Difference-quotient norm bounds #
The two-directional bound of the difference-quotient method: a function with an
L² weak derivative has L²-bounded difference quotients (direction i), and a
uniform bound on the difference quotients yields a weak derivative (direction ii).
See Evans, Partial Differential Equations (2nd ed.), §5.8.2.
The direction-i bound is proved here for a smooth, compactly supported
representative φ, by the fundamental theorem of calculus along the segment
t ↦ x + t • (h eₖ), a one-variable Cauchy-Schwarz bound on [0, 1]
(MeasureTheory.sq_intervalIntegral_le), a Tonelli swap of the order of
integration, and translation invariance of the Lebesgue integral.
g has whole-space weak k-derivative g' in L²: for every smooth compactly
supported test function φ, ∫ g ∂ₖφ = -∫ g' φ.
Equations
- One or more equations did not get rendered due to their size.
Instances For
Smooth compactly supported case #
Difference-quotient bound in the smooth compactly supported case. For φ smooth with
compact support, the L² norm of the difference quotient Dₖʰφ is bounded by the L²
norm of the k-th classical partial derivative: ‖Dₖʰφ‖ ≤ ‖∂ₖφ‖. The argument is the
fundamental-theorem-of-calculus proof of Evans, Partial Differential Equations (2nd
ed.), §5.8.2, specialised to the single coordinate direction k, so no operator-norm
loss to the full gradient is incurred.
Strong L² convergence of the difference quotient to the derivative #
Continuity of translation in L². The map w ↦ τ_w ψ is continuous from the space of
shifts into L²(ℝⁿ). This is the strong continuity of the translation group on L²(ℝⁿ),
obtained from joint continuity of composition with a measure-preserving family (Evans,
Partial Differential Equations (2nd ed.), §5.8.2).
Strong L² continuity of translation at the origin. As the shift w → 0, the
translated function τ_w ψ converges to ψ in L².
Strong L² convergence of the difference quotient to the derivative. For φ smooth
with compact support and a sequence of nonzero steps η m → 0, the difference quotients
Dₖ^{η m} φ converge in L² to the classical partial derivative ∂ₖφ. This is the strong
convergence input passed to the limit in the weak-derivative identification (Evans, Partial
Differential Equations (2nd ed.), §5.8.2).
Weak sequential compactness and the converse #
Weak sequential compactness of bounded sequences in L². A sequence bounded by M in
the separable Hilbert space EucL2 d has a subsequence converging weakly to a limit g' with
‖g'‖ ≤ M. Assembled from the sequential Banach-Alaoglu theorem on the weak dual
(WeakDual.isSeqCompact_closedBall), the Riesz self-duality of the Hilbert space
(InnerProductSpace.toDual), and the closed-ball membership of the weak-* limit.
Difference-quotient weak-limit converse (Evans §5.8.2, direction ii). If the
difference quotients Dₖʰ g are uniformly L²-bounded by M over all h ≠ 0, then g has a
weak k-derivative g' in L² with ‖g'‖ ≤ M. The sequence Dₖ^{1/(m+1)} g is bounded, so
by weak sequential compactness a subsequence converges weakly to some g' with ‖g'‖ ≤ M;
passing to the limit in the discrete integration-by-parts identity
⟪Dₖʰ g, ζ⟫ = -⟪g, Dₖ^{-h} ζ⟫, using the strong L² convergence Dₖ^{-hₘ} ζ → ∂ₖζ for a
test function ζ, identifies g' as the weak derivative (Evans, Partial Differential
Equations (2nd ed.), §5.8.2, Theorem 3).
General weak-derivative direction-i bound #
Weak-derivative difference-quotient bound (Evans §5.8.2, direction i). A function g
with L² weak k-derivative g' has difference quotients bounded in L² by the derivative:
‖Dₖʰ g‖ ≤ ‖g'‖, uniformly in the step h. This is the general form of the tight single-direction
bound norm_diffQuot_le_of_contDiff, obtained by testing Dₖʰ g against the smooth compactly
supported functions (dense in L²), where the segment-integral representation gives the bound
⟪Dₖʰ g, ζ⟫ ≤ ‖g'‖ · ‖ζ‖, then passing to the limit along a smooth sequence converging to
Dₖʰ g itself (Evans, Partial Differential Equations (2nd ed.), §5.8.2, Theorem 3).