Dimension of p-harmonic measure in space Murat Akman Workshop on - PowerPoint PPT Presentation

Bourgain : H − dim ω ≤ n − τ whenever Ω ⊂ R n where τ = τ ( n ) > 0. Wolff : There exists a Wolff snowflake in R 3 for which H − dim ω < 2, and there is another one for which H − dim ω > 2. [B]: Jean Bourgain. On the Hausdorff dimension of harmonic measure in higher dimension. Inv. Math. , 87:477-483, 1987. [W]: Thomas Wolff, Counterexamples with harmonic gradients in R 3 , In Essays on Fourier analysis in honor of Elias M. Stein , 42:321-384, 1995. [LVV]: John L. Lewis, Gregory C. Verchota, and Andrew L. Vogel. Wolff snowflakes. Pacific J. Math. , 218 (2005), no. 1, 139166.

Bourgain : H − dim ω ≤ n − τ whenever Ω ⊂ R n where τ = τ ( n ) > 0. Wolff : There exists a Wolff snowflake in R 3 for which H − dim ω < 2, and there is another one for which H − dim ω > 2. Lewis-Verchota-Vogel : Wolff’s result holds in R n ; Harmonic measure on both sides of a Wolff snowflake, say ω + , ω − could have max( H − dim ω + , H − dim ω − ) < n − 1 or min( H − dim ω + , H − dim ω − ) > n − 1 . [B]: Jean Bourgain. On the Hausdorff dimension of harmonic measure in higher dimension. Inv. Math. , 87:477-483, 1987. [W]: Thomas Wolff, Counterexamples with harmonic gradients in R 3 , In Essays on Fourier analysis in honor of Elias M. Stein , 42:321-384, 1995. [LVV]: John L. Lewis, Gregory C. Verchota, and Andrew L. Vogel. Wolff snowflakes. Pacific J. Math. , 218 (2005), no. 1, 139166.

Results of interest for p-harmonic measure For general p � = 2, we call µ as p-harmonic measure associated with a p-harmonic function. [BL]: Bj¨ orn Bennewitz and John Lewis. On the dimension of p-harmonic measure. Ann. Acad. Sci. Fenn. Math. , 30(2):459505, 2005. [LNV]: John Lewis, Kaj Nystr¨ om, and Andrew Vogel. On the dimension of p-harmonic measure in space. J. Eur. Math. Soc. (JEMS) 15 (2013), no. 6, 21972256.

Results of interest for p-harmonic measure For general p � = 2, we call µ as p-harmonic measure associated with a p-harmonic function. Bennewitz-Lewis : If ∂ Ω ⊂ R 2 is a quasi circle in the plane then H − dim µ ≥ 1 when 1 < p < 2 while H − dim µ ≤ 1 if 2 < p < ∞ . Moreover, strict inequality holds for H − dim µ when ∂ Ω is the Von Koch snowflake. [BL]: Bj¨ orn Bennewitz and John Lewis. On the dimension of p-harmonic measure. Ann. Acad. Sci. Fenn. Math. , 30(2):459505, 2005. [LNV]: John Lewis, Kaj Nystr¨ om, and Andrew Vogel. On the dimension of p-harmonic measure in space. J. Eur. Math. Soc. (JEMS) 15 (2013), no. 6, 21972256.

Results of interest for p-harmonic measure For general p � = 2, we call µ as p-harmonic measure associated with a p-harmonic function. Bennewitz-Lewis : If ∂ Ω ⊂ R 2 is a quasi circle in the plane then H − dim µ ≥ 1 when 1 < p < 2 while H − dim µ ≤ 1 if 2 < p < ∞ . Moreover, strict inequality holds for H − dim µ when ∂ Ω is the Von Koch snowflake. om-Vogel : Lewis-Nystr¨ • µ is concentrated on a set of σ − finite H n − 1 measure when ∂ Ω is sufficiently “flat” in the sense of Reifenberg and p ≥ n . • All examples produced by Wolff snowflake has H − dim µ < n − 1 when p ≥ n . • There is a Wolff snowflake for which H − dim µ > n − 1 when p > 2, near enough 2 [BL]: Bj¨ orn Bennewitz and John Lewis. On the dimension of p-harmonic measure. Ann. Acad. Sci. Fenn. Math. , 30(2):459505, 2005. [LNV]: John Lewis, Kaj Nystr¨ om, and Andrew Vogel. On the dimension of p-harmonic measure in space. J. Eur. Math. Soc. (JEMS) 15 (2013), no. 6, 21972256.

To state our recent work we need a notion of n capacity. If K ⊂ B ( x , r ) is a compact set, define n capacity of K as � |∇ ψ | n d x Cap ( K , B ( x , 2 r )) = inf R n where the infimum is taken over all infinitely differentiable ψ with compact support in B ( x , 2 r ) and ψ ≡ 1 on K .

