Capacity and perimeter from logarithmic fractional Sobolev spaces on Carnot groups

Abstract In this paper, we introduce the logarithmic fractional Sobolev space on Carnot groups and investigate the various functional-geometric aspects of the corresponding functional capacity. Moreover, we also investigate the logarithmic fractional perimeters and establish some fundamental properties of this perimeter, the coarea formula for logarithmic fractional Sobolev functions. Finally, we characterize the continuous embedding of the logarithmic fractional Sobolev space into the space L p ( μ ) in terms of the logarithmic fractional Sobolev capacity or the associated variational functional of a non-negative Radon measure μ or the logarithmic fractional perimeter.