Do non-continuous Sobolev maps pull back closed forms to weakly closed forms?












5












$begingroup$


$newcommand{R}{mathbb R}$
$newcommand{N}{mathbb N}$
$newcommand{de}{delta}$
$newcommand{sig}{sigma}$
$newcommand{Average}[1]{leftlangle#1rightrangle} $
$newcommand{IP}[2]{Average{#1,#2}}$



Let $n,d in mathbb{N}$, and let $Omega subseteq R^n$ be open.
Let $f in W^{1,k}(Omega,R^d)$. Let $omega in Omega^k(R^d)$ be a smooth closed $k$-form, such that $omega$ and its derivative $Tomega$ are both uniformly bounded globally; Is it true that $f^*omega$ is weakly closed? i.e. does
$$int_{Omega} IP{f^* omega}{de sig}=0$$
hold for every compactly-supported $k+1$-form $sigma in Omega^{k+1}(Omega)$?



I have read that this should be true, but I am only able to show this in two special cases:




  1. The form $omega$ is constant.


  2. $f$ is continuous.



Does this hold for non-continuous Sobolev maps in general?




Here is the problem as I see it: We approximate $f$ via smooth functions; suppose that $f_n in C^{infty}(Omega,R^d)$ satisfy $f_n to f$ in $W^{1,k}(Omega,R^d)$. Then
$$
int_{Omega} IP{f^* omega}{de sig}=lim_{n to infty} int_{Omega} IP{ f_n^* omega}{de sig}=lim_{n to infty} int_{Omega} IP{d f_n^* omega}{ sig}=0.
$$

Now, we need to justify the passage to the limit $int_{Omega} IP{f^* omega}{de sig}=lim_{n to infty} int_{Omega} IP{ f_n^* omega}{de sig}$.



If $omega$ is constant, that is $omega_q= alpha$ independently of $q in R^d$, where $alpha$ is a fixed element in $bigwedge^k (R^d)^* $, then we have
$$
|f^* omega-f_n^* omega| le |alpha| , left|bigwedge^{k} df-bigwedge^{k} df_nright|_{op},
$$

thus
$$
left|int_{Omega} IP{f^* omega}{de sig}- IP{ f_n^* omega}{de sig}right| le |alpha| |de sig|_{sup} int_{Omega} left|bigwedge^{k} df-bigwedge^{k} df_nright|.
$$

and The RHS tends to zero since Sobolev approximation lifts to exterior powers.



When $omega$ is not constant, we have a problem that the point of evaluation "moves with the function" that pulls back, that is
$$
begin{split}
& |f^* omega-f_n^* omega|(p)= \
&left|omega_{f(p)} circ bigwedge^{k} df_p -omega_{f_n(p)} circ bigwedge^{k} (df_n)_p right| le \
&left|big(omega_{f(p)}-omega_{f_n(p)}) circ bigwedge^{k} df_p +omega_{f_n(p)} circ big( bigwedge^{k} df_p-bigwedge^{k} (df_n)_p) right| le \
&|omega_{f(p)}-omega_{f_n(p)}| , , cdot , , left| bigwedge^{k} df_pright| +|omega_{f_n(p)} | , , cdot , , left| bigwedge^{k} df_p-bigwedge^{k} (df_n)_p right|
end{split}
$$



So, we we need to estimate $omega_{f(p)}-omega_{f_n(p)}$. When $f$ is continuous, we can take $f_n$ which converges uniformly to $f$, and thus we can continue the estimate:



$$
begin{split}
&|omega_{f(p)}-omega_{f_n(p)}| , , cdot , , left| bigwedge^{k} df_pright| +|omega_{f_n(p)} | , , cdot , , left| bigwedge^{k} df_p-bigwedge^{k} (df_n)_p right| le \
&|Tomega|_{sup} , cdot , |f(p)-f_n(p)| , cdot , left| bigwedge^{k} df_pright| +|omega |_{sup} , , cdot , , left| bigwedge^{k} df_p-bigwedge^{k} (df_n)_p right| le \
&|Tomega|_{sup} , cdot , |f-f_n|_{sup} , cdot , left| bigwedge^{k} df_pright| +|omega |_{sup} , , cdot , , left| bigwedge^{k} df_p-bigwedge^{k} (df_n)_p right| stackrel{L^1}{to} 0,
%&|alpha circ brk{bigwedge^{k} df_p-bigwedge^{k} (df_n)_p} | le |alpha| cdot |bigwedge^{k} df_p-bigwedge^{k} (df_n)_p|_{op}.
end{split}
$$