To state our recent work we need a notion of n capacity. If K ⊂ B ( x , r ) is a compact set, define n capacity of K as � |∇ ψ | n d x Cap ( K , B ( x , 2 r )) = inf R n where the infimum is taken over all infinitely differentiable ψ with compact support in B ( x , 2 r ) and ψ ≡ 1 on K . A compact set E ⊂ R n is said to be locally ( n , r 0 ) uniformly fat or locally uniformly ( n , r 0 ) thick provided there exists r 0 , β > 0 such that whenever x ∈ E , 0 < r ≤ r 0 Cap ( E ∩ B ( x , r ) , B ( x , 2 r )) ≥ β.

Let O ⊂ R n be an open set and ˆ z ∈ ∂ O , ρ > 0.

Let O ⊂ R n be an open set and ˆ z ∈ ∂ O , ρ > 0. Let u > 0 be p-harmonic in O ∩ B (ˆ z , ρ ) with continuous zero boundary values on ∂ O ∩ B (ˆ z , ρ ).

Let O ⊂ R n be an open set and ˆ z ∈ ∂ O , ρ > 0. Let u > 0 be p-harmonic in O ∩ B (ˆ z , ρ ) with continuous zero boundary values on ∂ O ∩ B (ˆ z , ρ ). Extend u to all B (ˆ z , ρ ) by defining u ≡ 0 on B (ˆ z , ρ ) \ O . Then u is p-harmonic in B (ˆ z , ρ ).

Let O ⊂ R n be an open set and ˆ z ∈ ∂ O , ρ > 0. Let u > 0 be p-harmonic in O ∩ B (ˆ z , ρ ) with continuous zero boundary values on ∂ O ∩ B (ˆ z , ρ ). Extend u to all B (ˆ z , ρ ) by defining u ≡ 0 on B (ˆ z , ρ ) \ O . Then u is p-harmonic in B (ˆ z , ρ ). Let µ be the p-harmonic measure associated with u .

ˆ z

ˆ z ρ

△ p u = 0 u=0 u=0 u=0 u=0 u > 0 ˆ z u=0 u=0 ρ

△ p u = 0 u=0 u=0 u=0 u=0 u > 0 z ˆ u=0 u=0 ρ

New result for p-harmonic measure in space Theorem A (A.-Lewis-Vogel) If p > n then µ is concentrated on a set of σ − finite H n − 1 measure. Same result holds when p = n provided that ∂ O ∩ B (ˆ z , ρ ) is locally uniformly fat in the sense of n − capacity. [ALV]: Murat Akman, John Lewis, and Andrew Vogel, Hausdorff dimension and σ − finiteness of p − harmonic measures in space when p ≥ n . a rXiv:1306.5617, submitted.

New result for p-harmonic measure in space Theorem A (A.-Lewis-Vogel) If p > n then µ is concentrated on a set of σ − finite H n − 1 measure. Same result holds when p = n provided that ∂ O ∩ B (ˆ z , ρ ) is locally uniformly fat in the sense of n − capacity. H − dim µ ≤ n − 1 when p ≥ n . [ALV]: Murat Akman, John Lewis, and Andrew Vogel, Hausdorff dimension and σ − finiteness of p − harmonic measures in space when p ≥ n . a rXiv:1306.5617, submitted.

New result for p-harmonic measure in space Theorem A (A.-Lewis-Vogel) If p > n then µ is concentrated on a set of σ − finite H n − 1 measure. Same result holds when p = n provided that ∂ O ∩ B (ˆ z , ρ ) is locally uniformly fat in the sense of n − capacity. H − dim µ ≤ n − 1 when p ≥ n . The main idea for our proof comes from the 1993 paper of Wolff mentioned earlier. [ALV]: Murat Akman, John Lewis, and Andrew Vogel, Hausdorff dimension and σ − finiteness of p − harmonic measures in space when p ≥ n . a rXiv:1306.5617, submitted.

Some Remarks If w ∈ ∂ O and B ( w , 4 r ) ⊂ B (ˆ z , ρ ) then there exists c = c ( p , n ) ≥ 1 with 1 B ( w , r ) u p − 1 ≤ cr p − n µ ( B ( w , 2 r )) . c r p − n µ ( B ( w , r / 2)) ≤ max

Some Remarks If w ∈ ∂ O and B ( w , 4 r ) ⊂ B (ˆ z , ρ ) then there exists c = c ( p , n ) ≥ 1 with 1 B ( w , r ) u p − 1 ≤ cr p − n µ ( B ( w , 2 r )) . c r p − n µ ( B ( w , r / 2)) ≤ max The left-hand side is true for any open set O and p ≥ n .

