Differentiability Properties of the Symbol of a Generalized Riesz Potential with Homogeneous Characteristic

Let f be a positive homogeneous function of degree 0 defined on the sphere Σ of the space ℝn, and let Φα be the symbol of the integral operator∫ℝnf((x−y)/|x−y|)|x−y|n−αu(y)dy,u∈C0∞(ℝn), with 0 < α < n. We study differentiability properties of the restriction of Φα to the unit sphere Σ in the spaces Hpl(Σ) for p ∈ (1,∞), where Hpl(Σ) denotes the space of Bessel potentials with the norm ‖f‖Hpl(Σ)=‖(δ+I)l/2f‖Lp(Σ) and δ is the Beltrami operator on the sphere. We prove that if f ∈ Lp(Σ), then Φα|Σ∈Hpl(Σ) for any l ≤ n/2 − α − |p−1 − 2−1|(n − 2). Conversely, if Φα|Σ∈Hpl(Σ) with |l ≥ n/2 − α+|p−1 − 2−1|(n − 2), then f ∈ Lp(Σ). The results are sharp. © 2019, Springer Science+Business Media, LLC, part of Springer Nature.