The Riesz–Thorin interpolation theorem #
Let T be a complex-linear operator from simple functions on (α, μ) to measurable functions on
(β, ν), bounded from L^{p₀} to L^{q₀} with norm M₀ and from L^{p₁} to L^{q₁} with
norm M₁. For 0 < θ < 1 define the intermediate exponents by
1 / p = (1 - θ) / p₀ + θ / p₁ and 1 / q = (1 - θ) / q₀ + θ / q₁.
Then ‖T f‖_q ≤ M₀ ^ (1 - θ) M₁ ^ θ ‖f‖_p for every simple f
(TauCeti.eLpNorm_le_rpow_mul_rpow_mul_eLpNorm). The exponents are arbitrary in (0, ∞] on
the source side (an exponent 0 makes the statement trivial) and in [1, ∞] on the target
side, and the constant is the weighted geometric mean of the two endpoint norms. The scalars
are complex: for real scalars the bound holds in general only up to a factor 2. The target
measure is σ-finite, an assumption only needed for an infinite target exponent q; for q < ∞
the theorem holds on an arbitrary measure space
(TauCeti.eLpNorm_le_rpow_mul_rpow_mul_eLpNorm_of_ne_top).
Compared with Marcinkiewicz interpolation, the theorem asks for strong-type bounds at both
endpoints, but it loses nothing in the constant and lets the source and target exponents move
independently. Its standard consequences, the Hausdorff–Young inequality and Young's convolution
inequality, both use an endpoint ∞, which is why the exponents are not restricted to be finite.
Operators on L² #
A bounded operator T on L² with ‖T‖ ≤ M₁ that also satisfies ‖T f‖_∞ ≤ M₀ ‖f‖₁ for
integrable f ∈ L², such as a unitary group with a dispersive estimate, satisfies
‖T f‖_q ≤ M₀ ^ (1 - θ) M₁ ^ θ ‖f‖_p
for conjugate exponents p, q with 2 ≤ q ≤ ∞, θ = 2 / q, and every f ∈ L² ∩ Lᵖ
(ContinuousLinearMap.eLpNorm_apply_le_rpow_mul_rpow_mul_eLpNorm). On simple functions this is
the theorem above, applied to a linear extension of T from the simple functions in L² to all
simple functions; it passes to L² ∩ Lᵖ by approximating f with simple functions in L² and
Lᵖ at once and applying Fatou's lemma.
The proof #
This is Thorin's complex method. By duality against simple functions
(MeasureTheory.MemLp.eLpNorm_le_of_forall_enorm_integral_mul_le), it suffices to bound
|∫ T f · g| for simple f with ‖f‖_p ≤ 1 and simple g with ‖g‖_{q'} ≤ 1, where q' is
the conjugate exponent. Embed f and g in the analytic families
f_z = sgn(f) |f| ^ {P(z)}, g_z = sgn(g) |g| ^ {Q(z)},
with affine exponents P, Q normalized so that P(θ) = Q(θ) = 1. Because f and g take
finitely many values and T is linear, Φ(z) = ∫ T f_z · g_z is a finite sum of exponentials in
z; it is entire and bounded on the strip 0 ≤ Re z ≤ 1. On the line Re z = 0 the functions
f_z and g_z have norm at most one in L^{p₀} and L^{q₀'}, so |Φ| ≤ M₀ there by Hölder's
inequality, and likewise |Φ| ≤ M₁ on Re z = 1. Hadamard's three-lines theorem
(Complex.HadamardThreeLines.norm_le_interp_of_mem_verticalClosedStrip') gives
|∫ T f · g| = |Φ(θ)| ≤ M₀ ^ (1 - θ) M₁ ^ θ.
References #
- M. Riesz, Sur les maxima des formes bilinéaires et sur les fonctionnelles linéaires, Acta Math. 49 (1927).
- G. O. Thorin, An extension of a convexity theorem due to M. Riesz, Kungl. Fysiogr. Sällsk. i Lund Förh. 8 (1939).
- L. Grafakos, Classical Fourier Analysis, Theorem 1.3.4.
- E. M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Chapter V, Theorem 1.3.
The Riesz–Thorin interpolation theorem. Let T be a complex-linear map from simple
functions on (α, μ) to almost-everywhere classes on a σ-finite (β, ν) with
‖T f‖_{q₀} ≤ M₀ ‖f‖_{p₀} and ‖T f‖_{q₁} ≤ M₁ ‖f‖_{p₁} for all simple f, where
p₀, p₁ ∈ [0, ∞] and 1 ≤ q₀, q₁ ≤ ∞. For 0 < θ < 1 and the intermediate exponents
1 / p = (1 - θ) / p₀ + θ / p₁, 1 / q = (1 - θ) / q₀ + θ / q₁,
‖T f‖_q ≤ M₀ ^ (1 - θ) M₁ ^ θ ‖f‖_p
for every simple f. σ-finiteness of ν is only used when q = ∞; for q < ∞ see
TauCeti.eLpNorm_le_rpow_mul_rpow_mul_eLpNorm_of_ne_top.
The Riesz–Thorin interpolation theorem for a finite target exponent, on an arbitrary
measure space (β, ν): under the hypotheses of TauCeti.eLpNorm_le_rpow_mul_rpow_mul_eLpNorm,
if the intermediate exponent q is finite then ‖T f‖_q ≤ M₀ ^ (1 - θ) M₁ ^ θ ‖f‖_p for every
simple f, without assuming ν σ-finite.
Operators on L² bounded from L¹ to L^∞ #
Interpolation between the L¹ → L^∞ and L² → L² bounds. Let T be a bounded operator
on L² with ‖T‖ ≤ M₁, such that ‖T f‖_∞ ≤ M₀ ‖f‖₁ for every integrable f ∈ L². For
conjugate exponents p and q with 2 ≤ q ≤ ∞, and θ = 2 / q,
‖T f‖_q ≤ M₀ ^ (1 - θ) M₁ ^ θ ‖f‖_p
for every f ∈ L² ∩ Lᵖ. This is the Riesz–Thorin theorem between the endpoints
(p, q) = (1, ∞) and (2, 2). It turns the dispersive estimate of a unitary group into its
Lᵖ → L^q decay estimates.