Product of functions in $L^p$ is in $L^p$? [closed]
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Is it true that if $f,gin L^p (mathbbR^n)$, then $fcdot gin L^p (mathbbR^n)$ ?
measure-theory lebesgue-measure
edited Sep 6 at 6:35
Devashish Kaushik
29314
asked Sep 6 at 4:43
Bogdan
63649
closed as off-topic by Xander Henderson, user99914, Guy Fsone, José Carlos Santos, Jyrki Lahtonen Sep 6 at 7:58
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." РXander Henderson, Community, Guy Fsone, Jos̩ Carlos Santos, Jyrki Lahtonen
add a comment |Â
up vote
-3
down vote
favorite
Is it true that if $f,gin L^p (mathbbR^n)$, then $fcdot gin L^p (mathbbR^n)$ ?
measure-theory lebesgue-measure
edited Sep 6 at 6:35
Devashish Kaushik
29314
asked Sep 6 at 4:43
Bogdan
63649
closed as off-topic by Xander Henderson, user99914, Guy Fsone, José Carlos Santos, Jyrki Lahtonen Sep 6 at 7:58
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." РXander Henderson, Community, Guy Fsone, Jos̩ Carlos Santos, Jyrki Lahtonen
1
Hello Bogdan, it would be nice if you tried to explain what you have tried until here!
– Robson
Sep 6 at 4:49
1
No. Choose $f=g$ in $L^pL^2p$.
– Kusma
Sep 6 at 5:24
Please show some of the things you have tried or provide your thoughts about the question. This will help others to give answers appropriate to your experience and skill level. As it stands, your answer is likely to be considered lacking in effort by the community and may be downvoted and closed. For more information, see the help center. Also see - how to ask a good question.
– Devashish Kaushik
Sep 6 at 5:55
add a comment |Â
up vote
-3
down vote
favorite
up vote
-3
down vote
favorite
Is it true that if $f,gin L^p (mathbbR^n)$, then $fcdot gin L^p (mathbbR^n)$ ?
measure-theory lebesgue-measure
edited Sep 6 at 6:35
Devashish Kaushik
29314
asked Sep 6 at 4:43
Bogdan
63649
Is it true that if $f,gin L^p (mathbbR^n)$, then $fcdot gin L^p (mathbbR^n)$ ?
measure-theory lebesgue-measure
measure-theory lebesgue-measure
edited Sep 6 at 6:35
Devashish Kaushik
29314
asked Sep 6 at 4:43
Bogdan
63649
edited Sep 6 at 6:35
Devashish Kaushik
29314
asked Sep 6 at 4:43
Bogdan
63649
edited Sep 6 at 6:35
Devashish Kaushik
29314
edited Sep 6 at 6:35
Devashish Kaushik
29314
edited Sep 6 at 6:35
Devashish Kaushik
29314
29314
asked Sep 6 at 4:43
Bogdan
63649
asked Sep 6 at 4:43
Bogdan
63649
asked Sep 6 at 4:43
Bogdan
63649
63649
closed as off-topic by Xander Henderson, user99914, Guy Fsone, José Carlos Santos, Jyrki Lahtonen Sep 6 at 7:58
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." РXander Henderson, Community, Guy Fsone, Jos̩ Carlos Santos, Jyrki Lahtonen
closed as off-topic by Xander Henderson, user99914, Guy Fsone, José Carlos Santos, Jyrki Lahtonen Sep 6 at 7:58
This question appears to be off-topic. The users who voted to close gave this specific reason:
- "This question is missing context or other details: Please improve the question by providing additional context, which ideally includes your thoughts on the problem and any attempts you have made to solve it. This information helps others identify where you have difficulties and helps them write answers appropriate to your experience level." РXander Henderson, Community, Guy Fsone, Jos̩ Carlos Santos, Jyrki Lahtonen
1
Hello Bogdan, it would be nice if you tried to explain what you have tried until here!
– Robson
Sep 6 at 4:49
1
No. Choose $f=g$ in $L^pL^2p$.
– Kusma
Sep 6 at 5:24
Please show some of the things you have tried or provide your thoughts about the question. This will help others to give answers appropriate to your experience and skill level. As it stands, your answer is likely to be considered lacking in effort by the community and may be downvoted and closed. For more information, see the help center. Also see - how to ask a good question.
– Devashish Kaushik
Sep 6 at 5:55
add a comment |Â
1
Hello Bogdan, it would be nice if you tried to explain what you have tried until here!
