We prove the $L^p$ boundedness of the Marcinkiewicz integral operators ${\mu }_{\Omega }$ on ${ℝ}^{n₁}\times \cdots \times {ℝ}^{n_k}$ under the condition that $\Omega \in L{\left(logL\right)}^{k/2}{(}^{n₁-1}\times \cdots {\times }^{n_k-1})$. The exponent k/2 is the best possible. This answers an open question posed by Y. Ding.

Let $f\in V_r\left(\right){\cup }_r\left(\right)$, where, for 1 ≤ r < ∞, $V_r\left(\right)$ (resp., ${}_r\left(\right)$) denotes the class of functions (resp., bounded functions) g: → ℂ such that g has bounded r-variation (resp., uniformly bounded r-variations) on (resp., on the dyadic arcs of ). In the author’s recent article [New York J. Math. 17 (2011)] it was shown that if is a super-reflexive space, and E(·): ℝ → () is the spectral decomposition of a trigonometrically well-bounded operator U ∈ (), then over a suitable non-void open interval of r-values, the condition...

Martingale Hardy spaces and BMO spaces generated by an operator T are investigated. An atomic decomposition of the space $H_p^T$ is given if the operator T is predictable. We generalize the John-Nirenberg theorem, namely, we prove that the $BM{O}_q$ spaces generated by an operator T are all equivalent. The sharp operator is also considered and it is verified that the $L_p$ norm of the sharp operator is equivalent to the $H_p^T$ norm. The interpolation spaces between the Hardy and BMO spaces are identified by the real method....

In this paper we give some criteria for the existence of compactly supported $C^{k+\alpha }$-solutions ($k$ is an integer and $0\le \alpha <1$) of matrix refinement equations. Several examples are presented to illustrate the general theory.

In this paper we deal with several characterizations of the Hardy-Sobolev spaces in the unit ball of Cn, that is, spaces of holomorphic functions in the ball whose derivatives up to a certain order belong to the classical Hardy spaces. Some of our characterizations are in terms of maximal functions, area functions or Littlewood-Paley functions involving only complex-tangential derivatives. A special case of our results is a characterization of Hp itself involving only complex-tangential derivatives....

Let ω be a Békollé-Bonami weight. We give a complete characterization of the positive measures μ such that ${\int }_{}|M_{\omega }{f\left(z\right)|}^{q}d\mu \left(z\right)\le C({\int }_{}{\left|f\left(z\right)\right|}^p{\omega \left(z\right)dV\left(z\right))}^{q/p}$ and $\mu \left(z\in :Mf\left(z\right)>\lambda \right)\le C/\left({\lambda }^q\right)({\int }_{}|f\left(z\right){{|}^p\omega \left(z\right)dV\left(z\right))}^{q/p}$, where $M_{\omega }$ is the weighted Hardy-Littlewood maximal function on the upper half-plane and 1 ≤ p,q <; ∞.