The point-wise product of a function of bounded mean oscillation with a function of the Hardy space $H^1$ is not locally integrable in general. However, in view of the duality between $H^1$ and $BMO$, we are able to give a meaning to the product as a Schwartz distribution. Moreover, this distribution can be written as the sum of an integrable function and a distribution in some adapted Hardy-Orlicz space. When dealing with holomorphic functions in the unit disc, the converse is also valid: every holomorphic...

We establish Lp-boundedness for a class of product singular integral operators on spaces M = M1 x M2 x . . . x Mn. Each factor space Mi is a smooth manifold on which the basic geometry is given by a control, or Carnot-Carathéodory, metric induced by a collection of vector fields of finite type. The standard singular integrals on Mi are non-isotropic smoothing operators of order zero. The boundedness of the product operators is then a consequence of a natural Littlewood- Paley theory on M. This in...

This paper studies a new maximal operator introduced by Hytönen, McIntosh and Portal in 2008 for functions taking values in a Banach space. The $L^p$-boundedness of this operator depends on the range space; certain requirements on type and cotype are present for instance. The original Euclidean definition of the maximal function is generalized to σ-finite measure spaces with filtrations and the $L^p$-boundedness is shown not to depend on the underlying measure space or the filtration. Martingale techniques...

We characterize the range of some spaces of functions by the Fourier transform associated with the Riemann-Liouville operator ${ℛ}_{\alpha },\alpha \ge 0$ and we give a new description of the Schwartz spaces. Next, we prove a Paley-Wiener and a Paley-Wiener-Schwartz theorems.

Estimates of the strong means of Marcinkiewicz type with the Cesaro means of negative order in one of the variables instead of square partial sums are obtained by characteristics constructed on the basis of moduli of continuity.

We study the regularity properties of bilinear maximal operator. Some new bounds and continuity for the above operators are established on the Sobolev spaces, Triebel-Lizorkin spaces and Besov spaces. In addition, the quasicontinuity and approximate differentiability of the bilinear maximal function are also obtained.

In this paper we study the regularity properties of the one-dimensional one-sided Hardy-Littlewood maximal operators ${ℳ}^{+}$ and ${ℳ}^{-}$. More precisely, we prove that ${ℳ}^{+}$ and ${ℳ}^{-}$ map $W^{1,p}\left(ℝ\right)\to W^{1,p}\left(ℝ\right)$ with $1<p<\infty$, boundedly and continuously. In addition, we show that the discrete versions $M^{+}$ and $M^{-}$ map $\mathrm{BV}\left(\mathbb{Z}\right)\to \mathrm{BV}\left(\mathbb{Z}\right)$ boundedly and map $l^1\left(\mathbb{Z}\right)\to \mathrm{BV}\left(\mathbb{Z}\right)$ continuously. Specially, we obtain the sharp variation inequalities of $M^{+}$ and $M^{-}$, that is, $$\mathrm{Var}\left(M^{+}\left(f\right)\right)\le \mathrm{Var}\left(f\right)\text{and}\mathrm{Var}\left(M^{-}\left(f\right)\right)\le \mathrm{Var}\left(f\right)$$ if $f\in \mathrm{BV}\left(\mathbb{Z}\right)$, where $\mathrm{Var}\left(f\right)$ is the total variation of $f$ on $\mathbb{Z}$ and $\mathrm{BV}\left(\mathbb{Z}\right)$ is the set of all functions $f:\mathbb{Z}\to ℝ$ satisfying $\mathrm{Var}\left(f\right)<\infty$.

We consider the boundedness of certain singular integral operators that arose in the study of Sobolev spaces on Lipschitz curves, [P1]. The standard theory available (David and Journé's T1 Theorem, for instance; see [D]) does not apply to this case becuase the operators are not necessarily Calderón-Zygmund operators, [Ch]. One of these operators gives an explicit formula for the resolvent at λ = 1 of the dyadic paraproduct, [Ch].

Two-sided estimates of Schatten-von Neumann norms for weighted Volterra integral operators are established. Analogous problems for some potential-type operators defined on Rn are solved.

We give $A_p$ type conditions which are sufficient for two-weight, strong $(p,p)$ inequalities for Calderón-Zygmund operators, commutators, and the Littlewood-Paley square function $g_{\lambda }^{*}$. Our results extend earlier work on weak $(p,p)$ inequalities in [13].

The uniqueness theorem for the ergodic maximal operator is proved in the continous case.