Distributions 5 | Regular Distributions [dark version]

Regular distributions are exactly those distributions that can be represented by ordinary functions—more precisely, by locally integrable functions—so they inherit the “tame” continuity properties that fail for singular objects like the Dirac delta. The core result starts with a practical characterization of when a linear functional on test functions is continuous: a distribution T is continuous precisely when, on every compact set K, T(Φ) can be bounded by a finite combination of sup-norms of derivatives of Φ up to some order M.

Concretely, for each compact K ⊂ R^N there must exist an integer M ≥ 0 and a constant C > 0 such that whenever a test function Φ has support inside K, the value |T(Φ)| is controlled by C times the sum over multi-indices α with |α| ≤ M of the supremum norms of D^αΦ. This estimate is not just a technicality: it directly implies continuity. If Φ_k → Φ in the distribution-test-function sense (meaning all derivatives D^αΦ_k converge uniformly and the supports stay inside one compact set), then the difference Φ_k − Φ also satisfies the same support condition, and the bound forces T(Φ_k) → T(Φ).

The reverse direction is proved by negating the estimate and then constructing a sequence of test functions that converges to zero in the test-function topology while their images under T do not. If the required bound fails, there exists a compact K such that for every choice of M and C one can find a test function Φ with support in K whose derivatives up to order M cannot control |T(Φ)|. From these counterexamples, a scaled sequence Ψ_k is built by dividing Φ_k by |T(Φ_k)|. Because the original inequality failure guarantees that |T(Φ_k)| grows faster than the relevant derivative sup-norms, the scaled functions Ψ_k converge to 0 together with all derivatives (with supports still contained in K). Yet linearity makes |T(Ψ_k)| = 1 for all k, so T(Ψ_k) cannot converge to 0. That contradiction shows the continuity estimate must hold.

With this continuity criterion in place, the discussion turns to “regular distributions.” First comes the notion of a locally integrable function f: a measurable function is locally integrable if its absolute value is integrable over every compact subset of R^N. This includes many functions that are not integrable over the whole space (for example, x ↦ x^2 on R is locally integrable because it is integrable on bounded intervals).

Given such an f, one defines a distribution T_f by pairing f with test functions via the integral T_f(Φ) = ∫_{R^N} f(x)Φ(x) dx. The integral is well-defined because Φ has compact support, so the integration effectively happens over a compact set where f is integrable. The resulting map is continuous and satisfies the same type of estimate as in the characterization, with the bound reducing to the supremum norm of Φ (since only the case m = 0 is needed).

A distribution T is called regular if it equals T_f for some locally integrable f—meaning it behaves like an ordinary function under test-function pairing. The contrast is immediate: not every distribution is regular, and the Dirac delta is singled out as a key example of a non-regular distribution to be treated next.