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In this paper, we establish the sharp critical and subcritical traceTrudinger-Moser and Adams inequalities on the half spaces and prove theexistence of their extremals through the method based on the Fourierrearrangement, harmonic extension and scaling invariance. These traceTrudinger-Moser and Adams inequalities can be considered as the borderline caseof the Sobolev trace inequalities of first and higher orders. Furthermore, weshow the existence of the least energy solutions for a class of bi-harmonicequations with nonlinear Neumann boundary condition associated with the traceAdams inequalities.

Sharp critical and subcritical trace Trudinger-Moser and Adams inequalities on the upper half spaces