Marcinkiewicz Interpolation Theorem
by Daniel Baczkowski
1
Introduction
Let (U, µ) be a measure space. Let M denote the collection of all extended real valued
µ-measurable functions on U. Also, let M0 denote the class of functions from M that are
finite µ-almost everywhere. In this section, we assume that the functions are in M0 .
For a function f ∈ M0 (U, µ), we denote by m(y) = mf (y) its distribution function,
namely for y > 0 define
m(y) = mf (y) = µ x ∈ U : |f (x)| > y .
It is easy to see that m(y) is monotone decreasing and also true that it is continuous on
the right for all y.
The decreasing rearrangement of f is the function f ∗ defined on [0, ∞) by
f ∗ (t) = inf y : mf (y) ≤ t for t ≥ 0.
(1)
Using the convention that inf ∅ = ∞, notice that if mf (y) happens to be continuous and
strictly decreasing then f ∗ is the inverse of mf . The basic property of f ∗ is that (1) implies
that f ∗ is equimeasurable with |f |, namely,
mf (y) ≡ mf ∗ (y) for y > 0.
We say that T : Lp (U, µ) → Lq (V, ν) is of strong type (p, q) if T is bounded; i.e. if there
exists a c > 0 such that
kT f kq ≤ ckf kp for all f ∈ Lp .
For 1 ≤ p < ∞, we say that T : Lp (U, µ) → M0 (V, ν) is of weak type (p, p) if there
exists a c > 0 such that
kf kp
m(T f ) (y) = ν x ∈ V : |(T f )(x)| > y ≤ c p for y > 0.
y
We say a map T : A → B is subadditive if for any f and g ∈ A,
|T (f + g)| ≤ |T (f )| + |T (g)|
Theorem 1. (Marcinkiewicz)
Let 1 ≤ p0 < p1 ≤ ∞ and assume that T : Lp0 (U, µ) ∪ Lp1 (U, µ) → M0 (V, ν) is subadditive.
Also, suppose that T is of weak type (p0 , p0 ) and (p1 , p1 ). Then, for each p such that
p0 < p < p1 , the map T is of strong type (p, p).
Theorem 2. (Marcinkiewicz)
Let 1 ≤ p0 < ∞ and assume that T : Lp0 (U, µ) ∪ L∞ (U, µ) → M0 (V, ν) is subadditive. Also,
suppose that T is of weak type (p0 , p0 ) and of strong type (∞, ∞). Then, for each p such that
p0 < p < ∞, the map T is of strong type (p, p).
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Lemma 1. If f ∈ Lp , 1 ≤ p < ∞, then
mf (y) ≤
and
kf kpp
kf kpp
for y > 0
yp
Z
(2)
∞
tp−1 mf (t)dt.
=p
(3)
0
Proof. By using the convenient notation {|f | > y} to denote {x ∈ U : |f (x)| > y}, we have
Z
Z
p
p
|f (x)| dµ ≥
|f (x)|p dµ ≥ y p µ{|f | > y} = y p mf (y)
kf kp =
U
{|f |>y}
in which (2) easily follows. Next, we introduce a set A ⊂ U × [0, ∞) which we define by
A = {(x, s) : |f (x)|p > s}. Using Fubini’s Theorem, we obtain
Z
U
Z Z
p
|f (x)|p
Z
|f (x)| dµ =
1 dsdµ =
χA (x, s)dsdµ
U 0
U ×[0,∞)
Z ∞Z
Z ∞
=
χA (x, s)dµds =
µ{x : |f (x)|p > s}ds.
0
U
0
Use the substitution s = tp to obtain
Z
Z ∞
Z
p−1
p
t µ{x : |f (x)| > t}dt = p
|f (x)| dµ = p
U
∞
tp−1 mf (t)dt.
0
0
For 1 ≤ p < ∞, denote by L∗p , the weak Lp space which is the set of functions in
M0 (U, µ) such that mf (y) ≤ M/y p , for all y > 0 where M > 0 is some constant. Let
kf kL∗p = supy>0 y mf (y)1/p . Notice that by Lemma 1, L∗p ⊇ Lp with kf kL∗p ≤ kf kLp , but
L∗p * Lp .
