Marcinkiewicz interpolation between two finite exponents #
A sublinear operator T that is simultaneously of weak type (p₀, p₀) and of weak type
(p₁, p₁), for 0 < p₀ < p₁ < ∞, is bounded on L^p for every p₀ < p < p₁. This is the
Marcinkiewicz interpolation theorem in the case of two finite endpoints; the case p₀ = 1,
p₁ = ∞, where the second endpoint is an L^∞ bound rather than a weak-type bound, is
TauCeti.MeasureTheory.Integral.Marcinkiewicz.Basic.
Source and target exponents agree at both endpoints here: the hypotheses are weak type
(p₀, p₀) and (p₁, p₁), and the conclusion is strong type (p, p). The off-diagonal theorem,
which interpolates weak type (p₀, q₀) and (p₁, q₁) with pᵢ ≤ qᵢ into strong type (p, q)
along the line 1 / q = (1 - θ) / q₀ + θ / q₁, is
TauCeti.lintegral_rpow_le_of_rpow_mul_meas_lt_le_rpow in
TauCeti.MeasureTheory.Integral.Marcinkiewicz.OffDiagonal.
The statements proved here #
As in the diagonal case, the analytic engine is stated in terms of the distribution functions
of T f and of f. For each height t, splitting f at the height c * t and applying the two
endpoint bounds to the two pieces produces the single inequality
ν {T f > t} ≤ A₀ t ^ (-p₀) * ∫⁻ x in {‖f‖ > c * t}, ‖f‖ ^ p₀ ∂μ + A₁ t ^ (-p₁) * ∫⁻ x in {‖f‖ ≤ c * t}, ‖f‖ ^ p₁ ∂μ,
which says that the part of f above the height c * t can only push T f above t through the
first endpoint, and the part below it only through the second. That inequality is the hypothesis
of TauCeti.lintegral_rpow_le_of_meas_ofReal_lt_le, and its conclusion is the L^p bound
∫⁻ (T f) ^ p ∂ν ≤ (p c ^ (p₀ - p) / (p - p₀) * A₀ + p c ^ (p₁ - p) / (p₁ - p) * A₁) * ∫⁻ ‖f‖ ^ p ∂μ.
Both constants degenerate as p approaches an endpoint, exactly as they must: an operator of
weak type (p₀, p₀) need not be bounded on L^{p₀}. Keeping the two endpoint constants separate
records which endpoint each half of the bound comes from, and makes the splitting parameter c
available for optimizing the total constant.
The proof #
Both statements are the diagonal case qᵢ = pᵢ, γ = 1 of their off-diagonal counterparts in
TauCeti.MeasureTheory.Integral.Marcinkiewicz.OffDiagonal: the distributional estimate is
TauCeti.lintegral_rpow_le_of_meas_ofReal_lt_le_rpow and the truncation argument is
TauCeti.meas_ofReal_lt_le_add_setLIntegral_rpow at the level c * t. With these exponents the
outer powers qᵢ / pᵢ are 1, the height c * t ^ γ is c * t, and the constants
q c ^ (qᵢ (pᵢ - p) / pᵢ) / |q - qᵢ| become p c ^ (pᵢ - p) / |p - pᵢ|.
The layer cake formula, Minkowski's integral inequality and the elementary integrals in the height
variable t all enter through the off-diagonal estimate. For a fixed x, the integral in t over
the region where c * t stays below ‖f x‖ converges because p₀ < p, and the integral over its
complement converges because p < p₁. The two weighted truncated double integrals of the diagonal
case are also available on their own: TauCeti.lintegral_mul_setLIntegral_lt evaluates the upper
one and TauCeti.lintegral_mul_setLIntegral_le_le bounds the lower one.
Bounded linear operators #
The operator-level theorem above asks for T on all measurable functions. A bounded linear
operator T on L^{p₁} is defined only there, but splitting f ∈ L^{p₁} at a height keeps both
pieces in L^{p₁}, and the L^{p₁} bound implies weak type (p₁, p₁) by Chebyshev's inequality.
So if T is of weak type (p₀, p₀) on L^{p₁}, then ‖T f‖_p ≤ C ‖f‖_p for every f ∈ L^{p₁}
and every p₀ < p < p₁ (ContinuousLinearMap.eLpNorm_le_of_rpow_mul_meas_lt_le). This is the
form in which the weak type (1, 1) bound for a singular integral operator on L² becomes an
Lᵖ bound for 1 < p < 2.
