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Uncertainty principles for the windowed offset linear canonical transformJul 15 2019The windowed offset linear canonical transform (WOLCT) can be identified as a generalization of the windowed linear canonical transform (WLCT). In this paper, we generalize several different uncertainty principles for the WOLCT, including Heisenberg uncertainty ... More

Creating and Flattening Cusp Singularities by Deformations of Bi-conformal EnergyJul 15 2019Mappingsofbi-conformalenergyformthewidestclass of homeomorphisms that one can hope to build a viable extension of Geometric Function Theory with connections to mathematical models of Nonlinear Elasticity. Such mappings are exactly the ones with finite ... More

$C^1$-smooth dependence on initial conditions and delay: spaces of initial histories of Sobolev type, and differentiability of translation in $L^p$Jul 14 2019The objective of this paper is to clarify the relationship between the $C^1$-smooth dependence of solutions to delay differential equations (DDEs) on initial histories (i.e., initial conditions) and delay parameters. For this purpose, we consider a class ... More

Improved Bounds for Hermite-Hadamard Inequalities in Higher DimensionsJul 13 2019Let $\Omega \subset \mathbb{R}^n$ be a convex domain and let $f:\Omega \rightarrow \mathbb{R}$ be a positive, subharmonic function (i.e. $\Delta f \geq 0$). Then $$ \frac{1}{|\Omega|} \int_{\Omega}{f dx} \leq \frac{c_n}{ |\partial \Omega| } \int_{\partial ... More

Commutators of potential type operators with Lipschitz symbols on variable Lebesgue spaces with different weightsJul 12 2019We prove that a generalized Fefferman-Phong type condition on a pair of weights $u$ and $v$ is sufficient for the boundedness of the commutators of potential type operators from $L^{p(\cdot)}_v$ into $L^{q(\cdot)}_u$. We also give an improvement of this ... More

The spectral matrices associated with the stochastic Darboux transformations of random walks on the integersJul 12 2019We consider UL and LU stochastic factorizations of the transition probability matrix of a random walk on the integers, which is a doubly infinite tridiagonal stochastic Jacobi matrix. We give conditions on the free parameter of both factorizations in ... More

Averages along the Square Integers: $\ell^p$ improving and Sparse InequalitiesJul 12 2019Let $f\in \ell^2(\mathbb Z)$. Define the average of $ f$ over the square integers by $ A_N f(x):=\frac{1}{N}\sum_{k=1}^N f(x+k^2) $. We show that $ A_N$ satisfies a local scale-free $ \ell ^{p}$-improving estimate, for $ 3/2 < p \leq 2$: \begin{equation*} ... More

Some geometric properties of Riemann's non-differentiable functionJul 12 2019Riemann's non-differentiable function is a celebrated example of a continuous but almost nowhere differentiable function. There is strong numeric evidence that one of its complex versions represents a geometric trajectory in experiments related to the ... More

A Sturm Liouville theorem for quadratic operator pencilsJul 12 2019We establish a Sturm{Liouville theorem for quadratic operator pencils counting their unstable real roots, with applications to stability of waves. Such pencils arise, for example, in reduction of eigenvalue systems to higher-order scalar problems.

Sequences of positive homoclinic solutions to difference equations with variable exponentJul 11 2019We study the existence of infinitely many positive homoclinic solutions to a second-order difference equation on integers with $p_k$-Laplacian. To achieve our goal we use the critical point theory and the general variational principle of Ricceri.

Analytic and Probabilistic Problems in Discrete GeometryJul 11 2019The thesis concentrates on two problems in discrete geometry, whose solutions are obtained by analytic, probabilistic and combinatoric tools. The first chapter deals with the strong polarization problem. This states that for any sequence $u_1,\dots, u_n$ ... More

Continuous changes of variables and the Magnus expansionJul 11 2019In this paper, we are concerned with a formulation of Magnus and Floquet-Magnus expansions for general nonlinear differential equations. To this aim, we introduce suitable continuous variable transformations generated by operators. As an application of ... More

