
We consider the equation

This is joint work with Renato Iturriaga and Andrzej Szulkin.
Given a set A Ì F_{p}^{n} with at least dp^{n} 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_{p}^{n} : f_{1}(x) = ¼ = f_{R}(x) = 0} is the zero set of homogeneous polynomials f_{1},...,f_{R} all of fixed degree d.
This is joint work with Akos Magyar.
We compare several possible notions of HardySobolev 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 L^{1} Sobolev space M^{1}_{1}, 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 arclength 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. I_{0} sets are examples of sets which have zdhd. We study properties of these sets. Our motivation is to provide a new approach to two longstanding 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^{b1/2} (x+pq^{1/2}) f(x), and show that they are solutions to the generalized StieltjesWigert 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 l^{2} 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 CauchyClifford 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 halfspaces, in terms of the aforementioned SIO's.
Joint work with Steve Hofmann, Salvador PerezEsteva, Marius Mitrea and Michael Taylor.
Let n ³ 3 and B be the unit ball in R^{n}. 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 JohnNirenberg space with exponent p and denote by JN_{p}(Q), where Q is a finite cube in R^{n}. 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 JohnNirenberg lemma II on R^{n} 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 realvalued even function with its Fourier series



This is joint work with D. S. Yu and S. P. Zhou.
A measure m on R^{n} is called locally and uniformly hdimensional if m( B_{r}(x) ) £ h(r) for all x Î R^{n} and for all 0 < r < 1, where h is a real valued function. If f Î L^{2}(m) and F_{m}f denotes its Fourier transform with respect to m, it is not true that F_{m}f Î L^{2}.
We prove that, under certain hypothesis on h, for any f Î L^{2}(m) the L^{2}norm of its Fourier transform restricted to a ball of radius r has the same order of growth as r^{n} h(r^{1}), when r®¥. Moreover, we prove that the ratio between these quantities is bounded by the L^{2}(m)norm of f:

