Reducibility for a Class of Analytic Multipliers on the Dirichlet Space

It is proved that if \(\phi \) is a finite Blaschke product with four zeros, then \(M_\phi \) is reducible on the Dirichlet space with norm \(\Vert \ \Vert \) if and only if \(\phi =\phi _1\circ \phi _2\), where \(\phi _1, \phi _2\) are Blaschke products and \(\phi _2\) is equivalent to \(z^2\). Also, the same reducibility of \(M_\phi \) with finite Blaschke product \(\phi \) on the Dirichlet space under the equivalent norms \(\Vert \ \Vert _1\) and \(\Vert \ \Vert _0\) is given.     

Hilbert $$C^*$$-modules as a subcategory of operator systems and injectivity

In this paper, we study a category whose objects are Hilbert \(C^*\)-modules and whose morphisms are completely semi-\(\phi \)-maps. We give a characterization of injective objects in this category. In fact, we investigate extendability of completely semi-\(\phi \)-maps on Hilbert \(C^*\)-modules, leading to an analog of the Arveson’s extension theorem for completely semi-\(\phi \)-maps (in contrast with \(\phi \)-maps). This theorem together with previous results suggest that the completely semi-\(\phi \)-maps are proper generalizations of the completely positive maps.     

Complex symmetry of Toeplitz operators with continuous symbols

In this note, we consider the question of when a Toeplitz operator on the Hardy–Hilbert space \(H^2\) of the open unit disk \(\mathbb {D}\) is complex symmetric, focusing on symbols \(\phi :\mathbb {T}\rightarrow \mathbb {C}\) that are continuous on the unit circle \(\mathbb {T}=\partial \mathbb {D}\). A closed curve \(\phi \) is called nowhere winding if the winding number of \(\phi \) is 0 about every point not in the range of \(\phi \). It is then shown that if \(T_\phi \) is complex symmetric, then \(\phi \) must be nowhere winding. Hence if \(\phi \) is a simple closed curve, then \(T_\phi \) cannot be a complex symmetric operator. The spectrum and invertibility of complex symmetric Toeplitz operators with continuous symbols are then described. Finally, given any continuous curve \(\gamma :[a,b]\rightarrow \mathbb {C}\), it is shown that there exists a complex symmetric Toeplitz operator with continuous symbol whose spectrum is precisely the range of \(\gamma \).     

On the approximation of dynamical indicators in systems with nonuniformly hyperbolic behavior

Let f be a \(C^{1+\alpha }\) diffeomorphism of a compact Riemannian manifold and \(\mu \) an ergodic hyperbolic measure with positive entropy. We prove that for every continuous potential \(\phi \) there exists a sequence of basic sets \(\Omega _n\) such that the topological pressure \(P(f|\Omega _n,\phi )\) converges to the free energy \(P_{\mu }(\phi ) = h(\mu ) + \int \phi {d\mu }\). We also prove that for a suitable class of potentials \(\phi \) there exists a sequence of basic sets \(\Omega _n\) such that \(P(f|\Omega _n,\phi ) \rightarrow P(\phi )\).     

Lower order eigenvalues of a system of equations of the drifting Laplacian on the metric measure spaces

Let \(\Omega \) be a bounded domain with smooth boundary in an n-dimensional metric measure space \((\mathbb {R}^n, \langle ,\rangle , e^{-\phi }dv)\) and let \(\mathbf {u}=(u^1, \ldots , u^n)\) be a vector-valued function from \(\Omega \) to \(\mathbb {R}^n\). In this paper, we investigate the Dirichlet eigenvalue problem of a system of equations of the drifting Laplacian: \(\mathbb {L}_{\phi } \mathbf {u} + \alpha [ \nabla (\mathrm {div}\mathbf { u}) -\nabla \phi \mathrm {div} \mathbf {u}]= - \widetilde{\sigma } \mathbf {u}\), in \( \Omega \), and \(u|_{\partial \Omega }=0,\) where \(\mathbb {L}_{\phi } = \Delta - \nabla \phi \cdot \nabla \) is the drifting Laplacian and \(\alpha \) is a nonnegative constant. We establish some universal inequalities for lower order eigenvalues of this problem on the metric measure space \((\mathbb {R}^n, \langle ,\rangle , e^{-\phi }dv)\) and the Gaussian shrinking soliton \((\mathbb {R}^n, \langle ,\rangle _{\mathrm {can}}, e^{-\frac{|x|^2}{4}}dv, \frac{1}{2})\). Moreover, we give an estimate for the upper bound of the second eigenvalue of this problem in terms of its first eigenvalue on the gradient product Ricci soliton \((\Sigma \times \mathbb {R}, \langle ,\rangle , e^{-\frac{\kappa t^2}{2}}dv, \kappa )\), where \( \Sigma \) is an Einstein manifold with constant Ricci curvature \(\kappa \).     

