The slice bound of one field at every cell scale #
A cell estimate for the parabolic Morrey class needs the radius factor
r ^ (5 * (1 - P / κ)), which a slice bound at one fixed scale cannot produce:
one ball's slice bound, integrated in time, carries no decay in the cell
radius. What does produce it is the slice estimate applied at each cell's own
radius.
The statement below performs that transfer for one fixed space-time field. The field is tied to the pressure by being a weak spatial derivative on a common open carrier at almost every time of the cell's window; the cell datum is a slice bound for some weak spatial derivative on the cell's own ball. Almost everywhere uniqueness of weak partial derivatives then transports the cell's bound to the one field. Its conclusion is exactly the slice hypothesis of the time integration that turns slice bounds into the cell power integral.
The slice bound proved at a cell's own radius, transported to a fixed space-time field that is a weak spatial derivative of the pressure on a carrier containing that cell's ball.