Best recipient space and weak S1 boundedness for second-order triple operator integrals

Determine the optimal recipient space for triple operator integrals associated with second-order divided difference functions f^{[2]} acting on Schatten classes. In particular, ascertain whether for every f in C^2(R) the bilinear Schur multiplier M_{f^{[2]}} maps S_2 × S_2 boundedly into the weak trace class S_{1,∞}.

Background

The paper establishes new upper bounds for bilinear Schur multipliers associated with second-order divided difference functions and complementary lower bounds, sharpening earlier results on multiple operator integrals. In the extrapolation subsection, the authors prove that for every f in C^2(R), M_{f^{[2]}} maps S_2 × S_2 into the Marcinkiewicz space M_{1,∞}.

However, the precise endpoint target space remains unsettled. The authors highlight that it is unknown in general whether weak S1 (S_{1,∞}) is the correct recipient space for S_2 × S_2, although this is known to hold for the generalized absolute value f(s) = s|s| and minor variants, as shown in prior work. The problem seeks a definitive characterization of the optimal target space, with the concrete question of S_{1,∞}-boundedness as a key test case.