Removability of the logarithmic singularity for the elliptic PDEs with measurable coefficients and its consequences

This paper introduces the notion of log-regularity (or log-irregularity) of the boundary point \(\zeta \) (possibly \(\zeta =\infty \)) of the arbitrary open subset \(\Omega \) of the Greenian deleted neigborhood of \(\zeta \) in \({\mathbb {R}}^2\) concerning second order uniformly elliptic equations with bounded and measurable coefficients, according as whether the log-harmonic measure of \(\zeta \) is null (or positive). A necessary and sufficient condition for the removability of the logarithmic singularity, that is to say for the existence of a unique solution to the Dirichlet problem in \(\Omega \) in a class \(O(\log |\cdot - \zeta |)\) is established in terms of the Wiener test for the log-regularity of \(\zeta \). From a topological point of view, the Wiener test at \(\zeta \) presents the minimal thinness criteria of sets near \(\zeta \) in minimal fine topology. Precisely, the open set \(\Omega \) is a deleted neigborhood of \(\zeta \) in minimal fine topology if and only if \(\zeta \) is log-irregular. From the probabilistic point of view, the Wiener test presents asymptotic law for the log-Brownian motion near \(\zeta \) conditioned on the logarithmic kernel with pole at \(\zeta \).     

Solvability of Minimal Graph Equation Under Pointwise Pinching Condition for Sectional Curvatures

We study the asymptotic Dirichlet problem for the minimal graph equation on a Cartan–Hadamard manifold M whose radial sectional curvatures outside a compact set satisfy an upper bound

$$\begin{aligned} K(P)\le - \frac{\phi (\phi -1)}{r(x)^2} \end{aligned}$$

and a pointwise pinching condition

$$\begin{aligned} |K(P) |\le C_K|K(P') | \end{aligned}$$

for some constants \(\phi >1\) and \(C_K\ge 1\), where P and \(P'\) are any 2-dimensional subspaces of \(T_xM\) containing the (radial) vector \(\nabla r(x)\) and \(r(x)=d(o,x)\) is the distance to a fixed point \(o\in M\). We solve the asymptotic Dirichlet problem with any continuous boundary data for dimensions \(n=\dim M>4/\phi +1\).

     

Fully robust one-sided cross-validation for regression functions

Fully robust OSCV is a modification of the OSCV method that produces consistent bandwidths in the cases of smooth and nonsmooth regression functions. We propose the practical implementation of the method based on the robust cross-validation kernel \(H_I\) in the case when the Gaussian kernel \(\phi \) is used in computing the resulting regression estimate. The kernel \(H_I\) produces practically unbiased bandwidths in the smooth and nonsmooth cases and performs adequately in the data examples. Negative tails of \(H_I\) occasionally result in unacceptably wiggly OSCV curves in the neighborhood of zero. This problem can be resolved by selecting the bandwidth from the largest local minimum of the curve. Further search for the robust kernels with desired properties brought us to consider the quartic kernel for the cross-validation purposes. The quartic kernel is almost robust in the sense that in the nonsmooth case it substantially reduces the asymptotic relative bandwidth bias compared to \(\phi \). However, the quartic kernel is found to produce more variable bandwidths compared to \(\phi \). Nevertheless, the quartic kernel has an advantage of producing smoother OSCV curves compared to \(H_I\). A simplified scale-free version of the OSCV method based on a rescaled one-sided kernel is proposed.     

