A Weighted Estimate for the Square Function on the Unit Ball in \C^n

We show that the Lusin area integral or the square function on the unit ball of \C^n, regarded as an operator in weighted space $L^2(w)$ has a linear bound in terms of the invariant $A_2$ characteristic of the weight. We show a dimension-free estimate for the ``area-integral'' associated to the weighted $L^2(w)$ norm of the square function. We prove the equivalence of the classical and the invariant $A_2$ classes.Comment: 11 pages, to appear in Arkiv for Matemati

Why the Riesz transforms are averages of the dyadic shifts?

The first author showed in [18] that the Hilbert transform lies in the closed convex hull of dyadic singular operators -so-called dyadic shifts. We show here that the same is true in any $\mathbb{R}^n$ -the Riesz transforms can be obtained as the results of averaging of dyadic shifts. The goal of this paper is almost entirely methodological: we simplify the previous approach, rather than presenting the new one

On the A_2 inequality for Calderon-Zygmund Operators

We prove that for a Calderon-Zygmund operator and weight w\in A_2, that it satisfies the linear in A2 bound due to Hytonen. Our proof will appeal to a distributional inequality used by several authors, adapted Haar functions, and standard stopping times.Comment: 8 pages. v2: A hyperlink in the references is fixe

Why the Riesz transforms are averages of the dyadic shifts?

The first author showed in [18] that the Hilbert transform lies in the closed convex hull of dyadic singular operators -so-called dyadic shifts. We show here that the same is true in any Rn -the Riesz transforms can be obtained as the results of averaging of dyadic shifts. The goal of this paper is almost entirely methodological: we simplify the previous approach, rather than presenting the new one

Isoflurane Induces Endothelial Apoptosis of the Post-Hypoxic Blood-Brain Barrier in a Transdifferentiated Human Umbilical Vein Edothelial Cell Model

Isoflurane is a popular volatile anesthetic agent used in humans as well as in experimental animal research. In previous animal studies of the blood-brain barrier (BBB), observations towards an increased permeability after exposure to isoflurane are reported. In this study we investigated the effect of a 2-hour isoflurane exposure on apoptosis of the cerebral endothelium following 24 hours of hypoxia in an in vitro BBB model using astrocyte-conditioned human umbilical vein endothelial cells (AC-HUVECs). Apoptosis of AC-HUVECs was investigated using light microscopy of the native culture for morphological changes, Western blot (WB) analysis of Bax and Bcl-2, and a TUNEL assay. Treatment of AC-HUVECs with isoflurane resulted in severe cellular morphological changes and a significant dose-dependent increase in DNA fragmentation, which was observed during the TUNEL assay analysis. WB analysis confirmed increases in pro-apoptotic Bax levels at 4 hours and 24 hours and decreases in anti-apoptotic Bcl-2 in a dose-dependent manner compared with the control group. These negative effects of isoflurane on the BBB after a hypoxic challenge need to be taken into account not only in experimental stroke research, but possibly also in clinical practice

A rotation method which gives linear Lp-Estimates for powers of the Ahlfors-Beurling operator

"Vegeu el resum a l'inici del document del fitxer adjunt.

Sharp A₂ inequality for haar shift operators

"Vegeu el resum a l'inici del document del fitxer adjunt"

Optimal lower exponent for the higher gradient integrability of solutions to two-phase elliptic equations in two dimensions

We study the higher gradient integrability of distributional solutions u to the equation div(σ∇u) = 0 in dimension two, in the case when the essential range of σ consists of only two elliptic matrices, i.e., σ ∈ {σ1,σ2} a.e. in Ω. In [9], for every pair of elliptic matrices σ1 and σ2 exponents pσ1,σ2 ∈ (2,+∞) and qσ1,σ2 ∈ (1,2) have been found so that if u ∈ W1,qσ1,σ2(Ω) is solution to the elliptic equation then ∇u ∈ Lpσ1,σ2(Ω) and the optimality of the upper exponent pσ1,σ2 has been proved. In this paper we complement the above result by proving the optimality of the lower exponent qσ1,σ2. Precisely, we show that for every arbitrarily small δ, one can find a particular microgeometry, i.e. an arrangement of the sets σ-1(σ1) and σ-1(σ2), for which there exists a solution u to the corresponding elliptic equation such that ∇u ∈ Lqσ1,σ2-δ, but ∇u Ɇ Lqσ1,σ2-δ. The existence of such optimal microgeometries is achieved by convex integration methods, adapting to the present setting the geometric constructions provided in [2] for the isotropic case