Sharp maximal inequalities for continuous-path semimartingales

Weighted Norm Inequalities for the Local Sharp Maximal Function

Sharp Maximal Inequalities for Conditionally Symmetric Martingales and Brownian Motion

Let B = {Bt)t>0 be a standard Brownian motion. For c > 0, k > 0 , let T(c, k) = inî{t > 0: maxs<í Bs cBt > k} , T"(c,k)= inf{r>0: max^, \BS\ c\B,\ > k} . We show that for c > 0 and k > 0, both T(c, k) and T*{c, k) axe finite almost everywhere. Moreover, T(c, k) and T*(c, k) e L if and only if c < pKp 1) for p > 1 , and for all c > 0 when p < 1 . These results have analogues for simple random wa...

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.

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.