Essential bounds from multiplier estimates

If multiplication by a function a increases Lᵖ norms by at most a factor C, for some finite exponent p, then a is essentially bounded by C. This file proves two forms of this multiplier criterion on a sigma-finite measure space:

• MeasureTheory.eLpNorm_top_le_of_forall_eLpNorm_indicator_le: it suffices to test the estimate on indicators of sets of finite measure, that is, to know ‖a 𝟙_s‖_p ≤ C μ(s) ^ (1 / p) for all such sets s;
• MeasureTheory.eLpNorm_top_le_of_dense_of_eLpNorm_mul_le: it suffices to test the estimate ‖a f‖_p ≤ C ‖f‖_p on a dense set of functions f in Lᵖ, for 1 ≤ p < ∞.

No integrability of a is assumed beyond almost-everywhere strong measurability. In the second form, an indicator is approximated in Lᵖ by test functions, a subsequence converges almost everywhere, and Fatou's lemma for Lᵖ seminorms passes the estimate to the indicator.

A typical application is to trigonometric polynomials on the circle: a function whose multiplication operator is bounded on trigonometric polynomials in L² is essentially bounded. This identifies the operators on circle L² commuting with the bilateral shift as the multiplication operators by essentially bounded symbols.

theorem MeasureTheory.eLpNorm_top_le_of_forall_eLpNorm_indicator_le {α : Type u_1} [MeasurableSpace α] {μ : Measure α} {p : ENNReal} {ε : Type u_2} [TopologicalSpace ε] [ESeminormedAddMonoid ε] [SigmaFinite μ] {a : α → ε} (ha : AEStronglyMeasurable a μ) (hp : p ≠ 0) (hp_top : p ≠ ⊤) {C : ENNReal} (h : ∀ (s : Set α), MeasurableSet s → μ s < ⊤ → eLpNorm (s.indicator a) p μ ≤ C * μ s ^ (1 / p.toReal)) : eLpNorm a ⊤ μ ≤ C

Multiplier criterion, indicator form. On a sigma-finite measure space, if for some exponent 0 < p < ∞ every set s of finite measure satisfies ‖a 𝟙_s‖_p ≤ C μ(s) ^ (1 / p), then a is essentially bounded by C.

theorem MeasureTheory.eLpNorm_top_le_of_dense_of_eLpNorm_mul_le {α : Type u_1} [MeasurableSpace α] {μ : Measure α} {p : ENNReal} {𝕜 : Type u_2} [NormedRing 𝕜] [NormOneClass 𝕜] [SigmaFinite μ] [Fact (1 ≤ p)] (hp_top : p ≠ ⊤) {a : α → 𝕜} (ha : AEStronglyMeasurable a μ) {D : Set ↥(Lp 𝕜 p μ)} (hD : Dense D) {C : ENNReal} (h : ∀ f ∈ D, eLpNorm (a * ↑↑f) p μ ≤ C * eLpNorm (↑↑f) p μ) : eLpNorm a ⊤ μ ≤ C

Multiplier criterion. On a sigma-finite measure space, if for some exponent 1 ≤ p < ∞ multiplication by a satisfies ‖a f‖_p ≤ C ‖f‖_p for all f in a dense subset of Lᵖ, then a is essentially bounded by C.