Young’s theorem
The following result is due to William Henry Young.
Theorem 1
Observe the analogy with the similar result
with convolution replaced by ordinary (pointwise) product
,
where the requirement is —i.e.,
—instead of (1).
The cases
-
1.
,
-
2.
, ,
are the most widely known; for these we provide a proof, supposing . We shall use the following facts:
-
•
If are measurable, then is measurable.
-
•
For any , if , then belongs to as well, and its -norm is the same as ’s.
-
•
For any , if , then belongs to as well, and its -norm is the same as ’s.
Proof of case 1.
Suppose , with . Then
This holds for all , therefore as well.
Proof of case 2.
First, suppose .
We may suppose and are Borel measurable:
if they are not, we replace them with Borel measurable functions
and
which are equal to and , respectively,
outside of a set of Lebesgue measure zero;
apply the theorem
to , , and ;
and deduce the theorem for , , and .
By Tonelli’s theorem,
thus the function belongs to . By Fubini’s theorem, the function belongs to for almost all , and belongs to ; plus,
Suppose now ;
choose so that .
By the argument above,
belongs to for almost all :
for those , put
Then and with ,
so and
but , so point 1 of the theorem is proved.
By Hölder’s inequality
,
but we know that , so and point 2 is also proved. Finally,
but means and thus , so that point 3 is also proved.
References
- 1 G. Gilardi. Analisi tre. McGraw-Hill 1994.
- 2 W. Rudin. Real and complex analysis. McGraw-Hill 1987.
- 3 W. H. Young. On the multiplication of successions of Fourier constants. Proc. Roy. Soc. Lond. Series A 87 (1912) 331–339.
Title | Young’s theorem |
---|---|
Canonical name | YoungsTheorem |
Date of creation | 2013-03-22 18:17:44 |
Last modified on | 2013-03-22 18:17:44 |
Owner | Ziosilvio (18733) |
Last modified by | Ziosilvio (18733) |
Numerical id | 15 |
Author | Ziosilvio (18733) |
Entry type | Theorem |
Classification | msc 44A35 |