So, does the preservation of closedness hold in general (for non-continuous maps) or is there a counter-example?










share|cite|improve this question











$endgroup$

















    5












    $begingroup$


    $newcommand{R}{mathbb R}$
    $newcommand{N}{mathbb N}$
    $newcommand{de}{delta}$
    $newcommand{sig}{sigma}$
    $newcommand{Average}[1]{leftlangle#1rightrangle} $
    $newcommand{IP}[2]{Average{#1,#2}}$



    Let $n,d in mathbb{N}$, and let $Omega subseteq R^n$ be open.
    Let $f in W^{1,k}(Omega,R^d)$. Let $omega in Omega^k(R^d)$ be a smooth closed $k$-form, such that $omega$ and its derivative $Tomega$ are both uniformly bounded globally; Is it true that $f^*omega$ is weakly closed? i.e. does
    $$int_{Omega} IP{f^* omega}{de sig}=0$$
    hold for every compactly-supported $k+1$-form $sigma in Omega^{k+1}(Omega)$?



    I have read that this should be true, but I am only able to show this in two special cases:




    1. The form $omega$ is constant.


    2. $f$ is continuous.



    Does this hold for non-continuous Sobolev maps in general?




    Here is the problem as I see it: We approximate $f$ via smooth functions; suppose that $f_n in C^{infty}(Omega,R^d)$ satisfy $f_n to f$ in $W^{1,k}(Omega,R^d)$. Then
    $$
    int_{Omega} IP{f^* omega}{de sig}=lim_{n to infty} int_{Omega} IP{ f_n^* omega}{de sig}=lim_{n to infty} int_{Omega} IP{d f_n^* omega}{ sig}=0.
    $$

    Now, we need to justify the passage to the limit $int_{Omega} IP{f^* omega}{de sig}=lim_{n to infty} int_{Omega} IP{ f_n^* omega}{de sig}$.



    If $omega$ is constant, that is $omega_q= alpha$ independently of $q in R^d$, where $alpha$ is a fixed element in $bigwedge^k (R^d)^* $, then we have
    $$
    |f^* omega-f_n^* omega| le |alpha| , left|bigwedge^{k} df-bigwedge^{k} df_nright|_{op},
    $$

    thus
    $$
    left|int_{Omega} IP{f^* omega}{de sig}- IP{ f_n^* omega}{de sig}right| le |alpha| |de sig|_{sup} int_{Omega} left|bigwedge^{k} df-bigwedge^{k} df_nright|.
    $$

    and The RHS tends to zero since Sobolev approximation lifts to exterior powers.



    When $omega$ is not constant, we have a problem that the point of evaluation "moves with the function" that pulls back, that is
    $$
    begin{split}
    & |f^* omega-f_n^* omega|(p)= \
    &left|omega_{f(p)} circ bigwedge^{k} df_p -omega_{f_n(p)} circ bigwedge^{k} (df_n)_p right| le \
    &left|big(omega_{f(p)}-omega_{f_n(p)}) circ bigwedge^{k} df_p +omega_{f_n(p)} circ big( bigwedge^{k} df_p-bigwedge^{k} (df_n)_p) right| le \
    &|omega_{f(p)}-omega_{f_n(p)}| , , cdot , , left| bigwedge^{k} df_pright| +|omega_{f_n(p)} | , , cdot , , left| bigwedge^{k} df_p-bigwedge^{k} (df_n)_p right|
    end{split}
    $$