– Robson
Sep 6 at 4:49
1
No. Choose $f=g$ in $L^pL^2p$.
– Kusma
Sep 6 at 5:24
Please show some of the things you have tried or provide your thoughts about the question. This will help others to give answers appropriate to your experience and skill level. As it stands, your answer is likely to be considered lacking in effort by the community and may be downvoted and closed. For more information, see the help center. Also see - how to ask a good question.
– Devashish Kaushik
Sep 6 at 5:55
1
1
Hello Bogdan, it would be nice if you tried to explain what you have tried until here!
– Robson
Sep 6 at 4:49
Hello Bogdan, it would be nice if you tried to explain what you have tried until here!
– Robson
Sep 6 at 4:49
1
1
No. Choose $f=g$ in $L^pL^2p$.
– Kusma
Sep 6 at 5:24
No. Choose $f=g$ in $L^pL^2p$.
– Kusma
Sep 6 at 5:24
Please show some of the things you have tried or provide your thoughts about the question. This will help others to give answers appropriate to your experience and skill level. As it stands, your answer is likely to be considered lacking in effort by the community and may be downvoted and closed. For more information, see the help center. Also see - how to ask a good question.
– Devashish Kaushik
Sep 6 at 5:55
Please show some of the things you have tried or provide your thoughts about the question. This will help others to give answers appropriate to your experience and skill level. As it stands, your answer is likely to be considered lacking in effort by the community and may be downvoted and closed. For more information, see the help center. Also see - how to ask a good question.
– Devashish Kaushik
Sep 6 at 5:55
add a comment |Â
1 Answer
1
active
oldest
votes
up vote
2
down vote
accepted
If $-frac 1 p < alpha <-frac 1 2p$ and $f(x)= x^alpha$ for $0<x<1$ and $0$ for all other $x$ then $f in L^p$ but $f notin L^2p$
answered Sep 6 at 5:33
Kavi Rama Murthy
26.5k31438
aren't we talking about $L^p(mathbbR^n)$?
– Robson
Sep 6 at 5:46
1
To disprove a statement it is enough to give one counter-example. I have given a counter-example with $n=1$. Also, it is trivial to modify this for any $n$.
– Kavi Rama Murthy
Sep 6 at 5:48
You're right!..
– Robson
Sep 6 at 5:51
how would we modify $x^alpha$ for any $n$?
– Robson
Sep 6 at 5:53
Consider $x_1^alpha g(x_2,x_3,...,x_n)$ for and $g$ which has integrals of p-th and 2p-th powers finite and positive.
– Kavi Rama Murthy
Sep 6 at 5:55
add a comment |Â
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
2
down vote
accepted
If $-frac 1 p < alpha <-frac 1 2p$ and $f(x)= x^alpha$ for $0<x<1$ and $0$ for all other $x$ then $f in L^p$ but $f notin L^2p$
answered Sep 6 at 5:33
Kavi Rama Murthy
26.5k31438
aren't we talking about $L^p(mathbbR^n)$?
– Robson
Sep 6 at 5:46
1
To disprove a statement it is enough to give one counter-example. I have given a counter-example with $n=1$. Also, it is trivial to modify this for any $n$.
– Kavi Rama Murthy
Sep 6 at 5:48
You're right!..
– Robson
Sep 6 at 5:51
how would we modify $x^alpha$ for any $n$?
– Robson
Sep 6 at 5:53
Consider $x_1^alpha g(x_2,x_3,...,x_n)$ for and $g$ which has integrals of p-th and 2p-th powers finite and positive.
– Kavi Rama Murthy
Sep 6 at 5:55
add a comment |Â
up vote
2
down vote
accepted
If $-frac 1 p < alpha <-frac 1 2p$ and $f(x)= x^alpha$ for $0<x<1$ and $0$ for all other $x$ then $f in L^p$ but $f notin L^2p$
answered Sep 6 at 5:33
Kavi Rama Murthy
26.5k31438
aren't we talking about $L^p(mathbbR^n)$?
– Robson
Sep 6 at 5:46
1
To disprove a statement it is enough to give one counter-example. I have given a counter-example with $n=1$. Also, it is trivial to modify this for any $n$.
– Kavi Rama Murthy
Sep 6 at 5:48
You're right!..
– Robson
Sep 6 at 5:51
how would we modify $x^alpha$ for any $n$?
