Many Theorems and a Few Stories John Garnett Seoul 5/12/17 1 - PDF document

Many Theorems and a Few Stories John Garnett Seoul 5/12/17 1

OUTLINE I. Extension Theorems. II. BMO and A p Weights. III. Constructions with H ∞ Interpolation, ∂ , and BMO. IV. Corona Theorems and Problems. V. Harmonic Measure and Integral Mean Spectra. VI. Traveling Salesman Theorem. VII. Work with Bishop: Harmonic Measure and Kleinian Groups. VIII. Applied Mathematics. IX. Random Welding. 2

I. Extension Theorems for BMO and Sobolev Spaces. Ω ⊂ R d connected and open, ϕ : Ω → R , 1 � || ϕ || BMO (Ω) = sup | ϕ − ϕ Q | dx | Q | Q ⊂ Ω Q 1 � where Q is a | Q | = its measure, ϕ Q = Q ϕdx. | Q | Theorem 1: Every ϕ ∈ BMO (Ω) has extension in BMO ( R d ) if and only if for all x, y ∈ Ω ds ( z ) � log δ ( x ) | x − y | � � � � � inf δ ( z ) ≤ C � + C log 2 + , � � δ ( y ) δ ( x ) + δ ( y ) Ω ⊃ γ joins x,y γ where δ ( x ) = dist( x, ∂ Ω) . 3

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Corollary: Ω ⊂ R 2 and ∂ Ω = Γ a Jordan curve. Every ϕ ∈ BMO (Ω) has extension in BMO ( R 2 ) � | w 1 − w 3 | ≤ C | w 1 − w 2 | for w 1 , w 2 ∈ Γ and w 3 on the smaller arc ( w 1 , w 2 ) , i.e. if and only if Γ is a quasicircle. 4

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L p k (Ω) = { f ∈ L p (Ω) : | α | ≤ k = > D α f ∈ L p (Ω) } , for 1 ≤ p ≤ ∞ , k ∈ N . Theorem 2: (Acta 1981) For any k , and p there exists a bounded linear extension operator Λ k : L p k (Ω) → L p k ( R n ) if and only if ∃ ε > 0, 0 < δ ≤ ∞ so that Ω is an ( ε, δ ) domain : x, y ∈ Ω , | x − y | < δ ⇓ ε ∃ arc γ ⊂ Ω joining x, y with length( γ ) ≤ | x − y | and dist( z, ∂ Ω) ≥ ε | x − z || y − z | , ∀ z ∈ γ. | x − y | 5

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II. BMO and A p Weights. John-Nirenberg Theorem: ϕ ∈ BMO ( R d ) ⇔ ∃ c : 1 � e c | ϕ ( x ) − ϕ Q | dx < ∞ . sup (JN) | Q | Q Q Theorem 3: (Annals 1978) If ϕ ∈ BMO ( R d ), then g ∈ L ∞ || ϕ − g || BMO ∼ sup { c : JN holds } . inf 6

A weight w ≥ 0 on R n is an A p -weight 1 ≤ p < ∞ if � 1 �� 1 � � � − 1 sup wdx w p − 1 dx < ∞ . | Q | | Q | Q Q Q (holds if and only if singular integrals or H-L maximal operator is bounded on L p ( w ) . ) Theorem 4: (Annals 1980) w ∈ A p ⇔ w = w 1 w 1 − p , w 1 , w 2 ∈ A 1 . 2 ⇒ Theorem 3. Theorem 4 = See also J. L. Rubio de Francia, Annals of Math 1982, for an elegant non-constructive proof of Theorem 4. 7

Q ⊂ R d is a dyadic cube if ∃ n, k j ∈ Z so that d { k j 2 − n ≤ x j ≤ ( k j + 1)2 − n } . � Q = j =1 ϕ ∈ L 1 loc is BMO d if 1 � || ϕ || BMO d = sup | ϕ − ϕ Q | dx < ∞ . | Q | Q dyadic Q BMO ⊂ BMO d , BMO � = BMO d , but BMO d was a simpler space. Theorem 5: (Pacific J. 1982). Assume R d ∋ α → ϕ ( α ) ∈ BMO d is measurable, || ϕ ( α ) || BMO d ≤ 1, ϕ ( α ) Q o = 0 for a fixed Q o and all α . Then 1 � ϕ ( α ) ( x + α ) dα ϕ ( x ) = lim (2 N ) d N →∞ | α j |≤ N is BMO and || ϕ || BMO ≤ C d . Theorem 5 yields BMO theorems like Theorem 3 from their simpler dyadic counterparts. For related H 1 result, see B. Davis, TAMS 1980. 8

