Differentiability of $L^p$-valued functions

Clash Royale CLAN TAG#URR8PPP

up vote
0
down vote

favorite

1

Let

• $(Omega,mathcal A,mu)$ be a finite measure space


• $X$ be a metric $mathbb R$-vector space


• $hin X$


• $Lambdasubseteq X$ be open


• $Y$ be a $mathbb R$-Banach space


• $pge1$

If $f:OmegatimesLambdatomathbb R$ with

1. $f(omega,;cdot;)$ is Gâteaux differentiable in direction $h$ for $mu$-almost all $omegainOmega$


2. $f(;cdot;,x)$ is strongly $mathcal A$-measurable for all $xinLambda$


3. For all compact $KsubseteqLambda$, there is a $thetainmathcal L^p(mu)$ with $$sup_xin Kleft|rm D_hf(;cdot;,x)right|_Yletheta;;;mutext-almost everywhere,$$ where $rm D_hf(omega,x)$ denotes the Gâteaux derivative of $f(omega,;cdot;)$ at $xinLambda$ in direction $h$ for all $omegainOmega$

then $$left|fracf(;cdot;,x+th)-f(;cdot;,x)t-rm D_hf(;cdot;,x)right|_L^p(mu;:Y)xrightarrowtto00tag2$$ and hence (if $f(;cdot;,x)inmathcal L^p(mu;Y)$ for all $xinLambda$) $$Lambdato L^p(mu;Y);,;;;xmapsto f(;cdot;,x)tag3$$ is Gâteaux differentiable in direction $h$.

---

Now consider the opposite case and assume that $g:Lambdatomathcal L^p(mu;Y)$ is Gâteaux differentiable in direction $h$. What can we say about the Gâteaux differentiability of $g(;cdot;)(omega)$ for $omegainOmega$?

Clearly, if $x_0inLambda$ and $(t_n)_ninmathbb Nsubseteqmathbb Rsetminusleft0right$ with $$x_n:=x_0+t_nhinLambda;;;textfor all ninmathbb N$$ and $t_nxrightarrowntoinfty0$, then $$left|fracg(x_n)-g(x_0)t_n-rm D_hg(x_0)right|_L^p(mu;:Y)xrightarrowntoinfty0tag4$$ and hence there is an increasing $(n_k)_kinmathbb Nsubseteqmathbb N$ with $$left|fracg(x_n_k)-g(x_0)t_n_k-rm D_hg(x_0)right|_Yxrightarrowktoinfty0;;;mutext-almost everywheretag5.$$

Can we say anything more?

(Note that there is a slight issue above: $rm D_hg(x_0)$ is only uniquely determined up to a $mu$-null set.)

real-analysis measure-theory gateaux-derivative

share|cite|improve this question

asked Aug 5 at 22:08

0xbadf00d

2,04141028

add a comment | 

up vote
0
down vote

favorite

1

Let

• $(Omega,mathcal A,mu)$ be a finite measure space


• $X$ be a metric $mathbb R$-vector space


• $hin X$


• $Lambdasubseteq X$ be open


• $Y$ be a $mathbb R$-Banach space


• $pge1$

If $f:OmegatimesLambdatomathbb R$ with

1. $f(omega,;cdot;)$ is Gâteaux differentiable in direction $h$ for $mu$-almost all $omegainOmega$


2. $f(;cdot;,x)$ is strongly $mathcal A$-measurable for all $xinLambda$


3. For all compact $KsubseteqLambda$, there is a $thetainmathcal L^p(mu)$ with $$sup_xin Kleft|rm D_hf(;cdot;,x)right|_Yletheta;;;mutext-almost everywhere,$$ where $rm D_hf(omega,x)$ denotes the Gâteaux derivative of $f(omega,;cdot;)$ at $xinLambda$ in direction $h$ for all $omegainOmega$

then $$left|fracf(;cdot;,x+th)-f(;cdot;,x)t-rm D_hf(;cdot;,x)right|_L^p(mu;:Y)xrightarrowtto00tag2$$ and hence (if $f(;cdot;,x)inmathcal L^p(mu;Y)$ for all $xinLambda$) $$Lambdato L^p(mu;Y);,;;;xmapsto f(;cdot;,x)tag3$$ is Gâteaux differentiable in direction $h$.