Hodge-theoretic analysis on manifolds with boundary, heatable currents, and Onsager's conjecture in fluid dynamicsJul 11 2019We use Hodge theory and functional analysis to develop a clean approach to heat flows and Onsager's conjecture on Riemannian manifolds with boundary, where the weak solution lies in the trace-critical Besov space $B_{3,1}^{\frac{1}{3}}$. We also introduce ... More

Growth Equation of the General Fractional CalculusJul 11 2019We consider the Cauchy problem $(\mathbb D_{(k)} u)(t)=\lambda u(t)$, $u(0)=1$, where $\mathbb D_{(k)}$ is the general convolutional derivative introduced in the paper (A. N. Kochubei, Integral Equations Oper. Theory {\bf 71} (2011), 583--600), $\lambda ... More

Box-counting by Hölder's traveling salesmanJul 11 2019We provide a sufficient Dini-type condition for a subset of a complete, quasiconvex metric space to be covered by a H\"older curve. This implies in particular that if the upper box-counting dimension of a set in a quasiconvex metric space is less or equal ... More

Global solvability criteria for quaternionic Riccati equationsJul 11 2019Some global existence criteria for quaternionic Riccati equations are established. Two of them are used to prove a completely non conjugation theorem for solutions of linear systems of ordinary differential equations.

Collective dynamics of opposing groups with stochastic communicationJul 11 2019We propose models describing the collective dynamics of two opposing groups of individuals with stochastic communication. Individuals from the same group are assumed to align in a stochastic manner, while individuals from different groups are assumed ... More

Discrepancy of minimal Riesz energy pointsJul 10 2019We find upper bounds for the spherical cap discrepancy of the set of minimizers of the Riesz $s$-energy on the sphere $\mathbb S^d.$ Our results are based in bounds for a Sobolev discrepancy introduced by Thomas Wolff in an unpublished manuscript where ... More

Ultracontractive Properties for Directed Graph Semigroups with Applications to Coupled OscillatorsJul 10 2019It is now well known that ultracontractive properties of semigroups with infinitesimal generator given by an undirected graph Laplacian operator can be obtained through an understanding of the geometry of the underlying infinite weighted graph. The aim ... More

Endpoint estimates for the maximal function over prime numbersJul 10 2019Given an ergodic dynamical system $(X, \mathcal{B}, \mu, T)$, we prove that for each function $f$ belonging to the Orlicz space $L(\log L)^2(\log \log L)(X, \mu)$, the ergodic averages \[ \frac{1}{\pi(N)} \sum_{p \in \mathbb{P}_N} f\big(T^p x\big), \] ... More

Higher Trigonometry: A Class Of Nonlinear SystemsJul 10 2019We study the initial value problem '$s\,' = c^{p - 1}, \; c\,' = -s^{p - 1}; \; \; s(0) = 0, \; c(0) = 1$' (both as a real system and as a complex system) for each integer $p > 2$, considering separately the cases '$p$ even' and '$p$ odd'.

The Fox-Wright function near the singularity and branch cutJul 10 2019The Fox-Wright function is a further extension of the generalized hypergeometric function obtained by introducing arbitrary positive scaling factors into the arguments of the gamma functions in the summand. Its importance comes mostly from its role in ... More

Riesz bases of exponentials for convex polytopes with symmetric facesJul 10 2019We prove that for any convex polytope $\Omega \subset \mathbb{R}^d$ which is centrally symmetric and whose faces of all dimensions are also centrally symmetric, there exists a Riesz basis of exponential functions in the space $L^2(\Omega)$.

