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 to prove that the weighted L 2 (w)-norm of a paraproduct with complexity (m, n), associated to a function b ∈ BMO d , depends linearly on the A d 2-characteristic of the weight w, linearly on the BMO d-norm of b, and polynomially on the complexity. This argument provides a new proof of the linear bound for the dyadic paraproduct due to Beznosova. We also prove that the L 2-norm of a t-Haar multiplier for any t ∈ R and weight w is a multiple of the square root of the C d 2t-characteristic of w times the square root of the A d 2-characteristic of w 2t , and is polynomial in the complexity.

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We provide a quantitative two weight estimate for the dyadic paraproduct π b under certain conditions on a pair of weights (u, v) and b in Carlu,v, a new class of functions that we show coincides with BM O when u = v ∈ A d 2 . We obtain quantitative two weight estimates for the dyadic square function and the martingale transforms under the assumption that the maximal function is bounded from L 2 (u) into L 2 (v) and v ∈ RH d 1 . Finally we obtain a quantitative two weight estimate from L 2 (u) into L 2 (v) for the dyadic square function under the assumption that the pair (u, v) is in joint A d 2 and u −1 ∈ RH d 1 , this is sharp in the sense that when u = v the conditions reduce to u ∈ A d 2 and the estimate is the known linear mixed estimate.2010 Mathematics Subject Classification. Primary 42B20, 42B25 ; Secondary 47B38.where ∆ I v := m I + v − m I − v, and I ± are the right and left children of I.Then π b , the dyadic paraproduct associated to b, is bounded from L 2 (u) into L 2 (v). Moreover, there exists a constant C > 0 such that for all f ∈ L 2 (u)where π b f := I∈D m I f b, h I h I .

In football, many people argue that in knockout competitions the team that plays the second game in its stadium would have an advantage, a greater probability of victory in the final outcome of a two leg knockout match. The purpose of this study is to verify the veracity of this statement using data from the Brazil Cup. We find evidence that the ability spread between the teams participating in a match is the main factor that explains the qualification of one of the teams for the next round. Until 2018, there were three criteria for break the tie in the Brazil Cup, they were used in the respectively order: goal difference, away goals rule, penalty shootout (there is no extra-time in any play-off of the championship). It is estimated that 36% of the matches end tied and need a criterion to determine the winner. Of these, 51% use the goal difference decision, 29% use the away goals rule and 20% penalty shoot-out. When considering the championship in general there is evidence that the home team wins the match in approximately 63% of the matches, a significant advantage. However, in the confrontations that were decided by the away goals rule or the penalty shoot-out, the home team wins percentage is 20% lower, indicating that these criteria level the odds of both teams.