Logarithmic mean oscillation on the Polydisc, Multi-parameter paraproducts and iterated commutators

We introduce another notion of bounded logarithmic mean oscillation in the N-torus and give an equivalent definition in terms of boundedness of multi-parameter paraproducts from the dyadic little BMO of Cotlar-Sadosky to the product BMO of Chang-Fefferman. We also obtain a sufficient condition for the boundedness of iterated commutators with Hilbert transforms betweeen the strong notions of these two spaces.Comment: To appea

Weighted boundedness of maximal functions and fractional Bergman operators

The aim of this paper is to study two-weight norm inequalities for fractional maximal functions and fractional Bergman operator defined on the upper-half space. Namely, we characterize those pairs of weights for which these maximal operators satisfy strong and weak type inequalities. Our characterizations are in terms of Sawyer and B\'ekoll\'e-Bonami type conditions. We also obtain a $\Phi$-bump characterization for these maximal functions, where $\Phi$ is a Orlicz function. As a consequence, we obtain two-weight norm inequalities for fractional Bergman operators. Finally, we provide some sharp weighted inequalities for the fractional maximal functions

An embedding relation for bounded mean oscillation on rectangles

In the two-parameter setting, we say a function belongs to the mean little $BMO$, if its mean over any interval and with respect to any of the two variables has uniformly bounded mean oscillation. This space has been recently introduced by S. Pott and the author in relation with the multiplier algebra of the product $BMO$ of Chang-Fefferman. We prove that the Cotlar-Sadosky space of functions of bounded mean oscillation $bmo(\mathbb{T}^N)$ is a strict subspace of the mean little $BMO$.Comment: Publishe

On some equivalent definitions of ρ\rho- Carleson measures on the unit ball

We give in this paper some equivalent definitions of the so called $\rho$-Carleson measures when $\rho(t)=(\log(4/t))^p(\log\log(e^4/t))^q$, $0\le p,q<\infty$. As applications, we characterize the pointwise multipliers on $LMOA(\mathbb S^n)$ and from this space to $BMOA(\mathbb S^n)$. Boundedness of the Ces\`aro type integral operators on $LMOA(\mathbb S^n)$ and from $LMOA(\mathbb S^n)$ to $BMOA(\mathbb S^n)$ is considered as well.Comment: 23 page

Schatten class Toeplitz operators on weighted Bergman spaces of tube domains over symmetric cones

We prove some characterizations of Schatten class Toeplitz operators on Bergman spaces of tube domains over symmetric cones for small exponents

Weighted norm inequalities for fractional Bergman operators

We prove in this note one weight norm inequalities for some positive Bergman-type operators.Comment: 17 pages. arXiv admin note: substantial text overlap with arXiv:1703.0085

Bounded and invertible Toeplitz products on vector weighted Bergman spaces of the polydisc

We characterize bounded and invertible Toeplitz products on vector weighted Bergman spaces of the unit polydisc. For our purpose, we will need the notion of B\'ekoll\'e-Bonami weights in several parameters

Sharp off-diagonal weighted norm estimates for the Bergman projection

We prove that for $1<p\le q<\infty$, $qp\geq {p'}^2$ or $p'q'\geq q^2$, $\frac{1}{p}+\frac{1}{p'}=\frac{1}{q}+\frac{1}{q'}=1$, $$\|\omega P_\alpha(f)\|_{L^p(\mathcal{H},y^{\alpha+(2+\alpha)(\frac{q}{p}-1)}dxdy)}\le C_{p,q,\alpha}[\omega]_{B_{p,q,\alpha}}^{(\frac{1}{p'}+\frac{1}{q})\max\{1,\frac{p'}{q}\}}\|\omega f\|_{L^p(\mathcal{H},y^{\alpha}dxdy)}$$ where $P_\alpha$ is the weighted Bergman projection of the upper-half plane $\mathcal{H}$, and $$[\omega]_{B_{p,q,\alpha}}:=\sup_{I\subset \mathbb{R}}\left(\frac{1}{|I|^{2+\alpha}}\int_{Q_I}\omega^{q}dV_\alpha\right)\left(\frac{1}{|I|^{2+\alpha}}\int_{Q_I}\omega^{-p'}dV_\alpha\right)^{\frac{q}{p'}},$$ with $Q_I=\{z=x+iy\in \mathbb{C}: x\in I, 0<y<|I|\}$.Comment: This paper is not for publicatio

On two-weight norm estimates for multilinear fractional maximal function

We prove some Sawyer-type characterizations for multilinear fractional maximal function for the upper triangle case. We also provide some two-weight norm estimates for this operator. As one of the main tools, we use an extension of the usual Carleson Embedding that is an analogue of the P. L. Duren extension of the Carleson Embedding for measures

Φ\Phi-Carleson measures and multipliers between Bergman-Orlicz spaces of the unit ball of Cn\mathbb {C}^n

We define the notion of $\Phi$-Carleson measures where $\Phi$ is either a concave growth function or a convex growth function and provide an equivalent definition. We then characterize $\Phi$-Carleson measures for Bergman-Orlicz spaces, and apply them to characterize multipliers between Bergman-Orlicz spaces