Taiwanese Journal of Mathematics

Carleson Measures and Trace Theorem for $\beta$-harmonic Functions

Heping Liu, Haibo Yang, and Qixiang Yang

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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.

Article information

Taiwanese J. Math., Volume 22, Number 5 (2018), 1107-1138.

Received: 10 May 2017
Accepted: 5 December 2017
First available in Project Euclid: 16 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 45P05: Integral operators [See also 47B38, 47G10] 30H

$\beta$-harmonic function Carleson measures and local compact Carleson measures bounded $q$-mean oscillation spaces boundary distribution Meyer wavelet vaguelette


Liu, Heping; Yang, Haibo; Yang, Qixiang. Carleson Measures and Trace Theorem for $\beta$-harmonic Functions. Taiwanese J. Math. 22 (2018), no. 5, 1107--1138. doi:10.11650/tjm/171201. https://projecteuclid.org/euclid.twjm/1513393251

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