Kind of passage to the limit in the sense of distributions
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Suppose $B$ is a ball in $mathbb{R}^{n}$ with $n>1$, and $f$ a locally integrable function. Suppose $F$ is a closed set with empty interior and $M>0$ such that the distribution defined by $f$ satisfies
begin{equation}
int_{Bsetminus F}f(t)phi(t)dtleq M
end{equation} for all test function $phi$. We know that if we had
$$f(x)leq M$$ for all $xin Bsetminus F$, we could get the same inequality by passing to the limit (supposing that $f$ is continuous); is it possible, in the same way, to circumvent around $F$ in (1) and prove that (1) holds for integrals over $B$?
real-analysis integration distribution-theory
asked Nov 19 at 6:29
M. Rahmat
295211
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0
down vote
favorite
Suppose $B$ is a ball in $mathbb{R}^{n}$ with $n>1$, and $f$ a locally integrable function. Suppose $F$ is a closed set with empty interior and $M>0$ such that the distribution defined by $f$ satisfies
begin{equation}
int_{Bsetminus F}f(t)phi(t)dtleq M
end{equation} for all test function $phi$. We know that if we had
$$f(x)leq M$$ for all $xin Bsetminus F$, we could get the same inequality by passing to the limit (supposing that $f$ is continuous); is it possible, in the same way, to circumvent around $F$ in (1) and prove that (1) holds for integrals over $B$?
real-analysis integration distribution-theory
asked Nov 19 at 6:29
M. Rahmat
295211
1
Something is off in your formulation. The left-hand side of your inequality is linear with respect to $phi$, and the right-hand side is not. Then again, does $M$ depend on $x$?
– TZakrevskiy
Nov 21 at 13:19
You are right! I got rid of $x$. How about it now?
– M. Rahmat
Nov 22 at 4:40
it is better, but the point about $phi$ being only on the left-hand side still stands.
– TZakrevskiy
Nov 22 at 7:08
Yes. Constant $M$ depends on $varphi$.
– M. Rahmat
Nov 22 at 13:34
Ok, how does $M$ depend on $varphi$? This information is important. Imagine that $M = |f|_{L^1(B)} |varphi|_infty$, in this case your inequality has no new information at all.
– TZakrevskiy
Nov 22 at 13:38
add a comment |
up vote
0
down vote
favorite
up vote
0
down vote
favorite
Suppose $B$ is a ball in $mathbb{R}^{n}$ with $n>1$, and $f$ a locally integrable function. Suppose $F$ is a closed set with empty interior and $M>0$ such that the distribution defined by $f$ satisfies
begin{equation}
int_{Bsetminus F}f(t)phi(t)dtleq M
end{equation} for all test function $phi$. We know that if we had
$$f(x)leq M$$ for all $xin Bsetminus F$, we could get the same inequality by passing to the limit (supposing that $f$ is continuous); is it possible, in the same way, to circumvent around $F$ in (1) and prove that (1) holds for integrals over $B$?
real-analysis integration distribution-theory
asked Nov 19 at 6:29
M. Rahmat
295211
Suppose $B$ is a ball in $mathbb{R}^{n}$ with $n>1$, and $f$ a locally integrable function. Suppose $F$ is a closed set with empty interior and $M>0$ such that the distribution defined by $f$ satisfies
begin{equation}
int_{Bsetminus F}f(t)phi(t)dtleq M
end{equation} for all test function $phi$. We know that if we had
$$f(x)leq M$$ for all $xin Bsetminus F$, we could get the same inequality by passing to the limit (supposing that $f$ is continuous); is it possible, in the same way, to circumvent around $F$ in (1) and prove that (1) holds for integrals over $B$?
real-analysis integration distribution-theory
real-analysis integration distribution-theory
asked Nov 19 at 6:29
M. Rahmat
295211
asked Nov 19 at 6:29
M. Rahmat
295211
edited Nov 22 at 4:39
asked Nov 19 at 6:29
M. Rahmat
295211
asked Nov 19 at 6:29
M. Rahmat
295211
asked Nov 19 at 6:29
M. Rahmat
295211
295211
1
Something is off in your formulation. The left-hand side of your inequality is linear with respect to $phi$, and the right-hand side is not. Then again, does $M$ depend on $x$?