New $Ψ$-Hadamard type fractional integral and derivativesJul 10 2019This paper develops a generalization of Hadamard type fractional calculus which has been named as the $ \Psi $-Hadamard type fractional calculus. Conditions are given under which the $\Psi$-Hadamard type fractional integral is bounded in a generalized ... More

On $Ψ$-Laplace transform method and its applications to $Ψ$-fractional differential equationsJul 10 2019Motivated by some recent developments in $\Psi$-fractional calculus, in this paper some new properties and the uniqueness of $ \Psi $-Laplace transform in the settings of $ \Psi $-fractional calculus are established. The final goal of this research is ... More

Generalized substantial fractional operators and well-posedness of Cauchy problemJul 10 2019In this work we focus on substantial fractional integral and differential operators which play an important role in modeling anomalous diffusion. We introduce a new generalized substantial fractional integral. Generalizations of fractional substantial ... More

Complete monotonicity of a ratio of gamma functions and some combinatorial inequalities for multinomial coefficientsJul 10 2019Let $m,n\in \mathbb{N}$. For all $i\in \{1,2,\dots,m\}$, $j\in \{1,2,\dots,n\}$, choose constants $X_i,Y_j > 0$, $M\in (0,1]$ and let $(x_{ij})\in (0,\infty)^{m\times n}$ be such that $\sum_{j=1}^n x_{ij} = X_i$, $\sum_{i=1}^m x_{ij} = Y_j$ and $\sum_{i=1}^m ... More

Spectral sets in $\Z_{p^2qr}$ tileJul 09 2019We prove the every spectral set in $\Z_{p^2qr}$ tiles, where $p$, $q$ and $r$ are primes, which is a special case of Fuglede's conjecture for cyclic groups.

Revisiting Biorthogonal Polynomials. An $LU$ factorization discussionJul 09 2019The Gauss-Borel or $LU$ factorization of Gram matrices of bilinear forms is the pivotal element in the discussion of the theory of biorthogonal polynomials. The construction of biorthogonal families of polynomials and its second kind functions, of the ... More

Approximation of Hausdorff operatorsJul 09 2019Truncating the Fourier transform averaged by means of a generalized Hausdorff operator, we approximate the adjoint to that Hausdorff operator of the given function. We find the formulas for the rate of approximation in various metrics in terms of the ... More

Estimates of the asymptotic Nikolskii constants for spherical polynomialsJul 08 2019Let $\Pi_n^d$ denote the space of spherical polynomials of degree at most $n$ on the unit sphere $\mathbb{S}^d\subset \mathbb{R}^{d+1}$ that is equipped with the surface Lebesgue measure $d\sigma$ normalized by $\int_{\mathbb{S}^d} \, d\sigma(x)=1$. This ... More

the center of distances for some multigeometric seriesJul 08 2019Basic properties of the center of distances of a set are investigated. Computation of the center for achievement sets is particularly aimed at. A new sufficient condition for the center of distances of the set of subsums of a fast convergent series to ... More

The Haar System in Triebel-Lizorkin Spaces: Endpoint ResultsJul 08 2019We determine for which parameters natural enumerations of the Haar system in $\mathbb{R}^d$ form a Schauder basis or basic sequence on Triebel-Lizorkin spaces. The new results concern the endpoint cases.

The Weyl formula for planar annuliJul 08 2019We study the zeros of cross-product of Bessel functions and obtain their approximations, based on which we reduce the eigenvalue counting problem for the Dirichlet Laplacian associated with a planar annulus to a lattice point counting problem associated ... More

Some classes of generating functions for generalized Hermite- and Chebyshev-type polynomials: Analysis of Euler's formulaJul 08 2019The aim of this paper is to construct generating functions for new families of special polynomials including the Appel polynomials, the Hermite-Kamp\`e de F\`eriet polynomials, the Milne-Thomson type polynomials, parametric kinds of Apostol type numbers ... More

Some discontinuous functional differential equation and its connection to smoothness of composition operators in $L^p$Jul 08 2019The objective of this paper is to deepen the understanding of the connection between the continuous and smooth dependence of solutions on initial conditions and the regularity of the history functionals for retarded functional differential equations. ... More

Remarks about the existence of conformable derivatives and some consequencesJul 08 2019Jul 09 2019The aim of the present paper is to make some notes to the newly introduced conformable derivative as a type local fractional derivative and to present a surprising result about the relation between the conformable derivatives and the usual integer order ... More

Szegő's Theorem for Canonical Systems: the Arov Gauge and a Sum RuleJul 07 2019We consider canonical systems and investigate the Szeg\H{o} class, which is defined via the finiteness of the associated entropy functional. Noting that the canonical system may be studied in a variety of gauges, we choose to work in the Arov gauge, in ... More