---

Now consider the opposite case and assume that $g:Lambdatomathcal L^p(mu;Y)$ is Gâteaux differentiable in direction $h$. What can we say about the Gâteaux differentiability of $g(;cdot;)(omega)$ for $omegainOmega$?

Clearly, if $x_0inLambda$ and $(t_n)_ninmathbb Nsubseteqmathbb Rsetminusleft0right$ with $$x_n:=x_0+t_nhinLambda;;;textfor all ninmathbb N$$ and $t_nxrightarrowntoinfty0$, then $$left|fracg(x_n)-g(x_0)t_n-rm D_hg(x_0)right|_L^p(mu;:Y)xrightarrowntoinfty0tag4$$ and hence there is an increasing $(n_k)_kinmathbb Nsubseteqmathbb N$ with $$left|fracg(x_n_k)-g(x_0)t_n_k-rm D_hg(x_0)right|_Yxrightarrowktoinfty0;;;mutext-almost everywheretag5.$$

Can we say anything more?

(Note that there is a slight issue above: $rm D_hg(x_0)$ is only uniquely determined up to a $mu$-null set.)

real-analysis measure-theory gateaux-derivative

share|cite|improve this question

asked Aug 5 at 22:08

0xbadf00d

2,04141028

add a comment | 

up vote
0
down vote

favorite

1

up vote
0
down vote

favorite

1

1

Let

• $(Omega,mathcal A,mu)$ be a finite measure space


• $X$ be a metric $mathbb R$-vector space


• $hin X$


• $Lambdasubseteq X$ be open


• $Y$ be a $mathbb R$-Banach space


• $pge1$

If $f:OmegatimesLambdatomathbb R$ with

1. $f(omega,;cdot;)$ is Gâteaux differentiable in direction $h$ for $mu$-almost all $omegainOmega$


2. $f(;cdot;,x)$ is strongly $mathcal A$-measurable for all $xinLambda$


3. For all compact $KsubseteqLambda$, there is a $thetainmathcal L^p(mu)$ with $$sup_xin Kleft|rm D_hf(;cdot;,x)right|_Yletheta;;;mutext-almost everywhere,$$ where $rm D_hf(omega,x)$ denotes the Gâteaux derivative of $f(omega,;cdot;)$ at $xinLambda$ in direction $h$ for all $omegainOmega$

then $$left|fracf(;cdot;,x+th)-f(;cdot;,x)t-rm D_hf(;cdot;,x)right|_L^p(mu;:Y)xrightarrowtto00tag2$$ and hence (if $f(;cdot;,x)inmathcal L^p(mu;Y)$ for all $xinLambda$) $$Lambdato L^p(mu;Y);,;;;xmapsto f(;cdot;,x)tag3$$ is Gâteaux differentiable in direction $h$.

---

Now consider the opposite case and assume that $g:Lambdatomathcal L^p(mu;Y)$ is Gâteaux differentiable in direction $h$. What can we say about the Gâteaux differentiability of $g(;cdot;)(omega)$ for $omegainOmega$?

Clearly, if $x_0inLambda$ and $(t_n)_ninmathbb Nsubseteqmathbb Rsetminusleft0right$ with $$x_n:=x_0+t_nhinLambda;;;textfor all ninmathbb N$$ and $t_nxrightarrowntoinfty0$, then $$left|fracg(x_n)-g(x_0)t_n-rm D_hg(x_0)right|_L^p(mu;:Y)xrightarrowntoinfty0tag4$$ and hence there is an increasing $(n_k)_kinmathbb Nsubseteqmathbb N$ with $$left|fracg(x_n_k)-g(x_0)t_n_k-rm D_hg(x_0)right|_Yxrightarrowktoinfty0;;;mutext-almost everywheretag5.$$

Can we say anything more?