Some Remarks If w ∈ ∂ O and B ( w , 4 r ) ⊂ B (ˆ z , ρ ) then there exists c = c ( p , n ) ≥ 1 with 1 B ( w , r ) u p − 1 ≤ cr p − n µ ( B ( w , 2 r )) . c r p − n µ ( B ( w , r / 2)) ≤ max The left-hand side is true for any open set O and p ≥ n . The right-hand side requires uniform fatness assumption when p = n and it is the only place this assumption is used.

Some Remarks If w ∈ ∂ O and B ( w , 4 r ) ⊂ B (ˆ z , ρ ) then there exists c = c ( p , n ) ≥ 1 with 1 B ( w , r ) u p − 1 ≤ cr p − n µ ( B ( w , 2 r )) . c r p − n µ ( B ( w , r / 2)) ≤ max The left-hand side is true for any open set O and p ≥ n . The right-hand side requires uniform fatness assumption when p = n and it is the only place this assumption is used. • Conjecture : Theorem A holds without uniform fatness assumption when p = n .

The tools we have used requires to find a PDE in divergence form for which u , u x k are both solutions and log |∇ u | is a sub solution for p ≥ n at points where ∇ u � = 0.

The tools we have used requires to find a PDE in divergence form for which u , u x k are both solutions and log |∇ u | is a sub solution for p ≥ n at points where ∇ u � = 0. It is known that if n ∂ � ( b ij ζ j ) where b ij = |∇ u | p − 4 [( p − 2) u x i u x j + δ ij |∇ u | 2 ] L ζ = ∂ x i i , j =1 then

The tools we have used requires to find a PDE in divergence form for which u , u x k are both solutions and log |∇ u | is a sub solution for p ≥ n at points where ∇ u � = 0. It is known that if n ∂ � ( b ij ζ j ) where b ij = |∇ u | p − 4 [( p − 2) u x i u x j + δ ij |∇ u | 2 ] L ζ = ∂ x i i , j =1 then n min( p − 1 , 1) | ξ | 2 |∇ u | p − 2 ≤ � b ik ξ i ξ k ≤ max(1 , p − 1) |∇ u | p − 2 | ξ | 2 i , k =1

The tools we have used requires to find a PDE in divergence form for which u , u x k are both solutions and log |∇ u | is a sub solution for p ≥ n at points where ∇ u � = 0. It is known that if n ∂ � ( b ij ζ j ) where b ij = |∇ u | p − 4 [( p − 2) u x i u x j + δ ij |∇ u | 2 ] L ζ = ∂ x i i , j =1 then n min( p − 1 , 1) | ξ | 2 |∇ u | p − 2 ≤ � b ik ξ i ξ k ≤ max(1 , p − 1) |∇ u | p − 2 | ξ | 2 i , k =1 ζ = u and ζ = u x k are both solutions for k = 1 , . . . , n to L ζ = 0.

The tools we have used requires to find a PDE in divergence form for which u , u x k are both solutions and log |∇ u | is a sub solution for p ≥ n at points where ∇ u � = 0. It is known that if n ∂ � ( b ij ζ j ) where b ij = |∇ u | p − 4 [( p − 2) u x i u x j + δ ij |∇ u | 2 ] L ζ = ∂ x i i , j =1 then n min( p − 1 , 1) | ξ | 2 |∇ u | p − 2 ≤ � b ik ξ i ξ k ≤ max(1 , p − 1) |∇ u | p − 2 | ξ | 2 i , k =1 ζ = u and ζ = u x k are both solutions for k = 1 , . . . , n to L ζ = 0. ζ = log |∇ u | is a sub solution to L ζ = 0 when p ≥ n and ∇ u � = 0.

The tools we have used requires to find a PDE in divergence form for which u , u x k are both solutions and log |∇ u | is a sub solution for p ≥ n at points where ∇ u � = 0. It is known that if n ∂ � ( b ij ζ j ) where b ij = |∇ u | p − 4 [( p − 2) u x i u x j + δ ij |∇ u | 2 ] L ζ = ∂ x i i , j =1 then n min( p − 1 , 1) | ξ | 2 |∇ u | p − 2 ≤ � b ik ξ i ξ k ≤ max(1 , p − 1) |∇ u | p − 2 | ξ | 2 i , k =1 ζ = u and ζ = u x k are both solutions for k = 1 , . . . , n to L ζ = 0. ζ = log |∇ u | is a sub solution to L ζ = 0 when p ≥ n and ∇ u � = 0. Is log |∇ u | a super solution when p < n and |∇ u | � = 0?

