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Definition of weak $L^p$ spaces
Clash Royale CLAN TAG#URR8PPP
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Today I came across the notation $L_textweak$ and I don't know what exactly it means. I can only find the weak Lp spaces
, but here they don't use the same notation. Is this the same thing?
I'm also a little bit confused by the norm given in the article: $$|f|_p,w:=sup_t>0tleft(muleft>trightright)^frac1p,$$
it doesn't seem to fit in the context of the computations in which I need it. Are the any common estimates (except for the bound by the $L^p$ norm) I don't see yet?
Thank you for any answer.
functional-analysis lp-spaces weak-lp-spaces
edited Jul 19 at 8:20
Davide Giraudo
121k15146249
asked Jul 18 at 22:59
jason paper
668
add a comment |Â
up vote
2
down vote
favorite
Today I came across the notation $L_textweak$ and I don't know what exactly it means. I can only find the weak Lp spaces
, but here they don't use the same notation. Is this the same thing?
I'm also a little bit confused by the norm given in the article: $$|f|_p,w:=sup_t>0tleft(muleft>trightright)^frac1p,$$
it doesn't seem to fit in the context of the computations in which I need it. Are the any common estimates (except for the bound by the $L^p$ norm) I don't see yet?
Thank you for any answer.
functional-analysis lp-spaces weak-lp-spaces
edited Jul 19 at 8:20
Davide Giraudo
121k15146249
asked Jul 18 at 22:59
jason paper
668
Can you give more background on $L_weak$ ? Where did you come across it
– Calvin Khor
Jul 19 at 8:03
In the paper read it is not defined, it is just used sometimes to show convergence of sequences, for example it says: From $rho _n righarrow rho$ in $C(I,L^s_weak(Omega))$ and $u_n rightharpoonup u$ in $L^2(I,W_0^1,2(Omega))$ it follows that $rho _n u_n rightarrow$ weakly star in $L^infty(0,T;L^frac2ss+1(Omega))$.
– jason paper
Jul 19 at 20:36
Sorry, I was too slow to edit, it should be: $rho _n rightarrow rho$ in $C(I,L^s_weak(Omega))$
– jason paper
Jul 19 at 20:43
1
I don't know how to prove your result but it seems like the correct context. The weak Lebesgue space is written $L^p_w = L^p,w=L^p,textweak$ etc, and also $L^p,infty$ because it coincides with the Lorenz space with those parameters
– Calvin Khor
Jul 20 at 7:29
Thanks for your answer. I couldn't find it out yet but then I will try further
– jason paper
Jul 20 at 22:59
add a comment |Â
up vote
2
down vote
favorite
up vote
2
down vote
favorite
Today I came across the notation $L_textweak$ and I don't know what exactly it means. I can only find the weak Lp spaces
, but here they don't use the same notation. Is this the same thing?
I'm also a little bit confused by the norm given in the article: $$|f|_p,w:=sup_t>0tleft(muleft>trightright)^frac1p,$$
it doesn't seem to fit in the context of the computations in which I need it. Are the any common estimates (except for the bound by the $L^p$ norm) I don't see yet?
Thank you for any answer.
functional-analysis lp-spaces weak-lp-spaces
edited Jul 19 at 8:20
Davide Giraudo
121k15146249
asked Jul 18 at 22:59
jason paper
668
Today I came across the notation $L_textweak$ and I don't know what exactly it means. I can only find the weak Lp spaces
, but here they don't use the same notation. Is this the same thing?
I'm also a little bit confused by the norm given in the article: $$|f|_p,w:=sup_t>0tleft(muleft>trightright)^frac1p,$$
it doesn't seem to fit in the context of the computations in which I need it. Are the any common estimates (except for the bound by the $L^p$ norm) I don't see yet?
Thank you for any answer.
functional-analysis lp-spaces weak-lp-spaces
edited Jul 19 at 8:20
Davide Giraudo
121k15146249
asked Jul 18 at 22:59
jason paper
668
edited Jul 19 at 8:20
Davide Giraudo
121k15146249
edited Jul 19 at 8:20
Davide Giraudo
121k15146249
edited Jul 19 at 8:20
Davide Giraudo
121k15146249
121k15146249
asked Jul 18 at 22:59
jason paper
668
asked Jul 18 at 22:59
jason paper
668
asked Jul 18 at 22:59
jason paper
668
668
Can you give more background on $L_weak$ ? Where did you come across it
– Calvin Khor
Jul 19 at 8:03
In the paper read it is not defined, it is just used sometimes to show convergence of sequences, for example it says: From $rho _n righarrow rho$ in $C(I,L^s_weak(Omega))$ and $u_n rightharpoonup u$ in $L^2(I,W_0^1,2(Omega))$ it follows that $rho _n u_n rightarrow$ weakly star in $L^infty(0,T;L^frac2ss+1(Omega))$.
