Origin of term Ahlfors-David regular












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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







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    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







    share|cite|improve this question

























      3












      3








      3







      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







      share|cite|improve this question













      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






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      mdr

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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.






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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.






            share|cite|improve this answer


























              3














              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.






              share|cite|improve this answer
























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                3








                3






                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.






                share|cite|improve this answer












                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.







                share|cite|improve this answer












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                share|cite|improve this answer










                answered 3 hours ago









                Josiah Park

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