If $fin C_0(mathbb R)cap C^2(mathbb R)$, is $bf'+frac12sigma^2f''in C_0(mathbb R)$?
$begingroup$
Let
$b,sigma:mathbb R$ be Lipschitz continuous with $$|b(x)|^2+|sigma(x)|^2le C(1+|x|^2);;;text{for all }xinmathbb Rtag1,$$ $sigmain C^2(mathbb R)$, $sigma''$ being bounded and $sigma(mathbb R)subseteqmathbb Rsetminusleft{0right}$
$C_0(mathbb R)$ denote the space of continuous functions vanishing at infinity
If $fin C_0(mathbb R)cap C^2(mathbb R)$, are we able to show that $$Lf:=bf'+frac12sigma^2f''in C_0(mathbb R)$$ or is there an example of such an $f$ with $Lfnotin C_0(mathbb R)$?
functional-analysis derivatives distribution-theory
edited Dec 27 '18 at 0:36
mathworker21
8,9421928
asked Dec 26 '18 at 22:57
0xbadf00d0xbadf00d
1,95841531
$endgroup$
add a comment |
$begingroup$
Let
$b,sigma:mathbb R$ be Lipschitz continuous with $$|b(x)|^2+|sigma(x)|^2le C(1+|x|^2);;;text{for all }xinmathbb Rtag1,$$ $sigmain C^2(mathbb R)$, $sigma''$ being bounded and $sigma(mathbb R)subseteqmathbb Rsetminusleft{0right}$
$C_0(mathbb R)$ denote the space of continuous functions vanishing at infinity
If $fin C_0(mathbb R)cap C^2(mathbb R)$, are we able to show that $$Lf:=bf'+frac12sigma^2f''in C_0(mathbb R)$$ or is there an example of such an $f$ with $Lfnotin C_0(mathbb R)$?
functional-analysis derivatives distribution-theory
edited Dec 27 '18 at 0:36
mathworker21
8,9421928
asked Dec 26 '18 at 22:57
0xbadf00d0xbadf00d
1,95841531
$endgroup$
add a comment |
$begingroup$
Let
$b,sigma:mathbb R$ be Lipschitz continuous with $$|b(x)|^2+|sigma(x)|^2le C(1+|x|^2);;;text{for all }xinmathbb Rtag1,$$ $sigmain C^2(mathbb R)$, $sigma''$ being bounded and $sigma(mathbb R)subseteqmathbb Rsetminusleft{0right}$
$C_0(mathbb R)$ denote the space of continuous functions vanishing at infinity
If $fin C_0(mathbb R)cap C^2(mathbb R)$, are we able to show that $$Lf:=bf'+frac12sigma^2f''in C_0(mathbb R)$$ or is there an example of such an $f$ with $Lfnotin C_0(mathbb R)$?
functional-analysis derivatives distribution-theory
edited Dec 27 '18 at 0:36
mathworker21
8,9421928
asked Dec 26 '18 at 22:57
0xbadf00d0xbadf00d
1,95841531
$endgroup$
Let
$b,sigma:mathbb R$ be Lipschitz continuous with $$|b(x)|^2+|sigma(x)|^2le C(1+|x|^2);;;text{for all }xinmathbb Rtag1,$$ $sigmain C^2(mathbb R)$, $sigma''$ being bounded and $sigma(mathbb R)subseteqmathbb Rsetminusleft{0right}$
$C_0(mathbb R)$ denote the space of continuous functions vanishing at infinity
If $fin C_0(mathbb R)cap C^2(mathbb R)$, are we able to show that $$Lf:=bf'+frac12sigma^2f''in C_0(mathbb R)$$ or is there an example of such an $f$ with $Lfnotin C_0(mathbb R)$?
functional-analysis derivatives distribution-theory
functional-analysis derivatives distribution-theory
edited Dec 27 '18 at 0:36
mathworker21
8,9421928
asked Dec 26 '18 at 22:57
0xbadf00d0xbadf00d
1,95841531
edited Dec 27 '18 at 0:36
mathworker21
8,9421928
asked Dec 26 '18 at 22:57
0xbadf00d0xbadf00d
1,95841531
edited Dec 27 '18 at 0:36
mathworker21
8,9421928
edited Dec 27 '18 at 0:36
mathworker21
8,9421928
edited Dec 27 '18 at 0:36
mathworker21
8,9421928
8,9421928
asked Dec 26 '18 at 22:57
0xbadf00d0xbadf00d
1,95841531
asked Dec 26 '18 at 22:57
0xbadf00d0xbadf00d
1,95841531
asked Dec 26 '18 at 22:57
0xbadf00d0xbadf00d
1,95841531
1,95841531
add a comment |
add a comment |
1 Answer
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$begingroup$
There are definitely examples of such $f$ with $Lf not in C_0(mathbb{R})$. Take some $f$ that does go to infinity but whose derivative is greater than $1$ in magnitude nearly always (one can visualize such a thing as a smoothed out version of a bunch of spikes with lengths tending to $0$). Then if we take $b equiv 1$ and $sigma$ some function that quickly goes to $0$ at infinity, we see $Lf approx f'$ does not go to $0$.
answered Dec 27 '18 at 0:45
mathworker21mathworker21
8,9421928
$endgroup$
$begingroup$
Could you provide a concrete example for such a $f$?
