Improved Bohr inequality for harmonic mappings

In order to improve the classical Bohr inequality, we explain some refined versions for a quasi-subordination family of functions in this paper, one of which is key to build our results. Using these investigations, we establish an improved Bohr inequality with refined Bohr radius under particular conditions for a family of harmonic mappings defined in the unit disk $\mathbb{D}$. Along the line of extremal problems concerning the refined Bohr radius, we derive a series of results. Here, the family of harmonic mappings has the form $f=h+\overline{g}$, where $g(0)=0$, the analytic part h is bounded by 1 and that $|g^{\prime }(z)|\le k|h^{\prime }(z)|$ in $\mathbb{D}$ and for some $k\in [0,1]$.     

Invariant hypercomplex structures and algebraic curves

We show that $U(k)$-invariant hypercomplex structures on (open subsets) of regular semisimple adjoint orbits in $\mathfrak{g}\mathfrak{l}(k,\mathbb{C})$ correspond to algebraic curves C of genus ${(k-1)}^2$, equipped with a flat projection $\pi :C\to {\mathbb{P}}^1$ of degree k, and an antiholomorphic involution $\sigma :C\to C$ covering the antipodal map on ${\mathbb{P}}^1$.     

The Bessel kernel determinant on large intervals and Birkhoff's ergodic theorem

The Bessel process models the local eigenvalue statistics near 0 of certain large positive definite matrices. In this work, we consider the probability $$\mathbb{P}\left(\exists \text{no points in the Bessel process on}(0,x_1)\cup (x_2,x_3)\cup \cdots \cup (x_{2g},x_{2g+1})\right),$$ where and $g\ge 0$ is any non-negative integer. We obtain asymptotics for this probability as the size of the intervals becomes large, up to and including the oscillations of order 1. In these asymptotics, the most intricate term is a one-dimensional integral along a linear flow on a g-dimensional torus, whose integrand involves ratios of Riemann θ-functions associated to a genus g Riemann surface. We simplify this integral in two generic cases: (a) If the flow is ergodic, we compute the leading term in the asymptotics of this integral explicitly using Birkhoff's ergodic theorem. (b) If the linear flow has certain “good Diophantine properties”, we obtain improved estimates on the error term in the asymptotics of this integral. In the case when the flow is both ergodic and has “good Diophantine properties” (which is always the case for $g=1$, and “almost always” the case for $g\ge 2$), these results can be combined, yielding particularly precise and explicit large gap asymptotics.     

A singular Liouville equation on planar domains

We show the existence of a solution for an equation where the nonlinearity is logarithmically singular at the origin, namely, $-\mathrm{\Delta }u=(\log u+f(u)){\chi }_{\{u>0\}}$ in $\mathrm{\Omega }\subset {\mathbb{R}}^2$ with Dirichlet boundary condition. The function f has exponential growth, which can be subcritical or critical with respect to the Trudinger–Moser inequality. We study the energy functional $I_{\epsilon }$ corresponding to the perturbed equation $-\mathrm{\Delta }u+g_{\epsilon }(u)=f(u)$, where $g_{\epsilon }$ is well defined at 0 and approximates $-\log u$. We show that $I_{\epsilon }$ has a critical point $u_{\epsilon }$ in $H_0^1(\mathrm{\Omega })$, which converges to a legitimate nontrivial nonnegative solution of the original problem as $\epsilon \to 0$. We also investigate the problem with $f(u)$ replaced by $\lambda f(u)$, when the parameter $\lambda >0$ is sufficiently large.     

Generalization of Hardy–Littlewood maximal inequality with variable exponent

Let $p(·)$ be a measurable function defined on ${\mathbb{R}}^d$ and $p_{-}:={inf}_{x\in {\mathbb{R}}^d}p(x)$. In this paper, we generalize the Hardy–Littlewood maximal operator. In the definition, instead of cubes or balls, we take the supremum over all rectangles the side lengths of which are in a cone-like set defined by a given function ψ. Moreover, instead of the integral means, we consider the $L_{q(·)}$-means. Let $p(·)$ and $q(·)$ satisfy the log-Hülder condition and $p(·)=q(·)r(·)$. Then, we prove that the maximal operator is bounded on $L_{p(·)}$ if and is bounded from $L_{p(·)}$ to the weak $L_{p(·)}$ if $1\le r_{-}\le \infty$. We generalize also the theorem about the Lebesgue points.     

