Mathematics

Let (X,d,μ) be a space of homogeneous type in the sense of Coifman and Weiss, i.e. d is a quasi metric on X and μ is a positive measure satisfying the doubling condition. Suppose that u and v are two locally finite positive Borel measures on (X,d,μ) . Subject to the pair of weights satisfying a side condition, we characterize the boundedness of a Calderón--Zygmund operator T from L 2 (u) to L 2 (v) in terms of the A 2 condition and two testing conditions. For every cube B⊂X , we have the following testing conditions, with 1 B taken as the indicator of B ∥T(u 1 B ) ∥ L 2 (B,v) ≤T∥ 1 B ∥ L 2 (u) , ∥ T ∗ (v 1 B ) ∥ L 2 (B,u) ≤T∥ 1 B ∥ L 2 (v) . The proof uses stopping cubes and corona decompositions originating in work of Nazarov, Treil and Volberg, along with the pivotal side condition.

An algebra denoted mH with three generators is introduced and shown to admit embeddings of the Hahn algebra and the rational Hahn algebra. It has a real version of the deformed Jordan plane as a subalgebra whose connection with Hahn polynomials is established. Representation bases corresponding to eigenvalue or generalized eigenvalue problems involving the generators are considered. Overlaps between these bases are shown to be bispectral orthogonal polynomials or biorthogonal rational functions thereby providing a unified description of these functions based on mH . Models in terms of differential and difference operators are used to identify explicitly the underlying special functions as Hahn polynomials and rational functions and to determine their characterizations. An embedding of mH in U( sl 2 ) is presented. A Padé approximation table for the binomial function is obtained as a by-product.

We study the equilibrium measure on the two dimensional sphere in the presence of an external field generated by r+1 equal point charges that are symmetrically located around the north pole. The support of the equilibrium measure is known as the droplet. The droplet has a motherbody which we characterize by means of a vector equilibrium problem (VEP) for r measures in the complex plane. The model undergoes two transitions which is reflected in the support of the first component of the minimizer of the VEP, namely the support can be a finite interval containing 0, the union of two intervals, or the full half-line. The two interval case corresponds to a droplet with two disjoint components, and it is analyzed by means of a genus one Riemann surface.

For the functions f , which can be represented in the form of the convolution f(x)= a 0 2 + 1 π ∫ −π π ∑ k=1 ∞ e −α k r cos(kt− βπ 2 )φ(x−t)dt , φ⊥1 , α>0, r∈(0,1) , β∈R , we establish the Lebesgue-type inequalities of the form ∥f− S n−1 (f) ∥ C ≤ e −α n r ( 4 π 2 ln n 1−r αr + γ n ) E n (φ ) C . These inequalities take place for all numbers n that are larger than some number n 1 = n 1 (α,r) , which constructively defined via parameters α and r . We prove that there exists a function, such that the sign " ≤ " in given estimate can be changed for " = ".

Let E??R 2 be a finite set, and let f:E?�[0,?? . In this paper, we address the algorithmic aspects of nonnegative C 2 interpolation in the plane. Specifically, we provide an efficient algorithm to compute a nonnegative C 2 ( R 2 ) extension of f with norm within a universal constant factor of the least possible. We also provide an efficient algorithm to approximate the trace norm.

Let H=−Δ+|x | 2 be the Hermite operator in R n . In this paper we study almost everywhere convergence of the Bochner-Riesz means associated with H which is defined by S λ R (H)f(x)= ∑ k=0 ∞ (1− 2k+n R 2 ) λ + P k f(x). Here P k f is the k -th Hermite spectral projection operator. For 2≤p<∞ , we prove that lim R→∞ S λ R (H)f=f a.e. for all f∈ L p ( R n ) provided that λ>λ(p)/2 and λ(p)=max{n(1/2−1/p)−1/2,0}. Conversely, we also show the convergence generally fails if λ<λ(p)/2 in the sense that there is an f∈ L p ( R n ) for 2n/(n−1)≤p such that the convergence fails. This is in surprising contrast with a.e. convergence of the classical Bochner-Riesz means for the Laplacian. For n≥2 and p≥2 our result tells that the critical summability index for a.e. convergence for S λ R (H) is as small as only the \emph{half} of the critical index for a.e. convergence of the classical Bochner-Riesz means. When n=1 , we show a.e. convergence holds for f∈ L p (R) with p≥2 whenever λ>0 . Compared with the classical result due to Askey and Wainger who showed the optimal L p convergence for S λ R (H) on R we only need smaller summability index for a.e. convergence.

We give a characterization of L p (σ) for uniformly rectifiable measures σ using Tolsa's α -numbers, by showing, for 1<p<∞ and f∈ L p (σ) , that ∥f ∥ L p (σ) ∼ ∥ ∥ ∥ ∥ ( ∫ ∞ 0 ( α fσ (x,r)+|f | x,r α σ (x,r)) 2 dr r ) 1 2 ∥ ∥ ∥ ∥ L p (σ) .

We introduce the Menchov-Rademacher operator for right triangles - a sample two-dimensional maximal operator, and prove an upper bound for its L 2 norm.

The biorthogonal rational functions of the 3 F 2 type on the uniform grid provide the simplest example of rational functions with bispectrality properties that are similar to those of classical orthogonal polynomials. These properties are described by three difference operators X,Y,Z which are tridiagonal with respect to three distinct bases of the relevant finite-dimensional space. The pairwise commutators of the operators X,Y,Z generate a quadratic algebra which is akin to the algebras of Askey-Wilson type attached to hypergeometric polynomials.

It is well-known that the functions f∈ L 1 ( R d ) whose translates along a lattice Λ form a tiling, can be completely characterized in terms of the zero set of their Fourier transform. We construct an example of a discrete set Λ⊂R (a small perturbation of the integers) for which no characterization of this kind is possible: there are two functions f,g∈ L 1 (R) whose Fourier transforms have the same set of zeros, but such that f+Λ is a tiling while g+Λ is not.