Mathematics

An open problem concerning Riemann sums, posed by O. Furdui, is considered.

We improve known upper bounds for the minimal dispersion of a point set in the unit cube and its inverse in both the periodic and non-periodic settings. Some of our bounds are sharp up to logarithmic factors.

We consider the bilinear pseudo-differential operators with symbols in the bilinear Hörmander classes B S m ?,? , 0<?<1 . In this paper, we show that the condition 1/p=1/ p 1 +1/ p 2 is necessary when we consider the boudnedness from H p 1 ? H p 2 to L p of those operators for the critical case.

We prove L p bounds for the Fourier extension operators associated to surfaces in R 3 with negative Gaussian curvatures for every p>3.25 .

Although fractional powers of non-negative operators have received much attention in recent years, there is still little known about their behavior if real-valued exponents are greater than one. In this article, we define and study the discrete fractional Laplace operator of arbitrary real-valued positive order. A series representation of the discrete fractional Laplace operator for positive non-integer powers is developed. Its convergence to a series representation of a known case of positive integer powers is proven as the power tends to the integer value. Furthermore, we show that the new representation for arbitrary real-valued positive powers of the discrete Laplace operator is consistent with existing theoretical results.

Let Ω⊂ R d , d≥2 , be a bounded convex domain and f:Ω→R be a non-negative subharmonic function. In this paper we prove the inequality 1 |Ω| ∫ Ω f(x)dx≤ d |∂Ω| ∫ ∂Ω f(x)dσ(x). Equivalently, the result can be stated as a bound for the gradient of the Saint Venant torsion function. Specifically, if Ω⊂ R d is a bounded convex domain and u is the solution of −Δu=1 with homogeneous Dirichlet boundary conditions, then ∥∇u ∥ L ∞ (Ω) <d |Ω| |∂Ω| . Moreover, both inequalities are sharp in the sense that if the constant d is replaced by something smaller there exist convex domains for which the inequalities fail. This improves upon the recent result that the optimal constant is bounded from above by d 3/2 due to Beck et al.

Given a bounded measurable function σ on R n , we let T σ be the operator obtained by multiplication on the Fourier transform by σ . Let 0< s 1 ≤ s 2 ≤⋯≤ s n <1 and ψ be a Schwartz function on the real line whose Fourier transform ψ ˆ is supported in [−2,−1/2]∪[1/2,2] and which satisfies ∑ j∈Z ψ ˆ ( 2 −j ξ)=1 for all ξ≠0 . In this work we sharpen the known forms of the Marcinkiewicz multiplier theorem by finding an almost optimal function space with the property that, if the function ( ξ 1 ,…, ξ n )↦ ∏ i=1 n (I− ∂ 2 i ) s i 2 [ ∏ i=1 n ψ ˆ ( ξ i )σ( 2 j 1 ξ 1 ,…, 2 j n ξ n )] belongs to it uniformly in j 1 ,…, j n ∈Z , then T σ is bounded on L p ( R n ) when | 1 p − 1 2 |< s 1 and 1<p<∞ . In the case where s i ≠ s i+1 for all i , it was proved in [Grafakos, Israel J. Math., to appear] that the Lorentz space L 1 s 1 ,1 ( R n ) is the function space sought. In this work we address the significantly more difficult general case when for certain indices i we might have s i = s i+1 . We obtain a version of the Marcinkiewicz multiplier theorem in which the space L 1 s 1 ,1 is replaced by an appropriate Lorentz space associated with a certain concave function related to the number of terms among s 2 ,…, s n that equal s 1 . Our result is optimal up to an arbitrarily small power of the logarithm in the defining concave function of the Lorentz space.

This is a tutorial on duality properties of special functions, mainly of orthogonal polynomials in the ( q -)Askey scheme. It is based on the first part of the 2017 R.P. Agarwal Memorial Lecture delivered by the author.

In this paper, we prove an extension theorem for spheres of square radii in F d q , which improves a result obtained by Iosevich and Koh (2010). Our main tool is a new point-hyperplane incidence bound which will be derived via a cone restriction theorem. We also will study applications on distance problems.

A Fuchsian system of rank 8 in 3 variables with 4 parameters is presented. The singular locus consists of six planes and a cubic surface. The restriction of the system onto the intersection of two singular planes is an ordinary differential equation of order four with three singular points. A middle convolution of this equation turns out to be the tensor product of two Gauss hypergeometric equation, and another middle convolution sends this equation to the Dotsenko-Fateev equation. Local solutions to these ordinary differential equations are found. Their coefficients are sums of products of the Gamma functions. These sums can be expressed as special values of the generalized hypergeometric series 4 F 3 at 1. Keywords: Fuchsian differential equation, hypergeometric differential equation, middle convolution, Dotsenko-Fateev equation, recurrence formula, series solution