where U is the class of all h∈ L2(R+,r−1dr) with ‖h‖L2(R+,r−1dr) ≤ 1. The operator Ω was introduced by Chen and Lin [7]. They showed that Ω is bounded on Lp(Rn) for all p > 2n/(2n− 1) provided that Ω ∈ (Sn−1). Recently, we have been able to show that the Lp(Rn) boundedness of Ω still holds for all p ≥ 2 if the condition Ω ∈ (Sn−1) is replaced by the more natural and weaker condition Ω ∈ L(logL)...

The relation between approach regions and singularities of nonnegative kernels Kt(x, y) is studied, where t ∈ (0,∞), x, y ∈ X, and X is a homogeneous space. For 1 ≤ p < q < ∞, a sufficient condition on approach regions Ωa (a ∈ X) is given so that the maximal function sup (x,t)∈Ωa ∫ X Kt(x, y)f(y) dσ(y) is weak-type (p, q) with respect to a pair of measures σ and ω. It is shown that this conditi...

A new proof of a weighted norm inequality for multilinear singular integrals of Calderón-Zygmund type is presented through a more general estimate involving a sharp maximal function. An application is given to the study of certain multilinear commutators.

In this paper some results on the structure of finite-dimensional Lie algebras are obtained by means of the concept of maximal abelian dimension. More concretely, a sufficient condition is given for the solvability in finite-dimensional Lie algebras by using maximal abelian dimensions. Besides, a necessary condition for the nilpotency is also stated for such Lie algebras. Finally, the maximal a...

For a family of weight functions, hκ, invariant under a finite reflection group on R, analysis related to the Dunkl transform is carried out for the weighted L spaces. Making use of the generalized translation operator and the weighted convolution, we study the summability of the inverse Dunkl transform, including as examples the Poisson integrals and the Bochner-Riesz means. We also define a m...

We study the regularity of the bilinear maximal operator when applied to Sobolev functions, proving that it maps W (R) × W (R) → W (R) with 1 < p, q < ∞ and r ≥ 1, boundedly and continuously. The same result holds on R when r > 1. We also investigate the almost everywhere and weak convergence under the action of the classical Hardy-Littlewood maximal operator, both in its global and local versi...

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...