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For a real analytic periodic function 𝜙: ℝ→ℝ, an integer 𝑏≥2 and 𝜆∈(1/𝑏,1), we prove the following dichotomy for the Weierstrass-type function 𝑊(𝑥)=∑_{𝑛≥0} 𝜆^𝑛 𝜙(𝑏^𝑛 𝑥): Either W(x) is real analytic, or the Hausdorff dimension of its graph is equal to 2+log𝑏𝜆. Furthermore, given b and 𝜙, the former alternative only happens for finitely many 𝜆 unless 𝜙 is constant.

We study L^p inequalities that sharpen the triangle inequality for sums of N functions in L^p.

We define piecewise continuous almost automorphic (p.c.a.a.) functions in the manners of Bochner, Bohr and Levitan, respectively, to describe almost automorphic motions in impulsive systems, and prove that with certain prefixed possible discontinuities they are equivalent to quasi-uniformly continuous Stepanov almost automorphic ones. Spatially almost automorphic sets on the line, which serve as suitable objects containing discontinuities of p.c.a.a. functions, are characterized in the manners of Bochner, Bohr and Levitan, respectively, and shown to be equivalent. Two Favard's theorems are established to illuminate the importance and convenience of p.c.a.a. functions in the study of almost periodically forced impulsive systems.

No abstract uploaded!

No abstract uploaded!

Motivated by the works of Liu, we provide a unified approach to find Appell-Lerch series and Hecke-type series representations for mock theta functions. We establish a number of parameterized identities with two parameters $a$ and $b$. Specializing the choices of $(a,b)$, we not only give various known and new representations for the mock theta functions of orders 2, 3, 5, 6 and 8, but also present many other interesting identities. We find that some mock theta functions of different orders are related to each other, in the sense that their representations can be deduced from the same $(a,b)$-parameterized identity. Furthermore, we introduce the concept of false Appell-Lerch series. We then express the Appell-Lerch series, false Appell-Lerch series and Hecke-type series in this paper using the building blocks $m(x,q,z)$ and $f_{a,b,c}(x,y,q)$ introduced by Hickerson and Mortenson, as well as $\overline{m}(x,q,z)$ and $\overline{f}_{a,b,c}(x,y,q)$ introduced in this paper. We also show the equivalences of our new representations for several mock theta functions and the known representations.

Let $f(x)=\sum_{n=0}^{\infty}\frac{1}{n!}q^{n(n-1)/2}x^n$ ($0<q<1$) be the deformed exponential function. It is known that the zeros of $f(x)$ are real and form a negative decreasing sequence $(x_k)$ ($k\ge 1$). We investigate the complete asymptotic expansion for $x_{k}$ and prove that for any $n\ge1$, as $k\to \infty$, \begin{align*} x_k=-kq^{1-k}\Big(1+\sum_{i=1}^{n}C_i(q)k^{-1-i}+o(k^{-1-n})\Big), \end{align*} where $C_i(q)$ are some $q$ series which can be determined recursively. We show that each $C_{i}(q)\in \mathbb{Q}[A_0,A_1,A_2]$, where $A_{i}=\sum_{m=1}^{\infty}m^i\sigma(m)q^m$ and $\sigma(m)$ denotes the sum of positive divisors of $m$. When writing $C_{i}$ as a polynomial in $A_0, A_1$ and $A_2$, we find explicit formulas for the coefficients of the linear terms by using Bernoulli numbers. Moreover, we also prove that $C_{i}(q)\in \mathbb{Q}[E_2,E_4,E_6]$, where $E_2$, $E_4$ and $E_6$ are the classical Eisenstein series of weight 2, 4 and 6, respectively.

We study the two main types of trajectories of the ABC flow in the near-integrable regime: spiral orbits and edge orbits. The former are helical orbits which are perturbations of similar orbits that exist in the integrable regime, while the latter exist only in the nonintegrable regime. We prove existence of ballistic (i.e., linearly growing) spiral orbits by using the contraction mapping principle in the Hamiltonian formulation, and we also find and analyze ballistic edge orbits. We discuss the relationship of existence of these orbits with questions concerning front propagation in the presence of flows, in particular the question of linear (i.e., maximal possible) front speed enhancement rate for ABC flows.

In this paper, we study the existence of homoclinic solutions for a class of fourth order differential equations. By using variational methods, the existence and the non-existence of nontrivial homoclinic solutions are obtained, depending on a parameter.

We study differentiability properties of a potential of the type K⋆μ, where μ is a finite Radon measure in RN and the kernel K satisfies |∇jK(x)|≤C|x|−(N−1+j),j=0,1,2. We introduce a notion of differentiability in the capacity sense, where capacity is classical capacity in the de la Vallée Poussin sense associated with the kernel |x|−(N−1). We require that the first order remainder at a point is small when measured by means of a normalized weak capacity “norm” in balls of small radii centered at the point. This implies weak LN/(N−1) differentiability and thus Lp differentiability in the Calderón–Zygmund sense for 1≤p<N/(N−1). We show that K⋆μ is a.e. differentiable in the capacity sense, thus strengthening a recent result by Ambrosio, Ponce and Rodiac. We also present an alternative proof of a quantitative theorem of the authors just mentioned, giving pointwise Lipschitz estimates for K⋆μ. As an application, we study level sets of newtonian potentials of finite Radon measures.

