Determine whether the epsilon-function Dini criterion yields tangent points in higher dimensions

Determine whether an analogue of Carleson’s \(\varepsilon^2\)-conjecture holds in higher dimensions when the higher-dimensional function \(\varepsilon_n(x,r)\) satisfies the corresponding Dini integrability condition, namely whether finiteness of \(\int_0^1\varepsilon_n(x,r)^2\,dr/r\) characterizes tangent points of the domain boundary almost everywhere.

Background

In the planar setting, Carleson’s ε2\varepsilon^2-conjecture characterizes tangent points through the Dini integral of the planar epsilon function, and that conjecture has been proved. The paper discusses εn\varepsilon_n as one proposed higher-dimensional analogue based on surface measure deviation from complementary hemispheres.

The authors explicitly state that it is not known whether the same tangent-point characterization should hold for εn\varepsilon_n. This issue is distinct from the paper’s main theorem, which uses the spectral function λ\lambda and a pointwise Hölder decay assumption rather than the Dini condition for εn\varepsilon_n.

References

Although the $\varepsilon_n$-function may initially seem like a more natural higher-dimensional candidate than the spectral function, $a(x,r)$ discussed above, it is not clear that an analogue of Conjecture \ref{c: carleson's conjecture} should hold if $\varepsilon_n$ satisfies the decay like in e: Dini epsilon.

e: Dini epsilon:

∫01ε(x,r)2drr<∞.\int_0^1 \varepsilon(x,r)^2\frac{dr}{r}<\infty.

— Characterizing higher-dimensional Hölder domains via spherical geometric functions  (2610.13866 - Casey et al., 9 Oct 2026) in Section 1, subsection “Related literature”