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  • 1. Liu, Congwen
    et al.
    Perälä, Antti
    Umeå University, Faculty of Science and Technology, Department of Mathematics and Mathematical Statistics.
    Si, Jiajia
    Weighted integrability of polyharmonic functions in the higher-dimensional case2021In: Analysis & PDE, ISSN 2157-5045, E-ISSN 1948-206X, Vol. 14, no 7, p. 2047-2068Article in journal (Refereed)
    Abstract [en]

    This paper is concerned with the L-p integrability of N-harmonic functions with respect to the standard weights (1 - vertical bar x vertical bar(2))(alpha) on the unit ball B of R-n, n >= 2. More precisely, our goal is to determine the real (negative) parameters alpha for which (1 - vertical bar x vertical bar(2))(alpha/p) u(x) is an element of L-p(B) implies that u equivalent to 0 whenever u is a solution of the N-Laplace equation on B. This question is motivated by the uniqueness considerations of the Dirichlet problem for the N-Laplacian Delta(N).

    Our study is inspired by a recent work of Borichev and Hedenmalm (Adv. Math. 264 (2014), 464-505), where a complete answer to the above question in the case n D 2 is given for the full scale 0 < p < infinity. When n >= 3, we obtain an analogous characterization for n-2/n-1 <= p < infinity and remark that the remaining case can be genuinely more difficult. Also, we extend the remarkable cellular decomposition theorem of Borichev and Hedenmalm to all dimensions.

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