Extending function from hyperplane segment to ball, estimating integral by integral on manifold
$begingroup$
Let $B:=B_R (0) subset mathbb{R}^d$ be the ball with radius $R$ at $0$ and $H subset mathbb{R}^d$ a hyperplane that satisfies $H cap B neq emptyset$ and $0 notin H$. Furthermore, let $f in C^1 (H)$ with $f geq 0$.
Consider the function $g : , B to mathbb{R}$ given by
begin{equation*}
g (x) =
begin{cases}
f(xi), & xi = (0,x) cap H neq emptyset \
0, & text{otherwise}
end{cases}
end{equation*}
where $(0,x)$ denotes an open line-segment. Note that $g$ is piecewise constant in $x-$direction. Basicly, $g$ extends $f$ from $H$ to $B$ by assigning $f(xi)$ along any $x-$direction from $H$ towards the boundary of $B$.
I am interested in deriving an estimate of the form
begin{equation*}
int_B g(x) , dx leq c , R int_{H cap B} f(xi) dxi ,
end{equation*}
where $c$ is a constant only depending on the dimension $d$, but am having trouble proving it (nicely). Here is my approach so far:
Transform integral to spherical coordinates with transformation $Psi$:
begin{equation*}
int_B g(x) , dx
= int_{S^{d-1}} int_{0}^{R} g (Psi^{-1}(r, s)) , r^{d-1} , dr , ds
end{equation*}Parametrize $H cap B$ over $S^{d-1}$: Let $U = lbrace s in S^{d-1}: , (0, 2 R s) cap H cap B neq emptyset rbrace$ and $h: , U to mathbb{R}$ a suitable $C^1 (U)$ function, such that $gamma :, U to H$ defines a parametrization of $H$ over $U subset S^{d-1}$ by $gamma (s) = Psi^{-1} (h(s), s)$. $H$ is the graph of $h$, so its first fundamental form satisfies $g^{gamma} (s) = 1 + Vert nabla h (s) Vert^2 geq 1$. Thus,
begin{align*}
& phantom{{}={}} int_{S^{d-1}} int_{0}^{R} g (Psi^{-1} (r, s)) , r^{d-1} , dr , ds
leq int_{S^{d-1}} frac{1}{d} R^{d} f left(Psi^{-1}(h (s), s) right) , ds \
& leq frac{1}{d} R int_{U} f (Psi^{-1} (h(s),s)) , sqrt{g^{gamma}} , ds
= frac{1}{d} R int_{H cap B} f (xi) , dxi
end{align*}
Any hints or comments are appreciated.
integration manifolds differential-forms integral-transforms
asked Jan 7 at 9:12
user404633user404633
63
$endgroup$
add a comment |
$begingroup$
Let $B:=B_R (0) subset mathbb{R}^d$ be the ball with radius $R$ at $0$ and $H subset mathbb{R}^d$ a hyperplane that satisfies $H cap B neq emptyset$ and $0 notin H$. Furthermore, let $f in C^1 (H)$ with $f geq 0$.
Consider the function $g : , B to mathbb{R}$ given by
begin{equation*}
g (x) =
begin{cases}
f(xi), & xi = (0,x) cap H neq emptyset \
0, & text{otherwise}
end{cases}
end{equation*}
where $(0,x)$ denotes an open line-segment. Note that $g$ is piecewise constant in $x-$direction. Basicly, $g$ extends $f$ from $H$ to $B$ by assigning $f(xi)$ along any $x-$direction from $H$ towards the boundary of $B$.
I am interested in deriving an estimate of the form
begin{equation*}
int_B g(x) , dx leq c , R int_{H cap B} f(xi) dxi ,
end{equation*}
where $c$ is a constant only depending on the dimension $d$, but am having trouble proving it (nicely). Here is my approach so far:
Transform integral to spherical coordinates with transformation $Psi$:
begin{equation*}
int_B g(x) , dx
= int_{S^{d-1}} int_{0}^{R} g (Psi^{-1}(r, s)) , r^{d-1} , dr , ds
end{equation*}Parametrize $H cap B$ over $S^{d-1}$: Let $U = lbrace s in S^{d-1}: , (0, 2 R s) cap H cap B neq emptyset rbrace$ and $h: , U to mathbb{R}$ a suitable $C^1 (U)$ function, such that $gamma :, U to H$ defines a parametrization of $H$ over $U subset S^{d-1}$ by $gamma (s) = Psi^{-1} (h(s), s)$. $H$ is the graph of $h$, so its first fundamental form satisfies $g^{gamma} (s) = 1 + Vert nabla h (s) Vert^2 geq 1$. Thus,
begin{align*}
& phantom{{}={}} int_{S^{d-1}} int_{0}^{R} g (Psi^{-1} (r, s)) , r^{d-1} , dr , ds
leq int_{S^{d-1}} frac{1}{d} R^{d} f left(Psi^{-1}(h (s), s) right) , ds \
& leq frac{1}{d} R int_{U} f (Psi^{-1} (h(s),s)) , sqrt{g^{gamma}} , ds
= frac{1}{d} R int_{H cap B} f (xi) , dxi
end{align*}
Any hints or comments are appreciated.
integration manifolds differential-forms integral-transforms
asked Jan 7 at 9:12
user404633user404633
63
$endgroup$
add a comment |
$begingroup$
Let $B:=B_R (0) subset mathbb{R}^d$ be the ball with radius $R$ at $0$ and $H subset mathbb{R}^d$ a hyperplane that satisfies $H cap B neq emptyset$ and $0 notin H$. Furthermore, let $f in C^1 (H)$ with $f geq 0$.
