Problems from probability theory
Problems on averages and lacunary maximal functions
We prove three results concerning convolution operators and lacunary maximal functions associated to dilates of measures. First we obtain an H¹ to bound for lacunary maximal operators under a dimensional assumption on the underlying measure and an assumption on an regularity bound for some p > 1. Secondly, we obtain a necessary and sufficient condition for L² boundedness of lacunary maximal operator associated to averages over convex curves in the plane. Finally we prove an regularity result...
Product rule and chain rule estimates for the Hajłasz gradient on doubling metric measure spaces
We use the Calderón Maximal Function to prove the Kato-Ponce Product Rule Estimate and the Christ-Weinstein Chain Rule Estimate for the Hajłasz gradient on doubling measure metric spaces.
Proof of a Conjecture on the Supports of Wigner Distributions.
Proper moving average representations and outer functions in two variables.
Properties of extremal sequences for the Bellman function of the dyadic maximal operator
We prove a necessary condition that has every extremal sequence for the Bellman function of the dyadic maximal operator. This implies the weak- uniqueness for such a sequence.
Quasihomographies in the theory of Teichmüller spaces [Book]
Quelques épilogues
Radial Functions and Regularity of Solutions to the Schrödinger Equation.
Radial maximal function characterizations for Hardy spaces on RD-spaces
An RD-space is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds. The authors prove that for a space of homogeneous type having “dimension” , there exists a such that for certain classes of distributions, the quasi-norms of their radial maximal functions and grand maximal functions are equivalent when . This result yields a radial maximal function characterization for Hardy spaces on .
Rationale trigonometrische Tschebyscheff-Approximation in zwei Variablen.
Rationalle Trigonometrische Tschebyscheff-Approximation In Zwei Variablen
Real analysis, quantitative topology, and geometric complexity.
In this paper, we give an overview of some topics involving behavior of homeomorphisms and ways in which real analysis can arise in geometric settings.
Rearrangement and continuity properties of functions on spaces of homogeneous type
Rearrangement estimates of the area integrals
We derive weighted rearrangement estimates for a large class of area integrals. The main approach used earlier to study these questions is based on distribution function inequalities.
Recent developments in the theory of function spaces with dominating mixed smoothness
The aim of these lectures is to present a survey of some results on spaces of functions with dominating mixed smoothness. These results concern joint work with Winfried Sickel and Miroslav Krbec as well as the work which has been done by Jan Vybíral within his thesis. The first goal is to discuss the Fourier-analytical approach, equivalent characterizations with the help of derivatives and differences, local means, atomic and wavelet decompositions. Secondly, on this basis we study approximation...
Recent progress on the Kakeya conjecture.
We survey recent developments on the Kakeya problem.[Proceedings of the 6th International Conference on Harmonic Analysis and Partial Differential Equations, El Escorial (Madrid), 2002].
Rectifiabilité quantifié et le problème du voyageur de commerce
Rectifiability, analytic capacity, and singular integrals.
Refined Hardy inequalities
The aim of this article is to present “refined” Hardy-type inequalities. Those inequalities are generalisations of the usual Hardy inequalities, their additional feature being that they are invariant under oscillations: when applied to highly oscillatory functions, both sides of the refined inequality are of the same order of magnitude. The proof relies on paradifferential calculus and Besov spaces. It is also adapted to the case of the Heisenberg group.