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Dispersive estimates and evolution
Organize the independent exterior resolvent, smoothing and Strichartz inputs with their dimensional, exponent and derivative-loss limits, then the negative-scale, causal L1 and real-linear perturbation transfers used by the article.
1 · Prerequisites
- Exterior Dirichlet realization, H² domain and form domain
- Positive spectral Hilbert scale: duality and interpolation
- Weak Leibniz rule with a smooth factor
- The Axiom of Choice
- Integer-order Sobolev spaces and their norms
- Interior mollification commutes with weak derivatives
- Local smooth approximation in integer-order Sobolev spaces
- Compactly supported Sobolev functions extend by zero in every integer order
- Complex Holder, Minkowski, and the quotient norm
- Test function cutoffs and euclidean localization
- Exterior and truncated obstacle domains
- Non-trapping obstacle
- Dirichlet spectral Sobolev scale and duality
- Continuous and mixed time-space function spaces
- Bochner-integrable function
- Holder's inequality for integrals, including the endpoint cases
- Continuous free convolution on a finite time interval
- On nonlinear Schrödinger equations in exterior domains · Section 2.3, Proposition 2.7, equations (2.14) and (2.15), and Remark 2.8; published page 302, PDF page 8
- Complex integer-order Sobolev notation
- A self-adjoint operator generates a strongly continuous unitary group
- Unbounded Borel functional calculus: domains, products, spectral mapping
- Tonelli and Fubini for the completed product, with only almost-everywhere section measurability
- Complex Lp completeness and almost-everywhere subsequences
- Heine-Borel by bisection: every closed bounded interval [a,b] is compact
- Strichartz estimates and the nonlinear Schrödinger equation on manifolds with boundary · Section 5, equations (54) and (55), and the argument following Lemma 5.2, PDF pages 18–21
- An explicit common interpolation model for the fractional Strichartz targets
- Well-defined strip norm and the action of a common endpoint operator
- Integer-order Sobolev spaces are Banach
- Bochner Lp is complete for a Banach value space
- Realification preserves a complex norm and its metric
- On nonlinear Schrödinger equations in exterior domains · Section 2.4, Propositions 2.10 and 2.11, equations (2.17), (2.18), (2.22), printed pages 304–306, PDF pages 10–12
- Resolvent and spectrum of an unbounded operator
- On nonlinear Schrödinger equations in exterior domains · Section 2.2, opening paragraph and Propositions 2.2 and 2.4, printed pp. 299–300, PDF pp. 5–6
- On nonlinear Schrödinger equations in exterior domains · Section 2.1, Dirichlet spectral scale and negative-order duality and interpolation, printed p. 299, PDF p. 5; Section 2.2, Proposition 2.5, equation (2.9), printed p. 300, PDF p. 6
- On nonlinear Schrödinger equations in exterior domains · Section 2.2, Remark 2.6, equation (2.10), printed p. 300, PDF p. 6
2 · Logical flowchart
Arrows show item dependencies. A grouped prerequisite node points to items that use one or more of its members; their exact links are listed above.
3 · Items
Lemma
Endpoint multiplier bounds extended to negative spectral orders
Proposition
Imported homogeneous and retarded exterior smoothing
Proposition
Three-dimensional Dirichlet energy-to-mixed-norm estimate
Proposition
Exterior Strichartz estimates with precise loss
Proposition