Carleson Measures and Trace Theorem for β-harmonic Functions

General harmonic extension has no uniqueness and harmonic functions may have different non-tangential boundary values in different convergence sense. In this paper, we establish first beta-harmonic functions in ultra-distribution frame. Further, we consider the characterization between Carleson measure space and boundary distribution space. For beta-harmonic functions with boundary distributions, there exists no maximum value principle. We apply Meyer wavelets to introduce basic harmonic functions and basic observers. We apply Meyer wavelets and vaguelette knowledge to prove the uniqueness of beta-harmonic extension and prove also that beta-harmonic function converges to boundary distribution in the relative norm sense.