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

Resumo

Let f be a positive homogeneous function of degree 0 defined on the sphere Σ of the space ℝⁿ, and let Φ_α be the symbol of the integral operator

\( \underset{{\mathrm{\mathbb{R}}}^n}{\int}\frac{f\left(\left(\mathrm{x}-y\right)/\left|\mathrm{x}-y\right|\right)}{{\left|\mathrm{x}-y\right|}^{n-\alpha }}u(y) dy,\kern1em u\in {C}_0^{\infty}\left({\mathrm{\mathbb{R}}}^n\right), \)

with 0 < α < n. We study differentiability properties of the restriction of Φ_α to the unit sphere Σ in the spaces \( {H}_p^l\left(\Sigma \right) \) for p ∈ (1,∞), where \( {H}_p^l\left(\Sigma \right) \) denotes the space of Bessel potentials with the norm \( {\left\Vert f\right\Vert}_{H_p^l\left(\Sigma \right)}={\left\Vert {\left(\delta +I\right)}^{l/2}f\right\Vert}_{L_p\left(\Sigma \right)} \) and δ is the Beltrami operator on the sphere. We prove that if f ∈ L_p(Σ), then \( {\left.{\Phi}_{\alpha}\right|}_{\Sigma}\in {H}_p^l\left(\Sigma \right) \) for any l ≤ n/2 − α − |p⁻¹ − 2⁻¹|(n − 2). Conversely, if \( {\left.{\Phi}_{\alpha}\right|}_{\Sigma}\in {H}_p^l\left(\Sigma \right) \) with |l ≥ n/2 − α+|p⁻¹ − 2⁻¹|(n − 2), then f ∈ L_p(Σ). The results are sharp.