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Let $2 < p < infty$ and $f, g in L^p(Omega, mathcalF, nu)$ be fixed elements of norm $1$, where $(Omega, mathcalF, nu)$ is any measure space. Consider the function $F$ defined on $mathbbR$ by
$$F(t) = int_Omega|f+t,g|^p dnu.$$
Is $F$ twice continuously differentiable on $mathbbR$, i.e. is $F$ of class $C^2$? Theorem 2.6 in Analysis by Lieb and Loss asserts that $F$ is differentiable at $t=0$, but I wanted to know if this holds for all $t$ and if any higher derivatives exist and are continuous. I know that $x mapsto |x|^p$ is $n = leftlfloorprightrfloor - 1$ times differentiable, but does not have a continuous $n$th derivative, so my guess is that the same holds here; $F$ is also $n$-times differentiable or $n-1$-times continuously differentiable. Do any additional assumptions on the measure space allow us to infer any more differentiability class properties of $F$?

measure-theory derivatives

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asked Mar 13 at 19:17

talfredtalfred

1007

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You need to make sure that integral that represents $F'''$ etc. remains defined.
  $endgroup$
  – Umberto P.
  Mar 13 at 19:31
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @UmbertoP. is that sufficient?
  $endgroup$
  – talfred
  Mar 13 at 20:00
  

  
  
  
  

add a comment | 

1

$begingroup$

Let $2 < p < infty$ and $f, g in L^p(Omega, mathcalF, nu)$ be fixed elements of norm $1$, where $(Omega, mathcalF, nu)$ is any measure space. Consider the function $F$ defined on $mathbbR$ by
$$F(t) = int_Omega|f+t,g|^p dnu.$$
Is $F$ twice continuously differentiable on $mathbbR$, i.e. is $F$ of class $C^2$? Theorem 2.6 in Analysis by Lieb and Loss asserts that $F$ is differentiable at $t=0$, but I wanted to know if this holds for all $t$ and if any higher derivatives exist and are continuous. I know that $x mapsto |x|^p$ is $n = leftlfloorprightrfloor - 1$ times differentiable, but does not have a continuous $n$th derivative, so my guess is that the same holds here; $F$ is also $n$-times differentiable or $n-1$-times continuously differentiable. Do any additional assumptions on the measure space allow us to infer any more differentiability class properties of $F$?

measure-theory derivatives

share|cite|improve this question

asked Mar 13 at 19:17

talfredtalfred

1007

$endgroup$

• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  You need to make sure that integral that represents $F'''$ etc. remains defined.
  $endgroup$
  – Umberto P.
  Mar 13 at 19:31
  

  
  
  
  


• 
  
  
  

  
  

  
  
  
  
  
  

  
  $begingroup$
  @UmbertoP. is that sufficient?
  $endgroup$
  – talfred
  Mar 13 at 20:00
  

  
  
  
  

add a comment | 

$begingroup$

Let $2 < p < infty$ and $f, g in L^p(Omega, mathcalF, nu)$ be fixed elements of norm $1$, where $(Omega, mathcalF, nu)$ is any measure space. Consider the function $F$ defined on $mathbbR$ by
$$F(t) = int_Omega|f+t,g|^p dnu.$$
Is $F$ twice continuously differentiable on $mathbbR$, i.e. is $F$ of class $C^2$? Theorem 2.6 in Analysis by Lieb and Loss asserts that $F$ is differentiable at $t=0$, but I wanted to know if this holds for all $t$ and if any higher derivatives exist and are continuous. I know that $x mapsto |x|^p$ is $n = leftlfloorprightrfloor - 1$ times differentiable, but does not have a continuous $n$th derivative, so my guess is that the same holds here; $F$ is also $n$-times differentiable or $n-1$-times continuously differentiable. Do any additional assumptions on the measure space allow us to infer any more differentiability class properties of $F$?

measure-theory derivatives

share|cite|improve this question

asked Mar 13 at 19:17

talfredtalfred

1007

$endgroup$

share|cite|improve this question

asked Mar 13 at 19:17

talfredtalfred

1007

share|cite|improve this question

asked Mar 13 at 19:17

talfredtalfred

1007

asked Mar 13 at 19:17

talfredtalfred

1007

asked Mar 13 at 19:17

talfredtalfred

1007