Origin of term Ahlfors-David regular
Much of the literature on analysis in metric spaces makes use of an assumption called Ahlfors regularity or Ahlfors-David regularity. Let $q>0$. A metric space $(X,d)$ is Ahlfors(-David) $q$-regular if there exists $Cgeq 1$ such that $C^{-1}r^q leq mathcal{H}^q(B(x,r)) leq Cr^q$ for all $x in X$ and $r in (0, text{diam } X)$. Here, $mathcal{H}^q$ denotes the $q$-dimensional Hausdorff measure. The term is so ubiquitous in the literature in this area that the origin seems impossible to trace down. I believe David refers to Guy David.
Can anyone fill me in on the origin of the terms and what exactly Ahlfors and David did using this type of condition?
reference-request mg.metric-geometry cv.complex-variables geometric-measure-theory
asked Jan 2 at 21:29
mdrmdr
1484
add a comment |
Much of the literature on analysis in metric spaces makes use of an assumption called Ahlfors regularity or Ahlfors-David regularity. Let $q>0$. A metric space $(X,d)$ is Ahlfors(-David) $q$-regular if there exists $Cgeq 1$ such that $C^{-1}r^q leq mathcal{H}^q(B(x,r)) leq Cr^q$ for all $x in X$ and $r in (0, text{diam } X)$. Here, $mathcal{H}^q$ denotes the $q$-dimensional Hausdorff measure. The term is so ubiquitous in the literature in this area that the origin seems impossible to trace down. I believe David refers to Guy David.
Can anyone fill me in on the origin of the terms and what exactly Ahlfors and David did using this type of condition?
reference-request mg.metric-geometry cv.complex-variables geometric-measure-theory
asked Jan 2 at 21:29
mdrmdr
1484
add a comment |
Much of the literature on analysis in metric spaces makes use of an assumption called Ahlfors regularity or Ahlfors-David regularity. Let $q>0$. A metric space $(X,d)$ is Ahlfors(-David) $q$-regular if there exists $Cgeq 1$ such that $C^{-1}r^q leq mathcal{H}^q(B(x,r)) leq Cr^q$ for all $x in X$ and $r in (0, text{diam } X)$. Here, $mathcal{H}^q$ denotes the $q$-dimensional Hausdorff measure. The term is so ubiquitous in the literature in this area that the origin seems impossible to trace down. I believe David refers to Guy David.
Can anyone fill me in on the origin of the terms and what exactly Ahlfors and David did using this type of condition?
reference-request mg.metric-geometry cv.complex-variables geometric-measure-theory
asked Jan 2 at 21:29
mdrmdr
1484
Much of the literature on analysis in metric spaces makes use of an assumption called Ahlfors regularity or Ahlfors-David regularity. Let $q>0$. A metric space $(X,d)$ is Ahlfors(-David) $q$-regular if there exists $Cgeq 1$ such that $C^{-1}r^q leq mathcal{H}^q(B(x,r)) leq Cr^q$ for all $x in X$ and $r in (0, text{diam } X)$. Here, $mathcal{H}^q$ denotes the $q$-dimensional Hausdorff measure. The term is so ubiquitous in the literature in this area that the origin seems impossible to trace down. I believe David refers to Guy David.
Can anyone fill me in on the origin of the terms and what exactly Ahlfors and David did using this type of condition?
reference-request mg.metric-geometry cv.complex-variables geometric-measure-theory
reference-request mg.metric-geometry cv.complex-variables geometric-measure-theory
asked Jan 2 at 21:29
mdrmdr
1484
asked Jan 2 at 21:29
mdrmdr
1484
asked Jan 2 at 21:29
mdrmdr
1484
asked Jan 2 at 21:29
mdrmdr
1484
asked Jan 2 at 21:29
mdrmdr
1484
1484
add a comment |
add a comment |
2 Answers
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To answer the last question, Calderón's problem was a question regarding mapping properties of the Cauchy integral
$$ C_{Gamma}f(z)=frac{1}{2pi i} intlimits_{Gamma} frac{f(xi)}{xi-z} dxi$$
namely, to determine the rectifiable Jordan curves $Gamma$ for which $C_{Gamma}$ gives rise to a bounded operator on $L^2(Gamma)$. This was solved by Guy David in 1984 who showed that $C_{Gamma}$ is bounded on $L^2(Gamma)$ precisely when $Gamma$ satisfies $ mathcal{H}(Gamma cap B(z_{0},r)) leq Cr$ for every $z_{0}inmathbb{C}$, $r>0$ and some constant $C$. This opened up a large study of (what was called then) Ahlfors regularity by David and Semmes. Some of the results of that study are collected in their monograph from the 90's, Analysis of and on Uniformly Rectifiable Sets.
answered Jan 2 at 23:07
Josiah ParkJosiah Park
1,264320
add a comment |
When $q=1$ this condition appears in Ahlfors' famous paper
Zur Theorie der Uberlagerungsflache, Acta Math., 65 (1935) for which he was awarded one of the two very first Fields medals. For long time it was known
to complex analysts just as the "Ahlfors condition".
answered Jan 3 at 5:27
Alexandre EremenkoAlexandre Eremenko
49.4k6137254
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
To answer the last question, Calderón's problem was a question regarding mapping properties of the Cauchy integral
$$ C_{Gamma}f(z)=frac{1}{2pi i} intlimits_{Gamma} frac{f(xi)}{xi-z} dxi$$
namely, to determine the rectifiable Jordan curves $Gamma$ for which $C_{Gamma}$ gives rise to a bounded operator on $L^2(Gamma)$. This was solved by Guy David in 1984 who showed that $C_{Gamma}$ is bounded on $L^2(Gamma)$ precisely when $Gamma$ satisfies $ mathcal{H}(Gamma cap B(z_{0},r)) leq Cr$ for every $z_{0}inmathbb{C}$, $r>0$ and some constant $C$. This opened up a large study of (what was called then) Ahlfors regularity by David and Semmes. Some of the results of that study are collected in their monograph from the 90's, Analysis of and on Uniformly Rectifiable Sets.