– Robson
Sep 6 at 5:53
Consider $x_1^alpha g(x_2,x_3,...,x_n)$ for and $g$ which has integrals of p-th and 2p-th powers finite and positive.
– Kavi Rama Murthy
Sep 6 at 5:55
add a comment |Â
up vote
2
down vote
accepted
up vote
2
down vote
accepted
If $-frac 1 p < alpha <-frac 1 2p$ and $f(x)= x^alpha$ for $0<x<1$ and $0$ for all other $x$ then $f in L^p$ but $f notin L^2p$
answered Sep 6 at 5:33
Kavi Rama Murthy
26.5k31438
If $-frac 1 p < alpha <-frac 1 2p$ and $f(x)= x^alpha$ for $0<x<1$ and $0$ for all other $x$ then $f in L^p$ but $f notin L^2p$
answered Sep 6 at 5:33
Kavi Rama Murthy
26.5k31438
answered Sep 6 at 5:33
Kavi Rama Murthy
26.5k31438
answered Sep 6 at 5:33
Kavi Rama Murthy
26.5k31438
answered Sep 6 at 5:33
Kavi Rama Murthy
26.5k31438
26.5k31438
aren't we talking about $L^p(mathbbR^n)$?
– Robson
Sep 6 at 5:46
1
To disprove a statement it is enough to give one counter-example. I have given a counter-example with $n=1$. Also, it is trivial to modify this for any $n$.
– Kavi Rama Murthy
Sep 6 at 5:48
You're right!..
– Robson
Sep 6 at 5:51
how would we modify $x^alpha$ for any $n$?
– Robson
Sep 6 at 5:53
Consider $x_1^alpha g(x_2,x_3,...,x_n)$ for and $g$ which has integrals of p-th and 2p-th powers finite and positive.
– Kavi Rama Murthy
Sep 6 at 5:55
add a comment |Â
aren't we talking about $L^p(mathbbR^n)$?
– Robson
Sep 6 at 5:46
1
To disprove a statement it is enough to give one counter-example. I have given a counter-example with $n=1$. Also, it is trivial to modify this for any $n$.
– Kavi Rama Murthy
Sep 6 at 5:48
You're right!..
– Robson
Sep 6 at 5:51
how would we modify $x^alpha$ for any $n$?
– Robson
Sep 6 at 5:53
Consider $x_1^alpha g(x_2,x_3,...,x_n)$ for and $g$ which has integrals of p-th and 2p-th powers finite and positive.
– Kavi Rama Murthy
Sep 6 at 5:55
aren't we talking about $L^p(mathbbR^n)$?
– Robson
Sep 6 at 5:46
aren't we talking about $L^p(mathbbR^n)$?
– Robson
Sep 6 at 5:46
1
1
To disprove a statement it is enough to give one counter-example. I have given a counter-example with $n=1$. Also, it is trivial to modify this for any $n$.
– Kavi Rama Murthy
Sep 6 at 5:48
To disprove a statement it is enough to give one counter-example. I have given a counter-example with $n=1$. Also, it is trivial to modify this for any $n$.
– Kavi Rama Murthy
Sep 6 at 5:48
You're right!..
– Robson
Sep 6 at 5:51
You're right!..
– Robson
Sep 6 at 5:51
how would we modify $x^alpha$ for any $n$?
– Robson
Sep 6 at 5:53
how would we modify $x^alpha$ for any $n$?
– Robson
Sep 6 at 5:53
Consider $x_1^alpha g(x_2,x_3,...,x_n)$ for and $g$ which has integrals of p-th and 2p-th powers finite and positive.
– Kavi Rama Murthy
Sep 6 at 5:55
Consider $x_1^alpha g(x_2,x_3,...,x_n)$ for and $g$ which has integrals of p-th and 2p-th powers finite and positive.
– Kavi Rama Murthy
Sep 6 at 5:55
add a comment |Â
1
Hello Bogdan, it would be nice if you tried to explain what you have tried until here!
– Robson
Sep 6 at 4:49
1
No. Choose $f=g$ in $L^pL^2p$.
– Kusma
Sep 6 at 5:24
Please show some of the things you have tried or provide your thoughts about the question. This will help others to give answers appropriate to your experience and skill level. As it stands, your answer is likely to be considered lacking in effort by the community and may be downvoted and closed. For more information, see the help center. Also see - how to ask a good question.
– Devashish Kaushik
Sep 6 at 5:55