    So, we we need to estimate $omega_{f(p)}-omega_{f_n(p)}$. When $f$ is continuous, we can take $f_n$ which converges uniformly to $f$, and thus we can continue the estimate:



    $$
    begin{split}
    &|omega_{f(p)}-omega_{f_n(p)}| , , cdot , , left| bigwedge^{k} df_pright| +|omega_{f_n(p)} | , , cdot , , left| bigwedge^{k} df_p-bigwedge^{k} (df_n)_p right| le \
    &|Tomega|_{sup} , cdot , |f(p)-f_n(p)| , cdot , left| bigwedge^{k} df_pright| +|omega |_{sup} , , cdot , , left| bigwedge^{k} df_p-bigwedge^{k} (df_n)_p right| le \
    &|Tomega|_{sup} , cdot , |f-f_n|_{sup} , cdot , left| bigwedge^{k} df_pright| +|omega |_{sup} , , cdot , , left| bigwedge^{k} df_p-bigwedge^{k} (df_n)_p right| stackrel{L^1}{to} 0,
    %&|alpha circ brk{bigwedge^{k} df_p-bigwedge^{k} (df_n)_p} | le |alpha| cdot |bigwedge^{k} df_p-bigwedge^{k} (df_n)_p|_{op}.
    end{split}
    $$



    So, does the preservation of closedness hold in general (for non-continuous maps) or is there a counter-example?










    share|cite|improve this question











    $endgroup$















      5












      5








      5


      1



      $begingroup$


      $newcommand{R}{mathbb R}$
      $newcommand{N}{mathbb N}$
      $newcommand{de}{delta}$
      $newcommand{sig}{sigma}$
      $newcommand{Average}[1]{leftlangle#1rightrangle} $
      $newcommand{IP}[2]{Average{#1,#2}}$



      Let $n,d in mathbb{N}$, and let $Omega subseteq R^n$ be open.
      Let $f in W^{1,k}(Omega,R^d)$. Let $omega in Omega^k(R^d)$ be a smooth closed $k$-form, such that $omega$ and its derivative $Tomega$ are both uniformly bounded globally; Is it true that $f^*omega$ is weakly closed? i.e. does
      $$int_{Omega} IP{f^* omega}{de sig}=0$$
      hold for every compactly-supported $k+1$-form $sigma in Omega^{k+1}(Omega)$?



      I have read that this should be true, but I am only able to show this in two special cases:




      1. The form $omega$ is constant.


      2. $f$ is continuous.



      Does this hold for non-continuous Sobolev maps in general?




      Here is the problem as I see it: We approximate $f$ via smooth functions; suppose that $f_n in C^{infty}(Omega,R^d)$ satisfy $f_n to f$ in $W^{1,k}(Omega,R^d)$. Then
      $$
      int_{Omega} IP{f^* omega}{de sig}=lim_{n to infty} int_{Omega} IP{ f_n^* omega}{de sig}=lim_{n to infty} int_{Omega} IP{d f_n^* omega}{ sig}=0.
      $$

      Now, we need to justify the passage to the limit $int_{Omega} IP{f^* omega}{de sig}=lim_{n to infty} int_{Omega} IP{ f_n^* omega}{de sig}$.



      If $omega$ is constant, that is $omega_q= alpha$ independently of $q in R^d$, where $alpha$ is a fixed element in $bigwedge^k (R^d)^* $, then we have
      $$
      |f^* omega-f_n^* omega| le |alpha| , left|bigwedge^{k} df-bigwedge^{k} df_nright|_{op},
      $$

      thus
      $$
      left|int_{Omega} IP{f^* omega}{de sig}- IP{ f_n^* omega}{de sig}right| le |alpha| |de sig|_{sup} int_{Omega} left|bigwedge^{k} df-bigwedge^{k} df_nright|.
      $$

      and The RHS tends to zero since Sobolev approximation lifts to exterior powers.