Single-peak solutions for a subcritical Schrödinger equation with non-power nonlinearity

This paper deals with the following slightly subcritical Schrödinger equation: $$-\mathrm{\Delta }u+V(x)u=f_{\epsilon }(u),u>0\text{in}{\mathbb{R}}^N,$$ where $V(x)$ is a nonnegative smooth function, $f_{\epsilon }(u)=\frac{u^p}{{[\ln (e+u)]}^{\epsilon }}$, $p=\frac{N+2}{N-2}$, $\epsilon >0$, $N\ge 7$. Most of the previous works for the Schrödinger equations were mainly investigated for power-type nonlinearity. In this paper, we will study the case when the nonlinearity $f_{\epsilon }(u)$ is a non-power nonlinearity. We show that, for ε small enough, there exists a family of single-peak solutions concentrating at the positive stable critical point of the potential $V(x)$.     

Multiple orthogonal polynomials associated with the exponential integral

We introduce a new family of multiple orthogonal polynomials satisfying orthogonality conditions with respect to two weights $(w_1,w_2)$ on the positive real line, with $w_1(x)=x^{\alpha }{e}^{-x}$ the gamma density and $w_2(x)=x^{\alpha }{E}_{\nu +1}(x)$ a density related to the exponential integral $E_{\nu +1}$. We give explicit formulas for the type I functions and type II polynomials, their Mellin transform, Rodrigues formulas, hypergeometric series, and recurrence relations. We determine the asymptotic distribution of the (scaled) zeros of the type II multiple orthogonal polynomials and make a connection to random matrix theory. Finally, we also consider two related families of mixed-type multiple orthogonal polynomials.     

Weyl families of transformed boundary pairs

Let $(\mathfrak{L},\mathrm{\Gamma })$ be an isometric boundary pair associated with a closed symmetric linear relation T in a Krein space $\mathfrak{H}$. Let $M_{\mathrm{\Gamma }}$ be the Weyl family corresponding to $(\mathfrak{L},\mathrm{\Gamma })$. We cope with two main topics. First, since $M_{\mathrm{\Gamma }}$ need not be (generalized) Nevanlinna, the characterization of the closure and the adjoint of a linear relation $M_{\mathrm{\Gamma }}(z)$, for some $z\in \mathbb{C}\setminus \mathbb{R}$, becomes a nontrivial task. Regarding $M_{\mathrm{\Gamma }}(z)$ as the (Shmul'yan) transform of $zI$ induced by Γ, we give conditions for the equality in $\overline{M_{\mathrm{\Gamma }}(z)}\subseteq \overline{M_{\overline{\mathrm{\Gamma }}}(z)}$ to hold and we compute the adjoint $M_{\overline{\mathrm{\Gamma }}}{(z)}^{\ast }$. As an application, we ask when the resolvent set of the main transform associated with a unitary boundary pair for $T^{+}$ is nonempty. Based on the criterion for the closeness of $M_{\mathrm{\Gamma }}(z)$, we give a sufficient condition for the answer. From this result it follows, for example, that, if T is a standard linear relation in a Pontryagin space, then the Weyl family $M_{\mathrm{\Gamma }}$ corresponding to a boundary relation Γ for $T^{+}$ is a generalized Nevanlinna family; a similar conclusion is already known if T is an operator. In the second topic, we characterize the transformed boundary pair $({\mathfrak{L}}^{\prime },{\mathrm{\Gamma }}^{\prime })$ with its Weyl family $M_{{\mathrm{\Gamma }}^{\prime }}$. The transformation scheme is either ${\mathrm{\Gamma }}^{\prime }=\mathrm{\Gamma }{V}^{-1}$ or ${\mathrm{\Gamma }}^{\prime }=V\mathrm{\Gamma }$ with suitable linear relations V. Results in this direction include but are not limited to: a 1-1 correspondence between $(\mathfrak{L},\mathrm{\Gamma })$ and $({\mathfrak{L}}^{\prime },{\mathrm{\Gamma }}^{\prime })$; the formula for $M_{{\mathrm{\Gamma }}^{\prime }}-M_{\mathrm{\Gamma }}$, for an ordinary boundary triple and a standard unitary operator V (first scheme); construction of a quasi boundary triple from an isometric boundary triple $(\mathfrak{L},{\mathrm{\Gamma }}_0,{\mathrm{\Gamma }}_1)$ with $ker\mathrm{\Gamma }=T$ and $T_0=T_0^{\ast }$ (second scheme, Hilbert space case).     