Let w ∈ L 1 ( R ) , w ≥ 0 . Then � 1 �� 1 1 � � � sup wdx wdx < ∞ ( A 2 ) | I | | I | I I I holds if and only if w satisfies the Helson-Szeg¨ o condition: u ∈ L ∞ , || v || ∞ < π w = e u +˜ v , 2 , (HS) because both hold ⇔ Hibert transform is L 2 ( w ) bounded. In dimension 1, A 2 and HS imply Theorem 3. ⇒ HS directly, without using the L 2 ( w ) bound- Problem: Prove A 2 = edness of H or M . 9

III. Constructions with H ∞ Interpolation, ∂ , and BMO. Let { z j } be a sequence in the upper half plane H = { x + iy : y > 0 } and H ∞ = { f : H → C : f is bounded and analytic } . Theorem (Carleson 1958) Every interpolation problem f ( z j ) = a j , j = 1 , 2 , . . . , ( a j ) ∈ ℓ ∞ (INT) has solution f ∈ H ∞ if and only if | z j − z k | ≥ c > 0 (hyperbolic separation) (i) inf k � = j y j and (ii) for all intervals I ⊂ R , � y j ≤ C | I | , x j ∈ I,y j < | I | (Carleson measure condition). 10

Problems: (already solved) Find constructive solutions to: (1) INT ⇒ ϕ = u + Hv, u, v ∈ L ∞ (2) ϕ ∈ BMO ( R ) = (3) µ Carleson measure on H : µ ( I × (0 , | I | ]) ≤ || µ || C | I | ⇓ ∂F = µ has solution on H which is bounded on R . Theorem 6: (Annals 1980) Constructive solutions to (2) and (3). Proof uses: (i) the J. P. Earl solution to (1), (ii) Approximation of Carleson measures by measures � y j δ z j from interpolating sequences { z j } , and (iii) a BMO extension theorem of Varopoulos. 11

For another construction, define for σ a measure on H : �� � K ( σ, z, ζ ) = 2 i Im ζ i i � � ζ − w − d | σ | ( w ) ( z − ζ )( z − ζ ) exp . z − w π Im w ≤ Im ζ Theorem 7: (Acta Math., 1983) If µ is a Carleson measure on H , then � µ dµ ( ζ ) ∈ L 1 � � S ( µ )( z ) = K , z, ζ loc || µ || C H satisfies ∂S ( µ ) = µ on H , and | S ( µ )( x ) | ≤ C 0 || µ || C . sup R 12

Theorem 8: Let { z j } ⊂ H satisfy | z j − z k | (i) inf k � = j ≥ c > 0 (hyperbolic separation) y j and (ii) for all intervals I ⊂ R , � x j ∈ I,y j < | I | y j ≤ C | I | . Define z − z k � B j ( z ) = k ; k � = j α k , z − z k where | α k | = 1 are convergence factors, and j | B j ( z j ) | > 0 . δ = inf Then � � y j − i y k � 2 � � F j ( z ) = γ j B j ( z ) exp , z − z j z − z k log 2 /δ y k ≤ y j in which � � − 4 i y k � γ j = B j ( z j ) exp , z j − z k log 2 /δ y k ≤ y j satisfies log 2 /δ � 4 F j ( z k ) = δ j,k and | F j ( z ) | ≤ C 0 . δ Paul Koosis called this “the Peter Jones mechanical interpolation for- mula”. 13

IV. Corona Theorems and Problems. When H ∞ (Ω) is the algebra of bounded analytic functions on a com- plex manifold Ω, the corona problem for Ω is: Given f 1 , . . . , f n ∈ H ∞ (Ω) such that for all z ∈ Ω , 1 ≤ j ≤ n | f j ( z ) | ≥ δ > 0 max are there g 1 , . . . , g n ∈ H ∞ (Ω) such that f 1 g 1 + . . . f n g n = 1? Ω = unit disc D , Yes, Carleson (1962). Ω a finite bordered Riemann surface, Yes, E. L. Stout (1964), many later proofs. Ω a Riemann surface, No, Brian Cole (ca 1970). Problem: Ω an infinitely connected plane domain. 14