      When $omega$ is not constant, we have a problem that the point of evaluation "moves with the function" that pulls back, that is
      $$
      begin{split}
      & |f^* omega-f_n^* omega|(p)= \
      &left|omega_{f(p)} circ bigwedge^{k} df_p -omega_{f_n(p)} circ bigwedge^{k} (df_n)_p right| le \
      &left|big(omega_{f(p)}-omega_{f_n(p)}) circ bigwedge^{k} df_p +omega_{f_n(p)} circ big( bigwedge^{k} df_p-bigwedge^{k} (df_n)_p) right| le \
      &|omega_{f(p)}-omega_{f_n(p)}| , , cdot , , left| bigwedge^{k} df_pright| +|omega_{f_n(p)} | , , cdot , , left| bigwedge^{k} df_p-bigwedge^{k} (df_n)_p right|
      end{split}
      $$



      So, we we need to estimate $omega_{f(p)}-omega_{f_n(p)}$. When $f$ is continuous, we can take $f_n$ which converges uniformly to $f$, and thus we can continue the estimate:



      $$
      begin{split}
      &|omega_{f(p)}-omega_{f_n(p)}| , , cdot , , left| bigwedge^{k} df_pright| +|omega_{f_n(p)} | , , cdot , , left| bigwedge^{k} df_p-bigwedge^{k} (df_n)_p right| le \
      &|Tomega|_{sup} , cdot , |f(p)-f_n(p)| , cdot , left| bigwedge^{k} df_pright| +|omega |_{sup} , , cdot , , left| bigwedge^{k} df_p-bigwedge^{k} (df_n)_p right| le \
      &|Tomega|_{sup} , cdot , |f-f_n|_{sup} , cdot , left| bigwedge^{k} df_pright| +|omega |_{sup} , , cdot , , left| bigwedge^{k} df_p-bigwedge^{k} (df_n)_p right| stackrel{L^1}{to} 0,
      %&|alpha circ brk{bigwedge^{k} df_p-bigwedge^{k} (df_n)_p} | le |alpha| cdot |bigwedge^{k} df_p-bigwedge^{k} (df_n)_p|_{op}.
      end{split}
      $$



      So, does the preservation of closedness hold in general (for non-continuous maps) or is there a counter-example?










      share|cite|improve this question











      $endgroup$




      $newcommand{R}{mathbb R}$
      $newcommand{N}{mathbb N}$
      $newcommand{de}{delta}$
      $newcommand{sig}{sigma}$
      $newcommand{Average}[1]{leftlangle#1rightrangle} $
      $newcommand{IP}[2]{Average{#1,#2}}$



      Let $n,d in mathbb{N}$, and let $Omega subseteq R^n$ be open.
      Let $f in W^{1,k}(Omega,R^d)$. Let $omega in Omega^k(R^d)$ be a smooth closed $k$-form, such that $omega$ and its derivative $Tomega$ are both uniformly bounded globally; Is it true that $f^*omega$ is weakly closed? i.e. does
      $$int_{Omega} IP{f^* omega}{de sig}=0$$
      hold for every compactly-supported $k+1$-form $sigma in Omega^{k+1}(Omega)$?



      I have read that this should be true, but I am only able to show this in two special cases:




      1. The form $omega$ is constant.


      2. $f$ is continuous.



      Does this hold for non-continuous Sobolev maps in general?




      Here is the problem as I see it: We approximate $f$ via smooth functions; suppose that $f_n in C^{infty}(Omega,R^d)$ satisfy $f_n to f$ in $W^{1,k}(Omega,R^d)$. Then
      $$
      int_{Omega} IP{f^* omega}{de sig}=lim_{n to infty} int_{Omega} IP{ f_n^* omega}{de sig}=lim_{n to infty} int_{Omega} IP{d f_n^* omega}{ sig}=0.
      $$

      Now, we need to justify the passage to the limit $int_{Omega} IP{f^* omega}{de sig}=lim_{n to infty} int_{Omega} IP{ f_n^* omega}{de sig}$.



      If $omega$ is constant, that is $omega_q= alpha$ independently of $q in R^d$, where $alpha$ is a fixed element in $bigwedge^k (R^d)^* $, then we have
      $$
      |f^* omega-f_n^* omega| le |alpha| , left|bigwedge^{k} df-bigwedge^{k} df_nright|_{op},
      $$

      thus
      $$
      left|int_{Omega} IP{f^* omega}{de sig}- IP{ f_n^* omega}{de sig}right| le |alpha| |de sig|_{sup} int_{Omega} left|bigwedge^{k} df-bigwedge^{k} df_nright|.
      $$

      and The RHS tends to zero since Sobolev approximation lifts to exterior powers.