Main declarations #
TauCeti.lintegral_rpow_le_of_meas_ofReal_lt_le: the interpolation estimate.TauCeti.meas_ofReal_lt_le_add_setLIntegral: the reusable truncation argument from subadditivity and the two weak-type endpoint bounds.TauCeti.lintegral_rpow_le_of_rpow_mul_meas_lt_le: operator-level Marcinkiewicz interpolation between two finite exponents.ContinuousLinearMap.eLpNorm_le_of_rpow_mul_meas_lt_le: a bounded linear operator onL^{p₁}of weak type(p₀, p₀)is of strong type(p, p)forp₀ < p < p₁.
References #
- L. Grafakos, Classical Fourier Analysis, Theorem 1.3.2.
- E. Stein, Singular Integrals and Differentiability Properties of Functions, Chapter I, §4.
The weighted mass of the part of f above the height c * t. For -1 < s the weight
t ^ s is integrable at the origin, and the double integral of the upper truncations of f ^ q
against it is exactly a multiple of ∫⁻ f ^ (q + (s + 1)).
The weighted mass of the part of f below the height c * t. For s < -1 the weight
t ^ s is integrable at infinity, and the double integral of the lower truncations of f ^ q
against it is at most a multiple of ∫⁻ f ^ (q + (s + 1)).
The bound is not an equality: where f is infinite the left-hand side sees nothing, because no
finite height c * t reaches it, while the right-hand side is infinite as soon as
0 < q + (s + 1).
Marcinkiewicz interpolation between two finite exponents, in distributional form.
Suppose that for every height t > 0 the superlevel set {u > t} obeys the two-sided bound
ν {u > t} ≤ A₀ t ^ (-p₀) * ∫⁻ x in {f > c * t}, f ^ p₀ ∂μ + A₁ t ^ (-p₁) * ∫⁻ x in {f ≤ c * t}, f ^ p₁ ∂μ,
in which the part of f above the height c * t is measured in L^{p₀} and the part below it
in L^{p₁}. Then for every p₀ < p < p₁
∫⁻ u ^ p ∂ν ≤ (p c ^ (p₀ - p) / (p - p₀) * A₀ + p c ^ (p₁ - p) / (p₁ - p) * A₁) * ∫⁻ f ^ p ∂μ.
Applied with u = T f for a sublinear T, the hypothesis is what the weak (p₀, p₀) and weak
(p₁, p₁) endpoint bounds for T give after splitting f at the height c * t, and the
conclusion is the strong type (p, p) bound. Each constant degenerates as p approaches its own
endpoint, as it must.
The reusable operator-level truncation argument for Marcinkiewicz interpolation between two finite exponents.
Suppose T is subadditive and of weak type (p₀, p₀) with constant A₀ and of weak type
(p₁, p₁) with constant A₁. Split v according to whether its extended norm exceeds c * t.
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 ^ (-p₀) t ^ (-p₀) * ∫⁻ x in {‖v‖ₑ > c * t}, ‖v x‖ₑ ^ p₀ ∂μ + A₁ e ^ (-p₁) t ^ (-p₁) * ∫⁻ x in {‖v‖ₑ ≤ c * t}, ‖v x‖ₑ ^ p₁ ∂μ.
The parameters c, d and e expose the choice of truncation rather than fixing the customary
c = 1, d = e = 1 / 2.
Marcinkiewicz interpolation at operator level, between two finite exponents.
A subadditive operator of weak type (p₀, p₀) with constant A₀ and of weak type (p₁, p₁)
with constant A₁ is of strong type (p, p) for every p₀ < p < p₁. The positive splitting
parameters c, d, e may be chosen arbitrarily subject to d + e ≤ 1.
Marcinkiewicz interpolation for a bounded linear operator on L^{p₁}. Let T be a
bounded linear operator on L^{p₁}, 1 ≤ p₁ < ∞, that is also of weak type (p₀, p₀) with
constant A on L^{p₁}, 0 < p₀ < p₁: for every g ∈ L^{p₁} and every height s,
s ^ p₀ · ν {‖T g‖ > s} ≤ A ∫ ‖g‖ ^ p₀.
Then for every p₀ < p < p₁ and every f ∈ L^{p₁},
‖T f‖_p ≤ (p / (p - p₀) · 2 ^ p₀ A + p / (p₁ - p) · 2 ^ p₁ ‖T‖ ^ p₁) ^ (1 / p) ‖f‖_p.
The bound is vacuous unless f also lies in L^p.