions are established and their zeros are found. We provide Python code for calculating sharp constants.

We correct some errors found in [T. Oikhberg and M.A. Tursi, Order extreme points and solid convex hulls, in ``The Mathematical Legacy of Victor Lomonosov'' (ed. R. Aron et.al.), de Gryuter, 2020, 297--315.]

Motivated by two seminal results of multivariate prediction theory by Helson and Lowdenslager and by Wiener and Masani we prove extremal representations of functions of matrices and derive their prediction-theoretic consequences. We also sketch a way to obtain matricial inequalities from our results. The main goal of the paper is the computation of the infimum of a set of values of the form $tr(A ΔA^*)$, where $Δ$ is a given non-negative Hermitian $n \times n$ matrix and the choices for $A$ exhauste a certain set of $n \times n$ matrices. In particular, we focus on norm-bounded unit spheres with certain types of properties of unitary invariance, what allows an application of the theory of majorization.

In 1829, Cauchy derived an upper bound for every root of a complex polynomial using the maximum of the absolute values of the coefficients. In 1931, Montel derived an upper bound using the sum of the absolute values of the coefficients. We derive noncommutative versions of the Cauchy and Montel bounds.

In this paper, we present several new results concerning multiparameter $C$-semigroups. We introduce and systematically analyze the class of multiparameter $C$-cosine functions, providing several new structural results and applications to abstract multiparameter Cauchy problems of first/second order in locally convex spaces. We also consider automatic extensions of multiparameter $C$-semigroups and multiparameter $C$-cosine functions.

We establish matrix versions of the comparisons between the $\ell^p$-norms or quasi-norms for sequences of complex numbers. For instance, given $1\ge q>0$, and a family of $m$ normal $d\times d$ matrices $A_1,\ldots, A_m$, we show that $$ \left|\sum_{k=1}^m A_k\right| \le \frac{1}{d}\sum_{i=1}^d V_i\left\{\sum_{k=1}^m |A_k|^{q}\right\}^{1/q}\!\!\!\!V_i^* $$ for some unitary $d\times d$ matrices $V_1,\ldots, V_d$. We also give applications to Olson's spectral order and to the comparison between the symmetric modulus and the quadratic symmetric modulus. In particular we show that the sum $A+B$ of two positive matrices submajorizes their Kato supremum $A\vee B$, thereby completing majorization results due to Ando.

We prove a Littlewood-Paley formula for the Hardy space of Dirichlet series $\mathscr{H}^p$ with $1\leq p<\infty$ in terms of almost every vertical limit function. This significantly strengthens previous results, which hold either only as an average over the vertical limit functions or under additional assumptions of uniform convergence. As part of our approach, we obtain a Hardy-Stein identity for the derivative of the $p$-mean of almost every vertical limit. We further show that the mean counting function exists for any $f$ in $\mathscr{H}^p$ in terms of almost all of its vertical limit functions. This is done by establishing a version of Jensen's formula in this setting. In the process, we also deduce ergodic versions of Fatou's lemma and the monotone and dominated convergence theorems for the Kronecker flow.

This work concerns regularity properties of weak solutions to elliptic equations in divergence form -div(a$\nabla$u) = div F , under low regularity assumptions on both the coefficient a and the source term F . We introduce generalized Morrey and Campanato spaces extending the classical definitions by replacing uniform boundedness requirements with suitable integrability conditions. Within this framework, we establish regularity estimates for the gradient of weak solutions in these generalized spaces. As applications, we recover classical H{ö}lder and Lebesgue estimates and derive fractional Sobolev regularity results. In particular, the proposed approach yields fractional Sobolev estimates in situations where the coefficient may be discontinuous and the gradient of the solution is not expected to be locally bounded.

We develop a harmonic-analytic method for constructing a generalized trigonometric system in noncommutative $L_p(\mathbb{T}^d_θ)$ spaces arising from the strongly continuous representation of $\mathbb{T}^d$ and show that the generalized trigonometric system is a Schauder basis in $L_p(\mathbb{T}^d_θ)$ for $1<p<\infty.$ In particular, we prove that this trigonometric system forms an RUC-basis in $L_p(\mathbb{T}^d_θ)$ for $2<p<\infty.$ Our results provide a noncommutative counterpart of the classical trigonometric basis in $L_p(\mathbb{T}^d)$. Further, we obtain a weak $(1,1)$ type estimate of partial sum operators associated with noncommutative trigonometric systems. This allows us to study uniformly boundedness of partial sum operators between pairs of symmetric spaces that do not necessarily possess nontrivial Boyd indices, extending known results in this direction to the setting of quasi-Banach symmetric spaces.

