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Journal of the London Mathematical Society 2004 70(1):182-198; doi:10.1112/S0024610704005265
© 2004 by London Mathematical Society
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© The London Mathematical Society

Harmonic Function Spaces on Groups

Cho-HO Chu

School of Mathematical Sciences, Queen Mary, University of London London E1 4NS, United Kingdom, c.chu{at}qmul.ac.uk

Received 24 June 2003. Revision received 11 December 2003.

A class of harmonic function spaces is introduced and studied, namely the spaces Formula(G) of {sigma}-harmonic Lp functions on a locally compact group G, for 1≤ p ≤ {infty} and a given complex measure {sigma} on G of unit norm. It is shown that there is a contractive projection from Lp(G) onto Formula(G), for 1< p≤ {infty}, and structural results for Formula(G) are deduced. Given an adapted probability measure {sigma} on G, a uniqueness result is proved, that the space Formula(G) contains only constant functions, for 1≤ p<{infty}. For any {sigma}, a result on the dimension of Formula (G) is proved.


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