$endgroup$
– 0xbadf00d
Dec 29 '18 at 16:22
$begingroup$
@0xbadf00d cmon. if you read/understand my solution, I won't need to
$endgroup$
– mathworker21
Dec 29 '18 at 23:10
add a comment |
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1 Answer
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1 Answer
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$begingroup$
There are definitely examples of such $f$ with $Lf not in C_0(mathbb{R})$. Take some $f$ that does go to infinity but whose derivative is greater than $1$ in magnitude nearly always (one can visualize such a thing as a smoothed out version of a bunch of spikes with lengths tending to $0$). Then if we take $b equiv 1$ and $sigma$ some function that quickly goes to $0$ at infinity, we see $Lf approx f'$ does not go to $0$.
answered Dec 27 '18 at 0:45
mathworker21mathworker21
8,9421928
$endgroup$
$begingroup$
Could you provide a concrete example for such a $f$?
$endgroup$
– 0xbadf00d
Dec 29 '18 at 16:22
$begingroup$
@0xbadf00d cmon. if you read/understand my solution, I won't need to
$endgroup$
– mathworker21
Dec 29 '18 at 23:10
add a comment |
$begingroup$
There are definitely examples of such $f$ with $Lf not in C_0(mathbb{R})$. Take some $f$ that does go to infinity but whose derivative is greater than $1$ in magnitude nearly always (one can visualize such a thing as a smoothed out version of a bunch of spikes with lengths tending to $0$). Then if we take $b equiv 1$ and $sigma$ some function that quickly goes to $0$ at infinity, we see $Lf approx f'$ does not go to $0$.
answered Dec 27 '18 at 0:45
mathworker21mathworker21
8,9421928
$endgroup$
$begingroup$
Could you provide a concrete example for such a $f$?
$endgroup$
– 0xbadf00d
Dec 29 '18 at 16:22
$begingroup$
@0xbadf00d cmon. if you read/understand my solution, I won't need to
$endgroup$
– mathworker21
Dec 29 '18 at 23:10
add a comment |
$begingroup$
There are definitely examples of such $f$ with $Lf not in C_0(mathbb{R})$. Take some $f$ that does go to infinity but whose derivative is greater than $1$ in magnitude nearly always (one can visualize such a thing as a smoothed out version of a bunch of spikes with lengths tending to $0$). Then if we take $b equiv 1$ and $sigma$ some function that quickly goes to $0$ at infinity, we see $Lf approx f'$ does not go to $0$.
answered Dec 27 '18 at 0:45
mathworker21mathworker21
8,9421928
$endgroup$
There are definitely examples of such $f$ with $Lf not in C_0(mathbb{R})$. Take some $f$ that does go to infinity but whose derivative is greater than $1$ in magnitude nearly always (one can visualize such a thing as a smoothed out version of a bunch of spikes with lengths tending to $0$). Then if we take $b equiv 1$ and $sigma$ some function that quickly goes to $0$ at infinity, we see $Lf approx f'$ does not go to $0$.
answered Dec 27 '18 at 0:45
mathworker21mathworker21
8,9421928
answered Dec 27 '18 at 0:45
mathworker21mathworker21
8,9421928
answered Dec 27 '18 at 0:45
mathworker21mathworker21
8,9421928
answered Dec 27 '18 at 0:45
mathworker21mathworker21
8,9421928
8,9421928
$begingroup$
Could you provide a concrete example for such a $f$?
$endgroup$
– 0xbadf00d
Dec 29 '18 at 16:22
$begingroup$
@0xbadf00d cmon. if you read/understand my solution, I won't need to
$endgroup$
– mathworker21
Dec 29 '18 at 23:10
add a comment |
$begingroup$
Could you provide a concrete example for such a $f$?
$endgroup$
– 0xbadf00d
Dec 29 '18 at 16:22
$begingroup$
@0xbadf00d cmon. if you read/understand my solution, I won't need to
$endgroup$
– mathworker21
Dec 29 '18 at 23:10
$begingroup$
Could you provide a concrete example for such a $f$?
$endgroup$
– 0xbadf00d
Dec 29 '18 at 16:22
$begingroup$
Could you provide a concrete example for such a $f$?
$endgroup$
– 0xbadf00d
Dec 29 '18 at 16:22
$begingroup$
@0xbadf00d cmon. if you read/understand my solution, I won't need to
$endgroup$
– mathworker21
Dec 29 '18 at 23:10
$begingroup$
@0xbadf00d cmon. if you read/understand my solution, I won't need to
$endgroup$
– mathworker21
Dec 29 '18 at 23:10
add a comment |
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