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Functional analysis exercise: $left(int_Omega|h|rho dxright)^pleq int_Omega|h|^prho dx$

0

$begingroup$

I have a doubt about this exercise:

Let $Omega subset mathbbR^n$ be an open set, let $h colon Omega rightarrow mathbbR$ and $rho colon Omega rightarrow [0, +infty)$ be Lebesgue measurable functions and suppose that $int_Omegarho dx =1$. Prove that for each $p in [1,+infty)$

beginequation
left(int_Omega|h|rho dxright)^pleq int_Omega|h|^prho dx.
endequation

My idea is to notice that if $dnu=rho dx$ i have that $nu(Omega)=1$ hence $nu$ is a probability measure. Now i can use the Jensen inequality, that is $left(mathbbE[|h|]right)^pleq mathbbE[|h|^p]$, and consequently follows

beginequation
left(int_Omega|h|dnuright)^pleq int_Omega|h|^pdnu,
endequation

and recalling $dnu=rho dx$ i have the desired conclusion.

functional-analysis inequality

share|cite|improve this question

edited Mar 31 at 7:40

Martin Sleziak

45k10122277

asked Mar 31 at 6:10

GiovanniGiovanni

409

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  This is correct.
  $endgroup$
  – Kavi Rama Murthy
  Mar 31 at 6:13
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thankyou so much!
  $endgroup$
  – Giovanni
  Mar 31 at 6:14
  

  
  
  
  

add a comment | 

0

$begingroup$

I have a doubt about this exercise:

Let $Omega subset mathbbR^n$ be an open set, let $h colon Omega rightarrow mathbbR$ and $rho colon Omega rightarrow [0, +infty)$ be Lebesgue measurable functions and suppose that $int_Omegarho dx =1$. Prove that for each $p in [1,+infty)$

beginequation
left(int_Omega|h|rho dxright)^pleq int_Omega|h|^prho dx.
endequation

My idea is to notice that if $dnu=rho dx$ i have that $nu(Omega)=1$ hence $nu$ is a probability measure. Now i can use the Jensen inequality, that is $left(mathbbE[|h|]right)^pleq mathbbE[|h|^p]$, and consequently follows

beginequation
left(int_Omega|h|dnuright)^pleq int_Omega|h|^pdnu,
endequation

and recalling $dnu=rho dx$ i have the desired conclusion.

functional-analysis inequality

share|cite|improve this question

edited Mar 31 at 7:40

Martin Sleziak

45k10122277

asked Mar 31 at 6:10

GiovanniGiovanni

409

$endgroup$

• 
  
  
  

  
  1
  

  
  
  
  
  
  

  
  $begingroup$
  This is correct.
  $endgroup$
  – Kavi Rama Murthy
  Mar 31 at 6:13
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  Thankyou so much!
  $endgroup$
  – Giovanni
  Mar 31 at 6:14
  

  
  
  
  

add a comment | 

$begingroup$

I have a doubt about this exercise:

Let $Omega subset mathbbR^n$ be an open set, let $h colon Omega rightarrow mathbbR$ and $rho colon Omega rightarrow [0, +infty)$ be Lebesgue measurable functions and suppose that $int_Omegarho dx =1$. Prove that for each $p in [1,+infty)$

beginequation
left(int_Omega|h|rho dxright)^pleq int_Omega|h|^prho dx.
endequation

My idea is to notice that if $dnu=rho dx$ i have that $nu(Omega)=1$ hence $nu$ is a probability measure. Now i can use the Jensen inequality, that is $left(mathbbE[|h|]right)^pleq mathbbE[|h|^p]$, and consequently follows

beginequation
left(int_Omega|h|dnuright)^pleq int_Omega|h|^pdnu,
endequation

and recalling $dnu=rho dx$ i have the desired conclusion.

functional-analysis inequality

share|cite|improve this question

edited Mar 31 at 7:40

Martin Sleziak

45k10122277

asked Mar 31 at 6:10

GiovanniGiovanni

409

$endgroup$

share|cite|improve this question

edited Mar 31 at 7:40

Martin Sleziak

45k10122277

asked Mar 31 at 6:10

GiovanniGiovanni

409

share|cite|improve this question

edited Mar 31 at 7:40

Martin Sleziak

45k10122277

asked Mar 31 at 6:10

GiovanniGiovanni

409

edited Mar 31 at 7:40

Martin Sleziak

45k10122277

edited Mar 31 at 7:40

Martin Sleziak

45k10122277

asked Mar 31 at 6:10

GiovanniGiovanni

409

asked Mar 31 at 6:10

GiovanniGiovanni

409