Abstract. We find explicitly the Bellman function for the dyadic maximal operator on L p as the solution of a Bellman PDE of Monge-Ampère type. This function has been previously found by A. Melas [M] in a different way, but it is our PDE-based approach that is of principal interest here. Clear and replicable, it holds promise as a unifying template for past and current Bellman function investigations.

We construct a family of polynomials with real coefficients that contains as a particular case the Fejér and Suffridge polynomials. These polynomials allow us to suggest a robust algorithm to search for cycles of arbitrary length in non-linear autonomous discrete dynamical systems. Numeric examples are included.

No abstract

No abstract

Abstract. Let B be a collection of measurable sets in R n . The associated geometric maximal operator M B is defined on L 1 (R n ) by M B f (x) = sup x∈R∈B 1 |R| R |f |. If α > 0, M B is said to satisfy a Tauberian condition with respect to α if there exists a finite constant C such that for all measurable sets E ⊂ R n the inequality |{x : M B χ E (x) > α}| ≤ C|E| holds. It is shown that if B is a homothecy invariant collection of convex sets in R n and the associated maximal operator M B satisfies a Tauberian condition with respect to some 0 < α < 1, then M B must satisfy a Tauberian condition with respect to γ for all γ > 0 and moreover M B is bounded on L p (R n ) for sufficiently large p. As a corollary of these results it is shown that any density basis that is a homothecy invariant collection of convex sets in R n must differentiate L p (R n ) for sufficiently large p.Let B be a collection of measurable sets in R n . We define the associated geometricThe operator M B is said to satisfy a Tauberian condition with respect to α if there exists a finite constant C such that for any measurable set E ⊂ R n the inequalityholds. This is a very weak condition on a maximal operator -weaker in fact than a restricted weak type (1,1) estimate. This is a useful condition on a maximal operator, however, as was shown by A. Córdoba and R. Fefferman in their work relating the L p bounds of certain multiplier operators to the weak type ( Now, suppose we are given a maximal operator M B satisfying a Tauberian condition such as, for instance,