On some results for a class of meromorphic functions having quasiconformal extension

We consider the class \(\Sigma (p)\) of univalent meromorphic functions f on \({\mathbb D}\) having a simple pole at \(z=p\in [0,1)\) with residue 1. Let \(\Sigma _k(p)\) be the class of functions in \(\Sigma (p)\) which have k-quasiconformal extension to the extended complex plane \({\hat{\mathbb C}}\), where \(0\le k < 1\). We first give a representation formula for functions in this class and using this formula, we derive an asymptotic estimate of the Laurent coefficients for the functions in the class \(\Sigma _k(p)\). Thereafter, we give a sufficient condition for functions in \(\Sigma (p)\) to belong to the class \(\Sigma _k(p).\) Finally, we obtain a sharp distortion result for functions in \(\Sigma (p)\) and as a consequence, we obtain a distortion estimate for functions in \(\Sigma _k(p).\)     

Additive iterative roots of identity and Hamel bases

We describe a class of discontinuous additive functions \({a:X\to X}\) on a real topological vector space X such that \({a^n={\rm id}_X}\) and \({a({\mathcal{H}}){\setminus} {\mathcal{H}}\neq\emptyset}\) for every infinite set \({{\mathcal{H}}\subset X}\) of vectors linearly independent over \({\mathbb{Q}}\). We prove the density of the family of all such functions in the linear topological space \({{\mathcal{A}}_X}\) of all additive functions \({a:X\to X}\) with the topology induced on \({{\mathcal{A}}_X}\) by the Tychonoff topology of the space X^X. Moreover, we consider additive functions \({a\in{\mathcal{A}}_X}\) satisfying \({a^n={\rm id}_X}\) and \({a({\mathcal{H}})= {\mathcal{H}}}\) for some Hamel basis \({{\mathcal{H}}}\) of X. We show that the class of all such functions is also dense in \({{\mathcal{A}}_X}\). The method is based on decomposition theorems for linear endomorphisms.     

Gauge functions,Eikonal equations and Bôcher’s theorem on stratified Lie groups

Let \({\mathcal{L} = \sum_{i=1}^m X_i^2}\) be a real sub-Laplacian on a Carnot group \({\mathbb{G}}\) and denote by \({\nabla_\mathcal{L} = (X_1,\ldots,X_m)}\) the intrinsic gradient related to \({\mathcal{L}}\). Our aim in this present paper is to analyze some features of the \({\mathcal{L}}\)-gauge functions on \({\mathbb{G}}\), i.e., the homogeneous functions d such that \({\mathcal{L}(d^\gamma) = 0}\) in \({\mathbb{G} \setminus \{0\}}\) , for some \({\gamma \in \mathbb{R} \setminus \{0\}}\). We consider the relation of \({\mathcal{L}}\)-gauge functions with: the \({\mathcal{L}}\)-Eikonal equation \({|\nabla_\mathcal{L} u| = 1}\) in \({\mathbb{G}}\); the Mean Value Formulas for the \({\mathcal{L}}\)-harmonic functions; the fundamental solution for \({\mathcal{L}}\); the Bôcher-type theorems for nonnegative \({\mathcal{L}}\)-harmonic functions in “punctured” open sets \({\dot \Omega:= \Omega \setminus \{x_0\}}\).     

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.     

On Hecke groups,Schwarzian triangle functions and a class of hyper-elliptic functions

Let m be a positive integer \(\ge \)3 and \(\lambda =2\cos \frac{\pi }{m}\). The Hecke group \(\mathfrak {G}(\lambda )\) is generated by the fractional linear transformations \(\tau + \lambda \) and \(-\frac{1}{\tau }\) for \(\tau \) in the upper half plane \(\mathbb H\) of the complex plane \(\mathbb C\). We consider a set of functions \(\mathfrak {f}_0, \mathfrak {f}_i\) and \(\mathfrak {f}_{\infty }\) automorphic with respect to \(\mathfrak {G}(\lambda )\), constructed from the conformal mapping of the fundamental domain of \(\mathfrak {G}(\lambda )\) to the upper half plane \(\mathbb H\), and establish their connection with the Legendre functions and a class of hyper-elliptic functions. Many well-known classical identities associated with the cases of \(\lambda =1\) and 2 are preserved. As an application, we will establish a set of identities expressing the reciprocal of \(\pi \) in terms of the hypergeometric series.     

