On the Bourgain--Brezis--Mironescu spaces over Carleson tents

We introduce Carleson analogs of the Bourgain–Brezis–Mironescu spaces $B$ and $B_0$ by measuring mean oscillation over upper Carleson tents. For these spaces, denoted by $B_{\mathcal{C}}^p$ and $B_{\mathcal{C},0}^p$, we prove two types of structural results. First, we show that they contain several natural classes of functions, including BMO/VMO–Carleson spaces, tent-space potential classes, and fractional Sobolev classes. Second, motivated by Zhu’s structural theorem for BMO spaces induced by the Bergman metric, we establish decompositions of $B_{\mathcal{C}}^p$ and $B_{\mathcal{C},0}^p$ into bounded-oscillation and bounded-average components. We then revisit the Bourgain–Brezis–Mironescu rigidity phenomenon in the Carleson setting. Although the direct rigidity statement fails for $B_{\mathcal{C},0}^p$, we introduce a natural $B_{\mathcal{C}}^p$-trace and prove that the rigidity theorem survives at the level of traces.