and Applied Analysis 3 In two main theorems in 20 , the authors proved the following results, which we now incorporate in the next theorem. Theorem A. Assume p ≥ 1 and φ is a holomorphic self-map of Π . Then the following statements true hold. a The operator Cφ : H Π → A∞ Π is bounded if and only if sup z∈Π Im z ( Imφ z )1/p < ∞. 1.8 b The operator Cφ : H Π → B∞ Π is bounded if and only if sup ...

Baum and Katz (Trans. Am. Math. Soc. 120:108-123, 1965) obtained convergence rates in the Marcinkiewicz-Zygmund law of large numbers. Their result has already been extended to the short-range dependent linear processes by many authors. In this paper, we extend the result of Baum and Katz to the long-range dependent linear processes. As a corollary, we obtain convergence rates in the Marcinkiewi...

This result is applied to weighted inequalities. In particular, it implies (i) the “twoweight conjecture” by D. Cruz-Uribe and C. Pérez in full generality; (ii) a simplification of the proof of the “A2 conjecture”; (iii) an extension of certain mixed Ap-Ar estimates to general Calderón-Zygmund operators; (iv) an extension of sharp A1 estimates (known for T ) to the maximal Calderón-Zygmund oper...

Let T be the Calderón-Zygmund operator, Coifman, Rochberg and Weiss (see [4]) proves that the commutator [b, T ](f) = bT (f) − T (bf)(where b ∈ BMO(R)) is bounded on L(R) for 1 < p <∞. Chanillo (see [2]) proves a similar result when T is replaced by the fractional operators. In [8, 16], Janson and Paluszynski study these results for the Triebel-Lizorkin spaces and the case b ∈ Lipβ(R), where Li...

The continuity for some multilinear operators related to certain fractional singular integral operators on Triebel-Lizorkin spaces is obtained. The operators include Calderon-Zygmund singular integral operator and fractional integral operator. 1. Introduction. Let T be a Calderon-Zygmund singular integral operator; a well-known result of Coifman et al. (see [6]) states that the commutator [b, T...

In global seismology Earth’s properties of fractal nature occur. Zygmund classes appear as the most appropriate and systematic way to measure this local fractality. For the purpose of seismic wave propagation, we model the Earth’s properties as Colombeau generalized functions. In one spatial dimension, we have a precise characterization of Zygmund regularity in Colombeau algebras. This is made ...

We prove that the weak Morrey space WM is contained in $$M_{{q_1}}^p$$ for 1 ≤ q1 < q p ∞. As applications, we show if commutator [b, T] bounded from Lp to Lp,∞ some ∈ (1, ∞), then b BMO, where T a Calderón-Zygmund operator. Also, ∞, BMO and only [6, M . For belonging Lipschitz class, obtain similar results.

Let p ∈ (0, 1]. In this paper, the authors prove that a sublinear operator T (which is originally defined on smooth functions with compact support) can be extended as a bounded sublinear operator from product Hardy spacesH(R × R) to some quasi-Banach space B if and only if T maps all (p, 2, s1, s2)-atoms into uniformly bounded elements of B. Here s1 ≥ ⌊n(1/p− 1)⌋ and s2 ≥ ⌊m(1/p− 1)⌋. As usual,...