– jason paper
Jul 19 at 20:36
Sorry, I was too slow to edit, it should be: $rho _n rightarrow rho$ in $C(I,L^s_weak(Omega))$
– jason paper
Jul 19 at 20:43
1
I don't know how to prove your result but it seems like the correct context. The weak Lebesgue space is written $L^p_w = L^p,w=L^p,textweak$ etc, and also $L^p,infty$ because it coincides with the Lorenz space with those parameters
– Calvin Khor
Jul 20 at 7:29
Thanks for your answer. I couldn't find it out yet but then I will try further
– jason paper
Jul 20 at 22:59
add a comment |Â
Can you give more background on $L_weak$ ? Where did you come across it
– Calvin Khor
Jul 19 at 8:03
In the paper read it is not defined, it is just used sometimes to show convergence of sequences, for example it says: From $rho _n righarrow rho$ in $C(I,L^s_weak(Omega))$ and $u_n rightharpoonup u$ in $L^2(I,W_0^1,2(Omega))$ it follows that $rho _n u_n rightarrow$ weakly star in $L^infty(0,T;L^frac2ss+1(Omega))$.
– jason paper
Jul 19 at 20:36
Sorry, I was too slow to edit, it should be: $rho _n rightarrow rho$ in $C(I,L^s_weak(Omega))$
– jason paper
Jul 19 at 20:43
1
I don't know how to prove your result but it seems like the correct context. The weak Lebesgue space is written $L^p_w = L^p,w=L^p,textweak$ etc, and also $L^p,infty$ because it coincides with the Lorenz space with those parameters
– Calvin Khor
Jul 20 at 7:29
Thanks for your answer. I couldn't find it out yet but then I will try further
– jason paper
Jul 20 at 22:59
Can you give more background on $L_weak$ ? Where did you come across it
– Calvin Khor
Jul 19 at 8:03
Can you give more background on $L_weak$ ? Where did you come across it
– Calvin Khor
Jul 19 at 8:03
In the paper read it is not defined, it is just used sometimes to show convergence of sequences, for example it says: From $rho _n righarrow rho$ in $C(I,L^s_weak(Omega))$ and $u_n rightharpoonup u$ in $L^2(I,W_0^1,2(Omega))$ it follows that $rho _n u_n rightarrow$ weakly star in $L^infty(0,T;L^frac2ss+1(Omega))$.
– jason paper
Jul 19 at 20:36
In the paper read it is not defined, it is just used sometimes to show convergence of sequences, for example it says: From $rho _n righarrow rho$ in $C(I,L^s_weak(Omega))$ and $u_n rightharpoonup u$ in $L^2(I,W_0^1,2(Omega))$ it follows that $rho _n u_n rightarrow$ weakly star in $L^infty(0,T;L^frac2ss+1(Omega))$.
– jason paper
Jul 19 at 20:36
Sorry, I was too slow to edit, it should be: $rho _n rightarrow rho$ in $C(I,L^s_weak(Omega))$
– jason paper
Jul 19 at 20:43
Sorry, I was too slow to edit, it should be: $rho _n rightarrow rho$ in $C(I,L^s_weak(Omega))$
– jason paper
Jul 19 at 20:43
1
1
I don't know how to prove your result but it seems like the correct context. The weak Lebesgue space is written $L^p_w = L^p,w=L^p,textweak$ etc, and also $L^p,infty$ because it coincides with the Lorenz space with those parameters
– Calvin Khor
Jul 20 at 7:29
I don't know how to prove your result but it seems like the correct context. The weak Lebesgue space is written $L^p_w = L^p,w=L^p,textweak$ etc, and also $L^p,infty$ because it coincides with the Lorenz space with those parameters
– Calvin Khor
Jul 20 at 7:29
Thanks for your answer. I couldn't find it out yet but then I will try further
– jason paper
Jul 20 at 22:59
Thanks for your answer. I couldn't find it out yet but then I will try further
– jason paper
Jul 20 at 22:59
add a comment |Â
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Can you give more background on $L_weak$ ? Where did you come across it
– Calvin Khor
Jul 19 at 8:03
In the paper read it is not defined, it is just used sometimes to show convergence of sequences, for example it says: From $rho _n righarrow rho$ in $C(I,L^s_weak(Omega))$ and $u_n rightharpoonup u$ in $L^2(I,W_0^1,2(Omega))$ it follows that $rho _n u_n rightarrow$ weakly star in $L^infty(0,T;L^frac2ss+1(Omega))$.
– jason paper
Jul 19 at 20:36
Sorry, I was too slow to edit, it should be: $rho _n rightarrow rho$ in $C(I,L^s_weak(Omega))$
– jason paper
Jul 19 at 20:43
1
I don't know how to prove your result but it seems like the correct context. The weak Lebesgue space is written $L^p_w = L^p,w=L^p,textweak$ etc, and also $L^p,infty$ because it coincides with the Lorenz space with those parameters
– Calvin Khor
Jul 20 at 7:29
Thanks for your answer. I couldn't find it out yet but then I will try further
– jason paper
Jul 20 at 22:59