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We prove a trace Hardy type inequality with the best constant on thepolyhedral convex cones which generalizes recent results of Alvino et al. andof Tzirakis on the upper half space. We also prove some traceHardy-Sobolev-Maz'ya type inequalities which generalize the recent results ofFilippas et al.. In applications, we derive some Hardy type inequalities andHardy-Sobolev-Maz'ya type inequalities for fractional Laplacian. Finally, weprove the logarithmic Sobolev trace inequalities and logarithmic Hardy traceinequalities on the upper half spaces. The best constants in these inequalitiesare explicitly computed in the radial case.

Some trace Hardy type inequalities and trace Hardy-Sobolev-Maz'ya type inequalities