The critical heat estimate #
This file proves lem:pv-heat-critical: for b ∈ L³,
‖S(·)b‖_{L⁵(Q_τ)} ≤ C‖b‖₃, and for b ∈ L² ∩ L³,
‖S(·)b‖_{L⁴(Q_τ)} ≤ C(‖b‖₂ + ‖b‖₃), with C independent of τ. The
smooth compact case is heatOrbit_five_smooth and
heatOrbit_tenThirds_smooth; approximation in Lᵖ extends both, and the
L⁴ bound follows by Hölder interpolation on Q_τ.
Every Lᵖ vector field, 1 ≤ p < ∞, is an Lᵖ limit of smooth compactly
supported vector fields.
Approximation step of lem:pv-heat-critical: a space-time bound for smooth
compact data in terms of the Lᵖ norm extends to every Lᵖ datum, together
with the measurability of its heat orbit on Q_τ.
Hölder interpolation ‖f‖₄ ≤ ‖f‖_{10/3}^{1/2} ‖f‖₅^{1/2} on any measure
space; the constant is one.
The heat orbit of an L³ datum is almost everywhere strongly measurable on
every slab Q_τ, and satisfies the critical L⁵ bound with the constant of
heatOrbit_five_smooth.
lem:pv-heat-critical: for b ∈ L³ and τ > 0,
‖S(·)b‖_{L⁵(Q_τ)} ≤ C‖b‖₃, and if also b ∈ L², then
‖S(·)b‖_{L⁴(Q_τ)} ≤ C(‖b‖₂ + ‖b‖₃); the constant does not depend on τ.
The heat orbit of an L³ datum lies in L⁵(Q_τ), as used by the
local solution of sec:pv-trace.
The heat orbit of an L² ∩ L³ datum lies in L⁴(Q_τ), as used by the
local solution of sec:pv-trace.