Kantorovich's Mass Transport Problem for CapacitiesJul 07 2019The aim of the present paper is to extend Kantorovich's mass transport problem to the framework of upper continuous capacities and to prove the cyclic monotonicity of the supports of optimal solutions. As in the probabilistic case, this easily yields ... More

A new family of series expansions for $1/π$ and a binomial identityJul 06 2019A doubly infinite set of series expansion for $1/\pi$ are reported. They follow trivially from a formal expansion for the quotient of the values taken by the gamma function for two (complex) arguments differing by an integer plus one half, obtained by ... More

Riemann-Hilbert Problem for the Matrix Laguerre Biorthogonal Polynomials: Eigenvalue Problems and the Matrix Discrete Painlevé IVJul 06 2019In this paper the Riemann-Hilbert problem, with jump supported on a appropriate curve on the complex plane with a finite endpoint at the origin, is used for the study of corresponding matrix biorthogonal polynomials associated with Laguerre type matrices ... More

Large and small values of quadratic Weyl sumsJul 06 2019We prove that for almost all (in the sense of Lebesgue measure) $(x_1, x_2)\in [0,1)^2$ one has $$ \overline{\lim}_{N\to \infty} \left| \sum_{n=1}^N \exp (2\pi i (x_1n+x_2n^2)) \right|N^{-1/2} (\log \log N)^{-1/6}=\infty. $$ For the lower limit we show ... More

A new study on the mild solution for impulsive fractional evolution equationsJul 06 2019In this article, we consider mild solutions to a class of impulsive fractional evolution equations of order $0<\alpha<1$. After analyzing analytic results reported in the literature using Mittag-Leffer function, $\alpha$-resolvent operator theory, we ... More

Analyticity of the spectrum and Dirichlet-to-Neumann operator technique for quantum graphsJul 05 2019In some previous works, the analytic structure of the spectrum of a quantum graph operator as a function of the vertex conditions and other parameters of the graph was established. However, a specific local coordinate chart on the Grassmanian of all possible ... More

Nikishin systems on star-like sets: Ratio asymptotic formulae for the associated multiple orthogonal polynomialsJul 05 2019In this paper we continue the investigations initiated in \cite{LopLopstar} on ratio asymptotics of multiple orthogonal polynomials and functions of the second kind associated with Nikishin systems on star-like sets. We describe in detail the limiting ... More

Constructive proof of Herschfeld's Convergence TheoremJul 05 2019We provide a constructive proof of Herschfeld's Convergence Theorem. We also discuss the role of the Monotone Convergence Theorem in Herschfeld's original argument, and speculate on whether there are general principles for constructivising arguments that ... More

Characterizations of Hardy spaces for Fourier integral operatorsJul 05 2019We prove several characterizations of the Hardy spaces for Fourier integral operators $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$, for $1<p<\infty$. First we characterize $\mathcal{H}^{p}_{FIO}(\mathbb{R}^{n})$ in terms of $L^{p}(\mathbb{R}^{n})$-norms of ... More

The Newton integral and the Stirling formulaJul 04 2019We present details of logically simplest integral sufficient for deducing the Stirling asymptotic formula for n!. It is the Newton integral, defined as the difference of values of any primitive at the endpoints of the integration interval. We review in ... More

Singular integrals along lacunary directions in $\mathbb{R}^n$Jul 04 2019A recent result by Parcet and Rogers is that finite order lacunarity characterizes the boundedness of the maximal averaging operator associated to an infinite set of directions in $\mathbb{R}^n$. Their proof is based on geometric-combinatorial coverings ... More

Morrey spaces for Schrödinger operators with nonnegative potentials, fractional integral operators and the Adams inequality on the Heisenberg groupsJul 04 2019Let $\mathcal L=-\Delta_{\mathbb H^n}+V$ be a Schr\"odinger operator on the Heisenberg group $\mathbb H^n$, where $\Delta_{\mathbb H^n}$ is the sublaplacian on $\mathbb H^n$ and the nonnegative potential $V$ belongs to the reverse H\"older class $RH_s$ ... More

Lineability and modes of convergenceJul 03 2019In this paper we look for the existence of large linear and algebraic structures of sequences of measurable functions with different modes of convergence. Concretely, the algebraic size of the family of sequences that are convergent in measure but not ... More

Mild and strong solutions for Hilfer evolution equationJul 03 2019In this paper, we investigate the existence and uniqueness of mild and strong solutions of fractional semilinear evolution equations in the Hilfer sense, by means of Banach fixed point theorem and the Gronwall inequality.

Asymptotic expoansions of mathieu-Bessel series. IJul 03 2019Jul 07 2019We consider the asymptotic expansion of the Mathieu-Bessel series \[S_\nu(a,b)=\sum_{n=1}^\infty \frac{n^\gamma J_\nu(nb/a)}{(n^2+a^2)^\mu}, \qquad (\mu, b>0,\ \gamma, \nu\in {\bf R})\] as $a\to+\infty$ with the other parameters held fixed, where $J_\nu(x)$ ... More

Asymptotic expoansions of mathieu-Bessel series. IJul 03 2019We consider the asymptotic expansion of the Mathieu-Bessel series \[S_\nu(a,b)=\sum_{n=1}^\infty \frac{n^\gamma J_\nu(nb/a)}{(n^2+a^2)^\mu}, \qquad (\mu, b>0,\ \gamma, \nu\in {\bf R})\] as $a\to+\infty$ with the other parameters held fixed, where $J_\nu(x)$ ... More

Delayed Langevin type equations with two fractional derivativesJul 03 2019In this paper, we introduce a delayed Mittag-Leffler type function. With the help of the delayed Mittag-Leffler type functions, we give an explicit formula of solutions to linear nonhomogeneous fractional time-delay Langevin equations involving two Riemann-Liouville ... More

Zero Spacings of Paraorthogonal Polynomials on the Unit CircleJul 02 2019We prove some new results about the spacing between neighboring zeros of paraorthogonal polynomials on the unit circle. Our methods also provide new proofs of some existing results. The main tool we will use is a formula for the phase of the appropriate ... More

Caracterization of Sobolev spaces on the sphereJul 02 2019We introduce a new characterization of the Sobolev spaces $H^\alpha$ on the unit sphere $\mathbb{S}^{d-1}$, where the smoothness index $\alpha$ is any positive real number and $d\geq 2$. This characterization is in terms of a square function, it avoids ... More

Characterization of Sobolev spaces on the sphereJul 02 2019Jul 04 2019We introduce a new characterization of the Sobolev spaces $H^\alpha$ on the unit sphere $\mathbb{S}^{d-1}$, where the smoothness index $\alpha$ is any positive real number and $d\geq 2$. This characterization is in terms of a square function, it avoids ... More

Semi-classical analysis of piecewise quasi-polynomial functionsJul 02 2019Motivated by applications to multiplicity formulas in index theory, we study a family of distributions $\Theta(m;k)$ associated to a piecewise quasi-polynomial function $m$. The family is indexed by an integer $k>0$, and admits an asymptotic expansion ... More

Trace formulas and continuous dependence of spectra for the periodic conservative Camassa-Holm flowJul 02 2019This article is concerned with the isospectral problem \[ -f'' + \frac{1}{4} f = z\omega f + z^2 \upsilon f \] for the periodic conservative Camassa-Holm flow, where $\omega$ is a periodic real distribution in $H^{-1}_{\mathrm{loc}}(\mathbb{R})$ and $\upsilon$ ... More

Absolute root separationJul 02 2019The absolute separation of a polynomial is the minimum nonzero difference between the absolute values of its roots. In the case of polynomials with integer coefficients, it can be bounded from below in terms of the degree and the height (the maximum absolute ... More

Control in the spaces of ensembles of pointsJul 01 2019We study the controlled dynamics of the {\it ensembles of points} of a Riemannian manifold $M$. Parameterized ensemble of points of $M$ is the image of a continuous map $\gamma:\Theta \to M$, where $\Theta$ is a compact set of parameters. The dynamics ... More

