Riesz-kernel Lᵖ bound for the Morrey embedding #
For p > d the (d-1)-Riesz potential of an Lᵖ function over a ball of radius R
centred at the base point is controlled by Cdp · R^{1-d/p} · ‖g‖_{Lᵖ}. The exponent
1 - d/p is the Morrey Hölder exponent, produced here from Hölder's inequality with the
conjugate exponent q = p/(p-1) together with the radial L^q norm of the singular kernel.
The kernel-norm computation is isolated in the private lemma setIntegral_ball_dist_rpow,
a closed-form value for the radial integral ∫_{B(x,R)} dist x y^s over a ball centred at the
singularity, valid for s > -d.
Riesz-kernel Lᵖ bound. For p > d there is a constant Cdp (depending only on
d, p) such that the (d-1)-Riesz potential of any Lᵖ function over a ball of radius
R centred at the base point is bounded by Cdp · R^{1-d/p} · ‖g‖_{Lᵖ}. The exponent
1 - d/p is precisely the Morrey Hölder exponent.
Smooth Morrey Hölder estimate on a ball. For p > d there is a constant C,
depending only on d and p, such that every smooth φ is Hölder continuous on ball c r
with exponent 1 - d/p and constant C · ‖∇φ‖_{Lᵖ(ball c r)}. This is Gilbarg–Trudinger
Theorem 7.19: the interior Hölder seminorm is controlled by the domain-restricted Lᵖ norm of
the gradient. The proof averages over the convex lens
ball x (2ρ) ∩ ball x' (2ρ) ∩ ball c r, which contains both points and stays inside the domain,
so the averaging never sees the gradient outside ball c r.
Operator norm of a derivative bounded by its coordinate partials. On
EuclideanSpace ℝ (Fin d) the operator norm of fderiv ℝ u y is bounded by the sum of the
absolute values of the coordinate partial derivatives partialD k u y. This converts the
‖∇u‖-shaped constant of exists_holder_smooth into the coordinate-partial sum.
Morrey on a ball in the smooth case. A smooth u with gradient components gₖ = ∂ₖu
in Lᵖ(ball c r) (p > d) is Hölder-(1−d/p) on the ball, with constant linear in
∑ₖ ‖gₖ‖_{Lᵖ(ball c r)}. This lifts exists_holder_smooth to the coordinate-partial form
consumed by the weak-gradient statement, by dominating the operator-norm Lᵖ seminorm of the
derivative by the sum of the coordinate-partial Lᵖ seminorms.
Gradient-convolution bridge. If g is the weak gradient of u on a measurable
set B and ρ is a normalised bump of outer radius ε centred at 0, then at every interior
point x with Metric.closedBall x ε ⊆ B the k-th partial of the mollification equals the
mollified gradient component: partialD k (uB ⋆ ρ) x = (gBk ⋆ ρ) x, where uB, gBk are the
extensions by zero off B. At B = Set.univ the support hypothesis is vacuous and the
indicators are the identity, which is the form the Lᵖ bootstrap consumes.
Morrey embedding on a ball (weak-gradient form). For p > d, a function u that is
integrable on Metric.ball c r with an Lᵖ weak gradient g there has a continuous
representative u' which is Hölder-(1 - d/p) on the ball, with constant linear in
∑ₖ ‖gₖ‖_{Lᵖ(ball c r)}. The representative is obtained by mollification: each mollification is
smooth and, by the smooth Morrey estimate applied on interior sub-balls, uniformly Hölder with the
target constant; the mollified gradients are bounded in Lᵖ by Young's inequality, and the
mollifications converge to u almost everywhere. The uniform-limit engine then produces u'.