A commuting triple of Hilbert space operators \((A,S,P)\) is said to be a \($\mathbb{P}\)-contraction if the closed pentablock \(\overline{\mathbb P}\) is a spectral set for \((A,S,P)\), where \[\mathbb{P}:=\{(a_{21}, \mbox{tr}(A_0), \mbox{det}(A_0))\colon A_0=[a_{ij}]_{2 \times 2}\ \text{and}\ \|A_0\| <1 \} \subseteq \mathbb{C}^3.\]A commuting triple of normal operators \((A, S, P)\) acting on a Hilbert space is said to be a \(\mathbb P\)-unitary if the joint spectrum \(\sigma_T(A, S, P)\) of \((A, S, P)\) is contained in the distinguished boundary \(b\mathbb{P}\) of \(\overline{\mathbb{P}}\). Also, \((A, S , P)\) is called a \(\mathbb P\)-isometry if it is the restriction of a \(\mathbb P\)-unitary \((\hat A, \hat S, \hat P)\) to a joint invariant subspace of \(\hat A,\hat S,\hat P\). We find several characterizations for the \(\mathbb P\)-unitaries and \(\mathbb P\)-isometries. We show that every \(\mathbb P\)-isometry admits a Wold type decomposition that splits it into a direct sum of a \(\mathbb P\)-unitary and a pure \(\mathbb P\)-isometry. Moving one step ahead we show that every \(\mathbb P\)-contraction \((A,S,P)\) possesses a canonical decomposition that orthogonally decomposes \((A,S,P)\) into a \(\mathbb P\)-unitary and a completely non-unitary \(\mathbb P\)-contraction. We find a necessary and sufficient condition such that a \(\mathbb P\)-contraction \((A, S, P)\) dilates to a \(\mathbb P\)-isometry \((X, T, V)\) with \(V\) being the minimal isometric dilation of \(P\). Then we show an explicit construction of such a conditional dilation. We show interplay between operator theory on the following three domains: the pentablock, the biball and the symmetrized bidisc.
We introduce the property (W-infinity) and the weakly infinity-compact approximation property (WICAP) of a Banach space X. We establish a characterization of the property (W-infinity) and relationships of the property (W-infinity), the approximate identities for the algebra W-infinity(X) and the WICAP. As a consequence, we obtain that both & ell;(p)(1 < p < infinity) and c(0) fail property (W-infinity). It is also shown that the WICAP is strictly stronger than the weakly compact approximation property.
We identify isometric isomorphisms of the space of Kurzweil-Henstock integrable functions as bi-absolutely-continuous changes of variable.
For every pair of non-constant holomorphic maps beta i : Si -+ S0 between noncompact Riemann surfaces, where i E {1, 2}, there exists an associated fiber product S1 & times;(beta 1,beta 2)S2 that has the structure of a singular Riemann surface, endowed with a canonical map beta to S0 satisfying beta io pi i = beta, where pi i is coordinate projection onto Si. This paper explores the relationship between the space of ends of this fiber product and the space of ends of its normal fiber product. In addition, we establish conditions on the maps beta 1 and beta 2 that ensure connectivity in the fiber product. Upon examination of these conditions, we establish a connection between the space of ends of the fiber product and the topological characteristics of the Riemann surfaces S1 and S2. Finally, we investigate the fiber product of infinite superelliptic curves by analyzing its connectedness and the space of ends.
In this paper, we establish a characterization of freely quasiconformal mappings in real Banach spaces. This characterization is in terms of the geometric moduli of rings which was introduced by Tukia and Va & uml;isa & uml;la & uml; in 2021. As an application, we obtain a generalization of geometric characterizations of quasiconformal mappings in Rn obtained by Gehring and Va & uml;isa & uml;la & uml;.
In this note, we present a logarithmic-type upper bound for weak subsolutions to a class of integro-differential problems, whose prototype is the Dirichlet problem for the fractional Laplacian. The bound is slightly smaller than the classical one in this field.
