Make sense of the off-diagonal RKHS operator composition

Determine how to define and control the off-diagonal composition H_{\mu_s,\mu_t}H_{\mu_t}^{-1} in Banach spaces constructed directly from the RKHS H_c, in order to obtain a remainder of order o(d_S^2(\mu,\nu)) in the local upper estimate for the Sinkhorn divergence.

Background

The proof of the local upper estimate for S_\varepsilon uses off-diagonal operators associated with pairs of measures along an interpolation. A sharper estimate in the d_S scale would require transferring the argument from measure variables to the common RKHS variable β.

The relevant operator composition behaves well on the diagonal, when the two interpolation parameters coincide, but the off-diagonal composition needed for the Sinkhorn-divergence representation does not admit an established interpretation in the required RKHS-based Banach spaces.

References

That is, we do not know how to make sense of H_{\mu_s,\mu_t}H_{\mu_t}{-1} in Banach spaces built directly on \Hil_c.

— Approximation and computation of the geodesic Sinkhorn distance  (2610.02007 - Lavenant et al., 1 Oct 2026) in Remark 4.5, “Why not a better remainder?”