Weighted rearrangement inequalities for local sharp maximal functions

Weighted Rearrangement Inequalities for Local Sharp Maximal Functions

Several weighted rearrangement inequalities for uncentered and centered local sharp functions are proved. These results are applied to obtain new weighted weak-type and strong-type estimates for singular integrals. A self-improving property of sharp function inequalities is established.

Weighted Norm Inequalities for the Local Sharp Maximal Function

Sharp Weighted Inequalities for the Vector–valued Maximal Function

We prove in this paper some sharp weighted inequalities for the vector–valued maximal function Mq of Fefferman and Stein defined by Mqf(x) = ( ∞ ∑ i=1 (Mfi(x)) q )1/q , where M is the Hardy–Littlewood maximal function. As a consequence we derive the main result establishing that in the range 1 < q < p < ∞ there exists a constant C such that ∫ Rn Mqf(x) p w(x)dx ≤ C ∫ Rn |f(x)|qM [ p q ]+1 w(x)d...

Sharp Inequalities for Maximal Functions Associated with General Measures

Sharp weak type (1, 1) and L p estimates in dimension one are obtained for uncentered maximal functions associated with Borel measures which do not necessarily satisfy a doubling condition. In higher dimensions uncentered maximal functions fail to satisfy such estimates. Analogous results for centered maximal functions are given in all dimensions.

Sharp Inequalities for Polygamma Functions

where μ is a nonnegative measure on [0,∞) such that the integral (2) converges for all x > 0. This means that a function f(x) is completely monotonic on (0,∞) if and only if it is a Laplace transform of the measure μ. The completely monotonic functions have applications in different branches of mathematical sciences. For example, they play some role in combinatorics, numerical and asymptotic an...