Alphabeta MathHow statement and proof provenance work
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Convolution of a tempered distribution with a schwartz function
Definition
For and , define the scalar function
For each fixed , the function is the reflection and translation of a Schwartz function, hence remains in by Basic operations are continuous on Schwartz space. The pairing with is therefore defined. This definition initially produces only a scalar function; its smoothness, derivative identities, polynomial growth, and regular-tempered interpretation are proved later. If or , the convolution is the zero function. No convolution of two arbitrary tempered distributions is defined, and no choice axiom is used.
Depends on
Used by
- Grand maximal test class of order N and the grand maximal function Definition
- Radial and nontangential maximal functions of a tempered distribution Definition
- Calderon reproducing pair and the telescoping identity in S' Lemma
- Cotlar's inequality for maximal truncations Lemma
- L2-normalised H1 atoms have uniformly bounded H1 norm Lemma
- Level decomposition of an Hᵖ distribution produces atoms Lemma
- ℓᵖ sums of atoms converge in S' and in Hᵖ Lemma
- Measurability and lower semicontinuity of the smooth maximal functions Lemma
- Sampling at a lattice produces periodisation of the spectrum over the dual lattice Lemma
- Schwartz approximate identities converge in the sense of tempered distributions Lemma
- The grand maximal function dominates every admissible radial and nontangential maximal function Lemma
- The grand maximal function is pointwise dominated by a tangential maximal function Lemma
- The Hilbert transform is the tempered convolution with pv(1/(pi x)) and has signum Fourier multiplier Lemma
- The Riesz transform is the principal value of its kernel, with the matching constant Lemma
- Tempered convolution is smooth with polynomial growth Theorem
- The Mihlin–Hörmander Fourier multiplier theorem Theorem
Dependency tree · two levels
10 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.
Sources
- Semyon Dyatlov, Lecture notes for 18.155 (2022) (standard reference, not scraped)
- Radu Gelca, Functional Analysis (standard reference, not scraped)