Dimension estimates for the boundary of planar Sobolev extension domains

We prove an asymptotically sharp dimension upper-bound for the boundary of bounded simply-connected planar Sobolev W-1,W-p-extension domains via the weak mean porosity of the boundary. The sharpness of our estimate is shown by examples.

Automated summary

We establish an asymptotically sharp dimension upper-bound for the boundary of bounded simply-connected planar Sobolev W-1, W-p-extension domains by using the weak mean porosity of the boundary. The sharpness of our estimate is demonstrated with examples.