Flat solutions to the Cauchy-Riemann Equations Yuan Zhang Joint with Y. Liu, Z. Chen and Y. Pan Indiana University - Purdue University Fort Wayne, USA Midwestern Workshop on Asymptotic Analysis Indiana University, Bloomington, IN October 9-11th, 2015 Yuan Zhang (IPFW) Flat solutions 1 / 16
Motivation - two Unique continuation Property (UCP) problems Definition: A smooth function (or map) f is said to be flat (at 0) if D α f (0) = 0 for all multi-indices α . Yuan Zhang (IPFW) Flat solutions 2 / 16
Motivation - two Unique continuation Property (UCP) problems Definition: A smooth function (or map) f is said to be flat (at 0) if D α f (0) = 0 for all multi-indices α . D R := { z ∈ C : | z | = R } . B R := { z ∈ C n : | z | = R } . Theorem (Chanillo-Sawyer) Let V ∈ L 2 ( D R ) and u : D R ⊂ R 2 → R N be smooth. If | ∆ u | ≤ V | ▽ u | , then UCP holds, i.e., u ≡ 0 on D R whenever u is flat. Theorem (Pan) Let V ∈ L 2 ( D R ) and v : D R ⊂ C → C M be smooth. If | ¯ ∂ v | ≤ V | v | , then UCP holds, i.e., v ≡ 0 on D R whenever v is flat. Yuan Zhang (IPFW) Flat solutions 2 / 16
Question ∂ -closed (0,1) form f near 0 ∈ C n and flat at 0, does Given a ¯ Question: there always exist a flat function u such that ¯ ∂ u = f locally? Yuan Zhang (IPFW) Flat solutions 3 / 16
Question ∂ -closed (0,1) form f near 0 ∈ C n and flat at 0, does Given a ¯ Question: there always exist a flat function u such that ¯ ∂ u = f locally? The global version of the question (on B R ): No! Yuan Zhang (IPFW) Flat solutions 3 / 16
Question ∂ -closed (0,1) form f near 0 ∈ C n and flat at 0, does Given a ¯ Question: there always exist a flat function u such that ¯ ∂ u = f locally? The global version of the question (on B R ): No! Examples suggested by Bo-Yong Chen. Yuan Zhang (IPFW) Flat solutions 3 / 16
Question ∂ -closed (0,1) form f near 0 ∈ C n and flat at 0, does Given a ¯ Question: there always exist a flat function u such that ¯ ∂ u = f locally? The global version of the question (on B R ): No! Examples suggested by Bo-Yong Chen. What happens in the sense of germs (where f cannot be trivially 0 near 0)? Yuan Zhang (IPFW) Flat solutions 3 / 16
Criterion in C Lemma Let f be flat at 0 ∈ C . The following two statements are equivalent: 1) ¯ ∂ u = fd ¯ z has a flat solution locally. 2) There exists some neighborhood U of 0 such that the following series ∞ �� f ( ξ ) � � ξ n +1 d ¯ z n ξ ∧ d ξ U n =0 is holomorphic near 0 . Yuan Zhang (IPFW) Flat solutions 4 / 16
Sketch of the proof f ( ζ ) ζ − z d ¯ Denote the Cauchy-Green operator by Tf ( z ) := − 1 � ζ ∧ d ζ. Then 2 π i D R ¯ ∂ Tf = fd ¯ z on D R . Higher order derivative formulas of T on D R : Theorem (Pan, preprint) Let f ∈ C k + α ( D R ) with 0 < α < 1 and k ∈ Z + ∪ { 0 } . Then ∂ k +1 T ( f )( z ) = − k ! f ( ζ ) − P k ( ζ, z ) � d ¯ ζ ∧ d ζ ( ξ − z ) k +2 2 π i D R on D R , where P k ( ζ, z ) is the Taylor expansion of f at z of degree k. See [Liu-Pan-Z., 2015, preprint] for the higher order derivative formulas of T on general domains. Yuan Zhang (IPFW) Flat solutions 5 / 16