Control in the spaces of ensembles of pointsJul 01 2019Jul 05 2019We study the controlled dynamics of the {\it ensembles of points} of a Riemannian manifold $M$. Parameterized ensemble of points of $M$ is the image of a continuous map $\gamma:\Theta \to M$, where $\Theta$ is a compact set of parameters. The dynamics ... More

Lipschitz one sets modulo sets of measure zeroJul 01 2019We denote the local ``little" and ``big" Lipschitz functions of a function $f: {{\mathbb R}}\to {{\mathbb R}}$ by $ {\mathrm {lip}}f$ and $ {\mathrm {Lip}}f$. In this paper we continue our research concerning the following question. Given a set $E {\subset} ... More

On pointwise $\ell^r$-sparse domination in a space of homogeneous typeJul 01 2019We prove a general sparse domination theorem in spaces of homogeneous type, in which we control a vector-valued operator pointwise by a positive, local expression called a sparse operator. We use the structure of the operator to get sparse domination ... More

Propagation of chaos for mean field rough differential equationsJul 01 2019We address propagation of chaos for large systems of rough differential equations associated with random rough differential equations of mean field type $$ dX_t = V(X_t,\mathcal{L}(X_t))dt + F(X_t,\mathcal{L}(X_t))dW_t $$ where $W$ is a random rough path ... More

A Boundedness Criterion for Singular Integral Operators of convolution type on the Fock SpaceJul 01 2019We show that for an entire function $\varphi$ belonging to the Fock space ${\mathscr F}^2(\mathbb{C}^n)$ on the complex Euclidean space $\mathbb{C}^n$, that the integral operator \begin{eqnarray*} S_{\varphi}F(z)= \frac{1}{\pi^n}\int_{\mathbb{C}^n} F(w) ... More

A Boundedness Criterion for Singular Integral Operators of convolution type on the Fock SpaceJul 01 2019Jul 12 2019We show that for an entire function $\varphi$ belonging to the Fock space ${\mathscr F}^2(\mathbb{C}^n)$ on the complex Euclidean space $\mathbb{C}^n$, that the integral operator \begin{eqnarray*} S_{\varphi}F(z)= \frac{1}{\pi^n}\int_{\mathbb{C}^n} F(w) ... More

Discrete analogues of maximally modulated singular integrals of Stein-Wainger type: $\ell^2(\mathbb{Z}^n)$ boundsJun 30 2019Consider the maximal operator $$\mathscr{C} f(x) = \sup_{\lambda\in\mathbb{R}}\Big|\sum_{\substack{y\in\mathbb{Z}^n\setminus\{0\}}} f(x-y) e^{i \lambda |y|^{2d}} K(y)\Big|,\quad (x\in\mathbb{Z}^n),$$ where $d$ is a positive integer, $K$ an appropriate ... More

New identities for some symmetric polynomials and their applicationsJun 30 2019We give new identities for some symmetric polynomials: elementary, complete homogeneous, power symmetric polynomials. As applications of these identities, we obtain some formulas for a higher order analogue of Fibonacci and Lucas numbers.

A Generalization of Fourier Series occurring in Atomic TheoryJun 29 2019A number of the Fourier Series which occur in the theory of the semi-classical atom due to Englert and Schwinger are generalized and presented.

Neural ODEs as the Deep Limit of ResNets with constant weightsJun 28 2019In this paper we prove that, in the deep limit, the stochastic gradient descent on a ResNet type deep neural network, where each layer share the same weight matrix, converges to the stochastic gradient descent for a Neural ODE and that the corresponding ... More

Fourier decay for self-similar measuresJun 28 2019We prove that, after removing a zero Hausdorff dimension exceptional set of parameters, all self-similar measures on the line have a power decay of the Fourier transform at infinity.

