Optimal DS_beta embedding for Fourier-constrained measure spaces

Characterize, for a translation- and dilation-invariant closed subspace W of S'(R^d,R^ell), the largest parameter beta such that W intersected with M(R^d,R^ell) embeds continuously into DS_beta, and, when W is defined by a Fourier constraint associated with a function Omega, determine whether this optimal beta admits an explicit expression in terms of Omega.

Background

The paper introduces the scale of spaces DS_beta(Rd), interpolating between finite measures at beta = 0 and the real Hardy space H_1 at beta = d. A key property is that measures in DS_beta have lower Hausdorff dimension at least beta, and the paper recalls known embeddings for divergence-free vector fields and gradients of distributions.

The unresolved problem asks for the strongest such dimensional embedding for an arbitrary translation- and dilation-invariant closed distributional constraint W. It further asks whether the optimal parameter can be computed explicitly when W arises from a smooth Fourier constraint Omega, thereby seeking an algebraic or geometric characterization of the dimension enforced by the constraint.

References

The natural question is, given some translation and dilation invariant closed subspace~$W \subset S'(Rd,R\ell)$, what is the largest possible~$\beta$ such that~$W \cap M(Rd,R\ell) \hookrightarrow \mathrm{DS}_\beta$? If~$W$ is defined by a Fourier constraint as in Subsection~\ref{sD2} in the appendix, can this optimal~$\beta$ be expressed explicitly in terms of the corresponding function~$\Omega$?

— Anisotropic Bourgain--Brezis inequalities  (2608.21135 - Stolyarov, 21 Aug 2026) in Section 4.3, “Reflection and possible further development,” paragraph “Relationship with DS_beta spaces”