Positive examples in C Example Let ϕ ∈ C ∞ ( R , C ) be flat at 0 and g be harmonic on D . Then ¯ ∂ u ( z ) = ϕ ( | z | ) g ( z ) d ¯ z always has a flat solution locally. Yuan Zhang (IPFW) Flat solutions 6 / 16
Counter-examples in C The construction is essentially motivated by Rosay and Coffman-Pan. s : a nondecreasing function on R + , s = 0 in [0 , 1 4 ], 0 < s < 1 on ( 1 4 , 3 4 ) and s = 1 on [ 3 4 , ∞ ); { r n } ∞ n =1 : a decreasing positive sequence, lim n →∞ r n = 0. ∆ r n := r n − r n +1 , annuli A n := { z ∈ C : r n +1 ≤ | z | ≤ r n } ; { p ( n ) } ∞ n =0 : an increasing positive integer sequence with p (0) = 0; { F ( n ) } ∞ n =0 : a positive sequence with F (0) = 1. Let g n ( z ) = F ( n ) z p ( n ) , X n = s ( |·|− r n +1 ) : A n → R , and ∆ r n g n ( z ) , z ∈ A n for odd n , f ( z ) = X n ( z ) g n − 1 ( z ) + (1 − X n ( z )) g n +1 ( z ) , z ∈ A n for even n , 0 , z = 0 . Yuan Zhang (IPFW) Flat solutions 7 / 16
Lemma of Coffman-Pan Lemma (Coffman-Pan) (∆ r n / r n ) If (∆ r n +2 ) / ( r n +2 ) is a bounded sequence and for each integer k ≥ 0 , F ( n + 1)( p ( n + 1)) k r p ( n +1) − 4 k n lim = 0 , (∆ r n / r n ) k n →∞ then f is smooth and flat at the origin. Yuan Zhang (IPFW) Flat solutions 8 / 16
The Family S (∆ r n / r n ) Denote by S the set of functions f such that (∆ r n +2 ) / ( r n +2 ) is bounded, F ( n + 1)( p ( n + 1)) k r p ( n +1) − 4 k n lim = 0 , (∆ r n / r n ) k n →∞ as well as either one of the following conditions: � p ( n ) lim F ( n )∆ r n r n +1 = ∞ , n →∞ � F ( n )(∆ r n − 1 ) 2 = ∞ , p ( n ) lim n →∞ � p ( n ) lim F ( n )∆ r n − 1 r n = ∞ , n →∞ � F ( n )(∆ r n +1 ) 2 = ∞ , p ( n ) lim n →∞ � p ( n ) lim F ( n )∆ r n +1 r n +2 = ∞ . n →∞ Yuan Zhang (IPFW) Flat solutions 9 / 16
Examples in S Example (Rosay) R = 1 , p ( n ) = n , r n = 2 − n +1 , F ( n ) = 2 n 2 / 2 . Yuan Zhang (IPFW) Flat solutions 10 / 16
Examples in S Example (Rosay) R = 1 , p ( n ) = n , r n = 2 − n +1 , F ( n ) = 2 n 2 / 2 . Example R = 1. p ( n ) , t ( n ) and q ( n ) are polynomials of degree d p , d t and d q with positive leading coefficients, t (1) = 0, d q > d p , d q > d t and d q < d p + d t . Let r n := 2 − t ( n ) , F ( n ) := 2 q ( n ) . Yuan Zhang (IPFW) Flat solutions 10 / 16
The main theorems Theorem For every f ∈ S , there does not exist a flat smooth u such that ¯ ∂ u = fd ¯ z near the origin. Yuan Zhang (IPFW) Flat solutions 11 / 16
The main theorems Theorem For every f ∈ S , there does not exist a flat smooth u such that ¯ ∂ u = fd ¯ z near the origin. Theorem There exists a family of germs of ¯ ∂ -closed (0,1) forms, flat at 0 ∈ C n , such that for every f in this family, the Cauchy-Riemann equation ¯ ∂ u = f has no flat solution in the sense of germs. Yuan Zhang (IPFW) Flat solutions 11 / 16
