Relate existing Dini results for the a-function and the lambda-function

Determine whether the Dini-type regularity results previously established for the characteristic-constant function \(a(x,r)\) remain valid when \(a(x,r)\) is replaced by the spherical eigenvalue function \(\lambda(x,r)\).

Background

The paper explains that, under pointwise smallness assumptions, the functions a(x,r)a(x,r) and λ(x,r)\lambda(x,r) control one another up to changes in constants and exponents. The main theorem is therefore formulated using λ\lambda, because eigenvalue estimates are more convenient for the proof.

However, the passage between the two functions relies on pointwise smallness. The authors explicitly leave unresolved whether results from the literature involving Dini-type control of a(x,r)a(x,r), particularly those of Fleschler–Tolsa–Villa, can be transferred to an assumption formulated using λ(x,r)\lambda(x,r).

References

However we should note that this passage from the $a(x,r)$-function to the $\lambda(x,r)$-function (and in reverse) relies on our assumption of pointwise smallness and thus it is not clear to us whether other results in the literature, e.g. the Dini-type results of , would work with the $a$-function replaced by the $\lambda$-function.

— Characterizing higher-dimensional Hölder domains via spherical geometric functions  (2610.13866 - Casey et al., 9 Oct 2026) in Section 2, subsection “Relationship to other spherical functions”