The off-diagonal Marcinkiewicz interpolation theorem #
Let T be a subadditive operator of weak type (p₀, q₀) and of weak type (p₁, q₁),
r ^ qᵢ * ν {T g > r} ≤ Aᵢ * (∫⁻ ‖g‖ ^ pᵢ ∂μ) ^ (qᵢ / pᵢ),
with finite exponents 0 < pᵢ ≤ qᵢ, p₀ ≠ p₁ and q₀ ≠ q₁. For 0 < θ < 1 define p and q
by 1 / p = (1 - θ) / p₀ + θ / p₁ and 1 / q = (1 - θ) / q₀ + θ / q₁. Then T is of strong
type (p, q):
∫⁻ (T v) ^ q ∂ν ≤ C * (∫⁻ ‖v‖ ^ p ∂μ) ^ (q / p),
with C = q / |q - q₀| * A₀ * d ^ (-q₀) + q / |q - q₁| * A₁ * e ^ (-q₁) for any splitting
parameters d, e > 0 with d + e ≤ 1 (TauCeti.lintegral_rpow_le_of_rpow_mul_meas_lt_le_rpow).
The source and target exponents move along two different segments; when p₀ = q₀ and p₁ = q₁
they coincide, and that diagonal case is also
TauCeti.lintegral_rpow_le_of_rpow_mul_meas_lt_le, whose weak-type hypotheses carry no outer
power. The hypothesis pᵢ ≤ qᵢ is what makes the exponent qᵢ / pᵢ at least 1 in the
argument below. Typical applications are the Hardy–Littlewood–Sobolev inequality and other bounds
L^p → L^q with p < q for operators that are only known to be of weak type at the endpoints.
The argument #
Let p₀ < p₁. Split v at the height c * t ^ γ, where γ is fixed by
(q - q₀) / γ = q₀ (p - p₀) / p₀; the interpolation relations then also give
(q - q₁) / γ = q₁ (p - p₁) / p₁. Its part above the height lies in L^{p₀} and its part below
lies in L^{p₁}, and the two weak-type bounds give, for every t > 0,
ν {T v > t} ≤ A₀ (d t) ^ (-q₀) (∫⁻_{‖v‖ > c t ^ γ} ‖v‖ ^ p₀) ^ (q₀ / p₀) + A₁ (e t) ^ (-q₁) (∫⁻_{‖v‖ ≤ c t ^ γ} ‖v‖ ^ p₁) ^ (q₁ / p₁)
(TauCeti.meas_ofReal_lt_le_add_setLIntegral_rpow). Integrating against q t ^ (q - 1) with the
layer cake formula leaves the two integrals ∫ t ^ (q - qᵢ - 1) (∫⁻_{…} ‖v‖ ^ pᵢ) ^ (qᵢ / pᵢ) dt.
Since qᵢ / pᵢ ≥ 1, Minkowski's integral inequality, in the form
TauCeti.lintegral_mul_setLIntegral_rpow_le for weighted truncations, moves the power qᵢ / pᵢ
inside the integral over the space, and for a fixed point the integral in t is
elementary (TauCeti.lintegral_indicator_ofReal_mul_rpow_lt_le and
TauCeti.lintegral_indicator_le_ofReal_mul_rpow_le). The choice of γ makes both terms
integrals of ‖v‖ ^ p, and the result is the distributional estimate
TauCeti.lintegral_rpow_le_of_meas_ofReal_lt_le_rpow:
∫⁻ (T v) ^ q ≤ q c ^ (q₀ (p₀ - p) / p₀) / |q - q₀| · A₀' (∫⁻ ‖v‖ ^ p) ^ (q₀ / p₀) + q c ^ (q₁ (p₁ - p) / p₁) / |q - q₁| · A₁' (∫⁻ ‖v‖ ^ p) ^ (q₁ / p₁),
where A₀' = A₀ d ^ (-q₀) and A₁' = A₁ e ^ (-q₁).
Both terms become multiples of (∫⁻ ‖v‖ ^ p) ^ (q / p) for c = (∫⁻ ‖v‖ ^ p) ^ ((1 - γ) / p).
The case p₁ < p₀ follows by exchanging the endpoints.
With qᵢ = pᵢ and γ = 1 the distributional estimate and the truncation argument specialise to
their diagonal forms, and TauCeti.MeasureTheory.Integral.Marcinkiewicz.General derives the
diagonal theorem from them.
