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
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
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
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 6 hours ago
mdr
1063
1063
add a comment |
add a comment |
1 Answer
1
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.
add a comment |
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "504"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f319955%2forigin-of-term-ahlfors-david-regular%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
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.
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.
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.
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 4 hours ago
Josiah Park
1,037319
1,037319
add a comment |
add a comment |
Thanks for contributing an answer to MathOverflow!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
Some of your past answers have not been well-received, and you're in danger of being blocked from answering.
Please pay close attention to the following guidance:
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
To learn more, see our tips on writing great answers.
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmathoverflow.net%2fquestions%2f319955%2forigin-of-term-ahlfors-david-regular%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown