Is there some inequality which relates ∫ Ω | f | 2 and ...

Banguizb

Banguizb

Answered

2022-07-07

Is there some inequality which relates
Ω | f | 2  and  Ω | f | 2
for some f L 2 ( Ω ) L 2 ( Ω )?

Answer & Explanation

lywiau63

lywiau63

Expert

2022-07-08Added 13 answers

Let Ω be the unit disk in R 2 . The function f ( x , y ) = ( 1 x 2 y 2 ) 1 / 2 satisfies
Ω | f | 2 = π 2 and Ω | f | 2 = 0.
On the other hand the functions f n ( x , y ) = ( x 2 + y 2 ) n / 2 satisfy
Ω | f n | 2 = π n + 1 and Ω | f n | 2 = 2 π
so that in general the ratio
Ω | f | 2 Ω | f | 2
is bounded neither above nor below.

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