Differentiation in $L^p$ and Weak Differentiation.












2












$begingroup$


In Weiss and Stein's book on Fourier analysis in Euclidean spaces, they define the notion of a 'derivative in $L^p$'. To be precise, they define the difference operators



$$ (Delta_h f)(x) = frac{f(x + h) - f(x)}{h} $$



and say that $f$ has a derivative $g$ in $L^p(mathbf{R}^n)$ if $Delta_h f to g$ in the $L^p$ norm as $h to 0$. I haven't encountered this definition before. It seems it should be equivalent to some kind of weak differentiation, or at least related in some sense? However, I'm unable to find other references that talk about this kind of construction using the language that Stein and Weiss use. Is there another name for this notion, and how is it related to weak differentiation?






sobolev-spaces harmonic-analysis weak-derivatives







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  • 1




    $begingroup$
    This is usually called strong derivative and there is nothing weak about it.
    $endgroup$
    – Kavi Rama Murthy
    Dec 18 '18 at 6:41


















2












$begingroup$


In Weiss and Stein's book on Fourier analysis in Euclidean spaces, they define the notion of a 'derivative in $L^p$'. To be precise, they define the difference operators



$$ (Delta_h f)(x) = frac{f(x + h) - f(x)}{h} $$



and say that $f$ has a derivative $g$ in $L^p(mathbf{R}^n)$ if $Delta_h f to g$ in the $L^p$ norm as $h to 0$. I haven't encountered this definition before. It seems it should be equivalent to some kind of weak differentiation, or at least related in some sense? However, I'm unable to find other references that talk about this kind of construction using the language that Stein and Weiss use. Is there another name for this notion, and how is it related to weak differentiation?






sobolev-spaces harmonic-analysis weak-derivatives







share|cite|improve this question









$endgroup$








  • 1




    $begingroup$
    This is usually called strong derivative and there is nothing weak about it.
    $endgroup$
    – Kavi Rama Murthy
    Dec 18 '18 at 6:41
















2












2








2





$begingroup$


In Weiss and Stein's book on Fourier analysis in Euclidean spaces, they define the notion of a 'derivative in $L^p$'. To be precise, they define the difference operators



$$ (Delta_h f)(x) = frac{f(x + h) - f(x)}{h} $$



and say that $f$ has a derivative $g$ in $L^p(mathbf{R}^n)$ if $Delta_h f to g$ in the $L^p$ norm as $h to 0$. I haven't encountered this definition before. It seems it should be equivalent to some kind of weak differentiation, or at least related in some sense? However, I'm unable to find other references that talk about this kind of construction using the language that Stein and Weiss use. Is there another name for this notion, and how is it related to weak differentiation?






sobolev-spaces harmonic-analysis weak-derivatives







share|cite|improve this question









$endgroup$




In Weiss and Stein's book on Fourier analysis in Euclidean spaces, they define the notion of a 'derivative in $L^p$'. To be precise, they define the difference operators



$$ (Delta_h f)(x) = frac{f(x + h) - f(x)}{h} $$



and say that $f$ has a derivative $g$ in $L^p(mathbf{R}^n)$ if $Delta_h f to g$ in the $L^p$ norm as $h to 0$. I haven't encountered this definition before. It seems it should be equivalent to some kind of weak differentiation, or at least related in some sense? However, I'm unable to find other references that talk about this kind of construction using the language that Stein and Weiss use. Is there another name for this notion, and how is it related to weak differentiation?






sobolev-spaces harmonic-analysis weak-derivatives




sobolev-spaces harmonic-analysis weak-derivatives






share|cite|improve this question













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asked Dec 18 '18 at 6:21









Jacob DensonJacob Denson

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754314








  • 1




    $begingroup$
    This is usually called strong derivative and there is nothing weak about it.
    $endgroup$
    – Kavi Rama Murthy
    Dec 18 '18 at 6:41
















  • 1




    $begingroup$
    This is usually called strong derivative and there is nothing weak about it.
    $endgroup$
    – Kavi Rama Murthy
    Dec 18 '18 at 6:41










1




1




$begingroup$
This is usually called strong derivative and there is nothing weak about it.
$endgroup$
– Kavi Rama Murthy
Dec 18 '18 at 6:41






$begingroup$
This is usually called strong derivative and there is nothing weak about it.
$endgroup$
– Kavi Rama Murthy
Dec 18 '18 at 6:41












1 Answer
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$begingroup$

On page 372-373 from this lecture note, the relation between the weak derivative (in the distributional sense) and this strong derivative (as the limit in $L^p$ of difference quotient) is discussed.



For dimension $n=1$ and $p=2$, there's a relatively easy proof that the limit of the difference quotient converges to the weak derivative.






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    $begingroup$

    On page 372-373 from this lecture note, the relation between the weak derivative (in the distributional sense) and this strong derivative (as the limit in $L^p$ of difference quotient) is discussed.



    For dimension $n=1$ and $p=2$, there's a relatively easy proof that the limit of the difference quotient converges to the weak derivative.






    share|cite|improve this answer









    $endgroup$


















      1












      $begingroup$

      On page 372-373 from this lecture note, the relation between the weak derivative (in the distributional sense) and this strong derivative (as the limit in $L^p$ of difference quotient) is discussed.



      For dimension $n=1$ and $p=2$, there's a relatively easy proof that the limit of the difference quotient converges to the weak derivative.






      share|cite|improve this answer









      $endgroup$
















        1












        1








        1





        $begingroup$

        On page 372-373 from this lecture note, the relation between the weak derivative (in the distributional sense) and this strong derivative (as the limit in $L^p$ of difference quotient) is discussed.



        For dimension $n=1$ and $p=2$, there's a relatively easy proof that the limit of the difference quotient converges to the weak derivative.






        share|cite|improve this answer









        $endgroup$



        On page 372-373 from this lecture note, the relation between the weak derivative (in the distributional sense) and this strong derivative (as the limit in $L^p$ of difference quotient) is discussed.



        For dimension $n=1$ and $p=2$, there's a relatively easy proof that the limit of the difference quotient converges to the weak derivative.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 18 '18 at 8:11









        BigbearZzzBigbearZzz

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