Abstract
The paper contains the study of sharp weighted logarithmic estimates for maximal operators on probability spaces equipped with a tree-like structure. These inequalities can be regarded as LlogL versions of the classical estimates of Fefferman and Stein. The proof exploits the existence of a certain special function, enjoying appropriate majorization and concavity conditions.
Highlights
The dyadic maximal operator M on Rd is an operator acting by the formula ⎧
Where f is a locally integrable function on Rd and the dyadic cubes are those formed by the grids 2−N Zd, N = 0, 1, 2
This operator plays a prominent role in analysis and PDEs, and in applications it is often of interest to have optimal or at least tight bounds for its norms
Summary
A set T of measurable subsets of X will be called a tree if the following conditions are satisfied: 1. For any K > 0 and any measurable function f : X → R, we have (MT f )wdμ ≤ K |f | log |f | MT wdμ + L(K) MT wdμ, X By the optimality of L(K) we mean that for any L < L(K) and any probability space (X, μ) with a tree T , there is a weight w and a function f for which (1.3) does not hold.
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