We consider the equation

This is joint work with Renato Iturriaga and Andrzej Szulkin.

Given a set A Ì F_pⁿ with at least dpⁿ elements, d > 0, we will discuss finding triples {(x,x+d,x+2d) Î A×A×A : d Î V}, where V = {x Î F_pⁿ : f₁(x) = ¼ = f_R(x) = 0} is the zero set of homogeneous polynomials f₁,...,f_R all of fixed degree d.

This is joint work with Akos Magyar.

We compare several possible notions of Hardy-Sobolev spaces on a manifold with a doubling measure. In particular, we consider several characterizations of these spaces, in terms of maximal functions, atomic decompositions, and gradients, and apply them to the L¹ Sobolev space M¹₁, defined by Hajlasz.

Joint work with N. Badr.

We extend an affine invariant inequality previously established for vector polynomials to general rational functions. We then prove several estimates for problems in harmonic analysis defined with respect to the affine arc-length measure.

This is joint work with Spyridon Dendrinos (University of Bristol) and James Wright (University of Edinburgh).

The set E Í Z is said to have zero discrete harmonic density (zdhd) if for every open U Í T and discrete measure m, there is a discrete measure, n, supported on U with [^(m)] = [^(n)] on E. I₀ sets are examples of sets which have zdhd. We study properties of these sets. Our motivation is to provide a new approach to two long-standing problems involving Sidon sets.

This is joint work with Colin Graham.

For b Î R, p > 0, and 0 < q < 1 fixed, we characterize the integrable functions on (0,¥) satisfying the functional equation f(qx) = q^b-1/2 (x+pq^-1/2) f(x), and show that they are solutions to the generalized Stieltjes-Wigert moment problem.

Singular and maximal Radon transforms are generalizations of singular integrals and maximal functions, when averages are taken over curves and surfaces. If they are given in terms of integral polynomials, such transformations have natural discrete analogues whose l^p mapping properties are related to questions both in ergodic and number theory.

We plan to survey some past results in the discrete settings, as well as to discuss the l² bounds of singular averages over polynomial "curves" on the integral (3 by 3) upper triangular group.

This is joint work with A. Ionescu, E. M. Stein and S. Wainger.

We study the interplay between the geometry of Hardy spaces and functional analytic properties of singular integral operators (SIO's), such as the Riesz transforms as well as Cauchy-Clifford and harmonic double layer operator, on the one hand and, on the other hand, the regularity and geometric properties of domains of locally finite perimeter. Among other things, we give several characterizations of Euclidean balls, their complements, and half-spaces, in terms of the aforementioned SIO's.

Joint work with Steve Hofmann, Salvador Perez-Esteva, Marius Mitrea and Michael Taylor.

Let n ³ 3 and B be the unit ball in Rⁿ. Denote by GM(B) = {f_a^*} the group of all Möbius transformations of the unit ball onto itself and SH the class of subharmonic functions u : B®R. Consider

John and Nirenberg (1961) introduced the space BMO(Q) and a larger space which we call the John-Nirenberg space with exponent p and denote by JN_p(Q), where Q is a finite cube in Rⁿ. They proved two lemmas for functions in BMO(Q) and JN_p(Q) respectively. The first one characterizes functions in BMO(Q) in terms of the exponential decay of the distribution function of their oscillations. The second shows that any function in JN_p(Q) is in weak L^p(Q).

We first give a new proof for John-Nirenberg lemma II on Rⁿ by using a dyadic maximal operator and a good lambda inequality. Then, we discuss the space JN_p and the corresponding lemma in the context of a doubling metric measure space.

Joint work with D. Aalto, L. Berkovits, O. E. Maasalo.

Let f Î L_2p be a real-valued even function with its Fourier series

This is joint work with D. S. Yu and S. P. Zhou.