On the differentiability of Lipschitz functions with respect to measures in the Euclidean space

For every finite measure \({\mu}\) on \({{\mathbb{R}}^n}\) we define a decomposability bundle \({V(\mu,\,\cdot)}\) related to the decompositions of \({\mu}\) in terms of rectifiable one-dimensional measures. We then show that every Lipschitz function on \({{\mathbb{R}}^n}\) is differentiable at \({\mu}\)-a.e. \({x}\) with respect to the subspace \({V(\mu,\,x)}\), and prove that this differentiability result is optimal, in the sense that, following (Alberti et al., Structure of null sets, differentiability of Lipschitz functions, and other problems, 2016), we can construct Lipschitz functions which are not differentiable at \({\mu}\)-a.e. \({x}\) in any direction which is not in \({V(\mu,\,x)}\). As a consequence we obtain a differentiability result for Lipschitz functions with respect to (measures associated to) \({k}\)-dimensional normal currents, which we use to extend certain basic formulas involving normal currents and maps of class \({C^1}\) to Lipschitz maps.     

Local Maximum Modulus Property for Polyanalytic Functions

Let \(\Omega \subset {\mathbb {C}}\) be an open subset and let \({\mathcal {F}}\) be a space of functions defined on \(\Omega \). \({\mathcal {F}}\) is said to have the local maximum modulus property if: for every \(f\in {\mathcal {F}},p_0\in \Omega ,\) and for every sufficiently small domain \(D\subset \Omega ,\) with \(p_0\in D,\) it holds true that \(\max _{z\in \overline{D}}\left| f(z)\right| = \max _{z\in \Sigma \cup \partial D}\left| f(z)\right| ,\) where \(\Sigma \subset \Omega \) denotes the set of points at which \(\left| f\right| \) attains strict local maximum. This property fails for \({\mathcal {F}}=C^{\infty }.\) We verify it however for the set of complex-valued functions whose real and imaginary parts are real analytic. We show by example that the property cannot be improved upon whenever \({\mathcal {F}}\) is the set of n-analytic functions on \(\Omega \), \(n\ge 2,\) in the sense that locality cannot be removed as a condition and independently \(\Sigma \) cannot be removed from the conclusion.     

On discrete universality of the Riemann zeta-function with respect to uniformly distributed shifts

The Voronin universality theorem asserts that a wide class of analytic functions can be approximated by shifts \(\zeta (s+i\tau )\), \(\tau \in \mathbb {R}\), of the Riemann zeta-function. In the paper, we obtain a universality theorem on the approximation of analytic functions by discrete shifts \(\zeta (s+ix_kh)\), \(k\in \mathbb {N}\), \(h>0\), where \(\{x_k\}\subset \mathbb {R}\) is such that the sequence \(\{ax_k\}\) with every real \(a\ne 0\) is uniformly distributed modulo 1, \(1\le x_k\le k\) for all \(k\in \mathbb {N}\) and, for \(1\le k\), \(m\le N\), \(k\ne m\), the inequality \(|x_k-x_m| \ge y^{-1}_N\) holds with \(y_N> 0\) satisfying \(y_Nx_N\ll N\).     

Rational Approximation of Functions in Hardy Spaces

Subsequent to our recent work on Fourier spectrum characterization of Hardy spaces \(H^p({\mathbb {R}})\) for the index range \(1\le p\le \infty ,\) in this paper we prove further results on rational Approximation, integral representation and Fourier spectrum characterization of functions for the Hardy spaces \(H^p({\mathbb {R}}), 0 < p\le \infty ,\) with particular interest in the index range \( 0< p \le 1.\) We show that the set of rational functions in \( H^p({\mathbb {C}}_{+1}) \) with the single pole \(-i\) is dense in \( H^p({\mathbb {C}}_{+1}) \) for \(0<p<\infty .\) Secondly, for \(0<p<1\), through rational function approximation we show that any function f in \(L^p({\mathbb {R}})\) can be decomposed into a sum \(g+h\), where g and h are, in the \(L^p({\mathbb {R}})\) convergence sense, the non-tangential boundary limits of functions in, respectively, \( H^p({\mathbb {C}}_{+1})\) and \(H^{p}({\mathbb {C}}_{-1}),\) where \(H^p({\mathbb {C}}_k)\ (k=\pm 1) \) are the Hardy spaces in the half plane \( {\mathbb {C}}_k=\{z=x+iy: ky>0\}\). We give Laplace integral representation formulas for functions in the Hardy spaces \(H^p,\) \(0<p\le 2.\) Besides one in the integral representation formula we give an alternative version of Fourier spectrum characterization for functions in the boundary Hardy spaces \(H^p\) for \(0<p\le 1\).     

