Let H ∼= R ⋉ R be the Heisenberg group and let μt be the normalized surface measure for the sphere of radius t in R. Consider the maximal function defined by Mf = supt>0 |f ∗ μt|. We prove for n ≥ 2 that M defines an operator bounded on L(H) provided that p > 2n/(2n− 1). This improves an earlier result by Nevo and Thangavelu, and the range for L boundedness is optimal. We also extend the result...

Let μ be a Borel measure on Rd which may be nondoubling. The only condition that μ must satisfy is μ(Q) ≤ c0l(Q) for any cube Q ⊂ Rd with sides parallel to the coordinate axes and for some fixed n with 0 < n ≤ d. This paper is to establish the weighted norm inequality for commutators of Calderón-Zygmund operators with RBMO(μ) functions by an estimate for a variant of the sharp maximal function ...

A " k-rule " is a sequence A = ((A n , B n) : n < ω) of pairwise disjoint sets B n , each of cardinality ≤ k and subsets A n ⊆ B n. A subset X ⊆ ω (a " real ") follows a rule A if for infinitely many n ∈ ω, X ∩ B n = A n. There are obvious cardinal invariants resulting from this definition: the least number of reals needed to follow all k-rules, s k , and the least number of k-rules without a r...

New estimates on the maximal function associated to the linear Schrödinger equation are established.

In the paper we prove an extension theorem for matrices with entries in H (U) for U being a Riemann surface of a special type. One of the main components of the proof is a Grauert type theorem for “holomorphic” vector bundles defined over maximal ideal spaces of certain Banach algebras.

The large deviation problem for sums of i.i.d. random vectors is considered. It is assumed that the underlying distribution is absolutely continuous and its density is of regular variation. An asymptotic expression for the probability of large deviations is established in the case of a non-normal stable limit law. The role of the maximal summand is also emphasized. AMS Subject Classification: P...

In the theory of progressive enlargements of filtrations, the supermartingale Zt = P (g > t | Ft) associated with an honest time g, and its additive (Doob-Meyer) decomposition, play an essential role. In this paper, we propose an alternative approach, using a multiplicative representation for the supermartingale Zt, based on Doob’s maximal identity. We thus give new examples of progressive enla...

In this paper, related to RBMO, we prove the sharp maximal inequalities for the Morrey spaces with the measure μ satisfying the growth condition. As an application we obtain the boundedness of commutators for these spaces.

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.