Main declarations #
TauCeti.lintegral_rpow_le_of_meas_ofReal_lt_le_rpow: the distributional interpolation estimate between two off-diagonal endpoints.TauCeti.meas_ofReal_lt_le_add_setLIntegral_rpow: the truncation argument from subadditivity and two off-diagonal weak-type bounds.TauCeti.lintegral_rpow_le_of_rpow_mul_meas_lt_le_rpow: the off-diagonal Marcinkiewicz interpolation theorem for finite exponents.
References #
- L. Grafakos, Classical Fourier Analysis, 3rd ed., Section 1.4.4.
- E. M. Stein, G. Weiss, Introduction to Fourier Analysis on Euclidean Spaces, Chapter V, §2.
- A. Zygmund, Trigonometric Series, Vol. II, Chapter XII, §4.
Off-diagonal Marcinkiewicz interpolation, in distributional form.
Let 0 < p₀ < p < p₁, p₀ ≤ q₀ and p₁ ≤ q₁, and let γ and q satisfy
(q - q₀) / γ = q₀ (p - p₀) / p₀ and (q - q₁) / γ = q₁ (p - p₁) / p₁. Suppose that for every
height t > 0 the superlevel set {u > t} obeys the bound
ν {u > t} ≤ A₀ t ^ (-q₀) (∫⁻ x in {f > c t ^ γ}, f ^ p₀ ∂μ) ^ (q₀ / p₀) + A₁ t ^ (-q₁) (∫⁻ x in {f ≤ c t ^ γ}, f ^ p₁ ∂μ) ^ (q₁ / p₁),
in which the part of f above the height c t ^ γ is measured in L^{p₀} and the part below it
in L^{p₁}. Then
∫⁻ u ^ q ∂ν ≤ q c ^ (q₀ (p₀ - p) / p₀) / |q - q₀| · A₀ (∫⁻ f ^ p ∂μ) ^ (q₀ / p₀) + q c ^ (q₁ (p₁ - p) / p₁) / |q - q₁| · A₁ (∫⁻ f ^ p ∂μ) ^ (q₁ / p₁).
The two relations for γ say that 1 / p and 1 / q divide the segments from 1 / p₀ to
1 / p₁ and from 1 / q₀ to 1 / q₁ in the same ratio; they force γ ≠ 0 and put q strictly
between q₀ and q₁. Applied with u = T f for a subadditive T, the hypothesis is what two
off-diagonal weak-type bounds give after splitting f at the height c t ^ γ
(TauCeti.meas_ofReal_lt_le_add_setLIntegral_rpow). The diagonal case pᵢ = qᵢ, γ = 1 is
TauCeti.lintegral_rpow_le_of_meas_ofReal_lt_le.
The operator-level truncation argument for off-diagonal Marcinkiewicz interpolation.
Suppose T is subadditive, of weak type (p₀, q₀) with constant A₀ and of weak type (p₁, q₁)
with constant A₁. Split v according to whether its extended norm exceeds a level a. If
d + e ≤ 1, then T v > t forces the image of the high part to exceed d * t or the image of
the low part to exceed e * t, and applying the two weak-type bounds to the two pieces gives
ν {T v > t} ≤ A₀ d ^ (-q₀) t ^ (-q₀) (∫⁻ x in {‖v‖ₑ > a}, ‖v x‖ₑ ^ p₀ ∂μ) ^ (q₀ / p₀) + A₁ e ^ (-q₁) t ^ (-q₁) (∫⁻ x in {‖v‖ₑ ≤ a}, ‖v x‖ₑ ^ p₁ ∂μ) ^ (q₁ / p₁).
Off-diagonal interpolation takes the level a = c t ^ γ; the diagonal case is
TauCeti.meas_ofReal_lt_le_add_setLIntegral.
The off-diagonal Marcinkiewicz interpolation theorem. Let T be a subadditive operator of
weak type (p₀, q₀) with constant A₀ and of weak type (p₁, q₁) with constant A₁:
r ^ qᵢ * ν {T g > r} ≤ Aᵢ * (∫⁻ ‖g‖ ^ pᵢ ∂μ) ^ (qᵢ / pᵢ) for every g and every r,
with 0 < pᵢ ≤ qᵢ < ∞, p₀ ≠ p₁ and q₀ ≠ q₁. For 0 < θ < 1 let
1 / p = (1 - θ) / p₀ + θ / p₁ and 1 / q = (1 - θ) / q₀ + θ / q₁. Then T is of strong
type (p, q):
∫⁻ (T v) ^ q ∂ν ≤ (q / |q - q₀| · A₀ d ^ (-q₀) + q / |q - q₁| · A₁ e ^ (-q₁)) * (∫⁻ ‖v‖ ^ p ∂μ) ^ (q / p),
for any splitting parameters d, e > 0 with d + e ≤ 1.