Suppose Δ̃ is the Laplace-Beltrami operator on the sphere $S^{d-1},{\Delta }_{\rho }^{k}f\left(x\right)={\Delta }_{\rho }{\Delta }_{\rho }^{k-1}f\left(x\right)$ and ${\Delta }_{\rho }f\left(x\right)=f\left(\rho x\right)-f\left(x\right)$ where ρ ∈ SO(d). Then ${\omega }^m{(f,t)}_{L_p\left(S^{d-1}\right)}\equiv sup\parallel {\Delta }_{\rho }^{m}f{\parallel }_{L_p\left(S^{d-1}\right)}:\rho \in SO\left(d\right),ma{x}_{x\in S^{d-1}}\rho x·x\ge cost$ and $K̃ₘ{(f,t^m)}_p\equiv inf\parallel f-g{\parallel }_{L_p\left(S^{d-1}\right)}+t^m\parallel {(-\Delta ̃)}^{m/2}g{\parallel }_{L_p\left(S^{d-1}\right)}:g\in \left({(-\Delta ̃)}^{m/2}\right)$ are equivalent for 1 < p < ∞. We note that for even m the relation was recently investigated by the second author. The equivalence yields an extension of the results on sharp Jackson inequalities on the sphere. A new strong converse inequality for $L_p\left(S^{d-1}\right)$ given in this paper plays a significant role in the proof.

One-sided versions of maximal functions for suitable defined distributions are considered. Weighted norm equivalences of these maximal functions for weights in the Sawyer's Aq+ classes are obtained.

Recently, the weak Triebel-Lizorkin space was introduced by Grafakos and He, which includes the standard Triebel-Lizorkin space as a subset. The latter has a wide applications in aspects of analysis. In this paper, the authors firstly give equivalent quasi-norms of weak Triebel-Lizorkin spaces in terms of Peetre's maximal functions. As an application of those equivalent quasi-norms, an atomic decomposition of weak Triebel-Lizorkin spaces is given.

Jones and Rosenblatt started the study of an ergodic transform which is analogous to the martingale transform. In this paper we present a unified treatment of the ergodic transforms associated to positive groups induced by nonsingular flows and to general means which include the usual averages, Cesàro-α averages and Abel means. We prove the boundedness in $L^p$, 1 < p < ∞, of the maximal ergodic transforms assuming that the semigroup is Cesàro bounded in $L^p$. For p = 1 we find that the maximal ergodic...

On montre que les produits de Riesz sur le tore sont des mesures ergodiques sous une condition de lacunarité pour les fréquences, indépendamment de toute propriété arithmétique, et que cette condition est la meilleure possible de ce point de vue. On établit un critère analogue pour la propriété de pureté discutés précédemment par B. Host et l’auteur, ce qui fournit l’exemple d’une mesure pure étrangère à toutes ses translatées et en particulier non ergodique.

Generalizing the classical BMO spaces defined on the unit circle with vector or scalar values, we define the spaces $BM{O}_{{\psi }_q}\left(\right)$ and $BM{O}_{{\psi }_q}(,B)$, where ${\psi }_q\left(x\right)=e^{x^q}-1$ for x ≥ 0 and q ∈ [1,∞[, and where B is a Banach space. Note that $BM{O}_{{\psi }_1}\left(\right)=BMO\left(\right)$ and $BM{O}_{{\psi }_1}(,B)=BMO(,B)$ by the John-Nirenberg theorem. Firstly, we study a generalization of the classical Paley inequality and improve the Blasco-Pełczyński theorem in the vector case. Secondly, we compute the idempotent multipliers of $BM{O}_{{\psi }_q}\left(\right)$. Pisier conjectured that the supports of idempotent multipliers of $L^{{\psi }_q}\left(\right)$ form a Boolean...