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...

In this work we prove Hardy space estimates for bilinear Littlewood-Paley-Stein square function and Calderón-Zygmund operators. Sufficient Carleson measure type conditions are given for square functions to be bounded from H p1 ×H p2 into Lp for indices smaller than 1, and sufficient BMO type conditions are given for a bilinear Calderón-Zygmund operator to be bounded from H p1 ×H p2 into H p for...

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 ...