Mixing
Convergence in $L^{3/2}$ and in $L^2$
$begingroup$
Let $(f_n)_n$ be a bounded sequence in $L^3(mathbb R)$, such that $f_nrightarrow f$ in $L^{3/2}(mathbb R)$. Prove that $f_nrightarrow f$ converges in $L^2mathbb (R)$.
I have and idea to first use the H$ddot {o}$lder inequality:
$int_{Omega} |f_n-f|^2 dx leq ||f_n-f||_3||f_n-f||_{3/2}$
Since $3$ and ${3/2}$ are conjugates. The latter term is convergent, but $||f_n-f||_3$ had to be worked out:
$||f_n-f||_3 leq ||f_n||_3+ ||f||_3 $
How do I obtain the bound for $f$ in $L^3 (mathbb R)$ ?
I have the following idea:
$int_Omega |f|^3dx=int_Omega lim_{n to infty} |f_n|^3 dx$
But how do I get the limit outside the integral or how can I use Fatou lemma here?
convergence lp-spaces
asked Jan 27 at 9:06
JohnnyJohnny
406
$endgroup$
add a comment |
$begingroup$
Let $(f_n)_n$ be a bounded sequence in $L^3(mathbb R)$, such that $f_nrightarrow f$ in $L^{3/2}(mathbb R)$. Prove that $f_nrightarrow f$ converges in $L^2mathbb (R)$.
I have and idea to first use the H$ddot {o}$lder inequality:
$int_{Omega} |f_n-f|^2 dx leq ||f_n-f||_3||f_n-f||_{3/2}$
Since $3$ and ${3/2}$ are conjugates. The latter term is convergent, but $||f_n-f||_3$ had to be worked out:
$||f_n-f||_3 leq ||f_n||_3+ ||f||_3 $
How do I obtain the bound for $f$ in $L^3 (mathbb R)$ ?
I have the following idea:
$int_Omega |f|^3dx=int_Omega lim_{n to infty} |f_n|^3 dx$
But how do I get the limit outside the integral or how can I use Fatou lemma here?
convergence lp-spaces
asked Jan 27 at 9:06
JohnnyJohnny
406
$endgroup$
add a comment |
$begingroup$
Let $(f_n)_n$ be a bounded sequence in $L^3(mathbb R)$, such that $f_nrightarrow f$ in $L^{3/2}(mathbb R)$. Prove that $f_nrightarrow f$ converges in $L^2mathbb (R)$.
I have and idea to first use the H$ddot {o}$lder inequality:
$int_{Omega} |f_n-f|^2 dx leq ||f_n-f||_3||f_n-f||_{3/2}$
Since $3$ and ${3/2}$ are conjugates. The latter term is convergent, but $||f_n-f||_3$ had to be worked out:
$||f_n-f||_3 leq ||f_n||_3+ ||f||_3 $
How do I obtain the bound for $f$ in $L^3 (mathbb R)$ ?
I have the following idea:
$int_Omega |f|^3dx=int_Omega lim_{n to infty} |f_n|^3 dx$
But how do I get the limit outside the integral or how can I use Fatou lemma here?
convergence lp-spaces
asked Jan 27 at 9:06
JohnnyJohnny
406
$endgroup$
Let $(f_n)_n$ be a bounded sequence in $L^3(mathbb R)$, such that $f_nrightarrow f$ in $L^{3/2}(mathbb R)$. Prove that $f_nrightarrow f$ converges in $L^2mathbb (R)$.
I have and idea to first use the H$ddot {o}$lder inequality:
$int_{Omega} |f_n-f|^2 dx leq ||f_n-f||_3||f_n-f||_{3/2}$
Since $3$ and ${3/2}$ are conjugates. The latter term is convergent, but $||f_n-f||_3$ had to be worked out:
$||f_n-f||_3 leq ||f_n||_3+ ||f||_3 $
How do I obtain the bound for $f$ in $L^3 (mathbb R)$ ?
I have the following idea:
$int_Omega |f|^3dx=int_Omega lim_{n to infty} |f_n|^3 dx$
But how do I get the limit outside the integral or how can I use Fatou lemma here?
convergence lp-spaces
convergence lp-spaces
asked Jan 27 at 9:06
JohnnyJohnny
406
asked Jan 27 at 9:06
JohnnyJohnny
406
asked Jan 27 at 9:06
JohnnyJohnny
406
asked Jan 27 at 9:06
JohnnyJohnny
406
asked Jan 27 at 9:06
JohnnyJohnny
406
406
add a comment |
add a comment |
1 Answer
1
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votes
$begingroup$
EDIT:I see now you had already pointed out the correct way to end. Sorry I had not read well your post(feel free to mark down my answer so to get a real one).
You can say this to get a bound also for $f$:there is a subsequence $f_{n_k}$ which converges pointwise almost everywhere to $f$. Now, using Fatou lemma you could conclude.
answered Jan 27 at 9:19
Tommaso ScognamiglioTommaso Scognamiglio
527412
$endgroup$
add a comment |
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1 Answer
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1 Answer
1
active
oldest
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oldest
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votes
$begingroup$
EDIT:I see now you had already pointed out the correct way to end. Sorry I had not read well your post(feel free to mark down my answer so to get a real one).
You can say this to get a bound also for $f$:there is a subsequence $f_{n_k}$ which converges pointwise almost everywhere to $f$. Now, using Fatou lemma you could conclude.
answered Jan 27 at 9:19
Tommaso ScognamiglioTommaso Scognamiglio
527412
$endgroup$
add a comment |
$begingroup$
EDIT:I see now you had already pointed out the correct way to end. Sorry I had not read well your post(feel free to mark down my answer so to get a real one).
You can say this to get a bound also for $f$:there is a subsequence $f_{n_k}$ which converges pointwise almost everywhere to $f$. Now, using Fatou lemma you could conclude.
answered Jan 27 at 9:19
Tommaso ScognamiglioTommaso Scognamiglio
527412
$endgroup$
add a comment |
$begingroup$
EDIT:I see now you had already pointed out the correct way to end. Sorry I had not read well your post(feel free to mark down my answer so to get a real one).
You can say this to get a bound also for $f$:there is a subsequence $f_{n_k}$ which converges pointwise almost everywhere to $f$. Now, using Fatou lemma you could conclude.
answered Jan 27 at 9:19
Tommaso ScognamiglioTommaso Scognamiglio
527412
$endgroup$
EDIT:I see now you had already pointed out the correct way to end. Sorry I had not read well your post(feel free to mark down my answer so to get a real one).
You can say this to get a bound also for $f$:there is a subsequence $f_{n_k}$ which converges pointwise almost everywhere to $f$. Now, using Fatou lemma you could conclude.
answered Jan 27 at 9:19
Tommaso ScognamiglioTommaso Scognamiglio
527412
edited Jan 27 at 9:27
answered Jan 27 at 9:19
Tommaso ScognamiglioTommaso Scognamiglio
527412
answered Jan 27 at 9:19
Tommaso ScognamiglioTommaso Scognamiglio
527412
answered Jan 27 at 9:19
Tommaso ScognamiglioTommaso Scognamiglio
527412
527412
add a comment |
add a comment |
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