Sharp Lorentz Estimates for Dyadic-like Maximal Operators and Related Bellman Functions

We precisely evaluate Bellman-type functions for the dyadic maximal operator on \(\mathbb {R}^{n}\) and of maximal operators on martingales related to local Lorentz-type estimates. Using a type of symmetrization principle, introduced for the dyadic maximal operator in earlier works of the authors, we precisely evaluate the supremum of the Lorentz quasinorm of the maximal operator on a function \(\phi \) when the integral of \(\phi \) is fixed and also the same Lorentz quasinorm of \(\phi \) is fixed. Also we find the corresponding supremum when the integral of \(\phi \) is fixed and several weak type conditions are given.     

On the action of the $$U_ p$$U p o perator on the local (at p) re presentation attached to congruence level Siegel modular forms

In this article, we study the action of the \(U_p\) Hecke operator on the normalized spherical vector \(\phi \) in the representation of \({{\mathrm{GSp}}}_4(\mathbf {Q}_p)\) induced from a character on the Borel subgroup. We compute the Petersson norm of \(U_p \phi \) in terms of certain local L-values associated with \(\phi \).     

Some Ricci-flat ($$\alpha ,\beta $$)-metrics

In this paper, we study a special class of Finsler metrics, \((\alpha ,\beta )\)-metrics, defined by \(F=\alpha \phi (\beta /\alpha )\), where \(\alpha \) is a Riemannian metric and \(\beta \) is a 1-form. We find an equation that characterizes Ricci-flat \((\alpha ,\beta )\)-metrics under the condition that the length of \(\beta \) with respect to \(\alpha \) is constant.     

Minimal Translation Surfaces in Euclidean Space

A translation surface in Euclidean space is a surface that is the sum of two regular curves \(\alpha \) and \(\beta \). In this paper we characterize all minimal translation surfaces. In the case that \(\alpha \) and \(\beta \) are non-planar curves, we prove that the curvature \(\kappa \) and the torsion \(\tau \) of both curves must satisfy the equation \(\kappa ^2 \tau = C\) where C is constant. We show that, up to a rigid motion and a dilation in the Euclidean space and, up to reparametrizations of the curves generating the surfaces, all minimal translation surfaces are described by two real parameters \(a,b\in \mathbb {R}\) where the surface is of the form \(\phi (s,t)=\beta _{a,b}(s)+\beta _{a,b}(t)\).     

Bergman interpolation on finite Riemann surfaces. Part II: Poincaré-Hyperbolic Case

Let M be a 3-manifold with torus boundary components \(T_{1}\) and \(T_2\). Let \(\phi :T_{1} \rightarrow T_{2}\) be a homeomorphism, \(M_\phi \) the manifold obtained from M by gluing \(T_{1}\) to \(T_{2}\) via the map \(\phi \), and T the image of \(T_{1}\) in \(M_\phi \). We show that if \(\phi \) is “sufficiently complicated” then any incompressible or strongly irreducible surface in \(M_\phi \) can be isotoped to be disjoint from T. It follows that every Heegaard splitting of a 3-manifold admitting a “sufficiently complicated” JSJ decomposition is an amalgamation of Heegaard splittings of the components of the JSJ decomposition.     

Compact $$\lambda $$-translating Solitons with Boundary

A \(\lambda \)-translating soliton with density vector \(\mathbf {v}\) is a surface \(\varSigma \) in Euclidean space \(\mathbb {R}^3\) whose mean curvature H satisfies \(2H=2\lambda +\langle N,\mathbf {v}\rangle \), where N is the Gauss map of \(\varSigma \). In this article, we study the shape of a compact \(\lambda \)-translating soliton in terms of its boundary. If \(\varGamma \) is a given closed curve, we deduce under what conditions on \(\lambda \) there exists a compact \(\lambda \)-translating soliton \(\varSigma \) with boundary \(\varGamma \) and we provide estimates of the surface area depending on the height of \(\varSigma \). Finally, we study the shape of \(\varSigma \) related with the geometry of \(\varGamma \), in particular, we give conditions that assert that \(\varSigma \) inherits the symmetries of its boundary \(\varGamma \).     

Plethysm and Lattice Point Counting

We apply lattice point counting methods to compute the multiplicities in the plethysm of \(\textit{GL}(n)\). Our approach gives insight into the asymptotic growth of the plethysm and makes the problem amenable to computer algebra. We prove an old conjecture of Howe on the leading term of plethysm. For any partition \(\mu \) of 3, 4, or 5, we obtain an explicit formula in \(\lambda \) and k for the multiplicity of \(S^\lambda \) in \(S^\mu (S^k)\).     

