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Schwartz functions dense in weighted $L^p$ space?
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Upon learning about Schwartz functions, one result that is usually presented to students is that not only are Schwartz functions dense in $L^2(mathbbR^n)$ but in $L^p(mathbbR^n)$ for all $1 le p < infty$. Here, $L^p(mathbbR^n)$ is the standard $L^p$ space with respect to Lebesgue measure.
With how nicely Schwartz functions behave (with their smooth rapid decay), does this result extend to weighted $L^p$ spaces? For instance, suppose we have a weight $w in A_p$ ($1 < p < infty$) where $A_p$ is the class of Muckenhoupt weights of parameter $p$. Let $L^p(w)$ be the space of Lebesgue measurable functions over $mathbbR^n$ such that
$$ | f|_L^p(w) = left( int_mathbbR^n |f(x)|^p w(x) ,dx right)^1/p < infty$$
(so by taking $w=1$, we get back to standard $L^p$ space.)
Can we say Schwartz functions are still dense in $L^p(w)$?
real-analysis harmonic-analysis
asked Jul 27 at 16:24
welshman500
326213
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up vote
4
down vote
favorite
Upon learning about Schwartz functions, one result that is usually presented to students is that not only are Schwartz functions dense in $L^2(mathbbR^n)$ but in $L^p(mathbbR^n)$ for all $1 le p < infty$. Here, $L^p(mathbbR^n)$ is the standard $L^p$ space with respect to Lebesgue measure.
With how nicely Schwartz functions behave (with their smooth rapid decay), does this result extend to weighted $L^p$ spaces? For instance, suppose we have a weight $w in A_p$ ($1 < p < infty$) where $A_p$ is the class of Muckenhoupt weights of parameter $p$. Let $L^p(w)$ be the space of Lebesgue measurable functions over $mathbbR^n$ such that
$$ | f|_L^p(w) = left( int_mathbbR^n |f(x)|^p w(x) ,dx right)^1/p < infty$$
(so by taking $w=1$, we get back to standard $L^p$ space.)
Can we say Schwartz functions are still dense in $L^p(w)$?
real-analysis harmonic-analysis
asked Jul 27 at 16:24
welshman500
326213
add a comment |Â
up vote
4
down vote
favorite
up vote
4
down vote
favorite
Upon learning about Schwartz functions, one result that is usually presented to students is that not only are Schwartz functions dense in $L^2(mathbbR^n)$ but in $L^p(mathbbR^n)$ for all $1 le p < infty$. Here, $L^p(mathbbR^n)$ is the standard $L^p$ space with respect to Lebesgue measure.
With how nicely Schwartz functions behave (with their smooth rapid decay), does this result extend to weighted $L^p$ spaces? For instance, suppose we have a weight $w in A_p$ ($1 < p < infty$) where $A_p$ is the class of Muckenhoupt weights of parameter $p$. Let $L^p(w)$ be the space of Lebesgue measurable functions over $mathbbR^n$ such that
$$ | f|_L^p(w) = left( int_mathbbR^n |f(x)|^p w(x) ,dx right)^1/p < infty$$
(so by taking $w=1$, we get back to standard $L^p$ space.)
Can we say Schwartz functions are still dense in $L^p(w)$?
real-analysis harmonic-analysis
asked Jul 27 at 16:24
welshman500
326213
Upon learning about Schwartz functions, one result that is usually presented to students is that not only are Schwartz functions dense in $L^2(mathbbR^n)$ but in $L^p(mathbbR^n)$ for all $1 le p < infty$. Here, $L^p(mathbbR^n)$ is the standard $L^p$ space with respect to Lebesgue measure.
With how nicely Schwartz functions behave (with their smooth rapid decay), does this result extend to weighted $L^p$ spaces? For instance, suppose we have a weight $w in A_p$ ($1 < p < infty$) where $A_p$ is the class of Muckenhoupt weights of parameter $p$. Let $L^p(w)$ be the space of Lebesgue measurable functions over $mathbbR^n$ such that
$$ | f|_L^p(w) = left( int_mathbbR^n |f(x)|^p w(x) ,dx right)^1/p < infty$$
(so by taking $w=1$, we get back to standard $L^p$ space.)
Can we say Schwartz functions are still dense in $L^p(w)$?
real-analysis harmonic-analysis
asked Jul 27 at 16:24
welshman500
326213
asked Jul 27 at 16:24
welshman500
326213
asked Jul 27 at 16:24
welshman500
326213
asked Jul 27 at 16:24
welshman500
326213
326213
add a comment |Â
add a comment |Â
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