ormander’s L 2 theory H¨ Theorem (H¨ ormander, Acta. Math., 1965) Let Ω be a bounded pseudoconvex open set in C n , Let δ be the diameter of Ω , and let φ be a plurisubharmonic function in Ω . For every ¯ ∂ -closed (0 , q − 1) (Ω , φ ) satisfying ¯ f ∈ L 2 (0 , q ) (Ω , φ ) , q > 0 , one can find u ∈ L 2 ∂ u = f in Ω and � � | u | 2 e − φ dV ≤ e δ 2 | f | 2 e − φ dV . q Ω Ω Yuan Zhang (IPFW) Flat solutions 12 / 16
ormander’s L 2 theory H¨ Theorem (H¨ ormander, Acta. Math., 1965) Let Ω be a bounded pseudoconvex open set in C n , Let δ be the diameter of Ω , and let φ be a plurisubharmonic function in Ω . For every ¯ ∂ -closed (0 , q − 1) (Ω , φ ) satisfying ¯ f ∈ L 2 (0 , q ) (Ω , φ ) , q > 0 , one can find u ∈ L 2 ∂ u = f in Ω and � � | u | 2 e − φ dV ≤ e δ 2 | f | 2 e − φ dV . q Ω Ω When q = 1, a minimal solution to ¯ ∂ u = f on Ω is the solution that is orthogonal to the space of holomorphic functions with respect to L 2 (Ω , φ ) norm. Yuan Zhang (IPFW) Flat solutions 12 / 16
Is the restriction of a minimal solution minimal? Ω 1 , Ω 2 : smooth bounded pseudoconvex domains, Ω 2 ⊂ Ω 1 ; φ : a bounded plurisubharmonic function in Ω 1 ; f : a ¯ ∂ -closed (0,1) form in Ω 1 . Consider the minimal solution u 1 to ¯ ∂ u = f , Ω 1 with respect to L 2 (Ω 1 , φ ) norm and the minimal solution u 2 to ¯ ∂ u = f | Ω 2 , Ω 2 with respect to L 2 (Ω 2 , φ | Ω 2 ) norm. Question: Is u 2 = u 1 | Ω 2 ? Yuan Zhang (IPFW) Flat solutions 13 / 16
Is the restriction of a minimal solution minimal? Ω 1 , Ω 2 : smooth bounded pseudoconvex domains, Ω 2 ⊂ Ω 1 ; φ : a bounded plurisubharmonic function in Ω 1 ; f : a ¯ ∂ -closed (0,1) form in Ω 1 . Consider the minimal solution u 1 to ¯ ∂ u = f , Ω 1 with respect to L 2 (Ω 1 , φ ) norm and the minimal solution u 2 to ¯ ∂ u = f | Ω 2 , Ω 2 with respect to L 2 (Ω 2 , φ | Ω 2 ) norm. Question: Is u 2 = u 1 | Ω 2 ? In general, No! Yuan Zhang (IPFW) Flat solutions 13 / 16
Is the restriction of a minimal solution minimal? Ω 1 , Ω 2 : smooth bounded pseudoconvex domains, Ω 2 ⊂ Ω 1 ; φ : a bounded plurisubharmonic function in Ω 1 ; f : a ¯ ∂ -closed (0,1) form in Ω 1 . Consider the minimal solution u 1 to ¯ ∂ u = f , Ω 1 with respect to L 2 (Ω 1 , φ ) norm and the minimal solution u 2 to ¯ ∂ u = f | Ω 2 , Ω 2 with respect to L 2 (Ω 2 , φ | Ω 2 ) norm. Question: Is u 2 = u 1 | Ω 2 ? In general, No! Examples? Yuan Zhang (IPFW) Flat solutions 13 / 16
Examples Let ˜ f ∈ S and consider f ( z ) := ˜ f ( z 1 ) d ¯ z 1 . Examples: Conclusion: For every f above, any given bounded plurisubharmonic weight function φ on B 1 and positive decreasing sequence r n ( < 1) → 0, the minimal solution u n to ¯ ∂ u = f | B rn on B r n with respect to L 2 ( B r n , φ | B rn ) norm is not the restriction of u 1 onto B r n . Yuan Zhang (IPFW) Flat solutions 14 / 16
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