Morrey--Sobolev inequalities with power weights on the half-space
Jean Van Schaftingen, Leon Winter
TL;DR
The paper addresses Morrey–Sobolev inequalities for weighted Sobolev spaces on the half-space with power weights $z_n^\gamma$ and extends the framework to the hyperbolic space via the critical case $\gamma=p-n$. It develops a global, sharp modulus of continuity bound $|u(x)-u(y)| \le C\Theta_{1-1/p,1-n/p,1-(n+\gamma)/p}(x,y)\big\||\nabla u|\big|^p_{L^p(z_n^\gamma)}^{1/p}$, with distinct regimes depending on $\gamma$ relative to $p-n$, and proves the optimality of these bounds through explicit test functions, showing the modulus must be comparable to the corresponding $\Theta$ distance and can be taken as a true distance. In the critical hyperbolic case $\gamma=p-n$, the results yield a hyperbolic Morrey–Sobolev inequality with distance $d_{\mathbb{H}^n}$ and its sharp modulus, along with a compactly supported variant. The work thus unifies interior and boundary regularity for weighted half-space Sobolev spaces and provides sharp, geometry-driven optimality results, connecting Euclidean and hyperbolic Morrey–Sobolev theory.
Abstract
Morrey--Sobolev inequalities are established for functions in weighted Sobolev spaces on the $n$-dimensional half-space, where the weight is a power of the distance to the boundary, as well as for Sobolev spaces on the $n$-dimensional hyperbolic space. All the estimates are optimal up to a multiplicative constant.