Mixing
Doubt in variant of Holder inequality
0
$begingroup$
Let $p, q, r in [1, infty), r neq infty$ such that $1/p+1/q=1/r$.
If $f in {L(X)} ^ p$ and $g in {L (X)}^q$. Is it true that $|f|^{p/r}<|f|^p$? According to I do not, because if I consider $f = 1/2$ constant and $p = 4 = q$ and $r = 2$ all in space $X = [0,1]$ then $ int |f|^4 <infty$ but $|1/2|^2 <|1/2|^ {4}$ something absurd. It is right?
real-analysis measure-theory holder-inequality
asked Jan 11 at 23:43
eraldcoileraldcoil
395211
$endgroup$
add a comment |
0
$begingroup$
Let $p, q, r in [1, infty), r neq infty$ such that $1/p+1/q=1/r$.
If $f in {L(X)} ^ p$ and $g in {L (X)}^q$. Is it true that $|f|^{p/r}<|f|^p$? According to I do not, because if I consider $f = 1/2$ constant and $p = 4 = q$ and $r = 2$ all in space $X = [0,1]$ then $ int |f|^4 <infty$ but $|1/2|^2 <|1/2|^ {4}$ something absurd. It is right?
real-analysis measure-theory holder-inequality
asked Jan 11 at 23:43
eraldcoileraldcoil
395211
$endgroup$
1
$begingroup$
What you have done is right but you are very likely to have copied the statement wrongly from somewhere. You can never expect an inequality for powers of the function from integrability. The inequality probably was stated for some norms of $f$ in $L^{p}$ and $L^{q}$.
$endgroup$
– Kavi Rama Murthy
Jan 12 at 0:02
$begingroup$
Exactly the problem is: $p,q,rin [1,infty] with rneq infty$ such that $1/p+1/q=1/r$. If $fin L^p$ and $gin L^q$ then $|fg|_{r}leq |f|_p|g|_q$
$endgroup$
– eraldcoil
Jan 12 at 1:47
$begingroup$
math.stackexchange.com/questions/159887/…
$endgroup$
– d.k.o.
Jan 12 at 2:55
$begingroup$
Thanks but I already solved the problem. My question is the question on the subject.
$endgroup$
– eraldcoil
Jan 12 at 2:57
add a comment |
0
0
0
$begingroup$
Let $p, q, r in [1, infty), r neq infty$ such that $1/p+1/q=1/r$.
If $f in {L(X)} ^ p$ and $g in {L (X)}^q$. Is it true that $|f|^{p/r}<|f|^p$? According to I do not, because if I consider $f = 1/2$ constant and $p = 4 = q$ and $r = 2$ all in space $X = [0,1]$ then $ int |f|^4 <infty$ but $|1/2|^2 <|1/2|^ {4}$ something absurd. It is right?
real-analysis measure-theory holder-inequality
asked Jan 11 at 23:43
eraldcoileraldcoil
395211
$endgroup$
Let $p, q, r in [1, infty), r neq infty$ such that $1/p+1/q=1/r$.
If $f in {L(X)} ^ p$ and $g in {L (X)}^q$. Is it true that $|f|^{p/r}<|f|^p$? According to I do not, because if I consider $f = 1/2$ constant and $p = 4 = q$ and $r = 2$ all in space $X = [0,1]$ then $ int |f|^4 <infty$ but $|1/2|^2 <|1/2|^ {4}$ something absurd. It is right?
real-analysis measure-theory holder-inequality
real-analysis measure-theory holder-inequality
asked Jan 11 at 23:43
eraldcoileraldcoil
395211
asked Jan 11 at 23:43
eraldcoileraldcoil
395211
asked Jan 11 at 23:43
eraldcoileraldcoil
395211
asked Jan 11 at 23:43
eraldcoileraldcoil
395211
asked Jan 11 at 23:43
eraldcoileraldcoil
395211
395211
1
$begingroup$
What you have done is right but you are very likely to have copied the statement wrongly from somewhere. You can never expect an inequality for powers of the function from integrability. The inequality probably was stated for some norms of $f$ in $L^{p}$ and $L^{q}$.