Consider the function $g : , B to mathbb{R}$ given by
begin{equation*}
g (x) =
begin{cases}
f(xi), & xi = (0,x) cap H neq emptyset \
0, & text{otherwise}
end{cases}
end{equation*}
where $(0,x)$ denotes an open line-segment. Note that $g$ is piecewise constant in $x-$direction. Basicly, $g$ extends $f$ from $H$ to $B$ by assigning $f(xi)$ along any $x-$direction from $H$ towards the boundary of $B$.
I am interested in deriving an estimate of the form
begin{equation*}
int_B g(x) , dx leq c , R int_{H cap B} f(xi) dxi ,
end{equation*}
where $c$ is a constant only depending on the dimension $d$, but am having trouble proving it (nicely). Here is my approach so far:
Transform integral to spherical coordinates with transformation $Psi$:
begin{equation*}
int_B g(x) , dx
= int_{S^{d-1}} int_{0}^{R} g (Psi^{-1}(r, s)) , r^{d-1} , dr , ds
end{equation*}Parametrize $H cap B$ over $S^{d-1}$: Let $U = lbrace s in S^{d-1}: , (0, 2 R s) cap H cap B neq emptyset rbrace$ and $h: , U to mathbb{R}$ a suitable $C^1 (U)$ function, such that $gamma :, U to H$ defines a parametrization of $H$ over $U subset S^{d-1}$ by $gamma (s) = Psi^{-1} (h(s), s)$. $H$ is the graph of $h$, so its first fundamental form satisfies $g^{gamma} (s) = 1 + Vert nabla h (s) Vert^2 geq 1$. Thus,
begin{align*}
& phantom{{}={}} int_{S^{d-1}} int_{0}^{R} g (Psi^{-1} (r, s)) , r^{d-1} , dr , ds
leq int_{S^{d-1}} frac{1}{d} R^{d} f left(Psi^{-1}(h (s), s) right) , ds \
& leq frac{1}{d} R int_{U} f (Psi^{-1} (h(s),s)) , sqrt{g^{gamma}} , ds
= frac{1}{d} R int_{H cap B} f (xi) , dxi
end{align*}
Any hints or comments are appreciated.
integration manifolds differential-forms integral-transforms
asked Jan 7 at 9:12
user404633user404633
63
$endgroup$
Let $B:=B_R (0) subset mathbb{R}^d$ be the ball with radius $R$ at $0$ and $H subset mathbb{R}^d$ a hyperplane that satisfies $H cap B neq emptyset$ and $0 notin H$. Furthermore, let $f in C^1 (H)$ with $f geq 0$.
Consider the function $g : , B to mathbb{R}$ given by
begin{equation*}
g (x) =
begin{cases}
f(xi), & xi = (0,x) cap H neq emptyset \
0, & text{otherwise}
end{cases}
end{equation*}
where $(0,x)$ denotes an open line-segment. Note that $g$ is piecewise constant in $x-$direction. Basicly, $g$ extends $f$ from $H$ to $B$ by assigning $f(xi)$ along any $x-$direction from $H$ towards the boundary of $B$.
I am interested in deriving an estimate of the form
begin{equation*}
int_B g(x) , dx leq c , R int_{H cap B} f(xi) dxi ,
end{equation*}
where $c$ is a constant only depending on the dimension $d$, but am having trouble proving it (nicely). Here is my approach so far:
Transform integral to spherical coordinates with transformation $Psi$:
begin{equation*}
int_B g(x) , dx
= int_{S^{d-1}} int_{0}^{R} g (Psi^{-1}(r, s)) , r^{d-1} , dr , ds
end{equation*}Parametrize $H cap B$ over $S^{d-1}$: Let $U = lbrace s in S^{d-1}: , (0, 2 R s) cap H cap B neq emptyset rbrace$ and $h: , U to mathbb{R}$ a suitable $C^1 (U)$ function, such that $gamma :, U to H$ defines a parametrization of $H$ over $U subset S^{d-1}$ by $gamma (s) = Psi^{-1} (h(s), s)$. $H$ is the graph of $h$, so its first fundamental form satisfies $g^{gamma} (s) = 1 + Vert nabla h (s) Vert^2 geq 1$. Thus,
begin{align*}
& phantom{{}={}} int_{S^{d-1}} int_{0}^{R} g (Psi^{-1} (r, s)) , r^{d-1} , dr , ds
leq int_{S^{d-1}} frac{1}{d} R^{d} f left(Psi^{-1}(h (s), s) right) , ds \
& leq frac{1}{d} R int_{U} f (Psi^{-1} (h(s),s)) , sqrt{g^{gamma}} , ds
= frac{1}{d} R int_{H cap B} f (xi) , dxi
end{align*}
Any hints or comments are appreciated.
integration manifolds differential-forms integral-transforms
integration manifolds differential-forms integral-transforms
asked Jan 7 at 9:12
user404633user404633
63
asked Jan 7 at 9:12
user404633user404633
63
edited Jan 7 at 13:46
user404633
asked Jan 7 at 9:12
user404633user404633
63
asked Jan 7 at 9:12
user404633user404633
63
asked Jan 7 at 9:12
user404633user404633
63
63
add a comment |
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