An optimal decomposition formula for the norm in the Orlicz space L(log L) is given. New proofs of some results involving L(log L) spaces are given and the decomposition is applied to apriori estimates for elliptic partial differential equations with the right-hand side in Zygmund classes. 2000 Mathematics Subject Classification: 46E35, 46E39, 35J05.

A cardinal related to compositions of Sierpiński-Zygmund functions will be considered. A combinatorial characterization of the cardinal is given and is used to answer some questions of K. Ciesielski and T. Natkaniec. It is shown that the bounding number of the continuum may be strictly smaller than continuum.

We present a survey of mixed norm inequalities for several directional operators, namely, directional Hardy-Littlewood maximal functions and Hilbert transforms (both appearing in the method of rotations of Calderón and Zygmund), X-ray transforms, and directional fractional operators related to Riesz type potentials with variable kernel. In dimensions higher than two several interesting question...

On compact manifolds (M, g) without boundary of dimension n ≥ 2, the gradient estimates for unit band spectral projection operators χλ is proved for any second order elliptic differential operators L by maximum principle. A new proof of Hörmander Multiplier Theorem on the eigenfunction expansion of the operator L is given in this setting by using the gradient estimates and the Calderón-Zygmund ...

A well-known open problem of Muckenhoupt–Wheeden says that any Calderón–Zygmund singular integral operator T is of weak type (1,1) with respect to a couple of weights (w,Mw). In this paper, we consider a somewhat “dual” problem: sup λ>0 λw { x ∈Rn : |Tf (x)| Mw > λ } ≤ c ∫

Here we show that Lerner’s method of local mean oscillation gives a simple proof of theA2 conjecture for spaces of homogeneous type: that is, the linear dependence on the A2 norm for weighted L 2 Calderon-Zygmund operator estimates. In the Euclidean case, the result is due to Hytönen, and for geometrically doubling spaces, Nazarov, Rezinikov, and Volberg obtained the linear bound.

Abstract. Given a Calderón–Zygmund (C–Z for short) operator T , which satisfies Hörmander condition, we prove that: if T maps all the characteristic atoms to W L1, then T is continuous from Lp to Lp(1 < p < ∞). So the study of strong continuity on arbitrary function in Lp has been changed into the study of weak continuity on characteristic functions.

Real-variable methods are used to prove the Marcinkiewicz Interpolation Theorem, boundedness of the dyadic and Hardy-Littlewood maximal operators, and the Calderón-Zygmund Covering Lemma. The Hilbert transform is defined, and its boundedness is investigated. All results lead to a final theorem on the pointwise convergence of the truncated Hilbert transform

In this paper the boundedness for a large class of multisublinear operators is established on product generalized Morrey spaces with non-doubling measures. As special cases, the corresponding results for multilinear Calderón-Zygmund operators, multilinear fractional integrals and multi-sublinear maximal operators will be obtained.

Strong laws are established for linear statistics that are weighted sums of a random sample. We show extensions of the Marcinkiewicz-Zygmund strong laws under certain moment conditions on both the weights and the distribution. These not only generalize the result of Bai and Cheng (2000, Statist Probab Lett 46: 105-112) to rho*-mixing sequences of random variables, but also improve them.