Existence of a Lipschitz truncation operator for the energy-critical function spaces

Determine whether, for every spatial dimension d≥3 and the spaces X=L^r(0,T;L^q_x(ℝ^d)) and Y=L^r(0,T;L^{\tilde q}_x(ℝ^d)) with r=2d/(d−2), q=2d/(d−2+4/d), and \tilde q=2d/(d−4+4/d), there exists a mapping φ:X∩Y→X∩Y that is the identity on {f:‖f‖_Y≤1}, satisfies ‖φ(f)‖_Y≲1, and is Lipschitz in the X-norm, ‖φ(f)−φ(g)‖_X≲‖f−g‖_X, in order to enable a stochastic Kato-type proof of local well-posedness for the energy-critical stochastic nonlinear Schrödinger equation with fluctuating nonlinearity.

Background

The paper establishes local well-posedness in H^1 for the stochastic nonlinear Schrödinger equation with fluctuating nonlinearity in the full energy-subcritical range. Extending this result to energy-critical nonlinearities would require function spaces different from those used in the subcritical argument, following the deterministic critical-space approach of Christ and Weinstein.

The authors explain that the critical-space argument would require a truncation operator that simultaneously preserves functions with sufficiently small Y-norm, uniformly truncates the Y-norm, and remains Lipschitz with respect to the weaker X-norm. They state that it is unclear whether such an operator exists; an affirmative answer would allow the deterministic critical argument to be adapted and would yield local well-posedness for the stochastic equation with energy-critical nonlinearities.