Mean Hölder–Lipschitz potentials in curved Campanato–Radon spaces and equations $(-\Delta )^{\frac{\alpha }{2}}u=\mu = F_k[u]$

Abstract

This paper shows that under

$$\begin{aligned} {\left\{ \begin{array}{ll} 0<s<1;\\ 0<\alpha<n;\\ 0<\beta \le n;\\ {1\le \min \{p, q\}\le \max \{p,q\}<\beta p(n-\alpha p)^{-1}}<\infty ;\\ \lambda =q(np^{-1}-s-\alpha )+n-\beta , \end{array}\right. } \end{aligned}$$

if $\mu$ is a nonnegative Radon measure on ${\mathbb {R}}^n$ with the $\beta$-dimensional upper curvature $|||\mu |||_{\beta }<\infty$ then $I_{\alpha } \dot{\varLambda }_s^{p,\infty }$ (the mean Hölder–Lipschitz potential space on ${\mathbb {R}}^n$) embeds continuously into $\mathcal {L}^{q,\lambda }_{\mu }$ (the curved Campanato–Radon space on ${\mathbb {R}}^n$); and yet its converse is still valid with $\mu$ being admissible/doubling (cf. Theorem 1.1), thereby discovering the $\gamma$-Hölder–Lipschitz continuity of any duality solution to the $\alpha$-th Laplace equation $(-\varDelta )^{\frac{\alpha }{2}}u=\mu$ or the $[1,\frac{n}{2})\cap \{1,2,\ldots ,n\}\ni k$-th Hessian equation $F_k[u]=\mu$ under a suitable curvature condition $|||\mu |||_{\beta }<\infty$ (cf. Theorems 1.3 and 1.5).