A note on Lyapunov instability in Newtonian dynamicsJun 27 2019In the class of analytic potentials, we give a new sufficient condition for the Lyapunov instability of a local minimum of the potential. In contrast with similar analytical results concerning the first non zero jet of the potential, this new condition ... More

The continuous primitive integral in the planeJun 27 2019An integral is defined on the plane that includes the Henstock--Kurzweil and Lebesgue integrals (with respect to Lebesgue measure). A space of primitives is taken as the set of continuous real-valued functions $F(x,y)$ defined on the extended real plane ... More

Equi-distributed property and spectral set conjecture on $\mathbb{Z}_{p^2}\times \mathbb{Z}_{p}$Jun 27 2019In this paper, we show an equi-disctributed property in $2$-dimensional finite abelian groups $\mathbb{Z}_{p^2}\times \mathbb{Z}_{p}$ where $p$ is a prime number. By using this equi-disctributed property, we prove that Fuglede's spectral set conjecture ... More

Bi-parameter Carleson embeddings with product weightsJun 26 2019Lennart Carleson showed in 1974 that the natural generalization of Carleson measure from one parameter case (disc) to bi-parameter case (bi-disc) does not work. Sun-Yang A. Chang in 1979 found the necessary and sufficient condition for the Carleson embedding ... More

A combinatorial property of planar measures and bi-parameter Carleson embeddings with product weightsJun 26 2019Jun 28 2019Lennart Carleson showed in 1974 that the natural generalization, using a "box" condition, from the one parameter case (disc) to the bi-parameter case (bi-disc) of his embedding theorem does not work. Sun-Yang A. Chang in 1979 found the necessary and sufficient ... More

Counterexamples for bi-parameter Carleson embeddingJun 26 2019Jun 28 2019We build here several counterexamples for two weight bi-parameter Carleson embedding theorem.

Combinatorics of planar measures and bi-parameter Carleson embeddingJun 26 2019The main result below is Theorem 1.4 that shows an unexpected property of any positive planar measure. This property goes, on the first glance, against a famous Carleson's counterexample, and against the obvious geometric property of huge overlap among ... More

Assouad dimension of planar self-affine setsJun 26 2019We calculate the Assouad dimension of a planar self-affine set $X$ satisfying the strong separation condition and the projection condition and show that $X$ is minimal for the conformal Assouad dimension. Furthermore, we see that such a self-affine set ... More

Two-weight commutator estimates: general multi-parameter frameworkJun 26 2019We provide an explicit technical framework for proving very general two-weight commutator estimates in arbitrary parameters. The aim is to both clarify existing literature, which often explicitly focuses on two parameters only, and to extend very recent ... More

Singularities of rational inner functions in higher dimensionsJun 26 2019We study the boundary behavior of rational inner functions (RIFs) in dimensions three and higher from both analytic and geometric viewpoints. On the analytic side, we use the critical integrability of the derivative of a rational inner function of several ... More

$Ω$-symmetric measures and related singular integralsJun 26 2019Let $\mathbb{S} \subset \mathbb{C}$ be the circle in the plane, and let $\Omega: \mathbb{S} \to \mathbb{S}$ be an odd bi-Lipschitz map with constant $1+\delta_\Omega$, where $\delta_\Omega>0$ is small. Assume also that $\Omega$ is twice continuously differentiable. ... More

Geometry on all prime Three ManifoldsJun 26 2019The point of this work is to construct geometric structures on the oriented closed prime three manifolds that don't at present already have them. One knows these compound prime three manifolds, have canonically up to deformation from the identity, incompressible ... More

A limited-range Calderón-Zygmund theoremJun 25 2019We work with singular integral operators whose kernels satisfy a condition weaker than the typical H\"ormander smoothness estimate. We give two proofs of a weak-type $(q,q)$ inequality for these operators and, via interpolation, obtain $L^p(\mathbb{R}^n)$ ... More

Orthogonal polynomials in and on a quadratic surface of revolutionJun 25 2019We present explicit constructions of orthogonal polynomials inside quadratic bodies of revolution, including cones, hyperboloids, and paraboloids. We also construct orthogonal polynomials on the surface of quadratic surfaces of revolution, generalizing ... More

Intrinsic graphs and big pieces of parabolic Lipschitz imagesJun 24 2019Let $\mathbb{H}$ be the first Heisenberg group, $\mathbb{W} \subset \mathbb{H}$ a vertical subgroup, and $\alpha > 0$. I show that intrinsic graphs of compactly supported $C^{1,\alpha}_{\mathbb{H}}(\mathbb{W})$-functions have big pieces of bilipschitz ... More

Superdifferential of the Takagi functionJun 24 2019The Takagi function is a classical example of a continuous nowhere differentiable function. It has empty subdifferential except in a countable set where its subdifferential is $\mathbb{R}$. In this paper we characterize its superdifferential.