Density-potential inversion for periodic systems within Moreau-Yosida-regularised density-functional theory is investigated, both theoretically and numerically. We develop the framework in a periodic homogeneous Sobolev space and use it to recover the exchange-correlation potential of Kohn-Sham theory through a limiting procedure. A key analytical ingredient is the proof of lower semicontinuity of the non-interacting kinetic-energy functional in the chosen topology. The proximal mapping, together with its algorithmic evaluation, plays a central role in the resulting inversion scheme. Numerical experiments illustrate the performance and properties of the method for both the Kohn-Sham and Gross-Pitaevskii equations.

We study random Schrödinger operators on closed Riemannian manifolds with Anderson-type potentials. We prove high-probability spectral inclusion bounds showing that eigenvalues remain close to those of the Laplacian, with deviations controlled by a norm of the potential coefficients. Compared with deterministic bounds, this yields a square-root cancellation gain. The proof is based on a general principle showing that randomisation improves operator norm bounds for multiplier-type operators, which we formulate in both discrete and continuous settings.

In this paper, we introduce a new semi-discrete modulus of smoothness, which generalizes the definition given by Kolomoitsev and Lomako (KL) in 2023 (in the paper published in the J. Approx. Theory), and we establish very general one- and two- sided error estimates under non-restrictive assumptions for pointwise linear operators. The proposed results have been proved exploiting the regularization and approximation properties of certain Steklov integrals introduced by Sendov and Popov in 1983. By the definition of semi-discrete moduli of smoothness here proposed, we derive sharper estimates than those that can be achieved by the classical averaged moduli of smoothness ($τ$-moduli). Furthermore, a Rathore-type theorem is established, and a new notion of K-functional is also introduced showing its equivalence with the semi-discrete modulus of smoothness and its realization. One-sided estimates of approximation can be established for classical operators on bounded domains, such as the Bernstein polynomials. In the case of approximation operators on the whole real line, one-sided estimates can be achieved, e.g., for the Shannon sampling (cardinal) series, as well as for the so-called generalized sampling operators.

The article establishes a rigorous shift operator calculus for stationary time series modeling, addressing a certain gap in the literature. It provides proofs of existence and isometry for the transfer function operators $f(B)$ and $f(T)$ where $B$ is the bilateral shift operator and $T$ is the unilateral shift operator for different families of functions $f$. The article establishes convergence of the power series of $f(B)$ and $f(T)$ under the operator norm for the Wiener algebra $\mathbb{W}_+$, and convergence under strong operator topology for $f$ in $H^{\infty}$, based on the use of Abel sums. Based on this calculus, it unifies the notion of stationary process invertibility with the operator invertibility of the transfer function $f(T)$.

We study universality of Lipschitz quotients. First, we modify a construction of Johnson, Lindenstrauss, Preiss and Schechtman to obtain a complete separable metric space that has every complete separable metric space as a Lipschitz quotient. Our main result is in the compact setting, where we prove that no such universal metric space can exist. We deduce this impossibility result by studying the curve-flat index, an ordinal index which provides a measure of the complexity of the curve-fragment structure in a metric space. We show that Lipschitz quotients cannot increase this index in compact domains; while there exist compact spaces with arbitrarily high countable curve-flat index. The main technical part of the paper is dedicated to proving a strong version of the latter fact: for every ordinal $α$ and every compact metric space $M$, there exists a compact metric space $N$ such that the curve-flat quotient of $N$ of order $α$ is almost-isometric to $M$.

We introduce the notion of Karamata regular operators, which is a notion of regularity that is suitable for obtaining concrete convergence rates for common fixed point problems. This provides a broad framework that includes, but goes beyond, Hölderian error bounds and Hölder regular operators. By concrete, we mean that the rates we obtain are explicitly expressed in terms of a function of the iteration number $k$ instead, of say, a function of the iterate $x^k$. While it is well-known that under Hölderian-like assumptions many algorithms converge linearly/sublinearly (depending on the exponent), little it is known when the underlying problem data does not satisfy Hölderian assumptions, which may happen if a problem involves exponentials and logarithms. Our main innovation is the usage of the theory of regularly varying functions which we showcase by obtaining concrete convergence rates for quasi-cylic algorithms in non-Hölderian settings. This includes certain rates that are neither sublinear nor linear but sit somewhere in-between, including a case where the rate is expressed via the Lambert W function. Finally, we connect our discussion to o-minimal geometry and show that, under mild assumptions, definable operators in any o-minimal structure are always Karamata regular.