Maximal operators and differentiation of integrals associated to collections of shifted balls in I1gn (i.e., balls that may not contain the origin) have been studied by various authors. One of the motivations has been the intimate connection of these concepts with the boundary behaviour of Poisson integrals along regions more general than cones. Generalizing the corresponding results of Nagel and Stein, and Hagelstein and Parissis (established for the case of collections of balls) we give characterizations of the two classes of monotone collections S2 of shifted convex bodies in I1gn that to the means (1/|B|)B+x|f|(B E S2)) satisfies the weak type (1, 1) inequality; 2) the means over are defined by the following properties respectively: 1) the maximal operator associated to S2 (i.e., the sets B + x (B E S2) are a.e. convergent for the characteristic function of an arbitrary measurable subset of I1gn.
Let E be a Banach space (or a Banach lattice), let tau be a vector topology on E and let x be a sequence (or a positive sequence) in E not converging to zero with respect to tau. We show how to construct infinite dimensional Banach spaces (or Banach lattices) consisting, up to the origin, of sequences in E not converging to zero with respect to tau and containing a subsequence of x. Plenty of applications to Banach space theory and to Banach lattice theory are provided.
We provide weak-type bounds for a family of bilinear fractional integrals that arise in the study of Euler–Riesz systems. These bounds are uniform in the natural parameter that describes the family and are sharp, in the sense that they do not hold for any larger set of indices.
For Banach spaces of analytic functions on the disc for which the polynomials are dense and their pointt evaluations continuous, we prove the following: If they contain a function such that the limit superior of its modulus is infinite almost everywhere on the unit circle, then the same is true for a residual set of functions.
In 1971, Cunningham proved that every star-shaped Kakeya set \(E\subset\mathbb{R}^2\) satisfies \(|E| \geq \pi/108\). In this paper, we show that Cunningham's bound is not optimal and can be improved to \(|E| \geq \pi/98\).
We provide a new characterisation of the decades old open problem of extending bilipschitz mappings given on a Euclidean separated net. In particular, this allows for the complete positive solution of the open problem in dimension two. Along the way, we develop a set of tools for bilipschitz extensions of mappings between subsets of Euclidean spaces.
Given a self-similar set \(\Lambda\) that is the attractor of an iterated function system (IFS) \(\{f_1,\dots,f_N\}\), consider the following method for constructing a random subset of \(\Lambda\): Let \(\mathbf{p}=(p_1,\dots,p_N)\) be a probability vector, and label all edges of a full \(M\)-ary tree independently at random with a number from \(\{1,2,\dots,N\}\) according to \(\mathbf{p}\), where \(M\geq 2\) is an arbitrary integer. Then each infinite path in the tree starting from the root receives a random label sequence which is the coding of a point in \(\Lambda\). We let \(F\subset\Lambda\) denote the set of all points obtained in this way. This construction was introduced by Allaart and Jones (2025), who considered the case of a homogeneous IFS on \(\mathbb{R}\) satisfying the Open Set Condition (OSC) and proved non-trivial upper and lower bounds for the Hausdorff dimension of \(F\). We demonstrate that under the OSC, the Hausdorff (and box-counting) dimension of \(F\) is equal to the upper bound of Allaart and Jones, and extend the result to higher dimensions as well as to non-homogeneous self-similar sets.
Let X = D/Gamma be a Riemann surface with Gamma of the first kind. We establish a necessary and sufficient criterion for [f ] E T(X) to have a Teichmuller-type extremal map.
We show that the real-analytic matrix-weighted double fibration transform determines the analytic wavefront set of a vector-valued function. We apply this result to show that the matrix weighted ray transform is injective on a two-dimensional, non-trapping, real-analytic Riemannian manifold with strictly convex boundary. Additionally, we show that a real-analytic Higgs field can be uniquely determined from the nonabelian ray transform on real-analytic Riemannian manifolds of any dimension with a strictly convex boundary point.