The tools we have used requires to find a PDE in divergence form for which u , u x k are both solutions and log |∇ u | is a sub solution for p ≥ n at points where ∇ u � = 0. It is known that if n ∂ � ( b ij ζ j ) where b ij = |∇ u | p − 4 [( p − 2) u x i u x j + δ ij |∇ u | 2 ] L ζ = ∂ x i i , j =1 then n min( p − 1 , 1) | ξ | 2 |∇ u | p − 2 ≤ � b ik ξ i ξ k ≤ max(1 , p − 1) |∇ u | p − 2 | ξ | 2 i , k =1 ζ = u and ζ = u x k are both solutions for k = 1 , . . . , n to L ζ = 0. ζ = log |∇ u | is a sub solution to L ζ = 0 when p ≥ n and ∇ u � = 0. Is log |∇ u | a super solution when p < n and |∇ u | � = 0? • Conjecture : There is p 0 , 2 < p 0 < n , such that if p 0 ≤ p then H − dim µ ≤ n − 1.

Sketch of the Proof of Theorem A

Sketch of the Proof of Theorem A Proposition Let λ be a non decreasing function on [0 , 1] with λ ( t ) lim t n − 1 = 0 . t → 0 There exists c = c ( p , n ) and a set Q ⊂ ∂ O ∩ B (ˆ z , ρ ) such that µ ( ∂ O ∩ B (ˆ z , ρ ) \ Q ) = 0 and for every w ∈ Q there exists arbitrarily small r = r ( w ) > 0 and a compact set F = F ( w , r ) such that H λ ( F ) = 0 and µ ( B ( w , 100 r )) ≤ c µ ( F ) .

Sketch of the Proof of Theorem A Proposition Let λ be a non decreasing function on [0 , 1] with λ ( t ) lim t n − 1 = 0 . t → 0 There exists c = c ( p , n ) and a set Q ⊂ ∂ O ∩ B (ˆ z , ρ ) such that µ ( ∂ O ∩ B (ˆ z , ρ ) \ Q ) = 0 and for every w ∈ Q there exists arbitrarily small r = r ( w ) > 0 and a compact set F = F ( w , r ) such that H λ ( F ) = 0 and µ ( B ( w , 100 r )) ≤ c µ ( F ) . We first show how our result follows from this proposition.

First observation: H n − 1 ( P m ) < ∞ for each positive integer m where � µ ( B ( x , t )) > 1 � P m := x ∈ ∂ O ∩ B (ˆ z , ρ ) : lim sup . t n − 1 m t → 0

First observation: H n − 1 ( P m ) < ∞ for each positive integer m where � µ ( B ( x , t )) > 1 � P m := x ∈ ∂ O ∩ B (ˆ z , ρ ) : lim sup . t n − 1 m t → 0 Therefore, this set has � µ ( B ( x , t )) � P = x ∈ ∂ O ∩ B (ˆ z , ρ ) : lim sup > 0 . t n − 1 t → 0 has σ − finite H n − 1 measure.

First observation: H n − 1 ( P m ) < ∞ for each positive integer m where � µ ( B ( x , t )) > 1 � P m := x ∈ ∂ O ∩ B (ˆ z , ρ ) : lim sup . t n − 1 m t → 0 Therefore, this set has � µ ( B ( x , t )) � P = x ∈ ∂ O ∩ B (ˆ z , ρ ) : lim sup > 0 . t n − 1 t → 0 has σ − finite H n − 1 measure. Second observation: From Proposition and measure theoretic arguments there exists a Borel set Q 1 ⊂ Q with z , ρ ) \ Q 1 ) = 0 and H λ ( Q 1 ) = 0 . µ ( ∂ O ∩ B (ˆ

Third observation: µ ( Q \ P ) = 0.

Third observation: µ ( Q \ P ) = 0. Otherwise, there is a compact set K ⊂ Q \ P and a positive non decreasing λ 0 ( t ) λ 0 with lim t n − 1 = 0 satisfying t → 0 µ ( B ( x , t )) µ ( K ) > 0 and lim = 0 uniformly for x ∈ K . λ 0 ( t ) t → 0

Third observation: µ ( Q \ P ) = 0. Otherwise, there is a compact set K ⊂ Q \ P and a positive non decreasing λ 0 ( t ) λ 0 with lim t n − 1 = 0 satisfying t → 0 µ ( B ( x , t )) µ ( K ) > 0 and lim = 0 uniformly for x ∈ K . λ 0 ( t ) t → 0 This tells us that µ ≪ H λ 0 on K . Choose Q 1 relative to λ 0 to conclude that H λ 0 ( K ∩ Q 1 ) = 0 implies µ ( K ∩ Q 1 ) = µ ( K ) = 0 � .

