$S^1$-equivariant relative symplectic cohomology and relative symplectic capacities

In this paper, we construct an $S^1$-equivariant version of the relative symplectic cohomology developed by Varolgunes. As an application, we construct a relative version of Gutt-Hutchings capacities and a relative version of symplectic (co)homology capacity. We will see that these relative symplectic capacities can detect the diplaceability and the heaviness of a compact subset of a symplectic manifold. We compare the first relative Gutt-Hutchings capacity and the relative symplectic (co)homology capacity and prove that they are equal to each other under a convexity assumption.