Theorem: (Carleson (1983)) If C \ Ω = E ⊂ R and for all x ∈ E | E ∩ [ x − r, x + r ] | ≥ cr, then the corona theorem holds for Ω . Forelli Projection: Ω = D / Γ , (i) P : H ∞ ( D ) → H ∞ (Ω) = { f ∈ H ∞ ( D ) : f ◦ γ = f, ∀ γ ∈ Γ } ; (ii) || P ( f ) || ∞ ≤ C || f || ∞ ; f ∈ H ∞ (Ω); (iii) P ( fg ) = fP ( g ) , (iv) P (1) = 1 . Forelli Projection ⇒ corona theorem for Ω . Carleson built a Forelli Projection. Theorem 9: (Jones and Marshall) Let G ( z, ζ ) be Green’s function for Ω, fix z 0 and let { ζ k } be the critical points of G ( z 0 , ζ ) . If there is A > 0 such that all components of � { ζ ∈ Ω : G ( ζ, ζ k ) > A } k are simply connected, then Ω has a projection operator and the corona theorem holds for Ω . For C \ Ω ⊂ R , Theorem 9 = ⇒ Carleson’s theorem. 15

Theorem 10: If C \ Ω ⊂ R , the corona theorem holds for Ω . ⇒ H ∞ (Ω) trivial. Note: | E | = 0 ⇐ Proof of Theorem 10 uses constructions from both Theorem 6 and Theorem 8. Problem: Corona theorem for Ω = C \ E , E ⊂ Γ , a Lipschitz graph. Known if Γ is C 1+ ε , or if Λ 1 ( E ∩ B ( z, r )) ≥ cr ∀ z ∈ E. Problem: Corona theorem for C \ ( K × K ), K = 1 3 Cantor set. Problem: Which Ω ⊂ C have Forelli Projections? For C \ Ω = E ⊂ R , it holds ⇐ ⇒ | E ∩ [ x − r, x + r ] | ≥ cr ∀ x ∈ E. 16

V. Harmonic Measure and Integral Mean Spectra. Theorem: (Makarov, 1985) Let Ω be a simply connected plane do- main and ω harmonic measure for z 0 ∈ Ω. Then α < 1 ⇒ ω << Λ α α > 1 ⇒ ω ⊥ Λ α . For a bounded univalent function ϕ define � 2 π | ϕ ′ ( re iθ ) | t = O ((1 − r ) − β ) � � β ϕ ( t ) = inf β : 0 and the integral mean spectrum , � � B ( t ) = sup β ϕ ( t ) . ϕ Makarov’s Theorem is ⇔ B (0) = 0 . Brennan’s Conjecture is B ( − 2) = 1 17

With ϕ = � ∞ n =1 a n z n , write A n = sup || ϕ || ∞ ≤ 1 | a n | . Theorem 11: (Carleson-Jones, Duke J. 1992) For bounded ϕ the limit log A n γ = − lim log n n →∞ exists and there exists bounded ϕ 1 such that log a n γ = − lim log n . n →∞ Moreover, 1 − γ = B (1) . Carleson and Jones further conjectured γ = 3 4 , i.e. B (1) = 1 4 . Belyaev proved γ < . 78 , i.e. B (1) ≥ . 23 . Brennan-Carlson-Jones-Kraetzer Conjecture: B ( t ) = t 2 4 , | t | ≤ 2 . Theorem 12: (Jones-Makarov, Annals 1995) B ( t ) = t − 1 + O (( t − 2) 2 ) ( t → 2) . 18

For arbitrary plane domains Jones and Wolff proved: Theorem 13: (Acta 1988) Let Ω be a plane domain such that ∂ Ω has positive logarithmic capacity. Then there exists F ⊂ ∂ Ω of Hausdorff dimension ≤ 1 and ω ( z, F ) = 1 for z ∈ Ω . Proof uses classical potential theory and the formula 1 ∂G ∂n log ∂G � � ∂n dx = γ = G ( ζ j ) 2 π ∂ Ω ∇ G ( ζ j )=0 from Ahlfors used earlier by Carleson (to show dim ω stricly less than dim ∂ Ω in certain cases). 19