aa r X i v : . [ m a t h . C V ] J a n ON RADÓ’S THEOREM FOR POLYANALYTIC FUNCTIONS
ABTIN DAGHIGHI
Abstract.
We prove versions of Radó’s theorem for polyanalytic functionsin one variable and also on simply connected C -convex domains in C n . Let Ω ⊂ C be a bounded, simply connected domain and let q ∈ Z + . Suppose atleast one of the following conditions holds true: (i) g ∈ C q (Ω) . (ii) g ∈ C κ (Ω) , for κ = min { , q − } , such that g is q -analytic on Ω \ g − (0) and such that Re g ( Im g respectively) is a solutions to the p ′ -Laplace equation ( p ′′ -Laplaceequation respectively) on Ω \ g − (0) , for some p ′ , p ′′ > . Then g agrees(Lebesgue) a.e. with a function that is q -analytic on Ω . In the process we give a simple proof of the fact that if f ∈ C q (Ω) is q -analyticon Ω \ f − (0) then f is q -analytic on Ω . The extensions of the results to severalcomplex variables are straightforward using known techniques. Introduction
Radó’s theorem states that a continuous function on an open subset of C n thatis holomorphic off its zero set extends to a holomorphic function on the given openset. For the one-dimensional result see Radó [7], and for a generalization to severalvariables, see e.g. Cartan [4]. Definition 1.1.
Let Ω ⊂ C be an open subset. A function f on Ω is called polyharmonic of order q if ∆ q f = 0 on Ω , where ∆ denotes the Laplace operator. Definition 1.2.
Let Ω ⊆ R n be an open subset. For a fixed p > , the p -Laplace operator of a real-valued function u on Ω is defined as ∆ p := div ( |∇ u | p − ∇ u ) The operator can also be defined for p = 1 (it is then the negative of the so-calledmean curvature operator) and p = ∞ but we shall not concern ourselves with suchcases. Remark . Note the subtle similarity between the notation for the p -Laplaceoperator ∆ p = div ( |∇ u | p − ∇ u ) and that of the p :th power of the Laplace operator ∆ p . We have that ∆ = ∆ . More generally, we have ∆ p u = |∇ u | p − |∇ u | ∆ u + ( p − n X i,j =1 ∂ x i u · ∂ x j u · ∂ x i ∂ x j u Mathematics Subject Classification.
Primary 35G05, 35J62; 32A99, 32V25 .
Key words and phrases.
Radó’s theorem, Polyanalytic functions, zero sets, α -analyticfunctions.*Corresponding author: [email protected]. Note that ∆ p is quasilinear . At least they both share the property of being ellipticoperators. In the case of ∆ p this is a direct consequence of the fact that ∆ is a ellipticoperator and therefore any finite power is also, in particular the elliptic regularitytheorem applies to ∆ p and to ∆ p , and implies that any real-valued distributionsolution u to ∆ p u = 0 (or to ∆ p ) on a domain Ω ⊂ R n is Lebesgue a.e. equal to a C ∞ -smooth solution ˜ u to ∆ p ˜ u = 0 (or to ∆ p ˜ u = 0 ) on Ω . Kilpeläinen [5] proved the following.
Theorem 1.4. If ω ⊂ R is a domain and if u ∈ C (Ω) satisfies the p -Laplaceequation div ( |∇| p − ∇ u ) = 0 on Ω \ u − (0) then u is a solution to the p -Laplacianon Ω . We mention that, more recently, Tarkhanov & Ly [6] proved the following relatedresult in higher dimension.
Theorem 1.5.
Let Ω ⊆ R n be an open subset. If u ∈ C , p − (Ω) such thatdiv ( |∇| p − ∇ u ) = 0 on Ω \ u − (0) then this holds true on all of Ω . We shall use the result of Kilpeläinen [5] in order to prove a natural versionof Radó’s theorem for polyanalytic functions. Avanissian & Traoré [1], [2] intro-duced the following definition of polyanalytic functions of order α ∈ Z n + in severalvariables. Definition 1.6.
Let Ω ⊂ C n be a domain, let α ∈ Z n + and let z = x + iy denoteholomorphic coordinates in C n . A function f on Ω is called polyanalytic of order α if in a neighborhood of every point of Ω , (cid:16) ∂∂ ¯ z j (cid:17) α j f ( z ) = 0 , ≤ j ≤ n . Definition 1.7.
Let Ω ⊂ C n be an open subset and let ( z , . . . , z n ) denote holo-morphic coordinates for C n . A function f , on Ω , is said to be separately C k -smoothwith respect to the z j -variable , if for any fixed ( c , . . . , c n − ) ∈ C n − , chosen suchthat the function z j f ( c , . . . , c j − , z j , c j , . . . , c n − ) , is well-defined (i.e. such that ( c , . . . , c j − , z j , c j , . . . , c n − ) belongs to the domain of f ) is C k -smooth with respect to Re z j , Im z j . For α ∈ Z n + we say that f is separately α -smooth if f is separately C α j -smooth with respect to z j for each ≤ j ≤ n .We shall need the following result. Theorem 1.8. (See [2, Theorem 1.3, p. 264] ) Let Ω ⊂ C n be a domain and let z = ( z , . . . , z n ) , denote holomorphic coordinates in C n with Re z =: x, Im z = y .Let f be a function which, for each j , is polyanalytic of order α j in the variable z j = x j + iy j (in such case we shall simply say that f is separately polyanalytic oforder α ). Then f is jointly smooth with respect to ( x, y ) on Ω and furthermore ispolyanalytic of order α = ( α , . . . , α n ) in the sense of Definition 1.6. Statement and proof of the result
Let us make the following first observation.
