Non-degeneracy of the Schwartz pairing: if a Schwartz function ψ pairs to zero
against every Schwartz function via the canonical Schwartz-to-tempered-distribution
embedding, then ψ = 0.
Key inductive step for weak density: if f₀ already matches the distribution u
on a finite set S' and any ψ annihilated by tests vanishing on S' is also
annihilated by ι g, then ι f₀ ψ = u ψ.
Given any tempered distribution u and any finite set S of test functions,
there exists a Schwartz function f whose Schwartz-distribution image agrees with u
on every element of S.
The canonical embedding of Schwartz functions into tempered distributions has dense range with respect to the weak (pointwise) topology on distributions.
A tempered distribution u lies in the Sobolev space H^m if it is represented
by integration against an element of SobolevSpace n m.
Instances For
Action of a constant-coefficient differential operator P(D) on a distribution
represented by a Sobolev H^s function: the result is represented by an H^{s-k}
function, when the polynomial has degree at most k ≤ s.
A constant-coefficient differential operator P(D) of degree k maps the Sobolev
class H^s into H^{s-k} (Melrose, Section 10, Prop 10.2).
The tail sum from index 0 equals the total sum of the multi-index.
Auxiliary recursion computing the multi-index iterated partial derivative on
SobolevSpace n m' by processing coordinates starting at index k.
Instances For
The monomial x^α := ∏ x_i^{α_i} viewed as a multivariate polynomial.
Instances For
Iterated distributional derivatives of order α map H^m to H^{m - |α|}.
Iteratively multiplying by Fourier variables equals smul-left multiplication by
the symbol of the monomial x^α, when acting on a distribution.
Almost-everywhere uniqueness of the Sobolev L^2 witness: two L^2 functions
that pair identically against all Schwartz functions via the weighted integral are
equal a.e.
Distributional derivative in the j-th direction maps Sobolev H^s witnesses to
H^{s-1} witnesses, with the resulting L^2 representative bounded pointwise by
(2π) ‖g‖.
Iterated k-fold partial derivative in the j-th direction maps Sobolev H^s
witnesses to H^{s-k} witnesses with L^2 representative bounded by (2π)^k ‖g‖.
Sequentially applying the iterated partial derivatives D_i^{α_i} along a list
l of coordinates preserves the Sobolev structure with an L^2 witness bounded by
(2π)^{Σ α} ‖g‖.
Almost-everywhere pointwise bound: any L^2 witness g' for the iterated
distributional derivative D^α u satisfies ‖g' ξ‖ ≤ (2π)^{|α|} ‖g ξ‖ for the
witness g of u.
Quantitative Sobolev bound: the L^2 witness of D^α u has norm at most
(2π)^{|α|} times the norm of the witness of u.
Continuity of the multi-index derivative D^α : H^m → H^{m - |α|} with an
explicit positive operator-norm constant C = (2π)^{|α|}.
Combined statement: D^α maps H^m to H^{m - |α|} and is bounded as a linear
operator with explicit constant (2π)^{|α|}. This corresponds to Melrose Prop 10.2.