Lp$L^p$ bound for the Hilbert transform along variable non-flat curves

We prove the $L^p$ bound for the Hilbert transform along variable non-flat curves $(t,u(x){[t]}^{\alpha }+v(x){[t]}^{\beta })$, where α and β satisfy $\alpha \ne \beta ,\alpha \ne 1,\beta \ne 1$. Compared with the associated theorem in the work (Guo et al. Proc. Lond. Math. Soc. 2017) investigating the case $\alpha =\beta \ne 1$, our result is more general while the proof is more involved. To achieve our goal, we divide the frequency of the objective function into three cases and take different strategies to control these cases. Furthermore, we need to introduce a “short” shift maximal function ${\mathbf{M}}^{[n]}$ to establish some pointwise estimates.     

The Riemann problem for a generalized Burgers equation with spatially decaying sound speed. I Large-time asymptotics

In this paper, we consider the classical Riemann problem for a generalized Burgers equation, $$u_t+h_{\alpha }(x)u{u}_x=u_{xx},$$ with a spatially dependent, nonlinear sound speed, $h_{\alpha }(x)\equiv {(1+x^2)}^{-\alpha }$ with $\alpha >0$, which decays algebraically with increasing distance from a fixed spatial origin. When $\alpha =0$, this reduces to the classical Burgers equation. In this first part of a pair of papers, we focus attention on the large-time structure of the associated Riemann problem, and obtain its detailed structure, as $t\to \infty$, via the method of matched asymptotic coordinate expansions (this uses the classical method of matched asymptotic expansions, with the asymptotic parameters being the independent coordinates in the evolution problem; this approach is developed in detail in the monograph of Leach and Needham, as referenced in the text), over all parameter ranges. We identify a significant bifurcation in structure at $\alpha =\frac{1}{2}$.     

Existence of positive ground state solutions for the coupled Choquard system with potential

In this paper, we study the following coupled Choquard system in ${\mathbb{R}}^N$: where $\alpha \in (0,N)$ and , in which $2_{\ast }^{\alpha }$ denotes $\frac{N+\alpha }{N-2}$ if $N\ge 3$ and $2_{\ast }^{\alpha }:=\infty$ if $N=1,2$. The function $I_{\alpha }$ is a Riesz potential. By using Nehari manifold method, we obtain the existence of a positive ground state solution in the case of bounded potential and periodic potential, respectively. In particular, the nonlinear term includes the well-studied case $p=q$ and $u(x)=v(x)$, and the less-studied case $p\ne q$ and $u(x)\ne v(x)$. Moreover, it seems to be the first existence result for the case $p\ne q$.     

Hausdorff operators on compact abelian groups

Necessary and sufficient conditions are given for the boundedness of Hausdorff operators on the generalized Hardy spaces $H_E^p(G)$, real Hardy space $H_{\mathbb{R}}^1(G)$, $\text{BMO}(G)$, and $\text{BMOA}(G)$ for compact Abelian group G. Surprisingly, these conditions turned out to be the same for all groups and spaces under consideration. Applications to Dirichlet series are given. The case of the space of continuous functions on G and examples are also considered.