Mixing
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(x-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 9 hours ago
M. Rahmat
292211
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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(x-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 9 hours ago
M. Rahmat
292211
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(x-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 9 hours ago
M. Rahmat
292211
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(x-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 9 hours ago
M. Rahmat
292211
asked 9 hours ago
M. Rahmat
292211
asked 9 hours ago
M. Rahmat
292211
asked 9 hours ago
M. Rahmat
292211
asked 9 hours ago
M. Rahmat
292211
292211
add a comment |
add a comment |
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