A quasisymmetric graph is a curve whose projection onto a line is a quasisymmetric map. We show that this class of curves is related to solutions of the reduced Beltrami equation and to a generalization of the Zygmund class Λ∗. This relation makes it possible to use the tools of harmonic analysis to construct nontrivial examples of quasisymmetric graphs and of quasiconformal maps.

This paper contains the material covered in a minicourse given by the author at the Centre de Recerca Matemàtica in Barcelona during the period May 4–9, 2009. The course was an expanded version of a series of three lectures delivered by the author two weeks earlier (April 23–25) at the New Mexico Analysis Seminar held at the University of New Mexico in Albuquerque. The author would like to than...

Recently, norm equivalences between spherical polynomials and their sample values at scattered sites have been proved. These so-called MarcinkiewiczZygmund inequalities involve a parameter that characterizes the density of the sampling set and they are applicable to all polynomials whose degree does not exceed an upper bound that is determined by the density parameter. We show that if one is sa...

The classical Morse-Sard Theorem says that the set of critical values of f : R → R has Lebesgue measure zero if f ∈ Ck+1. We show the Ck+1 smoothness requirement can be weakened to Ck+Zygmund. This is corollary to the following theorem: For integers n > m > r > 0, let s = (n− r)/(m− r); if f : R → R belongs to the Lipschitz class Λs and E is a set of rank r for f , then f(E) has measure zero. 0...

Abstract In this paper we study boundedness and compactness characterizations of the commutators Cauchy type integrals on bounded strongly pseudoconvex domains D in $\mathbb C^{n}$ with boundaries $bD$ satisfying minimum regularity condition $C^{2}$ , based recent results Lanzani–Stein Duong et al. We point out that setting integral is sum essential part which a Calderón–Zygmund operator remain...

In a previous paper, we obtained several “compact versions” of Rubio de Francia’s weighted extrapolation theorem, which allowed us to extrapolate the compactness linear operators from just one space full range Lebesgue spaces, where these are bounded. this study for bilinear in terms Muckenhoupt weights. As applications, easily recover and improve earlier results on commutators Calderón–Zygmund...

Petrov (1996) proved the connection between general moment conditions and the applicability of the strong law of large numbers to a sequence of pairwise independent and identically distributed random variables. This note examines this connection to a sequence of pairwise negative quadrant dependent (NQD) and identically distributed random variables. As a consequence of the main theorem ...

Let $\alpha>0$, $\beta>\alpha$, and let $X_1,\ldots, X_q$ be $\mathscr{C}^{\alpha}_{\mathrm{loc}}$ vector fields on a $\mathscr{C}^{\alpha+1}$ manifold which span the tangent space at every point, where $\mathscr{C}^{s}$ denotes Zygmund-H\"older of order $s$. We give necessary sufficient conditions for when there is $\mathscr{C}^{\beta+1}$ structure manifold, compatible with its structure, resp...