We study the two-weighted off-diagonal compactness of commutators of rough singular integral operators Tc that are associated with a kernel S2 is an element of Lq(Sd-1). We establish a characterisation of compactness of the commutator [b, Tc] in terms of the function b belonging to a suitable space of functions with vanishing mean oscillation. Our results expand upon certain previous compactness characterisations in that the results do not require smoothness from the kernel of the singular integral operator. Additionally, we prove a matrix-weighted compactness result for [b, Tc] by applying the so-called matrix-weighted Kolmogorov-Riesz theorem.
We investigate the first-order differential calculus over extended metric-topological measure spaces. The latter are quartets X = (X, tau, d, m), given by an extended metric space (X, d) together with a weaker topology tau (satisfying suitable compatibility conditions) and a finite Radon measure m on (X, tau). The class of extended metric-topological measure spaces encompasses all metric-measure spaces and many infinite-dimensional metric-measure structures, such as abstract Wiener spaces. In this framework, we study the following classes of objects: center dot The Banach algebra Lipb(X, tau, d) of bounded tau-continuous d-Lipschitz functions on X. center dot Several notions of Lipschitz derivations on X, defined in duality with Lipb(X, tau, d). center dot The metric Sobolev space W1,p(X), defined in duality with Lipschitz derivations on X. Inter alia, we generalise both Weaver's and Di Marino's theories of Lipschitz derivations to the extended setting, and we discuss their connections. We also introduce a Sobolev space W1,p(X) via an integration-by-parts formula, along the lines of Di Marino's notion of Sobolev space, and we prove its equivalence with other approaches, studied in the extended setting by Ambrosio, Erbar and Savare. En route, we obtain some results of independent interest, among which are: center dot A Lipschitz-constant-preserving extension result for tau-continuous d-Lipschitz functions. center dot A novel and rather robust strategy for proving the equivalence of Sobolev-type spaces defined via an integration-by-parts formula and those obtained with a relaxation procedure. center dot A new description of an isometric predual of the metric Sobolev space W1,p(X).
Let $K_n=(V,E)$ be the complete graph with $n\geq 3$ vertices (here $V$ and $E$ denote the set of vertices and edges of $K_n$ respectively). We find the optimal value ${\bf{C}}_{n,p}$ such that the inequality $$\|f-m_f\|_p\le {\bf C}_{n,p}{\rm Var}_{p}f$$ holds for every $f:V\to \mathbb{R},$ where ${\rm Var}_p$ stands for the $p$-variation, and $m_f$ stands for the average value of $f$, for all $p\in[1,3+\delta^1_n)\cup (3+\delta^2_n,+\infty)$, for $\delta^1_n=\frac{1}{2n^2\log(n)}+O(1/n^3)$ and $\delta^2_n=\frac{2}{n}+O(1/n^2).$ Moreover, we characterize all the maximizer functions in that case. The behavior of the maximizers is different in each of the intervals $(1,2)$, $(2,3+\delta^{1}_n)$ and $(3+\delta^{2}_n,\infty).$
We study an inverse boundary value problem in corrosion detection. The model is based on a conductivity equation with nonlinear Robin boundary condition. We prove that the nonlinear Robin term can be identified locally from Cauchy data measurements on a subset of the boundary. A possible strategy for turning a local identification result into a global one is suggested, and a partial result is proved in this direction. The inversion method is an adaptation to this nonlinear Robin problem of a method originally developed for semilinear elliptic equations. The strategy is based on linearization and relies on parametrizing solutions of the nonlinear equation on solutions of the linearized equation.
A central problem in the study of critical circle dynamics is understanding the regularity of Yoccoz conjugators - circle homeomorphisms that conjugate critical circle maps with irrational rotation numbers to their corresponding rigid rotations. One can approach this problem from a different angle by studying the regularity of extensions of these maps to the unit disk. Of particular interest is the question of when such a conjugator admits a David extension. Building on the work of Petersen and Zakeri, we classify the David regularity of a specific extension process known as the Yoccoz extension.
