Local L^{3/2} bounds for Newtonian potentials of compactly supported data #
The Newtonian representation ext:newtonian of the paper writes the local pressure as a sum of
Newtonian potentials N * g and first-derivative potentials ∂ⱼN * g of compactly supported
data g. The Liouville step needs each of these to be L^{3/2} on every ball of
three-dimensional space.
This file supplies the near-field half of that statement: Young's inequality on a ball, with
the kernel truncated at the radius R + ρ beyond which it cannot be seen by data supported in
the closed ball of radius R.
Exponent arithmetic #
Young's inequality 1 + 1/r = 1/q + 1/s with target r = 3/2 reads 1/q + 1/s = 5/3.
The kernel N has the size ‖z‖⁻¹, so its truncation lies in L^s exactly for s < 3; this
allows every q ≥ 1. The kernel ∂ⱼN has the size ‖z‖⁻², so its truncation lies in L^s
exactly for s < 3/2; this allows exactly q > 1. Both ranges contain the symmetric choice
q = s = 6/5, which is the Young instance available as
eLpNorm_scalarConvolution_six_fifths_three_halves. Since the data is compactly supported,
L^q ⊆ L^{6/5} for every q ≥ 6/5, so all the estimates below are routed through q = 6/5
and therefore cover every exponent q ≥ 6/5; in particular q = 3/2 and the exponents
q > 5/2 of def:sws.
A measurable representative supported in the same ball #
The force datum of a suitable weak solution is only almost everywhere strongly measurable, while
Young's inequality is stated for measurable data. Replacing the datum by the indicator of the
support ball of a strongly measurable representative changes neither its L^p sizes nor any of
its Newtonian potentials.
Data that vanishes off a closed ball and is almost everywhere strongly measurable agrees almost everywhere with measurable data that vanishes off the same ball.
Young's inequality on a ball #
Data supported in the closed ball of radius R only sees the kernel on the ball of radius
R + ρ when the potential is evaluated on the ball of radius ρ, so the potential agrees there
with the convolution against the truncated kernel, to which the L^{6/5} * L^{6/5} → L^{3/2}
Young estimate applies.
Young's estimate on a ball for the Newtonian potential of data supported in the closed ball
of radius R: the near-field kernel is the Newtonian kernel truncated at radius R + ρ.
Young's estimate on a ball for the first-derivative Newtonian potential of data supported in
the closed ball of radius R.
Local L^{3/2} membership on the ambient balls #
The Newtonian potential of compactly supported L^{6/5} data is L^{3/2} on every ambient
ball about the origin.
The first-derivative Newtonian potential of compactly supported L^{6/5} data is L^{3/2}
on every ambient ball about the origin.