From L⁴_t L⁶_x smallness to L³ smallness on backward cylinders #
The hypothesis of the Gustafson–Kang–Tsai criterion as it is used at the end of the proof of
lem:aniso:closure controls ∫_{-r²}^0 ‖u(t)‖_{L⁶(B_r)}^4 dt. Two Hölder inequalities, in
space (L⁶(B_r) ⊂ L³(B_r) with the factor |B_r|^{1/2}) and then in time (exponent 3/4
against the interval of length r²), turn it into the scale-invariant bound
∫_{B_r × (-r², 0)} |u|³ ≤ (4π/3)^{1/2} r² (∫ ‖u(t)‖_{L⁶}^4 dt)^{3/4}. The last theorem packages
this as L³ smallness ≤ η r² on all small backward cylinders at the origin, which is the
velocity half of the smallness hypothesis of CKN's Theorem A on those cylinders.
The spatial Hölder inequality ∫ g³ ≤ (∫ g⁶)^{1/2} |s|^{1/2} for an arbitrary ℝ≥0∞-valued
function; no measurability is needed because the lower Lebesgue integral of g³ is attained by
a measurable minorant.
The temporal Hölder inequality ∫ G^{3/4} ≤ (∫ G)^{3/4} |s|^{1/4} for an arbitrary
ℝ≥0∞-valued function, by the same measurable-minorant argument.
The volume factor |(-r², 0)|^{1/4} |B_r|^{1/2} = (4π/3)^{1/2} r² of the mixed-norm Hölder
inequality.
The mixed-norm Hölder inequality on the backward cylinder B_r × (-r², 0) at the origin:
the L³ integral of u is at most (4π/3)^{1/2} r² times the 3/4 power of the L⁴_t L⁶_x
quantity, by Tonelli and the two one-variable Hölder inequalities above.
A backward cylinder of radius r < 1 at the origin lies in the unit cylinder.
If the L⁴_t L⁶_x quantity on backward cylinders at the origin tends to zero, then for
every η > 0 the L³ integral of u over every sufficiently small backward cylinder is at
most η r².