Schwartz space and its seminorms

Depends on

• Complex Lp classes and Euclidean test-function conventions
• $C^k$ maps and multi-index derivative notation in Euclidean space
• The spaces $C_c(\mathbb{R}^n)$ and $C_c^\infty(\mathbb{R}^n)$

Used by

• Hᵖ equals Lᵖ with equivalent norms for 1<p<∞ Corollary
• Hilbert transform does not map L-infinity to L-infinity Counterexample
• Hilbert transform is not strong type (1,1) Counterexample
• Pointwise restriction is not defined on Lp equivalence classes Counterexample
• Smooth and integrable does not imply Schwartz Counterexample
• Calderón–Zygmund kernels and their associated operators Definition
• Carleson operator and measurable linearisation Definition
• Carleson tiles wave packets and tile order Definition
• Fourier restriction and adjoint extension operators Definition
• Grand maximal test class of order N and the grand maximal function Definition
• Japanese-bracket and Laplacian Bessel-potential operators Definition
• Lp Fourier multiplier and its norm Definition
• Radial and nontangential maximal functions of a tempered distribution Definition
• Radon transform on Schwartz functions Definition
• Schwartz topology and convergence Definition
• Schwartz/distribution spaces and Hilbert Fock conventions Definition
• Tempered distribution Definition
• The Lusin area function for a fixed admissible kernel and aperture Definition
• Translation-invariant Fourier multiplier on the Schwartz core Definition
• A smooth scattered surface inherits the Matérn fill-distance rate Example
• Circular Radon inversion at the zero truth Example
• Hilbert transform of the line Poisson kernel Example
• Normalized Hermite Fourier eigenfunctions Example
• Polynomial Gaussians are Schwartz Example
• The regular translation system on L²(ℝⁿ): position, momentum and trivial stabilizer Example
• Atoms have uniformly bounded Hᵖ quasi-norm and uniformly bounded test pairings Lemma
• Calderon reproducing pair and the telescoping identity in S' Lemma
• Compact distribution convolution preserves schwartz and tempered spaces Lemma
• Cotlar's inequality for maximal truncations Lemma
• Deconvolution of a Schwartz function along the dyadic dilates of a fixed kernel Lemma
• Dilations and their normalisations preserve Schwartz space, with scaling identities Lemma
• Dyadic Mihlin pieces: uniform L1 and first-difference bounds Lemma
• Dyadic pieces have annular Fourier support and uniformly bounded rescaled kernels Lemma
• Existence of a smooth inhomogeneous dyadic frequency partition Lemma
• Fourier coefficients of a lattice periodisation Lemma
• Fourier series of the periodized Gaussian Lemma
• Invertible linear substitutions preserve Schwartz space Lemma
• Level decomposition of an Hᵖ distribution produces atoms Lemma
• Local polynomial projections matching moments through order s Lemma
• Measurability and lower semicontinuity of the smooth maximal functions Lemma

…and 24 more results.

Dependency tree · two levels

24 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, MIT 18.155 (2022) (standard reference, not scraped)