      When $omega$ is not constant, we have a problem that the point of evaluation "moves with the function" that pulls back, that is
      $$
      begin{split}
      & |f^* omega-f_n^* omega|(p)= \
      &left|omega_{f(p)} circ bigwedge^{k} df_p -omega_{f_n(p)} circ bigwedge^{k} (df_n)_p right| le \
      &left|big(omega_{f(p)}-omega_{f_n(p)}) circ bigwedge^{k} df_p +omega_{f_n(p)} circ big( bigwedge^{k} df_p-bigwedge^{k} (df_n)_p) right| le \
      &|omega_{f(p)}-omega_{f_n(p)}| , , cdot , , left| bigwedge^{k} df_pright| +|omega_{f_n(p)} | , , cdot , , left| bigwedge^{k} df_p-bigwedge^{k} (df_n)_p right|
      end{split}
      $$



      So, we we need to estimate $omega_{f(p)}-omega_{f_n(p)}$. When $f$ is continuous, we can take $f_n$ which converges uniformly to $f$, and thus we can continue the estimate:



      $$
      begin{split}
      &|omega_{f(p)}-omega_{f_n(p)}| , , cdot , , left| bigwedge^{k} df_pright| +|omega_{f_n(p)} | , , cdot , , left| bigwedge^{k} df_p-bigwedge^{k} (df_n)_p right| le \
      &|Tomega|_{sup} , cdot , |f(p)-f_n(p)| , cdot , left| bigwedge^{k} df_pright| +|omega |_{sup} , , cdot , , left| bigwedge^{k} df_p-bigwedge^{k} (df_n)_p right| le \
      &|Tomega|_{sup} , cdot , |f-f_n|_{sup} , cdot , left| bigwedge^{k} df_pright| +|omega |_{sup} , , cdot , , left| bigwedge^{k} df_p-bigwedge^{k} (df_n)_p right| stackrel{L^1}{to} 0,
      %&|alpha circ brk{bigwedge^{k} df_p-bigwedge^{k} (df_n)_p} | le |alpha| cdot |bigwedge^{k} df_p-bigwedge^{k} (df_n)_p|_{op}.
      end{split}
      $$



      So, does the preservation of closedness hold in general (for non-continuous maps) or is there a counter-example?







      dg.differential-geometry ap.analysis-of-pdes sobolev-spaces regularity exterior-algebra






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited 9 hours ago







      Asaf Shachar

















      asked 2 days ago









      Asaf ShacharAsaf Shachar

      2,2591746




      2,2591746






















          1 Answer
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          6












          $begingroup$

          This is I guess Lemma 4.1 in https://arxiv.org/pdf/1301.4978.pdf. The assumptions might be a bit different than yours, but it might be useful. I guess the assumptions in Lemma 4.1 are close to be optimal. I will expand my answer when I have time. Let me know if Lemma 4.1 suits you.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Thanks. This looks similar to my question; Our week formulations of $df^*omega$ are a bit different though-I used duality with the adjoint of the exterior derivative (which depends on the Riemannian structure, in general) while you used a more metric-free approach (which indeed seems more natural). I need to convince myself that one can move between these two pictures in such a weak setting. However, my main issue of concern is different:
            $endgroup$
            – Asaf Shachar
            9 hours ago












          • $begingroup$
            As I described in the question, the problem lies within the "approximation step" (in the question it is phrased as $n to infty$, in your formulation you take $epsilon to 0$). As I see it, since the "point of evaluation" of the pullback form "moves with the function used to pull back" , we need uniform convergence of the mollifiers to the limiting map. (or so it seems to me). You can see how I elaborated on this point in my question. I don't see how your argument (which is essentially identical to mine) extends above the continuous setting,...
            $endgroup$
            – Asaf Shachar
            9 hours ago