Algebraic and qualitative aspects of quadratic vector fields related with classical orthogonal polynomialsJun 24 2019This paper is a sequel of the reference \cite[\S 4.2, p.p. 1782--1783]{almp}, in where some families of quadratic polynomial vector fields related with orthogonal polynomials were studied. We extend such results that contain some details related with ... More

Stokes matrices for a class of reducible equationsJun 24 2019This paper is a continuation of our previous work \cite{St} where we have studied the Stokes phenomenon for a particular family of equation \eqref{initial} with \eqref{form-0}-\eqref{npe} from a perturbative point of view. Here we focus on the explicit ... More

On two-signed solutions to a second order semi-linear parabolic partial differential equation with non-Lipschitz nonlinearityJun 21 2019In this paper, we establish the existence of a 1-parameter family of spatially inhomogeneous radially symmetric classical self-similar solutions to a Cauchy problem for a semi-linear parabolic PDE with non-Lipschitz nonlinearity and trivial initial data. ... More

Failure of the matrix weighted bilinear Carleson embedding theoremJun 20 2019We prove failure of the natural formulation of a matrix weighted bilinear Carleson embedding theorem, featuring a matrix valued Carleson sequence as well as products of norms for the embedding. We show that assuming an A2 weight is also not sufficient. ... More

Exponential asymptotics for the eigenvalues in the broken PT-symmetric regionJun 19 2019Stemming from the seminal work of Bender & Boettcher in 1998 (Phys. Rev. Lett. vol. 80 pp. 5243-5246), there has been great interest in the study of PT-symmetric models of quantum mechanics, where the primary focus is with the study of non-Hermitian Hamiltonians ... More

The convergence of discrete Fourier-Jacobi seriesJun 19 2019The discrete counterpart of the problem related to the convergence of the Fourier-Jacobi series is studied. To this end, given a sequence, we construct the analogue of the partial sum operator related to Jacobi polynomials and characterize its convergence ... More

Discrete Harmonic Analysis associated with Jacobi expansions III: the Littlewood-Paley-Stein $g_{k}$-functions and the Laplace type multipliersJun 19 2019The research about Harmonic Analysis associated with Jacobi expansions carried out in \cite{ACL-JacI} and \cite{ACL-JacII} is continued in this paper. Given the operator $\mathcal{J}^{(\alpha,\beta)}=J^{(\alpha,\beta)}-I$, where $J^{(\alpha,\beta)}$ is ... More

A bilinear proof of decoupling for the cubic moment curveJun 19 2019Using a bilinear method that is inspired by the method of efficient congruencing of Wooley [Woo16], we prove a sharp decoupling inequality for the moment curve in $\mathbb{R}^3$.

A bilinear proof of decoupling for the cubic moment curveJun 19 2019Jun 26 2019Using a bilinear method that is inspired by the method of efficient congruencing of Wooley [Woo16], we prove a sharp decoupling inequality for the moment curve in $\mathbb{R}^3$.

On another extension of coherent pairs of measuresJun 18 2019Let $M$ and $N$ be fixed non-negative integer numbers and let $\pi_N$ be a polynomial of degree $N$. Suppose that $(P_n)_{n\geq0}$ and $(Q_n)_{n\geq0}$ are two orthogonal polynomial sequences such that %their derivatives of orders $k$ and $m$ (respectively) ... More

Some summation theorems for truncated Clausen series and applicationsJun 18 2019The main aim of this paper is to derive some new summation theorems for terminating and truncated Clausen's hypergeometric series with unit argument, when one numerator parameter and one denominator parameter are negative integers. Further, using our ... More