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Mean Hölder-Lipschitz potentials in curved Campanato-Radon spaces and equations \((-\Delta)^{\frac{\alpha}{2}}u=\mu = F_k[u]\). (English) Zbl 1425.35217

For a nonnegative Radon measure \(\mu\) on \(\mathbb{R}^n\) with finite \(\beta\)-dimensional upper curvature, the authors prove that the mean Hölder-Lipschitz potential space \(I_{\alpha}\dot\Lambda_s^{p,\infty}\) on \(\mathbb{R}^n\), is embeded continuously into the curved Campanato-Radon space \(\mathcal{L}_{\mu}^{q,\lambda}\) on \(\mathbb{R}^n\), where \(s\in (0,1)\), \(\alpha\in (0,n)\), \(\beta\in (0,n]\), \(\min\{p,q\}\ge 1\), \(\max\{p,q\}<\beta p(n-\alpha p)^{-1}<\infty\), and \(\lambda=q(np^{-1}-s-\alpha)+n-\beta\). The converse problem is also valid when \(\mu\) is an admissible or doubling measure. Then, under a suitable curvature condition for \(\mu\), they show the \(\gamma\)-Hölder-Lipschitz continuity of any duality solution to the \(\alpha\)-th Laplace equation \((-\Delta)^{\frac{\alpha}{2}}u=\mu\), and to the \(k\)-th Hessian equation \(F_{k}[u]=\mu\), where \(k\in [1,\frac{n}{2})\cap \{1,2,\ldots,n\}\), and \(F_k[u]=\sum_{1\le i_1<\cdots<i_k\le n}\lambda_{i_1}\cdots \lambda_{i_k}\), with \(\lambda_{i_1}\cdots \lambda_{i_k}\) are eigenvalues of the Hessian matrix \(D^2u\).

MSC:

35R11 Fractional partial differential equations
47G40 Potential operators
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
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