          • $begingroup$
            if the form that being pulled back is not constant. To conclude, I did not understand if you think that continuity is truly needed here or not. Thanks again for all your help. (By the way, I also assume that $omega$ and its derivative are uniformly bounded globally. When the Sobolev map is assumed to be continuous, this is not need, since the statement can be reduced to the local case).
            $endgroup$
            – Asaf Shachar
            9 hours ago












          • $begingroup$
            Finally, note that it suffices to require here $f in W^{1,k}$ (instead of $f in W^{1,k+1}$), since we are only discussing pull-backs of closed forms, and not arbitrary commutation of the exterior derivative and pull-backs. This is note crucial here, but turns out to be important in some applications.
            $endgroup$
            – Asaf Shachar
            8 hours ago











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          active

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          $begingroup$

          This is I guess Lemma 4.1 in https://arxiv.org/pdf/1301.4978.pdf. The assumptions might be a bit different than yours, but it might be useful. I guess the assumptions in Lemma 4.1 are close to be optimal. I will expand my answer when I have time. Let me know if Lemma 4.1 suits you.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Thanks. This looks similar to my question; Our week formulations of $df^*omega$ are a bit different though-I used duality with the adjoint of the exterior derivative (which depends on the Riemannian structure, in general) while you used a more metric-free approach (which indeed seems more natural). I need to convince myself that one can move between these two pictures in such a weak setting. However, my main issue of concern is different:
            $endgroup$
            – Asaf Shachar
            9 hours ago












          • $begingroup$
            As I described in the question, the problem lies within the "approximation step" (in the question it is phrased as $n to infty$, in your formulation you take $epsilon to 0$). As I see it, since the "point of evaluation" of the pullback form "moves with the function used to pull back" , we need uniform convergence of the mollifiers to the limiting map. (or so it seems to me). You can see how I elaborated on this point in my question. I don't see how your argument (which is essentially identical to mine) extends above the continuous setting,...
            $endgroup$
            – Asaf Shachar
            9 hours ago










          • $begingroup$
            if the form that being pulled back is not constant. To conclude, I did not understand if you think that continuity is truly needed here or not. Thanks again for all your help. (By the way, I also assume that $omega$ and its derivative are uniformly bounded globally. When the Sobolev map is assumed to be continuous, this is not need, since the statement can be reduced to the local case).
            $endgroup$
            – Asaf Shachar
            9 hours ago












          • $begingroup$
            Finally, note that it suffices to require here $f in W^{1,k}$ (instead of $f in W^{1,k+1}$), since we are only discussing pull-backs of closed forms, and not arbitrary commutation of the exterior derivative and pull-backs. This is note crucial here, but turns out to be important in some applications.
            $endgroup$
            – Asaf Shachar
            8 hours ago
















          6












          $begingroup$

          This is I guess Lemma 4.1 in https://arxiv.org/pdf/1301.4978.pdf. The assumptions might be a bit different than yours, but it might be useful. I guess the assumptions in Lemma 4.1 are close to be optimal. I will expand my answer when I have time. Let me know if Lemma 4.1 suits you.






          share|cite|improve this answer











          $endgroup$













          • $begingroup$
            Thanks. This looks similar to my question; Our week formulations of $df^*omega$ are a bit different though-I used duality with the adjoint of the exterior derivative (which depends on the Riemannian structure, in general) while you used a more metric-free approach (which indeed seems more natural). I need to convince myself that one can move between these two pictures in such a weak setting. However, my main issue of concern is different:
            $endgroup$
            – Asaf Shachar
            9 hours ago












          • $begingroup$
            As I described in the question, the problem lies within the "approximation step" (in the question it is phrased as $n to infty$, in your formulation you take $epsilon to 0$). As I see it, since the "point of evaluation" of the pullback form "moves with the function used to pull back" , we need uniform convergence of the mollifiers to the limiting map. (or so it seems to me). You can see how I elaborated on this point in my question. I don't see how your argument (which is essentially identical to mine) extends above the continuous setting,...
            $endgroup$
            – Asaf Shachar
            9 hours ago










          • $begingroup$
            if the form that being pulled back is not constant. To conclude, I did not understand if you think that continuity is truly needed here or not. Thanks again for all your help. (By the way, I also assume that $omega$ and its derivative are uniformly bounded globally. When the Sobolev map is assumed to be continuous, this is not need, since the statement can be reduced to the local case).
            $endgroup$
            – Asaf Shachar
            9 hours ago