Third observation: µ ( Q \ P ) = 0. Otherwise, there is a compact set K ⊂ Q \ P and a positive non decreasing λ 0 ( t ) λ 0 with lim t n − 1 = 0 satisfying t → 0 µ ( B ( x , t )) µ ( K ) > 0 and lim = 0 uniformly for x ∈ K . λ 0 ( t ) t → 0 This tells us that µ ≪ H λ 0 on K . Choose Q 1 relative to λ 0 to conclude that H λ 0 ( K ∩ Q 1 ) = 0 implies µ ( K ∩ Q 1 ) = µ ( K ) = 0 � . µ is concentrated on P which has σ − finite H n − 1 measure. This finishes the proof of our result assuming Proposition.

Sketch of the Proof of Proposition Translation, dilation invariance of the p-Laplacian and a measure theoretic argument to reduce the proof of Proposition to the situation when w = 0, B (0 , 100) ⊂ B (ˆ z , ρ ).

Sketch of the Proof of Proposition Translation, dilation invariance of the p-Laplacian and a measure theoretic argument to reduce the proof of Proposition to the situation when w = 0, B (0 , 100) ⊂ B (ˆ z , ρ ). There is some c = c(p, n) and 2 ≤ t ≤ 50 such that 1 B (0 , t ) u ≤ c µ ( B (0 , 100)) ≤ c 2 . c ≤ µ ( B (0 , 1)) ≤ max B (0 , 2) u ≤ max

Sketch of the Proof of Proposition Translation, dilation invariance of the p-Laplacian and a measure theoretic argument to reduce the proof of Proposition to the situation when w = 0, B (0 , 100) ⊂ B (ˆ z , ρ ). There is some c = c(p, n) and 2 ≤ t ≤ 50 such that 1 B (0 , t ) u ≤ c µ ( B (0 , 100)) ≤ c 2 . c ≤ µ ( B (0 , 1)) ≤ max B (0 , 2) u ≤ max To finish the proof of Proposition, it suffices to show for given small ǫ, τ > 0 that there exists a Borel set E ⊂ ∂ O ∩ B (0 , 20) and c = c ( p , n ) ≥ 1 with τ ( E ) ≤ ǫ and µ ( E ) ≥ 1 φ λ c .

A stopping time argument Let M a large positive number and s < e − M . For each z ∈ ∂ O ∩ B (0 , 15) there is t = t ( z ), 0 < t < 1 with either ( α ) µ ( B ( z , t )) = Mt n − 1 , t > s or ( β ) t = s .

A stopping time argument Let M a large positive number and s < e − M . For each z ∈ ∂ O ∩ B (0 , 15) there is t = t ( z ), 0 < t < 1 with either ( α ) µ ( B ( z , t )) = Mt n − 1 , t > s or ( β ) t = s . Use the Besicovitch covering theorem to get a covering B ( z j , t j ) N 1 of ∂ O ∩ B (0 , 15) where t j = t ( z j ) is the maximal for which either ( α ) or ( β ) holds.

N � Ω := O ∩ B (0 , 15) \ B ( z i , t i ) and D := Ω \ B (˜ z , 2 r 1 ) i =1 △ p ˆ u = 0 B ( z i , t i ) u > 0 ˆ Let ˆ u be the p-harmonic function in D with continuous boundary values, u = ˆ min u on ∂ B (˜ z , 2 r 1 ) and ˆ u = 0 on ∂ Ω. Let ˆ µ be the p-harmonic B (˜ z , 2 r 1 ) measure associated with ˆ u .

u ≤ u in D . ˆ

u ≤ u in D . ˆ ∂ Ω is smooth except for a set of finite H n − 2

u ≤ u in D . ˆ ∂ Ω is smooth except for a set of finite H n − 2 Using some barrier type estimate one can also show 1 p − 1 in D |∇ ˆ u | ≤ cM

u ≤ u in D . ˆ ∂ Ω is smooth except for a set of finite H n − 2 Using some barrier type estimate one can also show 1 p − 1 in D |∇ ˆ u | ≤ cM and B ( z j , 2 t j ) u p − 1 ≤ c 2 t 1 − n µ ( B ( z j , t j )) ≤ ct 1 − p t 1 − n ˆ max µ ( B ( z j , 4 t j )) . j j j

For a given A >> 1, { 1 , . . . , N } can be divided into disjoint subsets G , B , U as  G := { j : t j > s }  u | p − 1 ≥ M − A for some x ∈ ∂ Ω ∩ ∂ B ( z j , t j ) } B := { j : t j = s and |∇ ˆ U := { j : j is not in G or B} 