– TZakrevskiy
Nov 21 at 13:19
You are right! I got rid of $x$. How about it now?
– M. Rahmat
Nov 22 at 4:40
it is better, but the point about $phi$ being only on the left-hand side still stands.
– TZakrevskiy
Nov 22 at 7:08
Yes. Constant $M$ depends on $varphi$.
– M. Rahmat
Nov 22 at 13:34
Ok, how does $M$ depend on $varphi$? This information is important. Imagine that $M = |f|_{L^1(B)} |varphi|_infty$, in this case your inequality has no new information at all.
– TZakrevskiy
Nov 22 at 13:38
add a comment |
1
Something is off in your formulation. The left-hand side of your inequality is linear with respect to $phi$, and the right-hand side is not. Then again, does $M$ depend on $x$?
– TZakrevskiy
Nov 21 at 13:19
You are right! I got rid of $x$. How about it now?
– M. Rahmat
Nov 22 at 4:40
it is better, but the point about $phi$ being only on the left-hand side still stands.
– TZakrevskiy
Nov 22 at 7:08
Yes. Constant $M$ depends on $varphi$.
– M. Rahmat
Nov 22 at 13:34
Ok, how does $M$ depend on $varphi$? This information is important. Imagine that $M = |f|_{L^1(B)} |varphi|_infty$, in this case your inequality has no new information at all.
– TZakrevskiy
Nov 22 at 13:38
1
1
Something is off in your formulation. The left-hand side of your inequality is linear with respect to $phi$, and the right-hand side is not. Then again, does $M$ depend on $x$?
– TZakrevskiy
Nov 21 at 13:19
Something is off in your formulation. The left-hand side of your inequality is linear with respect to $phi$, and the right-hand side is not. Then again, does $M$ depend on $x$?
– TZakrevskiy
Nov 21 at 13:19
You are right! I got rid of $x$. How about it now?
– M. Rahmat
Nov 22 at 4:40
You are right! I got rid of $x$. How about it now?
– M. Rahmat
Nov 22 at 4:40
it is better, but the point about $phi$ being only on the left-hand side still stands.
– TZakrevskiy
Nov 22 at 7:08
it is better, but the point about $phi$ being only on the left-hand side still stands.
– TZakrevskiy
Nov 22 at 7:08
Yes. Constant $M$ depends on $varphi$.
– M. Rahmat
Nov 22 at 13:34
Yes. Constant $M$ depends on $varphi$.
– M. Rahmat
Nov 22 at 13:34
Ok, how does $M$ depend on $varphi$? This information is important. Imagine that $M = |f|_{L^1(B)} |varphi|_infty$, in this case your inequality has no new information at all.
– TZakrevskiy
Nov 22 at 13:38
Ok, how does $M$ depend on $varphi$? This information is important. Imagine that $M = |f|_{L^1(B)} |varphi|_infty$, in this case your inequality has no new information at all.
– TZakrevskiy
Nov 22 at 13:38
add a comment |
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1
Something is off in your formulation. The left-hand side of your inequality is linear with respect to $phi$, and the right-hand side is not. Then again, does $M$ depend on $x$?
– TZakrevskiy
Nov 21 at 13:19
You are right! I got rid of $x$. How about it now?
– M. Rahmat
Nov 22 at 4:40
it is better, but the point about $phi$ being only on the left-hand side still stands.
– TZakrevskiy
Nov 22 at 7:08
Yes. Constant $M$ depends on $varphi$.
– M. Rahmat
Nov 22 at 13:34
Ok, how does $M$ depend on $varphi$? This information is important. Imagine that $M = |f|_{L^1(B)} |varphi|_infty$, in this case your inequality has no new information at all.
– TZakrevskiy
Nov 22 at 13:38