          • $begingroup$
            Finally, note that it suffices to require here $f in W^{1,k}$ (instead of $f in W^{1,k+1}$), since we are only discussing pull-backs of closed forms, and not arbitrary commutation of the exterior derivative and pull-backs. This is note crucial here, but turns out to be important in some applications.
            $endgroup$
            – Asaf Shachar
            8 hours ago














          6












          6








          6





          $begingroup$

          This is I guess Lemma 4.1 in https://arxiv.org/pdf/1301.4978.pdf. The assumptions might be a bit different than yours, but it might be useful. I guess the assumptions in Lemma 4.1 are close to be optimal. I will expand my answer when I have time. Let me know if Lemma 4.1 suits you.






          share|cite|improve this answer











          $endgroup$



          This is I guess Lemma 4.1 in https://arxiv.org/pdf/1301.4978.pdf. The assumptions might be a bit different than yours, but it might be useful. I guess the assumptions in Lemma 4.1 are close to be optimal. I will expand my answer when I have time. Let me know if Lemma 4.1 suits you.







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited 2 days ago









          Martin Sleziak

          2,96532028




          2,96532028










          answered 2 days ago









          Piotr HajlaszPiotr Hajlasz

          6,67142457




          6,67142457












          • $begingroup$
            Thanks. This looks similar to my question; Our week formulations of $df^*omega$ are a bit different though-I used duality with the adjoint of the exterior derivative (which depends on the Riemannian structure, in general) while you used a more metric-free approach (which indeed seems more natural). I need to convince myself that one can move between these two pictures in such a weak setting. However, my main issue of concern is different:
            $endgroup$
            – Asaf Shachar
            9 hours ago












          • $begingroup$
            As I described in the question, the problem lies within the "approximation step" (in the question it is phrased as $n to infty$, in your formulation you take $epsilon to 0$). As I see it, since the "point of evaluation" of the pullback form "moves with the function used to pull back" , we need uniform convergence of the mollifiers to the limiting map. (or so it seems to me). You can see how I elaborated on this point in my question. I don't see how your argument (which is essentially identical to mine) extends above the continuous setting,...
            $endgroup$
            – Asaf Shachar
            9 hours ago










          • $begingroup$
            if the form that being pulled back is not constant. To conclude, I did not understand if you think that continuity is truly needed here or not. Thanks again for all your help. (By the way, I also assume that $omega$ and its derivative are uniformly bounded globally. When the Sobolev map is assumed to be continuous, this is not need, since the statement can be reduced to the local case).
            $endgroup$
            – Asaf Shachar
            9 hours ago












          • $begingroup$
            Finally, note that it suffices to require here $f in W^{1,k}$ (instead of $f in W^{1,k+1}$), since we are only discussing pull-backs of closed forms, and not arbitrary commutation of the exterior derivative and pull-backs. This is note crucial here, but turns out to be important in some applications.
            $endgroup$
            – Asaf Shachar
            8 hours ago


















          • $begingroup$
            Thanks. This looks similar to my question; Our week formulations of $df^*omega$ are a bit different though-I used duality with the adjoint of the exterior derivative (which depends on the Riemannian structure, in general) while you used a more metric-free approach (which indeed seems more natural). I need to convince myself that one can move between these two pictures in such a weak setting. However, my main issue of concern is different:
            $endgroup$
            – Asaf Shachar
            9 hours ago












          • $begingroup$
            As I described in the question, the problem lies within the "approximation step" (in the question it is phrased as $n to infty$, in your formulation you take $epsilon to 0$). As I see it, since the "point of evaluation" of the pullback form "moves with the function used to pull back" , we need uniform convergence of the mollifiers to the limiting map. (or so it seems to me). You can see how I elaborated on this point in my question. I don't see how your argument (which is essentially identical to mine) extends above the continuous setting,...
            $endgroup$
            – Asaf Shachar
            9 hours ago










