Well-definedness of a fundamental solution by proofing that a set has measure zero with Cavalieri's principle












0












$begingroup$


I am currently walking my way through a paper from the year 2009, where the Malgrange-Ehrenpreis Theorem ("Every linear partial differential operator with constant coefficients has a fundamental solution.") is proved constructively, i.e. the fundamental solution $Einmathcal{D}'(mathbb{R}^n)$ is given explicitly.



The first part of the proof consists of showing well-definedness and in doing so one has to prove that $N={xiinmathbb{R}^nlvert P(ixi + lambdaeta)=0}$, where $lambdainmathbb{R}$, $etainmathbb{R}^n$ are fixed and $Pinmathbb{C}[xi]$, is a set of Lebesgue measure zero. This is done by Cavalieri's principle, i.e. showing that $N_{xi'}={xi_1inmathbb{R}lvert (xi_1,xi')in N}$ has measure zero for every $xi'inmathbb{R}^{n-1}$ (this is clear by the fundamental theorem of algebra, meaning that $N_{xi'}$ is finite) and then obtaining



begin{equation}
int_N dxi = int_{mathbb{R}^{n-1}}underbrace{bigg(int_{N_{xi'}}dxi_1bigg)}_{=0}dxi'=0.
end{equation}



So far so good. However before this is done in the paper, a linear transform is performed previously. The assumptions of the statement one finds $P_m(eta)neq 0$, where $P_m$ is the principal value of $P$. He needs this since $frac{1}{overline{P_m(eta)}}$ occurs in the definition of $E$. Then he writes the following:



"After a linear change of coordinates we can assume $P_m(1,0,...,0)neq 0$ and then $int_N dxi = int_{mathbb{R}^{n-1}}bigg(int_{N_{xi'}}dxi_1bigg)dxi'=0$ by Fubinis Theorem and since the sets $N_{xi'}$ are finite for every $xi'inmathbb{R}^{n-1}$."



This is where I have my problems:



(i) Why does he need to do a linear change of coordinates? From my understanding the Cavalieri principle works just fine without it. Does this have anything to do with the change of coordinates theorem?



(ii)Why does he need to make the assumption $P_m(1,0,...,0)neq 0$?



Thanks in advance and greetings from Munich. :)






polynomials lebesgue-measure distribution-theory change-of-variable product-measure







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    $begingroup$


    I am currently walking my way through a paper from the year 2009, where the Malgrange-Ehrenpreis Theorem ("Every linear partial differential operator with constant coefficients has a fundamental solution.") is proved constructively, i.e. the fundamental solution $Einmathcal{D}'(mathbb{R}^n)$ is given explicitly.



    The first part of the proof consists of showing well-definedness and in doing so one has to prove that $N={xiinmathbb{R}^nlvert P(ixi + lambdaeta)=0}$, where $lambdainmathbb{R}$, $etainmathbb{R}^n$ are fixed and $Pinmathbb{C}[xi]$, is a set of Lebesgue measure zero. This is done by Cavalieri's principle, i.e. showing that $N_{xi'}={xi_1inmathbb{R}lvert (xi_1,xi')in N}$ has measure zero for every $xi'inmathbb{R}^{n-1}$ (this is clear by the fundamental theorem of algebra, meaning that $N_{xi'}$ is finite) and then obtaining



    begin{equation}
    int_N dxi = int_{mathbb{R}^{n-1}}underbrace{bigg(int_{N_{xi'}}dxi_1bigg)}_{=0}dxi'=0.
    end{equation}



    So far so good. However before this is done in the paper, a linear transform is performed previously. The assumptions of the statement one finds $P_m(eta)neq 0$, where $P_m$ is the principal value of $P$. He needs this since $frac{1}{overline{P_m(eta)}}$ occurs in the definition of $E$. Then he writes the following:



    "After a linear change of coordinates we can assume $P_m(1,0,...,0)neq 0$ and then $int_N dxi = int_{mathbb{R}^{n-1}}bigg(int_{N_{xi'}}dxi_1bigg)dxi'=0$ by Fubinis Theorem and since the sets $N_{xi'}$ are finite for every $xi'inmathbb{R}^{n-1}$."



    This is where I have my problems:



    (i) Why does he need to do a linear change of coordinates? From my understanding the Cavalieri principle works just fine without it. Does this have anything to do with the change of coordinates theorem?



    (ii)Why does he need to make the assumption $P_m(1,0,...,0)neq 0$?