We study double-well systems with strong magnetic fields and deep potential wells. We present lower bounds on tunneling rates for generic values of the coupling constant. This result was recently announced and complements our recent counter-example construction which exhibits vanishing tunneling for specially-constructed double-well potentials.

In this paper, we investigate the existence of nontrivial least-energy solutions for the super-Liouville equation with positive coefficient functions on the two-dimensional sphere. Firstly, we derive a global Pohozaev-type identity by analyzing the behavior of solutions under conformal transformations, which generalizes the classical Kazdan-Warner obstruction for the two-dimensional Nirenberg problem. Secondly, by exploiting conformal symmetry, we establish a pointwise estimate that bounds the norm of the spinor component by the scalar component, and show that the $H^1 \times H^{1/2}$ energy of the spinor part remains uniformly bounded. As a byproduct of our analysis, parallel techniques are applied to the Dirac-Einstein equations on the 3-sphere, demonstrating that nontrivial solutions are uniformly bounded away from the trivial solution in the $H^1 \times H^{1/2}$ norm. Moreover, the compactness of the solution space is also analyzed from two perspectives: in the low-energy regime, and modulo the action of the Möbius group. Finally, by introducing a new natural constraint $\mathcal{A}$ and employing variational methods, we obtain a supersymmetric generalization of the Moser-Trudinger-Onofri inequality and establish the existence of least-energy solutions for even coefficient functions. In particular, these solutions are shown to be nontrivial provided that a certain spectral parameter associated with the coefficients satisfies $λ_1(h_2, h_1) < 1$. Concurrently, we provide a complete classification of nontrivial least-energy solutions in the case of positive constant coefficients.

For $p\in [1,\infty]$, we define a smooth manifold structure on the set $AC_{L^p}([a,b],N)$ of absolutely continuous functions $γ\colon [a,b]\to N$ with $L^p$-derivatives for all real numbers $a<b$ and each smooth manifold $N$ modeled on a sequentially complete locally convex topological vector space, such that $N$ admits a local addition. Smoothness of natural mappings between spaces of absolutely continuous functions is discussed, like superposition operators $AC_{L^p}([a,b],N_1)\to AC_{L^p}([a,b],N_2)$, $η\mapsto f\circ η$, for a smooth map $f\colon N_1\to N_2$. For $1\leq p <\infty$ and $r\in \mathbb{N}$ we show that the right half-Lie groups $\text{Diff}_K^r(\mathbb{R})$ and $\text{Diff}^r(M)$ are $L^p$-semiregular. Here $K$ is a compact subset of $\mathbb{R}$ and $M$ is a compact smooth manifold. An $L^p$-semiregular half-Lie group $G$ admits an evolution map $\text{Evol}:L^p([0,1],T_e G)\to AC_{L^p}([0,1],G)$, where $e$ is the neutral element of $G$. For the preceding examples, the evolution map $\text{Evol}$ is continuous.

It is well-known that the Sobolev spaces $W^{k,p}(\mathbb R^d)$ are vector lattices with respect to the pointwise almost everywhere order if $k \in \{0,1\}$, but not if $k \ge 2$. In this note, we consider negative $k$ and show that the span of the positive cone in $W^{k,p}(\mathbb R^d)$ is a vector lattice in this case. We also prove a related abstract result: if $(T(t))_{t \in [0,\infty)}$ is a positive $C_0$-semigroup on a Banach lattice $X$ with order continuous norm, then the span of the cone $X_{-1,+}$ in the extrapolation space $X_{-1}$ is a vector lattice. This complements results obtained by Bátkai, Jacob, Wintermayr, and Voigt in the context of perturbation theory and provides additional context for the theory of infinite-dimensional positive systems.

We analyze smooth nonlinear mappings for Hilbert and Banach spaces that carry small balls to convex sets, provided that the radius of the balls is small enough. Being focused on the study of new and mild sufficient conditions for a nonlinear mapping of Hilbert and Banach spaces to be locally convex, we address a suitably reformulated local convexity problem analyzed within the Leray-Schauder homotopy method approach for Hilbert spaces, and within the Lipscitz smoothness condition both for Hilbert and Banach spaces. Some of the results presented in the work prove to be interesting and novel even for finite-dimensional problems. Open problems related to the local convexity property for nonlinear mapping of Banach spaces are also formulated.