Brezis excercise. Does $hin L_p(Omega)$?
$begingroup$
Given two functions $f,gin L_p(Omega)$ define $h(x)=max{f(x),g(x)}$ prove that $hin L_p(Omega)$.
I know that $|h(x)|^pgeq |f(x)|^p$ and $|h(x)|^pgeq |g(x)|^p$ but this seems to go nowhere... any hints?
functional-analysis lp-spaces
asked Jan 7 at 21:54
AlfdavAlfdav
1027
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add a comment |
$begingroup$
Given two functions $f,gin L_p(Omega)$ define $h(x)=max{f(x),g(x)}$ prove that $hin L_p(Omega)$.
I know that $|h(x)|^pgeq |f(x)|^p$ and $|h(x)|^pgeq |g(x)|^p$ but this seems to go nowhere... any hints?
functional-analysis lp-spaces
asked Jan 7 at 21:54
AlfdavAlfdav
1027
$endgroup$
add a comment |
$begingroup$
Given two functions $f,gin L_p(Omega)$ define $h(x)=max{f(x),g(x)}$ prove that $hin L_p(Omega)$.
I know that $|h(x)|^pgeq |f(x)|^p$ and $|h(x)|^pgeq |g(x)|^p$ but this seems to go nowhere... any hints?
functional-analysis lp-spaces
asked Jan 7 at 21:54
AlfdavAlfdav
1027
$endgroup$
Given two functions $f,gin L_p(Omega)$ define $h(x)=max{f(x),g(x)}$ prove that $hin L_p(Omega)$.
I know that $|h(x)|^pgeq |f(x)|^p$ and $|h(x)|^pgeq |g(x)|^p$ but this seems to go nowhere... any hints?
functional-analysis lp-spaces
functional-analysis lp-spaces
asked Jan 7 at 21:54
AlfdavAlfdav
1027
asked Jan 7 at 21:54
AlfdavAlfdav
1027
asked Jan 7 at 21:54
AlfdavAlfdav
1027
asked Jan 7 at 21:54
AlfdavAlfdav
1027
asked Jan 7 at 21:54
AlfdavAlfdav
1027
1027
add a comment |
add a comment |
1 Answer
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$begingroup$
Hint: for all $xinOmega$, the following inequalities hold:
$$
leftlvert h(x)rightrvert^pleqslant maxleft{leftlvert f(x)rightrvert^p,leftlvert g(x)rightrvert^pright}leqslant leftlvert f(x)rightrvert^p+leftlvert g(x)rightrvert^p.
$$
answered Jan 7 at 22:20
Davide GiraudoDavide Giraudo
128k17154268
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1 Answer
1
active
oldest
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1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
Hint: for all $xinOmega$, the following inequalities hold:
$$
leftlvert h(x)rightrvert^pleqslant maxleft{leftlvert f(x)rightrvert^p,leftlvert g(x)rightrvert^pright}leqslant leftlvert f(x)rightrvert^p+leftlvert g(x)rightrvert^p.
$$
answered Jan 7 at 22:20
Davide GiraudoDavide Giraudo
128k17154268
$endgroup$
add a comment |
$begingroup$
Hint: for all $xinOmega$, the following inequalities hold:
$$
leftlvert h(x)rightrvert^pleqslant maxleft{leftlvert f(x)rightrvert^p,leftlvert g(x)rightrvert^pright}leqslant leftlvert f(x)rightrvert^p+leftlvert g(x)rightrvert^p.
$$
answered Jan 7 at 22:20
Davide GiraudoDavide Giraudo
128k17154268
$endgroup$
add a comment |
$begingroup$
Hint: for all $xinOmega$, the following inequalities hold:
$$
leftlvert h(x)rightrvert^pleqslant maxleft{leftlvert f(x)rightrvert^p,leftlvert g(x)rightrvert^pright}leqslant leftlvert f(x)rightrvert^p+leftlvert g(x)rightrvert^p.
$$
answered Jan 7 at 22:20
Davide GiraudoDavide Giraudo
128k17154268
$endgroup$
Hint: for all $xinOmega$, the following inequalities hold:
$$
leftlvert h(x)rightrvert^pleqslant maxleft{leftlvert f(x)rightrvert^p,leftlvert g(x)rightrvert^pright}leqslant leftlvert f(x)rightrvert^p+leftlvert g(x)rightrvert^p.
$$
answered Jan 7 at 22:20
Davide GiraudoDavide Giraudo
128k17154268
answered Jan 7 at 22:20
Davide GiraudoDavide Giraudo
128k17154268
answered Jan 7 at 22:20
Davide GiraudoDavide Giraudo
128k17154268
answered Jan 7 at 22:20
Davide GiraudoDavide Giraudo
128k17154268
128k17154268
add a comment |
add a comment |
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