Harmonic measure, Riesz transforms, and uniform rectifiability Xavier Tolsa 12 May 2017 X. Tolsa (ICREA / UAB) Harmonic measure, Riesz transforms, and rectifiability 12 May 2017 1 / 19
Rectifiability We say that E ⊂ R d is rectifiable if it is H 1 -a.e. contained in a countable union of curves of finite length. E is n -rectifiable if it is H n -a.e. contained in a countable union of C 1 (or Lipschitz) n -dimensional manifolds. X. Tolsa (ICREA / UAB) Harmonic measure, Riesz transforms, and rectifiability 12 May 2017 2 / 19
Rectifiability We say that E ⊂ R d is rectifiable if it is H 1 -a.e. contained in a countable union of curves of finite length. E is n -rectifiable if it is H n -a.e. contained in a countable union of C 1 (or Lipschitz) n -dimensional manifolds. E is n -AD-regular if H n ( B ( x , r ) ∩ E ) ≈ r n for all x ∈ E , 0 < r ≤ diam( E ). E is uniformly n -rectifiable if it is n -AD-regular and there are M , θ > 0 such that for all x ∈ E , 0 < r ≤ diam( E ), there exists a Lipschitz map g : R n ⊃ B n (0 , r ) → R d , �∇ g � ∞ ≤ M , such that H n � ≥ θ r n . � E ∩ B ( x , r ) ∩ g ( B n (0 , r )) X. Tolsa (ICREA / UAB) Harmonic measure, Riesz transforms, and rectifiability 12 May 2017 2 / 19
Rectifiability We say that E ⊂ R d is rectifiable if it is H 1 -a.e. contained in a countable union of curves of finite length. E is n -rectifiable if it is H n -a.e. contained in a countable union of C 1 (or Lipschitz) n -dimensional manifolds. E is n -AD-regular if H n ( B ( x , r ) ∩ E ) ≈ r n for all x ∈ E , 0 < r ≤ diam( E ). E is uniformly n -rectifiable if it is n -AD-regular and there are M , θ > 0 such that for all x ∈ E , 0 < r ≤ diam( E ), there exists a Lipschitz map g : R n ⊃ B n (0 , r ) → R d , �∇ g � ∞ ≤ M , such that H n � ≥ θ r n . � E ∩ B ( x , r ) ∩ g ( B n (0 , r )) Uniform n -rectifiability is a quantitative version of n -rectifiability introduced by David and Semmes (originating from P. Jones TST). X. Tolsa (ICREA / UAB) Harmonic measure, Riesz transforms, and rectifiability 12 May 2017 2 / 19
The Riesz and Cauchy transforms Let µ be a Borel measure in R d . Example: the Hausdorff measure H n E , where H n ( E ) < ∞ . X. Tolsa (ICREA / UAB) Harmonic measure, Riesz transforms, and rectifiability 12 May 2017 3 / 19
The Riesz and Cauchy transforms Let µ be a Borel measure in R d . Example: the Hausdorff measure H n E , where H n ( E ) < ∞ . In R d , the n -dimensional Riesz transform of f ∈ L 1 loc ( µ ) is R µ f ( x ) = lim ε ց 0 R µ,ε f ( x ), where � x − y R µ,ε f ( x ) = | x − y | n +1 f ( y ) d µ ( y ) . | x − y | >ε X. Tolsa (ICREA / UAB) Harmonic measure, Riesz transforms, and rectifiability 12 May 2017 3 / 19
The Riesz and Cauchy transforms Let µ be a Borel measure in R d . Example: the Hausdorff measure H n E , where H n ( E ) < ∞ . In R d , the n -dimensional Riesz transform of f ∈ L 1 loc ( µ ) is R µ f ( x ) = lim ε ց 0 R µ,ε f ( x ), where � x − y R µ,ε f ( x ) = | x − y | n +1 f ( y ) d µ ( y ) . | x − y | >ε In C , the Cauchy transform of f ∈ L 1 loc ( µ ) is C µ f ( z ) = lim ε ց 0 C µ,ε f ( z ), where � f ( ξ ) C µ,ε f ( z ) = z − ξ d µ ( ξ ) . | z − ξ | >ε X. Tolsa (ICREA / UAB) Harmonic measure, Riesz transforms, and rectifiability 12 May 2017 3 / 19
