Necessity of Triebel–Lizorkin endpoint index conditions

Determine whether the Triebel–Lizorkin secondary-index conditions stated in the endpoint estimates for positive-frequency toroidal pseudodifferential operators in the class S^m_{\rho,\delta,\triangle}, with 0<\rho<1 and 0\leq\delta<\rho, are necessary for endpoint boundedness at the critical loss d(1-\rho)|1/p-1/2|, including the corresponding holomorphic function spaces.

Background

The paper establishes sharpness of the critical derivative loss for positive-cone and holomorphic Besov and Triebel–Lizorkin spaces, and proves necessity of the Besov fine-index condition q≤t at the endpoint. For Triebel–Lizorkin spaces, however, the endpoint proposition gives different sufficient index conditions depending on p: p≤t when p<2, q≤2≤t when p=2, and q≤p when p>2.

The authors explicitly note that their sharpness argument does not prove necessity of all these Triebel–Lizorkin secondary-index alternatives. Resolving the issue would require adapting the spatial-overlap and randomization constructions from the cited Euclidean examples to the positive lattice, thereby determining whether the listed endpoint restrictions are merely sufficient or are also necessary.