$endgroup$
– Kavi Rama Murthy
Jan 12 at 0:02
$begingroup$
Exactly the problem is: $p,q,rin [1,infty] with rneq infty$ such that $1/p+1/q=1/r$. If $fin L^p$ and $gin L^q$ then $|fg|_{r}leq |f|_p|g|_q$
$endgroup$
– eraldcoil
Jan 12 at 1:47
$begingroup$
math.stackexchange.com/questions/159887/…
$endgroup$
– d.k.o.
Jan 12 at 2:55
$begingroup$
Thanks but I already solved the problem. My question is the question on the subject.
$endgroup$
– eraldcoil
Jan 12 at 2:57
add a comment |
1
$begingroup$
What you have done is right but you are very likely to have copied the statement wrongly from somewhere. You can never expect an inequality for powers of the function from integrability. The inequality probably was stated for some norms of $f$ in $L^{p}$ and $L^{q}$.
$endgroup$
– Kavi Rama Murthy
Jan 12 at 0:02
$begingroup$
Exactly the problem is: $p,q,rin [1,infty] with rneq infty$ such that $1/p+1/q=1/r$. If $fin L^p$ and $gin L^q$ then $|fg|_{r}leq |f|_p|g|_q$
$endgroup$
– eraldcoil
Jan 12 at 1:47
$begingroup$
math.stackexchange.com/questions/159887/…
$endgroup$
– d.k.o.
Jan 12 at 2:55
$begingroup$
Thanks but I already solved the problem. My question is the question on the subject.
$endgroup$
– eraldcoil
Jan 12 at 2:57
1
1
$begingroup$
What you have done is right but you are very likely to have copied the statement wrongly from somewhere. You can never expect an inequality for powers of the function from integrability. The inequality probably was stated for some norms of $f$ in $L^{p}$ and $L^{q}$.
$endgroup$
– Kavi Rama Murthy
Jan 12 at 0:02
$begingroup$
What you have done is right but you are very likely to have copied the statement wrongly from somewhere. You can never expect an inequality for powers of the function from integrability. The inequality probably was stated for some norms of $f$ in $L^{p}$ and $L^{q}$.
$endgroup$
– Kavi Rama Murthy
Jan 12 at 0:02
$begingroup$
Exactly the problem is: $p,q,rin [1,infty] with rneq infty$ such that $1/p+1/q=1/r$. If $fin L^p$ and $gin L^q$ then $|fg|_{r}leq |f|_p|g|_q$
$endgroup$
– eraldcoil
Jan 12 at 1:47
$begingroup$
Exactly the problem is: $p,q,rin [1,infty] with rneq infty$ such that $1/p+1/q=1/r$. If $fin L^p$ and $gin L^q$ then $|fg|_{r}leq |f|_p|g|_q$
$endgroup$
– eraldcoil
Jan 12 at 1:47
$begingroup$
math.stackexchange.com/questions/159887/…
$endgroup$
– d.k.o.
Jan 12 at 2:55
$begingroup$
math.stackexchange.com/questions/159887/…
$endgroup$
– d.k.o.
Jan 12 at 2:55
$begingroup$
Thanks but I already solved the problem. My question is the question on the subject.
$endgroup$
– eraldcoil
Jan 12 at 2:57
$begingroup$
Thanks but I already solved the problem. My question is the question on the subject.
$endgroup$
– eraldcoil
Jan 12 at 2:57
add a comment |
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1
$begingroup$
What you have done is right but you are very likely to have copied the statement wrongly from somewhere. You can never expect an inequality for powers of the function from integrability. The inequality probably was stated for some norms of $f$ in $L^{p}$ and $L^{q}$.
$endgroup$
– Kavi Rama Murthy
Jan 12 at 0:02
$begingroup$
Exactly the problem is: $p,q,rin [1,infty] with rneq infty$ such that $1/p+1/q=1/r$. If $fin L^p$ and $gin L^q$ then $|fg|_{r}leq |f|_p|g|_q$
$endgroup$
– eraldcoil
Jan 12 at 1:47
$begingroup$
math.stackexchange.com/questions/159887/…
$endgroup$
– d.k.o.
Jan 12 at 2:55
$begingroup$
Thanks but I already solved the problem. My question is the question on the subject.
$endgroup$
– eraldcoil
Jan 12 at 2:57