Measuring Triebel–Lizorkin fractional smoothness on domains in terms of first-order differences

Abstract

In this note, we give equivalent characterizations for a fractional Triebel–Lizorkin space (Formula presented.) in terms of first-order differences in a uniform domain (Formula presented.). The characterization is valid for any positive, non-integer real smoothness (Formula presented.) and indices (Formula presented.), (Formula presented.) as long as the fractional part (Formula presented.) is greater than (Formula presented.).