Gowers $$U_3$$ norm of some classes of bent Boolean functions

The Gowers \(U_3\) norm of a Boolean function is a measure of its resistance to quadratic approximations. It is known that smaller the Gowers \(U_3\) norm for a Boolean function larger is its resistance to quadratic approximations. Here, we compute Gowers \(U_3\) norms for some classes of Maiorana–McFarland bent functions. In particular, we explicitly determine the value of the Gowers \(U_3\) norm of Maiorana–McFarland bent functions obtained by using APN permutations. We prove that this value is always smaller than the Gowers \(U_3\) norms of Maiorana–McFarland bent functions obtained by using differentially \(\delta \)-uniform permutations, for all \(\delta \ge 4\). We also compute the Gowers \(U_3\) norms for a class of cubic monomial functions, not necessarily bent, and show that for \(n=6\), these norm values are less than that of Maiorana–McFarland bent functions. Further, we computationally show that there exist 6-variable functions in this class which are not bent but achieve the maximum second-order nonlinearity for 6 variables.     

The Sharp Weighted Bound for Multilinear Maximal Functions and Calderón–Zygmund Operators

We investigate the weighted bounds for multilinear maximal functions and Calderón–Zygmund operators from \(L^{p_1}(w_1)\times \cdots \times L^{p_m}(w_m)\) to \(L^{p}(v_{\vec {w}})\), where \(1<p_1,\cdots ,p_m<\infty \) with \(1/{p_1}+\cdots +1/{p_m}=1/p\) and \(\vec {w}\) is a multiple \(A_{\vec {P}}\) weight. We prove the sharp bound for the multilinear maximal function for all such \(p_1,\ldots , p_m\) and prove the sharp bound for \(m\)-linear Calderón–Zymund operators when \(p\ge 1\).     

Reflection positivity on real intervals

We study functions \(f : (a,b) \rightarrow {{\mathbb {R}}}\) on open intervals in \({{\mathbb {R}}}\) with respect to various kinds of positive and negative definiteness conditions. We say that f is positive definite if the kernel \(f\big (\frac{x + y}{2}\big )\) is positive definite. We call f negative definite if, for every \(h > 0\), the function \(e^{-hf}\) is positive definite. Our first main result is a Lévy–Khintchine formula (an integral representation) for negative definite functions on arbitrary intervals. For \((a,b) = (0,\infty )\) it generalizes classical results by Bernstein and Horn. On a symmetric interval \((-a,a)\), we call f reflection positive if it is positive definite and, in addition, the kernel \(f\big (\frac{x - y}{2}\big )\) is positive definite. We likewise define reflection negative functions and obtain a Lévy–Khintchine formula for reflection negative functions on all of \({{\mathbb {R}}}\). Finally, we obtain a characterization of germs of reflection negative functions on 0-neighborhoods in \({{\mathbb {R}}}\).     