Efficient and robust tests for semiparametric models

In this paper, we investigate a hypothesis testing problem in regular semiparametric models using the Hellinger distance approach. Specifically, given a sample from a semiparametric family of \(\nu \)-densities of the form \(\{f_{\theta ,\eta }:\theta \in \Theta ,\eta \in \Gamma \},\) we consider the problem of testing a null hypothesis \(H_{0}:\theta \in \Theta _{0}\) against an alternative hypothesis \(H_{1}:\theta \in \Theta _{1},\) where \(\eta \) is a nuisance parameter (possibly of infinite dimensional), \(\nu \) is a \(\sigma \)-finite measure, \(\Theta \) is a bounded open subset of \(\mathbb {R}^{p}\), and \(\Gamma \) is a subset of some Banach or Hilbert space. We employ the Hellinger distance to construct a test statistic. The proposed method results in an explicit form of the test statistic. We show that the proposed test is asymptotically optimal (i.e., locally uniformly most powerful) and has some desirable robustness properties, such as resistance to deviations from the postulated model and in the presence of outliers.     

Exploiting algebraic structure in global optimization and the Belgian chocolate problem

The Belgian chocolate problem involves maximizing a parameter \(\delta \) over a non-convex region of polynomials. In this paper we detail a global optimization method for this problem that outperforms previous such methods by exploiting underlying algebraic structure. Previous work has focused on iterative methods that, due to the complicated non-convex feasible region, may require many iterations or result in non-optimal \(\delta \). By contrast, our method locates the largest known value of \(\delta \) in a non-iterative manner. We do this by using the algebraic structure to go directly to large limiting values, reducing the problem to a simpler combinatorial optimization problem. While these limiting values are not necessarily feasible, we give an explicit algorithm for arbitrarily approximating them by feasible \(\delta \). Using this approach, we find the largest known value of \(\delta \) to date, \(\delta = 0.9808348\). We also demonstrate that in low degree settings, our method recovers previously known upper bounds on \(\delta \) and that prior methods converge towards the \(\delta \) we find.     

Spectral density estimates with partial symmetries and an application to Bahri–Lions-type results

We are concerned with the existence of infinitely many solutions for the problem \(-\Delta u=|u|^{p-2}u+f\) in \(\Omega \), \(u=u_0\) on \(\partial \Omega \), where \(\Omega \) is a bounded domain in \(\mathbb {R}^N\), \(N\ge 3\). This can be seen as a perturbation of the problem with \(f=0\) and \(u_0=0\), which is odd in u. If \(\Omega \) is invariant with respect to a closed strict subgroup of O(N), then we prove infinite existence for all functions f and \(u_0\) in certain spaces of invariant functions for a larger range of exponents p than known before. In order to achieve this, we prove Lieb–Cwikel–Rosenbljum-type bounds for invariant potentials on \(\Omega \), employing improved Sobolev embeddings for spaces of invariant functions.     

An $$L^1$$-Estimate for Certain Spectral Multipliers Associated with the Ornstein–Uhlenbeck Operator

We study a class of spectral multipliers \(\phi (L)\) for the Ornstein–Uhlenbeck operator L arising from the Gaussian measure on \(\mathbb {R}^n\) and find a sufficient condition for integrability of \(\phi (L)f\) in terms of the admissible conical square function and a maximal function.     

Associate space with respect to a locally $$\sigma $$-finite measure on a $$\delta $$δ-ring and applications to spaces of integrable functions defined by a vector measure

We show that for a locally \(\sigma \)-finite measure \(\mu \) defined on a \(\delta \)-ring, the associate space theory can be developed as in the \(\sigma \)-finite case, and corresponding properties are obtained. Given a saturated \(\sigma \)-order continuous \(\mu \)-Banach function space E, we prove that its dual space can be identified with the associate space \(E ^\times \) if, and only if, \(E^\times \) has the Fatou property. Applying the theory to the spaces \(L^p (\nu )\) and \(L_w^p (\nu )\), where \(\nu \) is a vector measure defined on a \(\delta \)-ring \(\mathcal {R}\) and \(1 \le p < \infty \), we establish results corresponding to those of the case when the vector measure is defined on a \(\sigma \)-algebra.