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In 2001, the Chinese government began a strong, sustained push for Chinese companies to increase their outgoing investment. The expansion of China's "Going Out" or "Going Global" strategy has brought an increasing number of new domestic and international actors in China-Africa relations. Although scholars and journalists have reported on the role of Chinese ministries and state-owned enterprises (SOEs) in Africa, the role of China's provincial/local governments and provincial companies is seldom covered. Medical teams from Chinese provinces have long been officially "twinned" with particular African countries: Hunan Province serves Sierra Leone and Zimbabwe, for example. Yet unofficial "twinning" between Chinese provinces and African countries has also gradually taken shape in the last thirty years. This generally evolved as Chinese aid workers from particular provinces gained experience in particular African countries, and then introduced others from their province over time. This has laid the foundation for "clustering"-when a group of companies from one region in China all locate fairly close to each other in one African country. As China's "Going Out" policies have evolved, overseas business clusters are becoming officially recognized and promoted by Chinese government officials at all levels. This brief explores how "twinning" has led to "clustering" of provincial Chinese actors in Africa, and how twinning and clustering are related to China's foreign aid program.

The Hardy operator Ta on a tree Γ is defined byProperties of Ta as a map from L p (Γ) into itself are established for 1 ≤ p ≤ ∞. The main result is that, with appropriate assumptions on u and v, the approximation numbers an(Ta) of Ta satisfy ( * ) lim n→∞ nan(Ta) = αp Γ |uv|dt for a specified constant αp and 1 < p < ∞. This extends results of Naimark, Newman and Solomyak for p = 2. Hitherto, for p = 2, (*) was unknown even when Γ is an interval. Also, upper and lower estimates for the l q and weak-l q norms of {an(Ta)} are determined.

The relation between a Banach function space X and its subspaces formed by: (i) those functions with absolutely continuous norm; (ii) those with continuous norm; and (iii) the closure of the set of bounded functions are investigated in the case X = L p (x) .

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