Sampling and Cubature on Sparse Grids Based on a B-spline Quasi-Interpolation

Let \(X_n = \{x^j\}_{j=1}^n\) be a set of n points in the d-cube \({\mathbb {I}}^d:=[0,1]^d\), and \(\Phi _n = \{\varphi _j\}_{j =1}^n\) a family of n functions on \({\mathbb {I}}^d\). We consider the approximate recovery of functions f on \({{\mathbb {I}}}^d\) from the sampled values \(f(x^1), \ldots , f(x^n)\), by the linear sampling algorithm \( L_n(X_n,\Phi _n,f) := \sum _{j=1}^n f(x^j)\varphi _j. \) The error of sampling recovery is measured in the norm of the space \(L_q({\mathbb {I}}^d)\)-norm or the energy quasi-norm of the isotropic Sobolev space \(W^\gamma _q({\mathbb {I}}^d)\) for \(1 < q < \infty \) and \(\gamma > 0\). Functions f to be recovered are from the unit ball in Besov-type spaces of an anisotropic smoothness, in particular, spaces \(B^{\alpha ,\beta }_{p,\theta }\) of a “hybrid” of mixed smoothness \(\alpha > 0\) and isotropic smoothness \(\beta \in {\mathbb {R}}\), and spaces \(B^a_{p,\theta }\) of a nonuniform mixed smoothness \(a \in {\mathbb {R}}^d_+\). We constructed asymptotically optimal linear sampling algorithms \(L_n(X_n^*,\Phi _n^*,\cdot )\) on special sparse grids \(X_n^*\) and a family \(\Phi _n^*\) of linear combinations of integer or half integer translated dilations of tensor products of B-splines. We computed the asymptotic order of the error of the optimal recovery. This construction is based on B-spline quasi-interpolation representations of functions in \(B^{\alpha ,\beta }_{p,\theta }\) and \(B^a_{p,\theta }\). As consequences, we obtained the asymptotic order of optimal cubature formulas for numerical integration of functions from the unit ball of these Besov-type spaces.     

Traces of the Nevanlinna Class on Discrete Sequences

We show that a discrete sequence \(\Lambda \) of the unit disk is the union of n interpolating sequences for the Nevanlinna class \(\mathcal {N}\) if and only if the trace of \(\mathcal {N}\) on \(\Lambda \) coincides with the space of functions on \(\Lambda \) for which the divided differences of order \(n-1\) are uniformly controlled by a positive harmonic function.     

On the admissible sets of type $\mathbb{H}\mathbb{Y}\mathbb{P}(\mathfrak{M})$ over recursively saturated models

Some effective expression is obtained for the elements of an admissible set \(\mathbb{H}\mathbb{Y}\mathbb{P}(\mathfrak{M})\) as template sets. We prove the Σ-reducibility of \(\mathbb{H}\mathbb{Y}\mathbb{P}(\mathfrak{M})\) to \(\mathbb{H}\mathbb{F}(\mathfrak{M})\) for each recursively saturated model \(\mathfrak{M}\) of a regular theory, give a criterion for uniformization in \(\mathbb{H}\mathbb{Y}\mathbb{P}(\mathfrak{M})\) for each recursively saturated model \(\mathfrak{M}\), and establish uniformization in \(\mathbb{H}\mathbb{Y}\mathbb{P}(\mathfrak{N})\) and \(\mathbb{H}\mathbb{Y}\mathbb{P}(\Re ')\), where \(\mathfrak{N}\) and \(\Re '\) are recursively saturated models of arithmetic and real closed fields. We also prove the absence of uniformization in \(\mathbb{H}\mathbb{F}(\mathfrak{M})\) and \(\mathbb{H}\mathbb{Y}\mathbb{P}(\mathfrak{M})\) for each countably saturated model \(\mathfrak{M}\) of an uncountably categorical theory, and give an example of this type of theory with definable Skolem functions. Furthermore, some example is given of a model of a regular theory with Σ-definable Skolem functions, but lacking definable Skolem functions in every extension by finitely many constants.     

On the Multiplication Tuples Related to Certain Reproducing Kernel Hilbert Spaces

We consider a class of subnormal operator tuples \(M_z\) consisting of multiplications by coordinate functions on a class of reproducing kernel Hilbert spaces associated with certain bounded domains \(\Omega \) in \({\mathbb {C}}^m\), with the closure \({{\bar{\Omega }}}\) of \(\Omega \) being the Taylor spectrum of \(M_z\) and the topological boundary \(\partial \Omega \) of \(\Omega \) being the Taylor essential spectrum of \(M_z\). If T is a subnormal operator tuple quasisimilar to \(M_z\), then we show that the Taylor spectrum of T is \({{\bar{\Omega }}}\) provided \({{\bar{\Omega }}}\) is polynomially convex and provided \(\Omega \) is either strictly pseudoconvex with \(C^2\) boundary or is starlike, and that the Taylor essential spectrum of T is \(\partial \Omega \) provided \(\Omega \) satisfies the Gleason property as well. This generalizes some previous work of the first-named author in the context of the unit ball and the unit polydisk. The relevant theory is then applied to the multiplication tuples on the Hardy and Bergman spaces of complex ellipsoids.     