We obtain improved fractional Poincaré inequalities in John domains of a metric space (X,d) endowed with a doubling measure μ under some mild regularity conditions on the measure μ. We also give sufficient conditions on a bounded domain to support fractional Poincaré type inequalities in this setting.

We give an explicit formula for the logarithmic potential of the asymptotic zero-counting measure of the sequence {(dn/dzn)(R(z)expT(z))}∞n=1. Here, R(z) is a rational function with at least two poles, all of which are distinct, and T(z) is a polynomial. This is an extension of a recent measure-theoretic refinement of Pólya’s Shire theorem for rational functions.

We prove a structure theorem for any n-rectifiable set E⊂Rn+1,n≥1, satisfying a weak version of the lower ADR condition, and having locally finite Hn (n-dimensional Hausdorff) measure. Namely, that Hn-almost all of E can be covered by a countable union of boundaries of bounded Lipschitz domains contained in Rn+1∖E. As a consequence, for harmonic measure in the complement of such a set E, we establish a non-degeneracy condition which amounts to saying that Hn|E is “absolutely continuous” with respect to harmonic measure in the sense that any Borel subset of E with strictly positive Hn measure has strictly positive harmonic measure in some connected component of Rn+1∖E. We also provide some counterexamples showing that our result for harmonic measure is optimal. Moreover, we show that if, in addition, a set E as above is the boundary of a connected domain Ω⊂Rn+1 which satisfies an infinitesimal interior thickness condition, then Hn|∂Ω is absolutely continuous (in the usual sense) with respect to harmonic measure for Ω. Local versions of these results are also proved: if just some piece of the boundary is n-rectifiable then we get the corresponding absolute continuity on that piece. As a consequence of this and recent results in “Rectifiability of harmonic measure” [Geom. Funct. Anal. 26 (2016), 703–728], we can decompose the boundary of any open connected set satisfying the previous conditions in two disjoint pieces: one that is n-rectifiable where Hausdorff measure is absolutely continuous with respect to harmonic measure and another purely n-unrectifiable piece having vanishing harmonic measure.

We show that sparse and Carleson coefficients are equivalent for every countable collection of Borel sets and hence, in particular, for dyadic rectangles, the case relevant to the theory of bi-parameter singular integrals. The key observation is that a dual refomulation by I. E. Verbitsky for Carleson coefficients over dyadic cubes holds also for Carleson coefficients over general sets.

We prove that almost all random subsets of a finite vector space are weak Salem sets (small Fourier coefficient), which extend a result of Hayes to a different probability model.

We improve the estimates in the restriction problem in dimension n⩾4. To do so, we establish a weak version of a k-linear restriction estimate for any k. The exponents in this weak k-linear estimate are sharp for all k and n.

We prove the boundedness of a class of tri-linear operators consisting of a quasi piece of bilinear Hilbert transform whose scale equals to or dominates the scale of its linear counter part. Such type of operators is motivated by the tri-linear Hilbert transform and its curved versions.

We introduce a class of tri-linear operators that combine features of the bilinear Hilbert transform and paraproduct. For two instances of these operators, we prove boundedness property in a large range D = { ( p 1 , p 2 , p 3 ) 3 : 1 &lt; p 1

We prove that for a large class of functions P and Q, the discrete bilinear operator T P, Q (f, g)(n)= m Z{0} f (n P (m)) g (n Q (m)) 1 m is bounded from l 2 l 2 into l 1+ , for any (0, 1].

We prove that, under certain conditions on the function pair arphi_1 and arphi_1 , bilinear average arphi_1 along curve arphi_1 satisfies certain decay estimate. As a consequence, Roth type theorems hold in the setting of finite fields. In particular, if arphi_1 with arphi_1 are linearly independent polynomials, then for any arphi_1 with arphi_1 , there are arphi_1 triplets arphi_1 . This extends a recent result of Bourgain and Chang who initiated this type of problems, and strengthens the bound in a result of Peluse, who generalized Bourgain and Chang's work. The proof uses discrete Fourier analysis and algebraic geometry.

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In this paper, we show that as τ → √−1∞, any zero of the Lam´e function converges to either ∞ or a finite point p satisfying Rep = 1 2 and e2πip being an algebraic number. Our proof is based on studying a special family of simply-periodic KdV potentials with period 1, i.e. algebro-geometric simply-periodic solutions of the KdV hierarchy. We show that except the pole 0, all other poles of such KdV potentials locate on the line Rez = 1 2. We also compute explicitly the eigenvalue setofthecorresponding L2[0,1] eigenvalueproblemforsuchKdVpotentials, thus extends Takemura’s works [26, 27]. Our main idea is to apply the classification result for simply-periodic KdV potentials by Gesztesy, Unterkofler and Weikard [11] and the Darboux transformation.

The classical Fej\'er's theorem is a criterion for pointwise convergence of Fourier series on the unit circle. We generalize it to locally compact groups.

We prove a mean ergodic theorem for amenable discrete quantum groups. As an application, we prove a Wiener type theorem for continuous measures on compact metrizable groups.