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The dyadic paraproduct is bounded in weighted Lebesgue spaces Lp(w) if and only if the weight w belongs to the Muckenhoupt class Adp. However, the sharp bounds on the norm of the dyadic paraproduct are not known even in the simplest L2(w) case. In this paper we prove the linear bound on the norm of the dyadic paraproduct in the weighted Lebesgue space L2(w)(More)
In this paper we present a brief survey on Haar multipliers, dyadic paraproducts, and recent results on their applications to deduce scalar and vector valued weighted inequalities. We present a new proof of the boundedness of a Haar multiplier in L p (R). The proof is based on a stopping time argument suggested by P. W. Jones for the case p = 2, that it is(More)
We show that if an operator T is bounded on weighted Lebesgue space L(w) and obeys a linear bound with respect to the A2 constant of the weight, then its commutator [b, T ] with a function b in BMO will obey a quadratic bound with respect to the A2 constant of the weight. We also prove that the kth-order commutator T k b = [b, T k−1 b ] will obey a bound(More)
We extend the definitions of dyadic paraproduct and t-Haar multipliers to dyadic operators that depend on the complexity (m, n), for m and n natural numbers. We use the ideas developed by Nazarov and Volberg in [NV] to prove that the weighted L(w)-norm of a paraproduct with complexity (m, n), associated to a function b ∈ BMO, depends linearly on the(More)
Decompressive craniectomy (DC) is a surgical practice that has been used since the late 19th century. The cerebral blood flow increase after the performance of a DC can delay and even prevent the development of cerebral circulatory arrest and brain death (BD). We aimed to determine the prevalence of BD, the use of DC, and the evolution to BD with versus(More)
The design and development of processes involving supercritical fluids depend on how easy the phase equilibrium can be accurately modelled and predicted. In the work described herein, the supercritical fluid-solid equilibrium has been considered. Modelling the fluid-solid equilibrium is associated with a number of drawbacks, even when it is possible to(More)