Ultradistributional boundary values of harmonic functions on the sphere

Abstract

Abstract. We present a theory of ultradistributional boundary values for harmonic functions defined on the Euclidean unit ball. We also give a characterization of ultradifferentiable functions and ultradistributions on the sphere in terms of their spherical harmonic expansions. To this end, we obtain explicit estimates for partial derivatives of spherical harmonics, which are of independent interest and refine earlier estimates by Calderón and Zygmund. We apply our results to characterize the support of ultradistributions on the sphere via Abel summability of their spherical harmonic expansions.

Rotation Invariant Ultradistributions

“…The spaces of ultradifferentiable functions and ultradistributions on S n−1 can be described in terms of spherical harmonic expansions. A proof of the following theorem will appear in our forthcoming paper [20], which also deals with ultradistributional boundary values of harmonic functions on the sphere. We point out that the distribution case goes back to Estrada and Kanwal [7].…”

“…It is important to point out that Theorem 1 as stated above does not reveal all topological information encoded by the spherical harmonic coefficients. Denote as E {Mp},h sh (S n−1 ) the Banach space of all (necessarily smooth) functions ϕ on S n−1 having spherical harmonic expansion with coefficients a k,j satisfying (2) for a given h. One can then show [20]…”

“…Spherical representations of distributions were studied by Drozhzhinov and Zav'yalov in [6]. We shall also exploit results on spherical harmonic expansions of ultradifferentiable functions and ultradistributions on the unit sphere S n−1 , recently obtained by us in [20]. We mention that the theory of spherical harmonic expansions of distributions was developed by Estrada and Kanwal in [7].…”

Boundary Values in Ultradistribution Spaces Related to Extended Gevrey Regularity

“…Even now this topic for ultradistribution spaces is interesting (cf. [15][16][17][18]). Especially, we have to mention [19].…”

Boundary values in ultradistribution spaces related to extended Gevrey regularity

“…Even now this topic for ultradistribution spaces is interesting cf. [3][4][5]22]. Especially, we have to mention [6].…”