Proposition 2.1.
Let Ω ⊂ C be a simply connected domain, let q ∈ Z + and let f ∈ C q (Ω) be a q -analytic function on Ω \ f − (0) . Then f is q -analytic on Ω . N RADÓ’S THEOREM FOR POLYANALYTIC FUNCTIONS 3
Proof. If f ≡ then we are done, so assume f . Since f is C q -smooth thefunction ∂ q ¯ z f is continuous. By assumption ∂ q ¯ z f = 0 on Ω \ f − (0) . Set Z :=( f − (0)) ◦ ( ◦ denoting the interior) and X := { f = 0 } ∪ Z. Now f | Z clearly satisfies ∂ q ¯ z f = 0 . Let p ∈ ∂X. If p is an isolated zero of f , then by continuity we have ∂ q ¯ z f ( p ) = 0 . Suppose p is a non-isolated zero. We have for each sufficiently large j ∈ Z + that {| z − p | < /j } ∩ X = ∅ . This implies that there exists a sequence { z j } j ∈ Z + of points z j ∈ X such that z j → p as j → ∞ . By continuity we have ∂ q ¯ z f ( p ) = lim j →∞ ∂ q ¯ z f ( z j ) = 0 This completes the proof. (cid:3)
Theorem 2.2.
Let Ω ⊂ C be a bounded, simply connected domain, let q ∈ Z + andlet f be a function q -analytic on Ω \ f − (0) . Suppose at least one of the followingconditions holds true:(i) f ∈ C κ (Ω) , for κ = min { , q − } , and Re f ( Im f respectively) is a solutions tothe p ′ -Laplace equation ( p ′′ -Laplace equation respectively) on Ω \ f − (0) , for some p ′ , p ′′ > .(ii) f ∈ C q (Ω) . Then f agrees (Lebesgue) a.e. with a function that is q -analytic on Ω . Proof.
The case (ii) follows from Proposition 2.1. So suppose (i) holds true. If q = 1 the theorem is well-known and due to Radó [7], so assume q ≥ . Let f = u + iv where u = Re f, v = Im f. Now f − (0) = u − (0) ∩ v − (0) , whence u (and v respectively) is a solution to the p ′ -Laplace equation ( p ′′ -Laplace equationrespectively) on Ω \ u − (0) ( Ω \ v − (0) respectively). If f ∈ C κ (Ω) and q ≥ then u and v respectively are at least C -smooth thus satisfy the conditions of Theorem1.4. Hence it follows that u ( v respectively) are solutions to the p ′ -Laplace equation( p ′′ -Laplace equation respectively) on all of Ω . By Remark 1.3 (in particular Ellipticregularity) it follows that u and v respectively agree (Lebesgue) a.e. on Ω with C ∞ -smooth functions ˜ u and ˜ v respectively. This implies that the function ˜ f := ˜ u + i ˜ v is C ∞ -smooth on Ω and agrees (Lebesgue) a.e. on Ω with f. Suppose there exists apoint p ∈ Ω such that ∂ q ¯ z ˜ f ( p ) = 0 . Set Z := ( f − (0)) ◦ and X := { f = 0 } ∪ Z. Bycontinuity there exists an open neighborhood U p of p in Ω such that ∂ q ¯ z ˜ f = 0 onthe open subset U p ∩ X. By the definition of ˜ f there exists a set E of zero measuresuch that on V p := ( X ∩ U p ) \ E we have that ∂ q ¯ z f exists (since X contains no pointof f − (0) \ Z ) and satisfies ∂ q ¯ z f = ∂ q ¯ z ˜ f on V p , which could only happen if V p is empty which is impossible since E cannot possess interior points. We concludethat ∂ q ¯ z ˜ f = 0 on Ω . This completes the proof. (cid:3)
Theorem 2.3 (Radó’s theorem for polyanalytic functions in several complex vari-ables) . Let Ω ⊂ C n be a bounded C -convex domain. Let α ∈ Z n + . Suppose f is α -analytic on Ω \ f − (0) such that one of the following conditions hold true:(i) For each j = 1 , . . . , n , the function f is separately C κ j -smooth with respectto z j (i.e. for each fixed value of the remaining variables z k , k = j , f becomesa C κ j -smooth function of z j ), κ j = min { , α j − } and Re f ( Im f respectively)are solutions to the p ′ -Laplace equation ( p ′′ -Laplace equation respectively) for some p ′ , p ′′ > . (ii) For each j = 1 , . . . , n , the function f is separately C α j -smooth with respect to ABTIN DAGHIGHI z j .Then f agrees (Lebesgue) a.e. with a function that is α -analytic on Ω .Proof. Denote for a fixed c ∈ C n − , Ω c,k := { z ∈ Ω : z j = c j , j < k, z j = c j − , j >k } . Since Ω is C -convex, Ω c,k is simply connected. Consider the function f c ( z k ) := f ( c , . . . , c k − , z k , c k , . . . , c n − ) . Clearly, f c is α k -analytic on Ω c,k \ f − (0) for any c ∈ C n − . Since f − c (0) ⊆ f − (0) , Theorem 2.2 applies to f c meaning that f agreesa.e. with a function ˜ f that is separately polyanalytic of order α j in the variable z j , ≤ j ≤ n . By Theorem 1.8 the function ˜ f must be polyanalytic of order α on Ω . This completes the proof. (cid:3) Corollary 2.4.
Let Ω ⊂ C be a bounded C -convex domain and let α ∈ Z n + . Suppose f is separately C α j -smooth with respect to z j , j = 1 , . . . , n. If f is α -analytic on Ω \ f − (0) , then f agrees (Lebesgue) a.e. with a function that is α -analytic on Ω . References
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