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Show that functions $a in C^infty(overlinemathbbB^n)$ with $a|_partialoverlinemathbbB^n = 0$ belong to $S^-1(mathbbR^n)$
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Let $mathbbB^n$ denote the open unit ball in $mathbbR^n$. I am reading a paper which claims that functions $a in C^infty(overlinemathbbB^n)$ such that $a|_partialoverlinemathbbB^n = 0$ belong to the symbol class $S^-1(mathbbR^n)$. The paper does not precisely define what $C^infty(overlinemathbbB^n)$ is, but I think it's meant to be
$$ain C^infty(overlinemathbbB^n) iff a in C^infty(mathbbB^n) text and all its partial derivatives have continuous extensions to $overlinemathbbB^n$.$$
On the other hand, the class $S^-1(mathbbR^n)$ is precisely defined to be
$$a in S^-1(mathbbR^n) iff a in C^infty(mathbbR^n) text and |partial^alpha a(x) | le C_alphalangle xrangle^alpha text for all multi-indices $alpha$. $$
Above, $langle x rangle = (1 + |x|^2)^frac12.$
To prove that functions $a in C^infty(overlinemathbbB^n)$, $a|_partialoverlinemathbbB^n = 0$, belong to $S^-1(mathbbR^n)$, one first must explain how $C^infty(overlinemathbbB^n)$ is identified as a subset of $C^infty(mathbbR)$. I think the author means to do it just using $a|_mathbbB^n$, via the map
$$mathbbR^n ni x mapsto varphi(x) = fracxlangle x rangle in mathbbB^n , quad C^infty(overlinemathbbB^n) ni a mapsto a(varphi(cdot)) in C^infty(mathbbR^n),$$
or something similar.
Now the goal is to show that $a in C^infty(overlinemathbbB^n)$, $a|_partialoverlinemathbbB^n = 0$, implies $ partial_x^alphaa(varphi(x)) le C_alpha langle x rangle^alpha$ for all $alpha$.
I am having trouble even showing the base case that $langle x rangle a(varphi(x))$ is bounded on $mathbbR^n.$ I wonder if a higher dimensional version of Taylor's formula would be useful but I'm unsure how to proceed.
real-analysis smooth-manifolds microlocal-analysis
asked Jul 27 at 7:15
JZShapiro
1,94311023
add a comment |Â
up vote
1
down vote
favorite
Let $mathbbB^n$ denote the open unit ball in $mathbbR^n$. I am reading a paper which claims that functions $a in C^infty(overlinemathbbB^n)$ such that $a|_partialoverlinemathbbB^n = 0$ belong to the symbol class $S^-1(mathbbR^n)$. The paper does not precisely define what $C^infty(overlinemathbbB^n)$ is, but I think it's meant to be
$$ain C^infty(overlinemathbbB^n) iff a in C^infty(mathbbB^n) text and all its partial derivatives have continuous extensions to $overlinemathbbB^n$.$$
On the other hand, the class $S^-1(mathbbR^n)$ is precisely defined to be
$$a in S^-1(mathbbR^n) iff a in C^infty(mathbbR^n) text and |partial^alpha a(x) | le C_alphalangle xrangle^alpha text for all multi-indices $alpha$. $$
Above, $langle x rangle = (1 + |x|^2)^frac12.$
To prove that functions $a in C^infty(overlinemathbbB^n)$, $a|_partialoverlinemathbbB^n = 0$, belong to $S^-1(mathbbR^n)$, one first must explain how $C^infty(overlinemathbbB^n)$ is identified as a subset of $C^infty(mathbbR)$. I think the author means to do it just using $a|_mathbbB^n$, via the map
$$mathbbR^n ni x mapsto varphi(x) = fracxlangle x rangle in mathbbB^n , quad C^infty(overlinemathbbB^n) ni a mapsto a(varphi(cdot)) in C^infty(mathbbR^n),$$
or something similar.
Now the goal is to show that $a in C^infty(overlinemathbbB^n)$, $a|_partialoverlinemathbbB^n = 0$, implies $ partial_x^alphaa(varphi(x)) le C_alpha langle x rangle^alpha$ for all $alpha$.
