Linear growth of local L^{3/2} norms from decay at infinity #
The Liouville step of the pressure identification needs its residual to satisfy
‖r‖_{L^{3/2}(B_ρ)} ≤ C * (1 + ρ) on the round balls euclideanBall 0 ρ. This
file turns the pointwise decay |h x| ≤ M * ‖x‖⁻¹ of a Newtonian potential of
compactly supported data into exactly that bound, with an explicit constant
built from the local norm near the origin and the universal ball constant
invNormBallConstant of CKN.Pressure.PotentialDecayShell.
This is the decay-at-infinity input to the uniqueness half of the Newtonian
representation ext:newtonian of the paper, whose proof applies Liouville's
theorem to a harmonic function that tends to zero in the L^{3/2} average
sense at infinity.
The round Euclidean ball sits inside the ambient ball of the same radius.
The L^{3/2} norm of the radial profile M * ‖x‖⁻¹ over the round ball of
radius ρ grows linearly in ρ.
Splitting estimate: a function that is L^{3/2} on the ambient ball of
radius 2 * R and decays like M * ‖x‖⁻¹ outside it has L^{3/2} norm at most
‖h‖_{L^{3/2}(B_{2R})} + M * invNormBallConstant ^ (2/3) * ρ on the round ball
of radius ρ.
The explicit growth constant attached to an inverse-distance decay estimate: the local norm near the origin plus the decay constant times the universal ball constant.
Equations
- CKN.invNormGrowthConstant h M R = MeasureTheory.lpNorm h (ENNReal.ofReal (3 / 2)) (MeasureTheory.volume.restrict (Metric.ball 0 (2 * R))) + M * CKN.invNormBallConstant ^ (2 / 3)
Instances For
Membership in L^{3/2} of every round ball, for a function that is
L^{3/2} near the origin and decays like M * ‖x‖⁻¹ at infinity.
Linear growth of the local L^{3/2} norms: this is the exact shape of the
growth hypothesis of the Liouville theorem for weakly harmonic functions.
The pair of hypotheses consumed by
weaklyHarmonicOn_eq_zero_of_lpNorm_linear_growth, produced from an
inverse-distance decay estimate centred at the origin.
The same conclusion from a decay estimate centred at an arbitrary point
x₀, obtained by re-centring at the origin.
Compactly supported and globally L^{3/2} data #
The remaining pieces of a pressure residual are either compactly supported
L^{3/2} functions (such as η p) or globally L^{3/2} functions (such as the
image of η U under the second-order singular integral). Both satisfy the
linear-growth bound with constant their global L^{3/2} norm.
Converting higher-order far-field decay to inverse-distance decay #
The first and second derivative potentials decay like ‖x - x₀‖⁻² and
‖x - x₀‖⁻³; on the far region 2 * R ≤ ‖x - x₀‖ these are stronger than the
inverse-distance decay required by the growth estimate.
A far-field bound of the shape A / ‖x - x₀‖ ^ 2 gives inverse-distance
decay with constant A / (2 * R).