Gilbert's Conjecture on Real H^p Theory for Clifford-Module-Valued Hardy Spaces

Determine whether, for every Clifford module 𝔥 over the real Clifford algebra Cℓ_n, there exist an element η in ℝ^n with η^2 = −1 and a subspace 𝔥_0 ⊆ 𝔥 such that: (i) 𝔥 decomposes as 𝔥 = 𝔥_0 ⊕ η 𝔥_0; (ii) for all p > 1 the Cauchy integral operator C: L^p(ℝ^{n−1}, 𝔥_0) → H^p(ℝ_+^n, 𝔥) is an isomorphism; and (iii) for all p > 1 the composition ℛ ∘ 𝔅, where 𝔅 is the boundary operator and ℛ is twice the orthogonal projection onto 𝔥_0, maps H^p(ℝ_+^n, 𝔥) continuously onto L^p(ℝ^{n−1}, 𝔥_0).

Background

The paper studies Hardy spaces in higher dimensions valued in Clifford modules and examines whether a higher-dimensional analogue of the classical real H^p boundary theory holds. In one complex dimension, real L^p functions arise as boundary values of holomorphic H^p functions via the Hilbert transform. In higher dimensions within Clifford analysis, the Cauchy integral and the Riesz transforms play analogous roles.

Gilbert proposed a conjecture asserting that for any Clifford module one can split the module by a vector η with η^2 = −1 into two complementary subspaces and identify the Hardy space with L^p on the boundary via the Cauchy and boundary operators, with an appropriate projection. This would provide a Real H^p theory for Clifford-module-valued Hardy spaces. The present paper claims to resolve the conjecture by showing it holds except in dimensions n ≡ 6, 7 (mod 8), where it fails.