(Note that there is a slight issue above: $rm D_hg(x_0)$ is only uniquely determined up to a $mu$-null set.)

real-analysis measure-theory gateaux-derivative

share|cite|improve this question

asked Aug 5 at 22:08

0xbadf00d

2,04141028

Let

• $(Omega,mathcal A,mu)$ be a finite measure space


• $X$ be a metric $mathbb R$-vector space


• $hin X$


• $Lambdasubseteq X$ be open


• $Y$ be a $mathbb R$-Banach space


• $pge1$

If $f:OmegatimesLambdatomathbb R$ with

1. $f(omega,;cdot;)$ is Gâteaux differentiable in direction $h$ for $mu$-almost all $omegainOmega$


2. $f(;cdot;,x)$ is strongly $mathcal A$-measurable for all $xinLambda$


3. For all compact $KsubseteqLambda$, there is a $thetainmathcal L^p(mu)$ with $$sup_xin Kleft|rm D_hf(;cdot;,x)right|_Yletheta;;;mutext-almost everywhere,$$ where $rm D_hf(omega,x)$ denotes the Gâteaux derivative of $f(omega,;cdot;)$ at $xinLambda$ in direction $h$ for all $omegainOmega$

then $$left|fracf(;cdot;,x+th)-f(;cdot;,x)t-rm D_hf(;cdot;,x)right|_L^p(mu;:Y)xrightarrowtto00tag2$$ and hence (if $f(;cdot;,x)inmathcal L^p(mu;Y)$ for all $xinLambda$) $$Lambdato L^p(mu;Y);,;;;xmapsto f(;cdot;,x)tag3$$ is Gâteaux differentiable in direction $h$.

---

Now consider the opposite case and assume that $g:Lambdatomathcal L^p(mu;Y)$ is Gâteaux differentiable in direction $h$. What can we say about the Gâteaux differentiability of $g(;cdot;)(omega)$ for $omegainOmega$?

Clearly, if $x_0inLambda$ and $(t_n)_ninmathbb Nsubseteqmathbb Rsetminusleft0right$ with $$x_n:=x_0+t_nhinLambda;;;textfor all ninmathbb N$$ and $t_nxrightarrowntoinfty0$, then $$left|fracg(x_n)-g(x_0)t_n-rm D_hg(x_0)right|_L^p(mu;:Y)xrightarrowntoinfty0tag4$$ and hence there is an increasing $(n_k)_kinmathbb Nsubseteqmathbb N$ with $$left|fracg(x_n_k)-g(x_0)t_n_k-rm D_hg(x_0)right|_Yxrightarrowktoinfty0;;;mutext-almost everywheretag5.$$

Can we say anything more?

(Note that there is a slight issue above: $rm D_hg(x_0)$ is only uniquely determined up to a $mu$-null set.)

real-analysis measure-theory gateaux-derivative

share|cite|improve this question

asked Aug 5 at 22:08

0xbadf00d

2,04141028

share|cite|improve this question

share|cite|improve this question

asked Aug 5 at 22:08

0xbadf00d

2,04141028

asked Aug 5 at 22:08

0xbadf00d

2,04141028

asked Aug 5 at 22:08

0xbadf00d

2,04141028

2,04141028

add a comment | 

add a comment | 

active

oldest

votes

Your Answer

StackExchange.ifUsing("editor", function ()
return StackExchange.using("mathjaxEditing", function ()
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
);
);
, "mathjax-editing");

StackExchange.ready(function()
var channelOptions =
tags: "".split(" "),
id: "69"
;
initTagRenderer("".split(" "), "".split(" "), channelOptions);

StackExchange.using("externalEditor", function()
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled)
StackExchange.using("snippets", function()
createEditor();
);

else
createEditor();

);

function createEditor()
StackExchange.prepareEditor(
heartbeatType: 'answer',
convertImagesToLinks: true,
noModals: false,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
);



);

 

draft saved

draft discarded

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name Email

StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2873392%2fdifferentiability-of-lp-valued-functions%23new-answer', 'question_page');

);

Post as a guest

Name Email

active

oldest

votes

active

oldest

votes

active

oldest

votes

active

oldest

votes

 

draft saved

draft discarded

 

draft saved

draft discarded

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name Email

StackExchange.ready(
function ()
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f2873392%2fdifferentiability-of-lp-valued-functions%23new-answer', 'question_page');

);

Post as a guest

Name Email

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name Email

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name Email

Sign up or log in

StackExchange.ready(function ()
StackExchange.helpers.onClickDraftSave('#login-link');
);

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Sign up using Google

Sign up using Facebook

Sign up using Email and Password

Post as a guest

Name Email

Name Email

Name

Email

Name Email

Name

Email