Juan José Marín, José María Martell, Marius Mitrea
We prove well-posedness results for the Dirichlet problem in $\mathbb{R}^{n}_{+}$ for homogeneous, second order, constant complex coefficient elliptic systems with boundary data in generalized Hölder spaces $\mathscr{C}^ω(\mathbb{R}^{n-1},\mathbb{C}^M)$ and in generalized Morrey-Campanato spaces $\mathscr{E}^{ω,p}(\mathbb{R}^{n-1},\mathbb{C}^M)$ under certain assumptions on the growth function $ω$. We also identify a class of growth functions $ω$ for which $\mathscr{C}^ω(\mathbb{R}^{n-1},\mathbb{C}^M)=\mathscr{E}^{ω,p}(\mathbb{R}^{n-1},\mathbb{C}^M)$ and for which the aforementioned well-posedness results are equivalent, in the sense that they have the same unique solution, satisfying natural regularity properties and estimates.
Mingming Cao, Juan José Marín, José María Martell
We generalize the extrapolation theory of Rubio de Francia to the context of Banach function spaces and modular spaces. Our results are formulated in terms of some natural weighted estimates for the Hardy-Littlewood maximal function and are stated in measure spaces and for general Muckenhoupt bases. Finally, we give several applications in analysis and partial differential equations.
Juan José Marín, José María Martell, Dorina Mitrea, Marius Mitrea
We prove several characterizations of $\mathscr{C}^{1,ω}$-domains (aka Lyapunov domains), where $ω$ is a growth function satisfying natural assumptions. For example, given an Ahlfors regular domain $Ω\subseteq{\mathbb{R}}^n$, we show that the modulus of continuity of the geometric measure theoretic outward unit normal $ν$ to $Ω$ is dominated by (a multiple of) $ω$ if and only if the action of each Riesz transform $R_j$ associated with $\partialΩ$ on the constant function $1$ has a modulus of continuity dominated by (a multiple of) $ω$. The proof of this result requires that we establish a higher-dimensional generalization of the classical Plemelj-Privalov theorem, identifying a large class of singular integral operators that are bounded on generalized Hölder spaces. This class includes the Cauchy-Clifford operator and the harmonic double layer operator, among others.