Sharp integral inequalities for the dyadic maximal operator and applications

Mathematische Zeitschrift - We prove a sharp integral inequality for the dyadic maximal function of $$\phi \in L^p$$ . This inequality connects certain quantities related to integrals of $$\phi $$...     

Extremal problems related to maximal dyadic-like operators

We obtain sharp estimates for the localized distribution function of the dyadic maximal function Md?, when ? belongs to Lp,∞. Using this we obtain sharp estimates for the quasi-norm of Md? in Lp,∞ given the localized L¹-norm and certain weak Lp-conditions.     

The Bellman function of the dyadic maximal operator related to Kolmogorov’s inequality

We precisely evaluate the Bellman function of two variables of the dyadic maximal operator related to Kolmogorov’s inequality, thus giving an alternative proof of the results in [3]. Additionally, we characterize the sequences of functions that are extremal for this Bellman function. More precisely, we prove that they behave approximately like eigenfunctions of the dyadic maximal operator, for a specific eigenvalue.     

Supersymmetric models with minimal flavor violation and their running

We revisit the formulation of the principle of minimal flavor violation (MFV) in the minimal supersymmetric extension of the standard model, both at moderate and large tan β, and with or without new CP-violating phases. We introduce a counting rule which keeps track of the highly hierarchical structure of the Yukawa matrices. In this manner, we are able to control systematically which terms can be discarded in the soft SUSY breaking part of the Lagrangian. We argue that for the implementation of this counting rule, it is convenient to introduce a new basis of matrices in which both the squark (and slepton) mass terms as well as the trilinear couplings can be expanded. We derive the RGE for the MFV parameters and show that the beta functions also respect the counting rule. For moderate tan β, we provide explicit analytic solutions of these RGE and illustrate their behavior by analyzing the neighborhood (also switching on new phases) of the SPS-1a benchmark point. We then show that even in the case of large tan β, the RGE remain valid and that the analytic solutions obtained for moderate tan β still allow us to understand the most important features of the running of the parameters, as illustrated with the help of the SPS-4 benchmark point.     

Local Lower Norm Estimates for Dyadic Maximal Operators and Related Bellman Functions

We provide lower \(L^{q}\) and weak \(L^{q}\)-bounds for the localized dyadic maximal operator on \(\mathbb {R}^{n}\), when the local \(L^{1}\) and the local \(L^{p}\) norms of the function are given. We actually do that in the more general context of homogeneous trees in probability spaces.     

Sharp Lorentz Estimates for Dyadic-like Maximal Operators and Related Bellman Functions

We precisely evaluate Bellman-type functions for the dyadic maximal operator on \(\mathbb {R}^{n}\) and of maximal operators on martingales related to local Lorentz-type estimates. Using a type of symmetrization principle, introduced for the dyadic maximal operator in earlier works of the authors, we precisely evaluate the supremum of the Lorentz quasinorm of the maximal operator on a function \(\phi \) when the integral of \(\phi \) is fixed and also the same Lorentz quasinorm of \(\phi \) is fixed. Also we find the corresponding supremum when the integral of \(\phi \) is fixed and several weak type conditions are given.     

On weak type inequalities for dyadic maximal functions

We obtain sharp estimates for the localized distribution function of the dyadic maximal function , given the local L¹ norms of ? and of G○? where G is a convex increasing function such that G(x)/x→+∞ as x→+∞. Using this we obtain sharp refined weak type estimates for the dyadic maximal operator.     

Sharp Weak Type Inequalities for the Dyadic Maximal Operator

We obtain sharp estimates for the localized distribution function of $\mathcal{M}\phi $ , when ? belongs to L ^p,∞ where $\mathcal{M}$ is the dyadic maximal operator. We obtain these estimates given the L ¹ and L ^q norm, q<p and certain weak-L ^p conditions.In this way we refine the known weak (1,1) type inequality for the dyadic maximal operator. As a consequence we prove that the inequality 0.1 is sharp allowing every possible value for the L ¹ and the L ^q norm for a fixed q such that 1<q<p, where ∥?∥ _p,∞ is the usual quasi norm on L ^p,∞.