New estimates for the maximal singular integral

In this paper we pursue the study of the problem of controlling the maximal singular integral $T^{*}f$ by the singular integral $Tf$. Here $T$ is a smooth homogeneous Calder\'on-Zygmund singular integral of convolution type. We consider two forms of control, namely, in the L^2(\Rn) norm and via pointwise estimates of $T^{*}f$ by $M(Tf)$ or $M^2(Tf)$, where $M$ is the Hardy-Littlewood maximal operator and $M^2=M \circ M$ its iteration. It is known that the parity of the kernel plays an essential role in this question. In a previous article we considered the case of even kernels and here we deal with the odd case. Along the way, the question of estimating composition operators of the type $T^\star \circ T$ arises. It turns out that, again, there is a remarkable difference between even and odd kernels. For even kernels we obtain, quite unexpectedly, weak $(1,1)$ estimates, which are no longer true for odd kernels. For odd kernels we obtain sharp weaker inequalities involving a weak $L^1$ estimate for functions in $L LogL$.Comment: v2: 56 pages, with small changes made after acceptance by International Math. Research Notice

Social Networks

We survey the literature on social networks by putting together the economics, sociological and physics/applied mathematics approaches, showing their similarities and differences. We expose, in particular, the two main ways of modeling network formation. While the physics/applied mathematics approach is capable of reproducing most observed networks, it does not explain why they emerge. On the contrary, the economics approach is very precise in explaining why networks emerge but does a poor job in matching real-world networks. We also analyze behaviors on networks, which take networks as given and focus on the impact of their structure on individuals’ outcomes. Using a game-theoretical framework, we then compare the results with those obtained in sociology.random graph, game theory, centrality measures, network formation, weak and strong ties

Identity and Social Distance in Friendship Formation

We analyze a network formation model where agents belong to different communities. Both individual benefits and costs depend on direct as well as indirect connections. Benefits of an indirect connection decrease with distance in the network while the cost of a link depends on the type of agents involved. Two individuals from the same community always face a low linking cost and the cost of forming a relationship for two individuals of different communities diminishes with the rate of exposure of each of them to the other community. We derive a number of results with regard to equilibrium networks. In particular, socialization among the same type of agents can be weak even if the within-type link cost is very low and oppositional identity patterns can arise for a wide range of parameters. Our model also suggests that policies aiming at reducing segregation are socially desirable only if they reduce the within-community cost differential by a sufficiently large amount.networks; identity; homophily; social norms

Social Networks

We survey the literature on social networks by putting together the economics, sociological and physics/applied mathematics approaches, showing their similarities and differences. We expose, in particular, the two main ways of modeling network formation. While the physics/applied mathematics approach is capable of reproducing most observed networks, it does not explain why they emerge. On the contrary, the economics approach is very precise in explaining why networks emerge but does a poor job in matching real-world networks. We also analyze behaviors on networks, which take networks as given and focus on the impact of their structure on individuals’ outcomes. Using a game-theoretical framework, we then compare the results with those obtained in sociology.Random Graph; Game Theory; Centrality Measures; Network Formation; Weak

The sources of long-run growth in Spain 1850-2000

Between 1850 and 2000, Spain s real income increased by about 40-fold, at an average rate of 2.5 percent. The sources of this long-run growth are investigated using Jorgenson-type growth accounting analysis. We find that growth upsurges are closely related to increases in TFP. Spanish economic growth went through three successive phases. The century before 1950 was characterized by slow growth driven by factor accumulation. TFP improvements pushed up explosive growth during the Golden Age and mitigated the deceleration during the transition to democracy years (1975-86). Since the accession to the European Union Spain has experienced a dramatic productivity slowdow

Wages and labor income in history : a survey

Income distribution has been a main topic in economics since the days of Gregory King and William Petty. In this paper some empirical issues in the study of labor income are surveyed in the light of economic history, including the hypothesis of the stability of factor shares across time and space and the relative importance of raw labor and human capital in labor income