    Thanks in advance and greetings from Munich. :)






    polynomials lebesgue-measure distribution-theory change-of-variable product-measure







    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$


      I am currently walking my way through a paper from the year 2009, where the Malgrange-Ehrenpreis Theorem ("Every linear partial differential operator with constant coefficients has a fundamental solution.") is proved constructively, i.e. the fundamental solution $Einmathcal{D}'(mathbb{R}^n)$ is given explicitly.



      The first part of the proof consists of showing well-definedness and in doing so one has to prove that $N={xiinmathbb{R}^nlvert P(ixi + lambdaeta)=0}$, where $lambdainmathbb{R}$, $etainmathbb{R}^n$ are fixed and $Pinmathbb{C}[xi]$, is a set of Lebesgue measure zero. This is done by Cavalieri's principle, i.e. showing that $N_{xi'}={xi_1inmathbb{R}lvert (xi_1,xi')in N}$ has measure zero for every $xi'inmathbb{R}^{n-1}$ (this is clear by the fundamental theorem of algebra, meaning that $N_{xi'}$ is finite) and then obtaining



      begin{equation}
      int_N dxi = int_{mathbb{R}^{n-1}}underbrace{bigg(int_{N_{xi'}}dxi_1bigg)}_{=0}dxi'=0.
      end{equation}



      So far so good. However before this is done in the paper, a linear transform is performed previously. The assumptions of the statement one finds $P_m(eta)neq 0$, where $P_m$ is the principal value of $P$. He needs this since $frac{1}{overline{P_m(eta)}}$ occurs in the definition of $E$. Then he writes the following:



      "After a linear change of coordinates we can assume $P_m(1,0,...,0)neq 0$ and then $int_N dxi = int_{mathbb{R}^{n-1}}bigg(int_{N_{xi'}}dxi_1bigg)dxi'=0$ by Fubinis Theorem and since the sets $N_{xi'}$ are finite for every $xi'inmathbb{R}^{n-1}$."



      This is where I have my problems:



      (i) Why does he need to do a linear change of coordinates? From my understanding the Cavalieri principle works just fine without it. Does this have anything to do with the change of coordinates theorem?



      (ii)Why does he need to make the assumption $P_m(1,0,...,0)neq 0$?



      Thanks in advance and greetings from Munich. :)






      polynomials lebesgue-measure distribution-theory change-of-variable product-measure







      share|cite|improve this question









      $endgroup$




      I am currently walking my way through a paper from the year 2009, where the Malgrange-Ehrenpreis Theorem ("Every linear partial differential operator with constant coefficients has a fundamental solution.") is proved constructively, i.e. the fundamental solution $Einmathcal{D}'(mathbb{R}^n)$ is given explicitly.



      The first part of the proof consists of showing well-definedness and in doing so one has to prove that $N={xiinmathbb{R}^nlvert P(ixi + lambdaeta)=0}$, where $lambdainmathbb{R}$, $etainmathbb{R}^n$ are fixed and $Pinmathbb{C}[xi]$, is a set of Lebesgue measure zero. This is done by Cavalieri's principle, i.e. showing that $N_{xi'}={xi_1inmathbb{R}lvert (xi_1,xi')in N}$ has measure zero for every $xi'inmathbb{R}^{n-1}$ (this is clear by the fundamental theorem of algebra, meaning that $N_{xi'}$ is finite) and then obtaining



      begin{equation}
      int_N dxi = int_{mathbb{R}^{n-1}}underbrace{bigg(int_{N_{xi'}}dxi_1bigg)}_{=0}dxi'=0.
      end{equation}



      So far so good. However before this is done in the paper, a linear transform is performed previously. The assumptions of the statement one finds $P_m(eta)neq 0$, where $P_m$ is the principal value of $P$. He needs this since $frac{1}{overline{P_m(eta)}}$ occurs in the definition of $E$. Then he writes the following:



      "After a linear change of coordinates we can assume $P_m(1,0,...,0)neq 0$ and then $int_N dxi = int_{mathbb{R}^{n-1}}bigg(int_{N_{xi'}}dxi_1bigg)dxi'=0$ by Fubinis Theorem and since the sets $N_{xi'}$ are finite for every $xi'inmathbb{R}^{n-1}$."



      This is where I have my problems:



      (i) Why does he need to do a linear change of coordinates? From my understanding the Cavalieri principle works just fine without it. Does this have anything to do with the change of coordinates theorem?



      (ii)Why does he need to make the assumption $P_m(1,0,...,0)neq 0$?



      Thanks in advance and greetings from Munich. :)






      polynomials lebesgue-measure distribution-theory change-of-variable product-measure




      polynomials lebesgue-measure distribution-theory change-of-variable product-measure






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      asked Jan 5 at 10:25









      Joseph ExpoJoseph Expo

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