The Riesz and Cauchy transforms Let µ be a Borel measure in R d . Example: the Hausdorff measure H n E , where H n ( E ) < ∞ . In R d , the n -dimensional Riesz transform of f ∈ L 1 loc ( µ ) is R µ f ( x ) = lim ε ց 0 R µ,ε f ( x ), where � x − y R µ,ε f ( x ) = | x − y | n +1 f ( y ) d µ ( y ) . | x − y | >ε In C , the Cauchy transform of f ∈ L 1 loc ( µ ) is C µ f ( z ) = lim ε ց 0 C µ,ε f ( z ), where � f ( ξ ) C µ,ε f ( z ) = z − ξ d µ ( ξ ) . | z − ξ | >ε The existence of principal values is not guarantied, in general. X. Tolsa (ICREA / UAB) Harmonic measure, Riesz transforms, and rectifiability 12 May 2017 3 / 19
The Riesz and Cauchy transforms We say that R µ is bounded in L 2 ( µ ) if the operators R µ,ε are bounded in L 2 ( µ ) uniformly on ε > 0. X. Tolsa (ICREA / UAB) Harmonic measure, Riesz transforms, and rectifiability 12 May 2017 4 / 19
The Riesz and Cauchy transforms We say that R µ is bounded in L 2 ( µ ) if the operators R µ,ε are bounded in L 2 ( µ ) uniformly on ε > 0. Analogously for C µ . X. Tolsa (ICREA / UAB) Harmonic measure, Riesz transforms, and rectifiability 12 May 2017 4 / 19
The Riesz and Cauchy transforms We say that R µ is bounded in L 2 ( µ ) if the operators R µ,ε are bounded in L 2 ( µ ) uniformly on ε > 0. Analogously for C µ . We also denote R µ = R µ 1 , C µ = C µ 1 . X. Tolsa (ICREA / UAB) Harmonic measure, Riesz transforms, and rectifiability 12 May 2017 4 / 19
Harmonic measure Ω ⊂ R n +1 open and connected. For p ∈ Ω, ω p is the harmonic measure in Ω with pole in p . X. Tolsa (ICREA / UAB) Harmonic measure, Riesz transforms, and rectifiability 12 May 2017 5 / 19
Harmonic measure Ω ⊂ R n +1 open and connected. For p ∈ Ω, ω p is the harmonic measure in Ω with pole in p . That is, for E ⊂ ∂ Ω, ω p ( E ) is the value at p of the harmonic extension of χ E to Ω. X. Tolsa (ICREA / UAB) Harmonic measure, Riesz transforms, and rectifiability 12 May 2017 5 / 19
Harmonic measure Ω ⊂ R n +1 open and connected. For p ∈ Ω, ω p is the harmonic measure in Ω with pole in p . That is, for E ⊂ ∂ Ω, ω p ( E ) is the value at p of the harmonic extension of χ E to Ω. Questions about the metric properties of harmonic measure: When H n ≈ ω p ? Which is the connection with rectifiability? Dimension of harmonic measure? X. Tolsa (ICREA / UAB) Harmonic measure, Riesz transforms, and rectifiability 12 May 2017 5 / 19
Metric properties of harmonic measure In the plane if Ω is simply connected and H 1 ( ∂ Ω) < ∞ , then H 1 ≈ ω p . (F.& M. Riesz) Many results in C using complex analysis (Makarov, Jones, Bishop, Wolff,...). The analogue of Riesz theorem fails in higher dimensions (counterexamples by Wu and Ziemer). In higher dimension, need real analysis techniques. Connection with uniform rectifiability studied recently by Hofmann, Martell, Uriarte-Tuero, Mayboroda, Azzam, Badger, Bortz, Toro, Akman, etc. X. Tolsa (ICREA / UAB) Harmonic measure, Riesz transforms, and rectifiability 12 May 2017 6 / 19
