The importance of the Hardy-Littlewood Maximal function The 2019 Stack Overflow Developer Survey Results Are InA question about the Hardy-Littlewood maximal function.If a function is radial, then its Hardy-Littlewood maximal function is radial as wellImprovement of weak type inequality for Hardy-Littlewood Maximal inequalityProve that the Hardy-Littlewood maximal function is bounded above and belowReal Analysis, Folland problem 3.3.23 Differentiation on Euclidean SpaceHardy-Littlewood maximal function of a probability densityA question about Hardy-Littlewood maximal function and a characterization of measurable sets.Hardy-Littlewood maximal function not integrable in B(0,1)Is the truncated Hardy-Littlewood maximal function of an $L^1$ function also in $L^1$?Hardy-Littlewood maximal function $f^*$ is greater than $f$ for measurable $f$
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The importance of the Hardy-Littlewood Maximal function
The 2019 Stack Overflow Developer Survey Results Are InA question about the Hardy-Littlewood maximal function.If a function is radial, then its Hardy-Littlewood maximal function is radial as wellImprovement of weak type inequality for Hardy-Littlewood Maximal inequalityProve that the Hardy-Littlewood maximal function is bounded above and belowReal Analysis, Folland problem 3.3.23 Differentiation on Euclidean SpaceHardy-Littlewood maximal function of a probability densityA question about Hardy-Littlewood maximal function and a characterization of measurable sets.Hardy-Littlewood maximal function not integrable in B(0,1)Is the truncated Hardy-Littlewood maximal function of an $L^1$ function also in $L^1$?Hardy-Littlewood maximal function $f^*$ is greater than $f$ for measurable $f$
$begingroup$
In my measure theory class, we've spent a good chunk of the last week or so studying the Hardy-Littlewood Maximal function defined for a function $fin L_loc^1 (mathbbR^d)$ as $$ Mf(x) = sup_xin B frac1m(B) int_B |f(y)| dy $$
We've shown that in general, $Mfnotin L^1(mathbbR^d)$, unless $f = 0$ almost everywhere.
My question is, what motivates the study of such a function? Why do we care about $Mf$? And how did somebody (I'm presuming Hardy and/or Littlewood) come up with such a function in the first place?
real-analysis
asked Mar 30 at 21:15
NateNate
335
$endgroup$
add a comment |
$begingroup$
In my measure theory class, we've spent a good chunk of the last week or so studying the Hardy-Littlewood Maximal function defined for a function $fin L_loc^1 (mathbbR^d)$ as $$ Mf(x) = sup_xin B frac1m(B) int_B |f(y)| dy $$
We've shown that in general, $Mfnotin L^1(mathbbR^d)$, unless $f = 0$ almost everywhere.
My question is, what motivates the study of such a function? Why do we care about $Mf$? And how did somebody (I'm presuming Hardy and/or Littlewood) come up with such a function in the first place?
real-analysis
asked Mar 30 at 21:15
NateNate
335
$endgroup$
$begingroup$
There a lot of applications listed here: en.wikipedia.org/wiki/Hardy%E2%80%93Littlewood_maximal_function
$endgroup$
– Lorenzo
Mar 30 at 21:20
add a comment |
$begingroup$
In my measure theory class, we've spent a good chunk of the last week or so studying the Hardy-Littlewood Maximal function defined for a function $fin L_loc^1 (mathbbR^d)$ as $$ Mf(x) = sup_xin B frac1m(B) int_B |f(y)| dy $$
We've shown that in general, $Mfnotin L^1(mathbbR^d)$, unless $f = 0$ almost everywhere.
My question is, what motivates the study of such a function? Why do we care about $Mf$? And how did somebody (I'm presuming Hardy and/or Littlewood) come up with such a function in the first place?
real-analysis
asked Mar 30 at 21:15
NateNate
335
$endgroup$
In my measure theory class, we've spent a good chunk of the last week or so studying the Hardy-Littlewood Maximal function defined for a function $fin L_loc^1 (mathbbR^d)$ as $$ Mf(x) = sup_xin B frac1m(B) int_B |f(y)| dy $$
We've shown that in general, $Mfnotin L^1(mathbbR^d)$, unless $f = 0$ almost everywhere.
My question is, what motivates the study of such a function? Why do we care about $Mf$? And how did somebody (I'm presuming Hardy and/or Littlewood) come up with such a function in the first place?
real-analysis
real-analysis
asked Mar 30 at 21:15
NateNate
335
asked Mar 30 at 21:15
NateNate
335
edited Mar 30 at 21:22
Nate
asked Mar 30 at 21:15
NateNate
335
asked Mar 30 at 21:15
NateNate
335
asked Mar 30 at 21:15
NateNate
335
335
$begingroup$
There a lot of applications listed here: en.wikipedia.org/wiki/Hardy%E2%80%93Littlewood_maximal_function
$endgroup$
– Lorenzo
Mar 30 at 21:20
add a comment |
$begingroup$
There a lot of applications listed here: en.wikipedia.org/wiki/Hardy%E2%80%93Littlewood_maximal_function
$endgroup$
– Lorenzo
Mar 30 at 21:20
$begingroup$
There a lot of applications listed here: en.wikipedia.org/wiki/Hardy%E2%80%93Littlewood_maximal_function
$endgroup$
– Lorenzo
Mar 30 at 21:20
$begingroup$
There a lot of applications listed here: en.wikipedia.org/wiki/Hardy%E2%80%93Littlewood_maximal_function
$endgroup$
– Lorenzo
Mar 30 at 21:20
add a comment |
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$begingroup$
There a lot of applications listed here: en.wikipedia.org/wiki/Hardy%E2%80%93Littlewood_maximal_function
$endgroup$
– Lorenzo
Mar 30 at 21:20