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In this paper, using the method of blow-up analysis, we obtained aTrudinger-Moser inequality involving L p -norm on a closed Riemann surface andproved the existence of an extremal function for the correspondingTrudinger-Moser functional. This extends an early result of Yang [27]. Similarresult in the Euclidean space was also established by Lu-Yang [17].

A Trudinger-Moser inequality involving L^p -norm on a closed Riemann surface