For a given A >> 1, { 1 , . . . , N } can be divided into disjoint subsets G , B , U as  G := { j : t j > s }  u | p − 1 ≥ M − A for some x ∈ ∂ Ω ∩ ∂ B ( z j , t j ) } B := { j : t j = s and |∇ ˆ U := { j : j is not in G or B}  We define � E := ∂ O ∩ B ( z j , t j ) j ∈G∪B

For a given A >> 1, { 1 , . . . , N } can be divided into disjoint subsets G , B , U as  G := { j : t j > s }  u | p − 1 ≥ M − A for some x ∈ ∂ Ω ∩ ∂ B ( z j , t j ) } B := { j : t j = s and |∇ ˆ U := { j : j is not in G or B}  We define � E := ∂ O ∩ B ( z j , t j ) j ∈G∪B Easy to show φ λ τ ( E ) ≤ ǫ

We show � u | p − 1 | log |∇ ˆ u || d H n − 1 ≤ c ′ log M |∇ ˆ ∂ Ω

We show � u | p − 1 | log |∇ ˆ u || d H n − 1 ≤ c ′ log M |∇ ˆ ∂ Ω We use this to show u ( x ) | p − 1 ≤ M − A } ) � µ ( ∂ Ω ∩ ˆ B ( z j , t j )) ≤ ˆ µ ( { x ∈ ∂ Ω : |∇ ˆ j ∈U ≤ ( p − 1) � u || d H n − 1 ≤ c u | p − 1 | log |∇ ˆ |∇ ˆ ( AlogM ) A ∂ Ω

We show � u | p − 1 | log |∇ ˆ u || d H n − 1 ≤ c ′ log M |∇ ˆ ∂ Ω We use this to show u ( x ) | p − 1 ≤ M − A } ) � µ ( ∂ Ω ∩ ˆ B ( z j , t j )) ≤ ˆ µ ( { x ∈ ∂ Ω : |∇ ˆ j ∈U ≤ ( p − 1) � u || d H n − 1 ≤ c u | p − 1 | log |∇ ˆ |∇ ˆ ( AlogM ) A ∂ Ω A is ours to choose, and we choose it very large to make this set small.

We show � u | p − 1 | log |∇ ˆ u || d H n − 1 ≤ c ′ log M |∇ ˆ ∂ Ω We use this to show u ( x ) | p − 1 ≤ M − A } ) � µ ( ∂ Ω ∩ ˆ B ( z j , t j )) ≤ ˆ µ ( { x ∈ ∂ Ω : |∇ ˆ j ∈U ≤ ( p − 1) � u || d H n − 1 ≤ c u | p − 1 | log |∇ ˆ |∇ ˆ ( AlogM ) A ∂ Ω A is ours to choose, and we choose it very large to make this set small. Use this to prove µ ( E ) ≥ 1 / c .

Part II: Example of domain in R n for which H − dim µ < n − 1 [GM]: John B. Garnett and Donald E. Marshall, Harmonic Measure, volume 2 of New Mathematical Monographs. Cambridge University Press , Cambridge, 2008.

Part II: Example of domain in R n for which H − dim µ < n − 1 There is an unpublished result of Jones-Wolff in [GM, Chapter IX, Theorem 3.1]; Jones-Wolff : Let Ω = C ∪ {∞} \ C where C is a certain compact set. Then H − dim ω < 1. [GM]: John B. Garnett and Donald E. Marshall, Harmonic Measure, volume 2 of New Mathematical Monographs. Cambridge University Press , Cambridge, 2008.

Part II: Example of domain in R n for which H − dim µ < n − 1 There is an unpublished result of Jones-Wolff in [GM, Chapter IX, Theorem 3.1]; Jones-Wolff : Let Ω = C ∪ {∞} \ C where C is a certain compact set. Then H − dim ω < 1. We generalized this result to p-harmonic measure, µ , in R n for p ≥ n ≥ 2 and for a certain domain. [GM]: John B. Garnett and Donald E. Marshall, Harmonic Measure, volume 2 of New Mathematical Monographs. Cambridge University Press , Cambridge, 2008.

Let S ′ be the square with side length 1 / 2 and center 0 in R n . Let C 0 = S ′ .