answered Jan 2 at 23:07
Josiah ParkJosiah Park
1,264320
add a comment |
To answer the last question, Calderón's problem was a question regarding mapping properties of the Cauchy integral
$$ C_{Gamma}f(z)=frac{1}{2pi i} intlimits_{Gamma} frac{f(xi)}{xi-z} dxi$$
namely, to determine the rectifiable Jordan curves $Gamma$ for which $C_{Gamma}$ gives rise to a bounded operator on $L^2(Gamma)$. This was solved by Guy David in 1984 who showed that $C_{Gamma}$ is bounded on $L^2(Gamma)$ precisely when $Gamma$ satisfies $ mathcal{H}(Gamma cap B(z_{0},r)) leq Cr$ for every $z_{0}inmathbb{C}$, $r>0$ and some constant $C$. This opened up a large study of (what was called then) Ahlfors regularity by David and Semmes. Some of the results of that study are collected in their monograph from the 90's, Analysis of and on Uniformly Rectifiable Sets.
answered Jan 2 at 23:07
Josiah ParkJosiah Park
1,264320
add a comment |
To answer the last question, Calderón's problem was a question regarding mapping properties of the Cauchy integral
$$ C_{Gamma}f(z)=frac{1}{2pi i} intlimits_{Gamma} frac{f(xi)}{xi-z} dxi$$
namely, to determine the rectifiable Jordan curves $Gamma$ for which $C_{Gamma}$ gives rise to a bounded operator on $L^2(Gamma)$. This was solved by Guy David in 1984 who showed that $C_{Gamma}$ is bounded on $L^2(Gamma)$ precisely when $Gamma$ satisfies $ mathcal{H}(Gamma cap B(z_{0},r)) leq Cr$ for every $z_{0}inmathbb{C}$, $r>0$ and some constant $C$. This opened up a large study of (what was called then) Ahlfors regularity by David and Semmes. Some of the results of that study are collected in their monograph from the 90's, Analysis of and on Uniformly Rectifiable Sets.
answered Jan 2 at 23:07
Josiah ParkJosiah Park
1,264320
To answer the last question, Calderón's problem was a question regarding mapping properties of the Cauchy integral
$$ C_{Gamma}f(z)=frac{1}{2pi i} intlimits_{Gamma} frac{f(xi)}{xi-z} dxi$$
namely, to determine the rectifiable Jordan curves $Gamma$ for which $C_{Gamma}$ gives rise to a bounded operator on $L^2(Gamma)$. This was solved by Guy David in 1984 who showed that $C_{Gamma}$ is bounded on $L^2(Gamma)$ precisely when $Gamma$ satisfies $ mathcal{H}(Gamma cap B(z_{0},r)) leq Cr$ for every $z_{0}inmathbb{C}$, $r>0$ and some constant $C$. This opened up a large study of (what was called then) Ahlfors regularity by David and Semmes. Some of the results of that study are collected in their monograph from the 90's, Analysis of and on Uniformly Rectifiable Sets.
answered Jan 2 at 23:07
Josiah ParkJosiah Park
1,264320
answered Jan 2 at 23:07
Josiah ParkJosiah Park
1,264320
answered Jan 2 at 23:07
Josiah ParkJosiah Park
1,264320
answered Jan 2 at 23:07
Josiah ParkJosiah Park
1,264320
1,264320
add a comment |
add a comment |
When $q=1$ this condition appears in Ahlfors' famous paper
Zur Theorie der Uberlagerungsflache, Acta Math., 65 (1935) for which he was awarded one of the two very first Fields medals. For long time it was known
to complex analysts just as the "Ahlfors condition".
answered Jan 3 at 5:27
Alexandre EremenkoAlexandre Eremenko
49.4k6137254
add a comment |
When $q=1$ this condition appears in Ahlfors' famous paper
Zur Theorie der Uberlagerungsflache, Acta Math., 65 (1935) for which he was awarded one of the two very first Fields medals. For long time it was known
to complex analysts just as the "Ahlfors condition".
answered Jan 3 at 5:27
Alexandre EremenkoAlexandre Eremenko
49.4k6137254
add a comment |
When $q=1$ this condition appears in Ahlfors' famous paper
Zur Theorie der Uberlagerungsflache, Acta Math., 65 (1935) for which he was awarded one of the two very first Fields medals. For long time it was known
to complex analysts just as the "Ahlfors condition".
answered Jan 3 at 5:27
Alexandre EremenkoAlexandre Eremenko
49.4k6137254
When $q=1$ this condition appears in Ahlfors' famous paper
Zur Theorie der Uberlagerungsflache, Acta Math., 65 (1935) for which he was awarded one of the two very first Fields medals. For long time it was known
to complex analysts just as the "Ahlfors condition".
answered Jan 3 at 5:27
Alexandre EremenkoAlexandre Eremenko
49.4k6137254
answered Jan 3 at 5:27
Alexandre EremenkoAlexandre Eremenko
49.4k6137254
answered Jan 3 at 5:27
Alexandre EremenkoAlexandre Eremenko
49.4k6137254
answered Jan 3 at 5:27
Alexandre EremenkoAlexandre Eremenko
49.4k6137254
49.4k6137254
add a comment |
add a comment |
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