Connection between harmonic measure and Riesz transform Let E ( x ) the fundamental solution of the Laplacian in R n +1 : 1 E ( x ) = c n | x | n − 1 . X. Tolsa (ICREA / UAB) Harmonic measure, Riesz transforms, and rectifiability 12 May 2017 7 / 19
Connection between harmonic measure and Riesz transform Let E ( x ) the fundamental solution of the Laplacian in R n +1 : 1 E ( x ) = c n | x | n − 1 . The kernel of the Riesz transform is x K ( x ) = | x | n +1 = c ∇E ( x ) . X. Tolsa (ICREA / UAB) Harmonic measure, Riesz transforms, and rectifiability 12 May 2017 7 / 19
Connection between harmonic measure and Riesz transform Let E ( x ) the fundamental solution of the Laplacian in R n +1 : 1 E ( x ) = c n | x | n − 1 . The kernel of the Riesz transform is x K ( x ) = | x | n +1 = c ∇E ( x ) . The Green function G ( · , · ) of Ω is � E ( x − y ) d ω p ( y ) . G ( x , p ) = E ( x − p ) − X. Tolsa (ICREA / UAB) Harmonic measure, Riesz transforms, and rectifiability 12 May 2017 7 / 19
Connection between harmonic measure and Riesz transform Let E ( x ) the fundamental solution of the Laplacian in R n +1 : 1 E ( x ) = c n | x | n − 1 . The kernel of the Riesz transform is x K ( x ) = | x | n +1 = c ∇E ( x ) . The Green function G ( · , · ) of Ω is � E ( x − y ) d ω p ( y ) . G ( x , p ) = E ( x − p ) − Therefore, for x ∈ Ω: � K ( x − y ) d ω p ( y ) . c ∇ x G ( x , p ) = K ( x − p ) − R ω p ( x ) = K ( x − p ) − c ∇ x G ( x , p ) . That is, X. Tolsa (ICREA / UAB) Harmonic measure, Riesz transforms, and rectifiability 12 May 2017 7 / 19
Riesz transforms and rectifiability Theorem (Nazarov, T., Volberg, 2012) Let E ⊂ R n +1 with H n ( E ) < ∞ , and µ = H n E . If R µ : L 2 ( µ ) → L 2 ( µ ) is bounded, then E n-rectifiable. If, additionally, E is n-AD-regular, then E is uniformly n-rectifiable. X. Tolsa (ICREA / UAB) Harmonic measure, Riesz transforms, and rectifiability 12 May 2017 8 / 19
Riesz transforms and rectifiability Theorem (Nazarov, T., Volberg, 2012) Let E ⊂ R n +1 with H n ( E ) < ∞ , and µ = H n E . If R µ : L 2 ( µ ) → L 2 ( µ ) is bounded, then E n-rectifiable. If, additionally, E is n-AD-regular, then E is uniformly n-rectifiable. David-Semmes problem. X. Tolsa (ICREA / UAB) Harmonic measure, Riesz transforms, and rectifiability 12 May 2017 8 / 19
Riesz transforms and rectifiability Theorem (Nazarov, T., Volberg, 2012) Let E ⊂ R n +1 with H n ( E ) < ∞ , and µ = H n E . If R µ : L 2 ( µ ) → L 2 ( µ ) is bounded, then E n-rectifiable. If, additionally, E is n-AD-regular, then E is uniformly n-rectifiable. David-Semmes problem. Case of lower density 0 by Eiderman-Nazarov-Volberg. X. Tolsa (ICREA / UAB) Harmonic measure, Riesz transforms, and rectifiability 12 May 2017 8 / 19
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