Estimates for the trajectory operator on Sobolev paths #
The trajectory operator
D_A u = u' + A u : W^{1,2}(ℝ, E) → L²(ℝ, E)
(TauCeti.WeakDerivStep.trajectoryOperator, for an L∞ class of A) is controlled on Sobolev
paths by any estimate it satisfies on compactly supported C¹ paths. An estimate
a ∫ ‖u'‖² + b ∫ ‖u‖² ≤ c ∫ ‖u' + A u‖² + d ∫_{[-R, R]} ‖u‖²
on such paths extends to W^{1,2}(ℝ, E), with the last term replaced by the squared norm of the
restriction to a slightly larger interval, because those paths are dense
(TauCeti.WeakDerivStep.dense_setOf_exists_contDiff_hasCompactSupport) and both sides are
continuous. Restriction from W^{1,2}(ℝ, E) to L²(-R, R) is compact (Rellich), so an estimate
with a, b > 0 is an estimate with a compact error term, the input of Peetre's lemma. This is how
the Fredholm property of D_A is proved in TauCeti.Analysis.ODE.EnergyEstimate.Lyapunov.
When A is a C¹ family of symmetric operators with a spectral gap and a small derivative on the
whole line, rather than only near its limits, the coercive estimate for test paths
TauCeti.integral_norm_deriv_sq_add_mul_integral_norm_sq_le has no error term, so D_A is
injective. This applies in particular to a constant invertible symmetric coefficient.
Main declarations #
TauCeti.WeakDerivStep.mul_norm_weakFDeriv_sq_add_mul_norm_prev_sq_le_of_forall_contDiff: an estimate on compactly supportedC¹paths extends toW^{1,2}(ℝ, E).TauCeti.WeakDerivStep.norm_weakFDeriv_sq_add_mul_norm_prev_sq_le_norm_trajectoryOperator_sq: for a family with a uniform spectral gapαand⟪v, A' v⟫ ≤ β ‖v‖², the estimate‖u'‖² + (α² - β) ‖u‖² ≤ ‖D_A u‖²without compact error, andTauCeti.WeakDerivStep.min_one_mul_norm_sq_le_norm_trajectoryOperator_sq: the same estimate as the lower boundmin 1 (α² - β) ‖u‖² ≤ ‖D_A u‖²in the norm ofW^{1,2}(ℝ, E), with the linear formTauCeti.WeakDerivStep.min_one_mul_norm_le_norm_trajectoryOperator.TauCeti.WeakDerivStep.ker_trajectoryOperator_eq_bot: for such a family withβ < α²,D_Ais injective.
References #
- D. McDuff and D. Salamon, J-holomorphic Curves and Symplectic Topology, 2nd ed., Appendix A.1 (estimates with a compact error and the Fredholm criterion).
- M. Schwarz, Morse Homology, Chapter 2 (the Fredholm property of linearized trajectory operators).
Restriction of a path with a smooth compactly supported representative #
Extending a test-path estimate to Sobolev paths. If every compactly supported C¹ path φ
satisfies a ∫ ‖φ'‖² + b ∫ ‖φ‖² ≤ c ∫ ‖φ' + A φ‖² + d ∫_{[-R, R]} ‖φ‖² with d ≥ 0, then
a ‖u'‖² + b ‖u‖² ≤ c ‖D_A u‖² + d ‖u‖²_{L²(-R', R')} for every u ∈ W^{1,2}(ℝ, E) and every
R' > R. The coefficient of the operator is any L∞ class AL of A.
The coercive trajectory estimate on Sobolev paths. If a C¹ family of symmetric operators
satisfies α ‖v‖ ≤ ‖A s v‖ and ⟪v, A' s v⟫ ≤ β ‖v‖² everywhere, then
‖u'‖² + (α² - β) ‖u‖² ≤ ‖D_A u‖² for every u ∈ W^{1,2}(ℝ, E), with no compact error term. The
coefficient of the operator is any L∞ class AL of A.
The coercive trajectory estimate in the full Sobolev norm. If a C¹ family of symmetric
operators satisfies α ‖v‖ ≤ ‖A s v‖ and ⟪v, A' s v⟫ ≤ β ‖v‖² everywhere, then
min 1 (α² - β) ‖u‖² ≤ ‖D_A u‖² for every u ∈ W^{1,2}(ℝ, E), where
‖u‖² = ‖u‖²_{L²} + ‖u'‖²_{L²}. The coefficient of the operator is any L∞ class AL of A.
The coercive trajectory estimate as a linear bound. If a C¹ family of symmetric
operators satisfies α ‖v‖ ≤ ‖A s v‖ and ⟪v, A' s v⟫ ≤ β ‖v‖² everywhere, then
min 1 (α² - β) ‖u‖ ≤ ‖D_A u‖ for every u ∈ W^{1,2}(ℝ, E). For β < α² this bounds the inverse
of D_A by (min 1 (α² - β))⁻¹. The coefficient of the operator is any L∞ class AL of A.
The trajectory operator of a coercive family is injective. If a C¹ family of symmetric
operators satisfies α ‖v‖ ≤ ‖A s v‖ and ⟪v, A' s v⟫ ≤ β ‖v‖² everywhere with β < α², then
u' + A u = 0 has no nonzero solution in W^{1,2}(ℝ, E).