Lᵖ bounds for an operator on L² from bounds for its adjoint

Let T : L²(μ; E) → L²(ν; F) be a bounded operator between L² spaces of Hilbert space valued functions, and let T† be its Hilbert space adjoint. If T† is bounded from L^{q'} to L^{p'} on L², that is ‖T† g‖_{p'} ≤ C ‖g‖_{q'} for every g ∈ L², then T is bounded from Lᵖ to L^q on L² with the same constant, where p, p' and q, q' are pairs of Hölder conjugate exponents and q < ∞.

This is the duality argument that transfers a bound in one range of exponents to the dual range: for instance the Calderón–Zygmund bound for 1 < p < 2 gives the bound for 2 < p < ∞. Neither ‖T f‖_q nor ‖f‖_p is assumed finite, so when C < ∞ the theorem in particular shows that T maps L² ∩ Lᵖ into L^q: the pairing of T f is only tested against bounded functions of finite-measure support (AEStronglyMeasurable.eLpNorm_le_of_forall_enorm_integral_inner_le).

Main declarations

• ContinuousLinearMap.eLpNorm_le_of_eLpNorm_adjoint_le: a L^{q'} → L^{p'} bound for the adjoint gives a Lᵖ → L^q bound for the operator.

References

• E. Stein, Singular Integrals and Differentiability Properties of Functions, Chapter II, §2.4.
• L. Grafakos, Classical Fourier Analysis, Section 5.3.

theorem ContinuousLinearMap.eLpNorm_le_of_eLpNorm_adjoint_le {α : Type u_1} {β : Type u_2} {𝕜 : Type u_3} {E : Type u_4} {F : Type u_5} [RCLike 𝕜] {mα : MeasurableSpace α} {mβ : MeasurableSpace β} {μ : MeasureTheory.Measure α} {ν : MeasureTheory.Measure β} [NormedAddCommGroup E] [InnerProductSpace 𝕜 E] [CompleteSpace E] [NormedAddCommGroup F] [InnerProductSpace 𝕜 F] [CompleteSpace F] [MeasureTheory.SigmaFinite ν] (T : ↥(MeasureTheory.Lp E 2 μ) →L[𝕜] ↥(MeasureTheory.Lp F 2 ν)) {p p' q q' : ENNReal} [p.HolderConjugate p'] [q.HolderConjugate q'] (hq : q ≠ ⊤) {C : ENNReal} (hT : ∀ (g : ↥(MeasureTheory.Lp F 2 ν)), MeasureTheory.eLpNorm (↑↑((adjoint T) g)) p' μ ≤ C * MeasureTheory.eLpNorm (↑↑g) q' ν) (f : ↥(MeasureTheory.Lp E 2 μ)) : MeasureTheory.eLpNorm (↑↑(T f)) q ν ≤ C * MeasureTheory.eLpNorm (↑↑f) p μ

Duality for operators on L². Let T : L²(μ; E) → L²(ν; F) be a bounded operator on a σ-finite target, let p, p' and q, q' be Hölder conjugate with q < ∞, and suppose the adjoint satisfies ‖T† g‖_{p'} ≤ C ‖g‖_{q'} for every g ∈ L². Then ‖T f‖_q ≤ C ‖f‖_p for every f ∈ L².