          • $begingroup$
            if the form that being pulled back is not constant. To conclude, I did not understand if you think that continuity is truly needed here or not. Thanks again for all your help. (By the way, I also assume that $omega$ and its derivative are uniformly bounded globally. When the Sobolev map is assumed to be continuous, this is not need, since the statement can be reduced to the local case).
            $endgroup$
            – Asaf Shachar
            9 hours ago












          • $begingroup$
            Finally, note that it suffices to require here $f in W^{1,k}$ (instead of $f in W^{1,k+1}$), since we are only discussing pull-backs of closed forms, and not arbitrary commutation of the exterior derivative and pull-backs. This is note crucial here, but turns out to be important in some applications.
            $endgroup$
            – Asaf Shachar
            8 hours ago
















          $begingroup$
          Thanks. This looks similar to my question; Our week formulations of $df^*omega$ are a bit different though-I used duality with the adjoint of the exterior derivative (which depends on the Riemannian structure, in general) while you used a more metric-free approach (which indeed seems more natural). I need to convince myself that one can move between these two pictures in such a weak setting. However, my main issue of concern is different:
          $endgroup$
          – Asaf Shachar
          9 hours ago






          $begingroup$
          Thanks. This looks similar to my question; Our week formulations of $df^*omega$ are a bit different though-I used duality with the adjoint of the exterior derivative (which depends on the Riemannian structure, in general) while you used a more metric-free approach (which indeed seems more natural). I need to convince myself that one can move between these two pictures in such a weak setting. However, my main issue of concern is different:
          $endgroup$
          – Asaf Shachar
          9 hours ago














          $begingroup$
          As I described in the question, the problem lies within the "approximation step" (in the question it is phrased as $n to infty$, in your formulation you take $epsilon to 0$). As I see it, since the "point of evaluation" of the pullback form "moves with the function used to pull back" , we need uniform convergence of the mollifiers to the limiting map. (or so it seems to me). You can see how I elaborated on this point in my question. I don't see how your argument (which is essentially identical to mine) extends above the continuous setting,...
          $endgroup$
          – Asaf Shachar
          9 hours ago




          $begingroup$
          As I described in the question, the problem lies within the "approximation step" (in the question it is phrased as $n to infty$, in your formulation you take $epsilon to 0$). As I see it, since the "point of evaluation" of the pullback form "moves with the function used to pull back" , we need uniform convergence of the mollifiers to the limiting map. (or so it seems to me). You can see how I elaborated on this point in my question. I don't see how your argument (which is essentially identical to mine) extends above the continuous setting,...
          $endgroup$
          – Asaf Shachar
          9 hours ago












          $begingroup$
          if the form that being pulled back is not constant. To conclude, I did not understand if you think that continuity is truly needed here or not. Thanks again for all your help. (By the way, I also assume that $omega$ and its derivative are uniformly bounded globally. When the Sobolev map is assumed to be continuous, this is not need, since the statement can be reduced to the local case).
          $endgroup$
          – Asaf Shachar
          9 hours ago






          $begingroup$
          if the form that being pulled back is not constant. To conclude, I did not understand if you think that continuity is truly needed here or not. Thanks again for all your help. (By the way, I also assume that $omega$ and its derivative are uniformly bounded globally. When the Sobolev map is assumed to be continuous, this is not need, since the statement can be reduced to the local case).
          $endgroup$
          – Asaf Shachar
          9 hours ago














          $begingroup$
          Finally, note that it suffices to require here $f in W^{1,k}$ (instead of $f in W^{1,k+1}$), since we are only discussing pull-backs of closed forms, and not arbitrary commutation of the exterior derivative and pull-backs. This is note crucial here, but turns out to be important in some applications.
          $endgroup$
          – Asaf Shachar
          8 hours ago




          $begingroup$
          Finally, note that it suffices to require here $f in W^{1,k}$ (instead of $f in W^{1,k+1}$), since we are only discussing pull-backs of closed forms, and not arbitrary commutation of the exterior derivative and pull-backs. This is note crucial here, but turns out to be important in some applications.
          $endgroup$
          – Asaf Shachar
          8 hours ago


















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