Pointwise bounds and support for normalized mollifications

Elementary stability properties of the normalized ContDiffBump mollification used throughout the Caffarelli–Kohn–Nirenberg argument.

• abs_mollify_le: a uniform pointwise bound is preserved by mollification.
• support_mollify_subset: mollification does not spread the support of a function beyond the closed ε-neighbourhood of its topological support.

theorem CKN.abs_mollify_le {d : ℕ} {u : Vec d → ℝ} (hu : Continuous u) {M : ℝ} (hM : ∀ (x : Vec d), |u x| ≤ M) {ε : ℝ} (hε : 0 < ε) (x : Vec d) : |mollify u ε hε x| ≤ M

Pointwise bound for a mollification. If u is continuous and bounded above in absolute value by M, then its normalized mollification at scale ε obeys the same bound at every point: |mollify u ε hε x| ≤ M. This is the elementary bound under the Caffarelli–Kohn–Nirenberg normalized kernel used in the slice-distribution estimates.

theorem CKN.support_mollify_subset {d : ℕ} {u : Vec d → ℝ} {ε : ℝ} (hε : 0 < ε) : Function.support (mollify u ε hε) ⊆ Metric.cthickening ε (tsupport u)

Support of a mollification. The support of mollify u ε hε is contained in the closed ε-neighbourhood of the topological support of u: outside cthickening ε (tsupport u) the mollification vanishes, because the Caffarelli–Kohn–Nirenberg kernel only sees the values of u on the radius-ε ball around the point.