I am having trouble even showing the base case that $langle x rangle a(varphi(x))$ is bounded on $mathbbR^n.$ I wonder if a higher dimensional version of Taylor's formula would be useful but I'm unsure how to proceed.
real-analysis smooth-manifolds microlocal-analysis
asked Jul 27 at 7:15
JZShapiro
1,94311023
add a comment |Â
up vote
1
down vote
favorite
up vote
1
down vote
favorite
Let $mathbbB^n$ denote the open unit ball in $mathbbR^n$. I am reading a paper which claims that functions $a in C^infty(overlinemathbbB^n)$ such that $a|_partialoverlinemathbbB^n = 0$ belong to the symbol class $S^-1(mathbbR^n)$. The paper does not precisely define what $C^infty(overlinemathbbB^n)$ is, but I think it's meant to be
$$ain C^infty(overlinemathbbB^n) iff a in C^infty(mathbbB^n) text and all its partial derivatives have continuous extensions to $overlinemathbbB^n$.$$
On the other hand, the class $S^-1(mathbbR^n)$ is precisely defined to be
$$a in S^-1(mathbbR^n) iff a in C^infty(mathbbR^n) text and |partial^alpha a(x) | le C_alphalangle xrangle^alpha text for all multi-indices $alpha$. $$
Above, $langle x rangle = (1 + |x|^2)^frac12.$
To prove that functions $a in C^infty(overlinemathbbB^n)$, $a|_partialoverlinemathbbB^n = 0$, belong to $S^-1(mathbbR^n)$, one first must explain how $C^infty(overlinemathbbB^n)$ is identified as a subset of $C^infty(mathbbR)$. I think the author means to do it just using $a|_mathbbB^n$, via the map
$$mathbbR^n ni x mapsto varphi(x) = fracxlangle x rangle in mathbbB^n , quad C^infty(overlinemathbbB^n) ni a mapsto a(varphi(cdot)) in C^infty(mathbbR^n),$$
or something similar.
Now the goal is to show that $a in C^infty(overlinemathbbB^n)$, $a|_partialoverlinemathbbB^n = 0$, implies $ partial_x^alphaa(varphi(x)) le C_alpha langle x rangle^alpha$ for all $alpha$.
I am having trouble even showing the base case that $langle x rangle a(varphi(x))$ is bounded on $mathbbR^n.$ I wonder if a higher dimensional version of Taylor's formula would be useful but I'm unsure how to proceed.
real-analysis smooth-manifolds microlocal-analysis
asked Jul 27 at 7:15
JZShapiro
1,94311023
Let $mathbbB^n$ denote the open unit ball in $mathbbR^n$. I am reading a paper which claims that functions $a in C^infty(overlinemathbbB^n)$ such that $a|_partialoverlinemathbbB^n = 0$ belong to the symbol class $S^-1(mathbbR^n)$. The paper does not precisely define what $C^infty(overlinemathbbB^n)$ is, but I think it's meant to be
$$ain C^infty(overlinemathbbB^n) iff a in C^infty(mathbbB^n) text and all its partial derivatives have continuous extensions to $overlinemathbbB^n$.$$
On the other hand, the class $S^-1(mathbbR^n)$ is precisely defined to be
$$a in S^-1(mathbbR^n) iff a in C^infty(mathbbR^n) text and |partial^alpha a(x) | le C_alphalangle xrangle^alpha text for all multi-indices $alpha$. $$
Above, $langle x rangle = (1 + |x|^2)^frac12.$
To prove that functions $a in C^infty(overlinemathbbB^n)$, $a|_partialoverlinemathbbB^n = 0$, belong to $S^-1(mathbbR^n)$, one first must explain how $C^infty(overlinemathbbB^n)$ is identified as a subset of $C^infty(mathbbR)$. I think the author means to do it just using $a|_mathbbB^n$, via the map
$$mathbbR^n ni x mapsto varphi(x) = fracxlangle x rangle in mathbbB^n , quad C^infty(overlinemathbbB^n) ni a mapsto a(varphi(cdot)) in C^infty(mathbbR^n),$$
or something similar.
Now the goal is to show that $a in C^infty(overlinemathbbB^n)$, $a|_partialoverlinemathbbB^n = 0$, implies $ partial_x^alphaa(varphi(x)) le C_alpha langle x rangle^alpha$ for all $alpha$.
I am having trouble even showing the base case that $langle x rangle a(varphi(x))$ is bounded on $mathbbR^n.$ I wonder if a higher dimensional version of Taylor's formula would be useful but I'm unsure how to proceed.
real-analysis smooth-manifolds microlocal-analysis
asked Jul 27 at 7:15
JZShapiro
1,94311023
asked Jul 27 at 7:15
JZShapiro
1,94311023
asked Jul 27 at 7:15
JZShapiro
1,94311023
asked Jul 27 at 7:15
JZShapiro
1,94311023
1,94311023
add a comment |Â
add a comment |Â
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