Example: Let f (x) = 1/x1/p where x ∈ (0, 1] and 1 ≤ p < ∞. Then,
1
if 0 < y ≤ 1
p
1/y if y > 1
1/p
= ∞. So, f ∈
/ Lp .
mf (y) = {x : 1/x1/p > y} =
and hence f ∈ L∗p (0, 1]. But, kf kp =
R1
0
1/x dx
Lemma 2. If f ∈ Lp where 1 ≤ p < ∞, then
Z
Z ∞
p
p
|f | dµ = mf (y)y + p
tp−1 mf (t)dt
{|f |>y}
y
Z
Z y
|f |p dµ = −mf (y)y p + p
tp−1 mf (t)dt
{|f |≤y}
0
2
(4)
(5)
Proof. First notice that (3) minus (5) yields (4), hence it suffice to prove (4). Proceeding
analogously as in the proof of Lemma 1,
Z
Z
p
Z
|f | dµ =
{|f |>y}
Z
Z
p
{|f |>y}
0
Z
Z
Z
Z
Z
|f (x)|p
1 dsdµ
yp
{|f |>y}
0
|f (x)|p
p
Z
χA (x, s)dsdµ
1 dsdµ = y mf (y) +
1 dµ +
{|f |>y}
yp
1 dsdµ +
1 dsdµ =
{|f |>y}
=y
|f (x)|p
{|f |>y}
U ×[y p ,∞)
yp
where A = {(x, s) : y p ≤ s ≤ |f (x)|p }. Now, using Fubini’s Theorem, then the substitution
s = tp , we get
Z ∞
Z ∞
Z
p
tp−1 µ{x : |f (x)|p > tp }dt
µ{x : |f (x)| > s}ds = p
χA (x, s)dsdµ =
p
p
y
y
U×[y ,∞)
Z ∞
tp−1 mf (t)dt
=p
y
which yields (4).
2
Proofs of the theorems
Proof of Theorem 1: We will consider for simplicity the case when (U, µ) = (V, ν) = ([a, b], µ)
where µ denotes the Lebesgue measure on the interval [a, b] (i.e. dµ = dx). For a given
y > 0, define
f (x) if |f (x)| ≤ y
g(x) = gy (x) =
0
if |f (x)| > y
0
if |f (x)| ≤ y
h(x) = hy (x) =
f (x) if |f (x)| > y
Since f (x) = g(x) + h(x) and T is subadditive, we have
x : |T f (x)| > y ⊂ x : |T g(x)| > y/2 ∪ x : |T h(x)| > y/2 .
So,
m |T f |, y ≤ m |T g|, y/2 + m |T h|, y/2
where m |F |, y = mF y .
Since T is of weak type (p1 , p1 ) and (p0 , p0 ), we have, respectively, that
p1 Z
Z
2
p1
p1 −p1
|g| dx = c1 2 y
|f |p1 dx
m |T g|, y/2 ≤ c1
y
[a,b]
{|f |≤y}
and
p0 Z
Z
2
p0
p0 −p0
m |T h|, y/2 ≤ c0
|h| dx = c0 2 y
|f |p0 dx .
y
[a,b]
{|f |>y}
3
(6)
For simplicity, use m(t) = mf (t). Now, using (2) through (6) it follows that
Z ∞
p
y p−1 m |T f |, y dy
kT f kp = p
0
Z y
Z ∞
p1 −1
p−p1 −1
p1
t
m(t)dt dy
y
≤ pc1 2 p1
0
0
Z ∞
Z ∞
Z ∞
p−1
p−p0 −1
p0 −1
p0
y m(y)dy + p0
t
t
m(t)dt dy
+ c0 2 p
0
0
y
Z ∞
Z ∞
p1
p1 −1
p−p1 −1
p
p0
t
m(t)
y
dy dt
= c0 2 kf kp + c1 pp1 2
0
t
Z t
Z ∞
p0 −1
p−p0 −1
p0
t
m(t)
y
dy dt
+ c0 pp0 2
0
0
Z ∞
Z ∞
1
1
p
p1
p0
p−1
p0
= c0 2 kf kp + c1 p1 2 p
t m(t)dt + c0 p0 2 p
tp−1 m(t)dt
p − p1
p − p0
0
0
p
= ckf kp .
Thus, T is of strong type (p, p), and the theorem follows in this case.