Let S ′ be the square with side length 1 / 2 and center 0 in R n . Let C 0 = S ′ . Let Q 11 , . . . , Q 14 be the squares of the four corners of C 0 of side length a 1 , 4 � 0 < α < a 1 < β < 1 / 4, and let C 1 = Q 1 i . i =1

Let S ′ be the square with side length 1 / 2 and center 0 in R n . Let C 0 = S ′ . Let Q 11 , . . . , Q 14 be the squares of the four corners of C 0 of side length a 1 , 4 � 0 < α < a 1 < β < 1 / 4, and let C 1 = Q 1 i . i =1 Let { Q 2 j } , j = 1 , . . . , 16 be the square of corners of each Q 1 i , i = 1 , . . . , 4 16 � of side length a 1 a 2 , α < a 2 < β . Let C 2 = Q 2 j . j =1 S ′ C 0 C 1 C 2

Let S ′ be the square with side length 1 / 2 and center 0 in R n . Let C 0 = S ′ . Let Q 11 , . . . , Q 14 be the squares of the four corners of C 0 of side length a 1 , 4 � 0 < α < a 1 < β < 1 / 4, and let C 1 = Q 1 i . i =1 Let { Q 2 j } , j = 1 , . . . , 16 be the square of corners of each Q 1 i , i = 1 , . . . , 4 16 � of side length a 1 a 2 , α < a 2 < β . Let C 2 = Q 2 j . j =1 S ′ C 0 C 1 C 2 Continuing recursively, at the m th step we get 4 m squares Q mj , 1 ≤ j ≤ 4 m 4 m � of side length a 1 a 2 . . . a m , α < a m < β and let C m = Q mj . j =1 Then C is obtained as the limit in the Hausdorff metric of C m as m → ∞

Let S = 2 S ′ ⊂ R n and let u be a p-harmonic function in S \ C with boundary values u = 1 on ∂ S and u = 0 on C . Let µ be the p-harmonic measure associated to u .

Let S = 2 S ′ ⊂ R n and let u be a p-harmonic function in S \ C with boundary values u = 1 on ∂ S and u = 0 on C . Let µ be the p-harmonic measure associated to u . Following Jones-Wolff argument, using sub solution estimates, a stopping time argument similar to the one we have used we show that

Let S = 2 S ′ ⊂ R n and let u be a p-harmonic function in S \ C with boundary values u = 1 on ∂ S and u = 0 on C . Let µ be the p-harmonic measure associated to u . Following Jones-Wolff argument, using sub solution estimates, a stopping time argument similar to the one we have used we show that Theorem B (A.-Lewis-Vogel) H − dim µ < n − 1 when p ≥ n .

Is there any other measure or PDE that one can study the same problem? [BL]: Bj¨ orn Bennewitz and John Lewis. On the dimension of p-harmonic measure. Ann. Acad. Sci. Fenn. Math. , 30(2):459505, 2005.

Is there any other measure or PDE that one can study the same problem? In [HKM, Chapter 21], it was shown that the measure associated with a positive weak solution u with 0 boundary values for a larger class of qusailinear elliptic PDEs exists; div A ( x , ∇ u ) = 0 where A : R n × R n → R n satisfies certain structural assumptions. The measure is so called A -harmonic measure. [BL]: Bj¨ orn Bennewitz and John Lewis. On the dimension of p-harmonic measure. Ann. Acad. Sci. Fenn. Math. , 30(2):459505, 2005.

Is there any other measure or PDE that one can study the same problem? In [HKM, Chapter 21], it was shown that the measure associated with a positive weak solution u with 0 boundary values for a larger class of qusailinear elliptic PDEs exists; div A ( x , ∇ u ) = 0 where A : R n × R n → R n satisfies certain structural assumptions. The measure is so called A -harmonic measure. If A ( ξ ) = | ξ | p − 2 ξ , then the above PDE becomes the usual p-Laplace equation. [BL]: Bj¨ orn Bennewitz and John Lewis. On the dimension of p-harmonic measure. Ann. Acad. Sci. Fenn. Math. , 30(2):459505, 2005.

Is there any other measure or PDE that one can study the same problem? In [HKM, Chapter 21], it was shown that the measure associated with a positive weak solution u with 0 boundary values for a larger class of qusailinear elliptic PDEs exists; div A ( x , ∇ u ) = 0 where A : R n × R n → R n satisfies certain structural assumptions. The measure is so called A -harmonic measure. If A ( ξ ) = | ξ | p − 2 ξ , then the above PDE becomes the usual p-Laplace equation. In [BL, Closing remarks 10], the authors pointed out this fact and asked for what PDE one can obtain dimension estimates on the associated measure. [BL]: Bj¨ orn Bennewitz and John Lewis. On the dimension of p-harmonic measure. Ann. Acad. Sci. Fenn. Math. , 30(2):459505, 2005.

