Maximal-operator estimate for the localized heat-kernel supremum

Establish an analogue, possibly for q>1, of the estimate in Lemma 2bis with the operator e^{-\omega\kappa(s)}T_{b,\kappa(s),d} replaced by the maximal heat-kernel operator T^{\omega}_{b,*,d}, in order to obtain a cleaner description of the space \mathring F^{\infty,q}_\alpha(\beta).

Background

The paper introduces spaces C^q(\mu,a) to formulate the localized mixed-norm estimates needed for the Triebel–Lizorkin spaces F^{\infty,q}_\alpha(\beta). Lemma 2bis proves a boundedness estimate for the scale-dependent operator e^{-\omega\kappa(t)}T_{b,\kappa(t),d}, where T_{b,t,d} is convolution with a Gaussian-type kernel.

The authors explain that replacing this scale-dependent operator by the supremum operator T^{\omega}_{b,*,d} would yield a cleaner description of \mathring F^{\infty,q}_\alpha(\beta), but they do not have a proof of the corresponding estimate. Thus, the unresolved problem is to establish that maximal-operator analogue, at least in the indicated range q>1.