Proof of Theorem 2: Now suppose that 1 ≤ p0 < ∞ and p1 = ∞. We will again consider
when (U, µ) = (V, ν) = ([a, b], µ) where µ denotes the Lebesgue measure. For a given y > 0,
define

y

f (x)
if |f (x)| ≤

2c1
g(x) = gy (x) =
y
y


sign f (x) if |f (x)| >
2c1
2c1
where c1 > 0 is from the definition of T being of weak type (p1 , p1 ). Let h(x) = hy (x) =
f (x) − g(x). Also by the definition of T ,
kT gk∞ ≤ c1 kgk∞ ≤ y/2
since kgk∞ ≤ y/(2c1 ). So, x : |(T g)(x)| > y/2 = ∅. As (6) also holds in this case, we have
from (1) that
p0
2
khkpp00
m |T f |, y = µ |T f | > y ≤ µ |T h| > y/2 ≤ c0
y
Z
p0
−p0
y
p0
p0
≤ c0 2
|f (x)| dx +
µ |f | > y/(2c1 ) y
2c1
{|f |>y/(2c1 )}
Z
p0
−p0
p0
−p0
≤ c0 2 y
|f (x)| dx + (2c1 ) mf y/(2c1 )
{|f |>y/(2c1 )}
One only needs now to proceed exactly in the same fashion as in the proof for Theorem 1.
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3
Application of Marcinkiewicz Interpolation Theorem
It is quite interesting that the Marcinkiewicz Interpolation Theorem is used to prove many
pertinent theorems in Approximation Theory and Analysis. In this section, we simply state
just one application of this powerful tool. We present a generalization of the HausdorffYoung inequality due to Payley. The main difference between the theorems being that Payley
introduced a weight function into his inequality and resorted to the theorem of Marcinkiewicz.
In what follows, we consider the measure space (Rn , µ) where µ denotes the Lebesgue measure
(dµ = dx). Let F denote the Fourier transform which is defined by
Z
ˆ
(Ff )(ξ) = f (ξ) = f (x) exp{−ihx, ξi}dx
where hx, ξi = x1 ξ1 + · · · + xn ξn (note here x = (x1 , . . . , xn ) and ξ = (ξ1 , . . . , ξn )). Also, we
assume that w(x) is a weight function, i.e. a positive measurable function on Rn . Letting
Lp (w) denote the Lp space with respect to wdx, the norm on Lp (w) is
Z
kf kLp (w) =
p
1/p
|f (x)| w(x)dx
.
Rn
Theorem 3. Assume that 1 ≤ p ≤ 2. Then,
kFf kL2 (|ξ|−n(2−p) ) ≤ Cp kf kLp .
Proof. Consider the map defined by (T f )(ξ) = |ξ|n fˆ(ξ). By Parseval’s formula,
kT f kL2 (|ξ|−2n ) = kfˆkL2 ≤ Ckf kL2 .
Simply by the definition of weak Lp spaces we have that kT f kL∗2 (|ξ|−2n ) ≤ kT f kL2 (|ξ|−2n ) .
Altogether, we have that T is of weak type (2, 2). We now work towards showing that T is
of weak type (1, 1). Thus, the Marcinkiewicz interpolation theorem implies the theorem.
Now, consider the set Ey = {ξ : |ξ|n |fˆ(ξ)| > y}. For simplicity, we let ν denote the
measure |ξ|−2n dξ and assume that kf kL1 = 1. Then, |fˆ(ξ)| ≤ 1. Therefore, for ξ ∈ Ey we
have y ≤ |ξ|n . This forces
Z
Z
−2n
m T f, y = ν Ey =
|ξ| dξ ≤
|ξ|−2n dξ.
{ξ:|ξ|n ≥y}
Ey
Substituting ξ = y 1/n u gives |ξ|−2n = y −2 un and dξ = y du; thus,
Z
Z
−2n
−1
|ξ|
dξ = y
|u|−2n du = Cy −1 .
{ξ:|ξ|n ≥y}
{u:|u|≥1}
So, we have that m T f, y ≤ Cy −1 . Now, we have shown that
y · m T f, y ≤ Ckf kL1
which implies T is of weak type (1, 1).
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References:
Bergh, J. and Löfström, J., An Introduction to Interpolation Spaces, Springer-Verlag, 1976.
Petrushev, P., handout on the Marcinkiewicz Interpolation Theorem.
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