Families of Strictly Pseudoconvex Domains and Peak Functions

We prove that given a family \((G_t)\) of strictly pseudoconvex domains varying in \(\mathcal {C}^2\) topology on domains, there exists a continuously varying family of peak functions \(h_{t,\zeta }\) for all \(G_t\) at every \(\zeta \in \partial G_t\).     

Linear extensions of some Baire-one functions

Let X be a Hausdorff topological space, and let \({\mathscr {B}}_1(X)\) denote the space of all real Baire-one functions defined on X. Let A be a nonempty subset of X endowed with the topology induced from X, and let \({\mathscr {F}}(A)\) be the set of functions \(A\rightarrow {\mathbb R}\) with a property \({\mathscr {F}}\) making \({\mathscr {F}}(A)\) a linear subspace of \({\mathscr {B}}_1(A)\). We give a sufficient condition for the existence of a linear extension operator \(T_A:{\mathscr {F}}(A)\rightarrow {\mathscr {F}}(X)\), where \({\mathscr {F}}\) means to be piecewise continuous on a sequence of closed and \(G_\delta \) subsets of X and is denoted by \({\mathscr {P}_0}\). We show that \(T_A\) restricted to bounded elements of \({\mathscr {F}}(A)\) endowed with the supremum norm is an isometry. As a consequence of our main theorem, we formulate the conclusion about existence of a linear extension operator for the classes of Baire-one-star and piecewise continuous functions.     

{{C^m}} Positive Eigenvectors for Linear Operators Arising in the Computation of Hausdorff Dimension

We consider a broad class of linear Perron–Frobenius operators \({\Lambda:X \rightarrow X}\), where \({X}\) is a real Banach space of \({C^m}\) functions. We prove the existence of a strictly positive \({C^m}\) eigenvector \({v}\) with eigenvalue \({r=r(\Lambda) =}\) the spectral radius of \({\Lambda}\). We prove (see Theorem 6.5 in Sect. 6 of this paper) that \({r(\Lambda)}\) is an algebraically simple eigenvalue and that, if \({\sigma(\Lambda)}\) denotes the spectrum of the complexification of \({\Lambda,\sigma(\Lambda) \backslash \{r(\Lambda)\}\subseteq \{\zeta \in \mathbb{C} \big| |\zeta| \le r_*\}}\), where \({r_* < r(\Lambda)}\). Furthermore, if \({u \in X}\) is any strictly positive function, \({(\frac 1r \Lambda)^k(u) \rightarrow s_u v}\) as \({k \rightarrow \infty}\), where \({s_u > 0}\) and convergence is in the norm topology on \({X}\). In applications to the computation of Hausdorff dimension, one is given a parametrized family \({\Lambda_s,s > s_*}\), of such operators and one wants to determine the (unique) value \({s_0}\) such that \({r(\Lambda_{s_0})=1}\). In another paper (Falk and Nussbaum in C\({^{\rm m}}\) Eigenfunctions of Perron–Frobenius operators and a new approach to numerical computation of Hausdorff dimension, submitted) we prove that explicit estimates on the partial derivatives of the positive eigenvector \({v_s}\) of \({\Lambda_s}\) can be obtained and that this information can be used to give rigorous, sharp upper and lower bounds for \({s_0}\).     

Toric Aspects of the First Eigenvalue

In this paper we study the smallest non-zero eigenvalue \(\lambda _1\) of the Laplacian on toric Kähler manifolds. We find an explicit upper bound for \(\lambda _1\) in terms of moment polytope data. We show that this bound can only be attained for \(\mathbb C\mathbb P^n\) endowed with the Fubini–Study metric and therefore \(\mathbb C\mathbb P^n\) endowed with the Fubini–Study metric is spectrally determined among all toric Kähler metrics. We also study the equivariant counterpart of \(\lambda _1\) which we denote by \(\lambda _1^T\). It is the smallest non-zero eigenvalue of the Laplacian restricted to torus-invariant functions. We prove that \(\lambda _1^T\) is not bounded among toric Kähler metrics thus generalizing a result of Abreu–Freitas on \(S^2\). In particular, \(\lambda _1^T\) and \(\lambda _1\) do not coincide in general.