Is there any other measure or PDE that one can study the same problem? In [HKM, Chapter 21], it was shown that the measure associated with a positive weak solution u with 0 boundary values for a larger class of qusailinear elliptic PDEs exists; div A ( x , ∇ u ) = 0 where A : R n × R n → R n satisfies certain structural assumptions. The measure is so called A -harmonic measure. If A ( ξ ) = | ξ | p − 2 ξ , then the above PDE becomes the usual p-Laplace equation. In [BL, Closing remarks 10], the authors pointed out this fact and asked for what PDE one can obtain dimension estimates on the associated measure. { Laplace } ⊆ { p-Laplace } ⊆ {A − Harmonic PDEs } . [BL]: Bj¨ orn Bennewitz and John Lewis. On the dimension of p-harmonic measure. Ann. Acad. Sci. Fenn. Math. , 30(2):459505, 2005.

Is there any other measure or PDE that one can study the same problem? In [HKM, Chapter 21], it was shown that the measure associated with a positive weak solution u with 0 boundary values for a larger class of qusailinear elliptic PDEs exists; div A ( x , ∇ u ) = 0 where A : R n × R n → R n satisfies certain structural assumptions. The measure is so called A -harmonic measure. If A ( ξ ) = | ξ | p − 2 ξ , then the above PDE becomes the usual p-Laplace equation. In [BL, Closing remarks 10], the authors pointed out this fact and asked for what PDE one can obtain dimension estimates on the associated measure. { Laplace } ⊆ { p-Laplace } ⊆ {△ f u = 0 } ⊆ {A − Harmonic PDEs } . [BL]: Bj¨ orn Bennewitz and John Lewis. On the dimension of p-harmonic measure. Ann. Acad. Sci. Fenn. Math. , 30(2):459505, 2005.

Introduction Let p be fixed, 1 < p < ∞ . Let f be a function with following properties;

Introduction Let p be fixed, 1 < p < ∞ . Let f be a function with following properties; (a) f : R n → (0 , ∞ ) is homogeneous of degree p . That is, f ( η ) = | η | p f ( η | η | ) > 0 when η ∈ R n \ { 0 } .

Introduction Let p be fixed, 1 < p < ∞ . Let f be a function with following properties; (a) f : R n → (0 , ∞ ) is homogeneous of degree p . That is, f ( η ) = | η | p f ( η | η | ) > 0 when η ∈ R n \ { 0 } . (b) f is uniformly convex in B (0 , 1) \ B (0 , 1 / 2). That is, D f is Lipschitz and ∃ c ≥ 1 such that for a.e. η ∈ R n , n ∂ 2 f 1 2 < | η | < 1 and all ξ ∈ R n we have c − 1 | ξ | 2 ≤ � ( η ) ξ j ξ k ≤ c | ξ | 2 . ∂η j η k j , k =1

We consider weak solutions, u , to the Euler Lagrange equation; n ∂ � ∂ f ( ∇ u ) � � △ f u := = 0 ∂ x i ∂η i i =1

We consider weak solutions, u , to the Euler Lagrange equation; n ∂ � ∂ f ( ∇ u ) � � △ f u := = 0 ∂ x i ∂η i i =1 in Ω ∩ N where N is an open neighborhood of ∂ Ω. Assume also that u > 0 in N ∩ Ω with continuous boundary values on ∂ Ω. Set u ≡ 0 in N \ Ω to have u ∈ W 1 , p ( N ) and △ f u = 0 weakly in N . Then, there exists a unique finite positive Borel measure µ f associated with u having support contained in ∂ Ω satisfying � � φ d µ f whenever φ ∈ C ∞ �∇ η f ( ∇ u ) , ∇ φ � d x = − 0 ( N ) .

We consider weak solutions, u , to the Euler Lagrange equation; n ∂ � ∂ f ( ∇ u ) � � △ f u := = 0 ∂ x i ∂η i i =1 in Ω ∩ N where N is an open neighborhood of ∂ Ω. Assume also that u > 0 in N ∩ Ω with continuous boundary values on ∂ Ω. Set u ≡ 0 in N \ Ω to have u ∈ W 1 , p ( N ) and △ f u = 0 weakly in N . Then, there exists a unique finite positive Borel measure µ f associated with u having support contained in ∂ Ω satisfying � � φ d µ f whenever φ ∈ C ∞ �∇ η f ( ∇ u ) , ∇ φ � d x = − 0 ( N ) . • f ( η ) = | η | 2 → Laplace equation, △ u = 0. • f ( η ) = | η | p , 1 < p < ∞ → p-Laplace equation, div( |∇ u | p − 2 ∇ u ) = 0.

If we define n ∂ � � � L ζ = f η i η j ζ j ∂ x i i =1 Then

If we define n ∂ � � � L ζ = f η i η j ζ j ∂ x i i =1 Then ζ = u is a weak solution to L ζ = 0

If we define n ∂ � � � L ζ = f η i η j ζ j ∂ x i i =1 Then ζ = u is a weak solution to L ζ = 0 ζ = u x k for k = 1 , . . . , n is weak solution to L ζ = 0