L p → L q norm estimates of Cauchy transforms on the Dirichlet problem and their applications
LL p → L q NORM ESTIMATES OF CAUCHY TRANSFORMS ONTHE DIRICHLET PROBLEM AND THEIR APPLICATIONS
JIAN-FENG ZHU AND ANTTI RASILA
Abstract.
Denote by C α ( D ) the space of the functions f on the unit disk D which are H¨older continuous with the exponent α , and denote by C ,α ( D ) thespace which consists of differentiable functions f such that their derivatives arein the space C α ( D ). Let C be the Cauchy transform of Dirichlet problem. Inthis paper, we obtain the norm estimates of (cid:107)C(cid:107) L p → L q , where 3 / < p < q = p/ ( p − / < p <
2, then u ∈ C µ ( D ),where µ = 2 /p −
1. We also show that if 2 < p < ∞ , then u ∈ C ,ν ( D ), where ν = 1 − /p . Finally, for the case p = ∞ , we show that u is not necessarily in C , ( D ), but its gradient, i.e., |∇ u | is Lipschitz continuous with respect to thepseudo-hyperbolic metric. This paper is inspired by [2, Chapter 4] and [10]. Introduction
The space L p (Ω) . Throughout this paper, we use D denote by the unit disk, and T the unit circle. Suppose Ω is a domain in the complex plane C . Denote by L p (Ω)the space of the complex-valued measurable functions on Ω with a finite integral (cid:107) f (cid:107) p = (cid:18)(cid:90) Ω | f ( z ) | p d A ( z ) (cid:19) p , ≤ p < ∞ , where d A ( z ) is the normalized area measure on Ω (cf. [7, Page 1]). For the case p = ∞ , we let L ∞ (Ω) denote the space of (essentially) bounded functions on Ω. For f ∈ L ∞ (Ω), we define (cid:107) f (cid:107) ∞ = ess sup {| f ( z ) | : z ∈ Ω } . It is known that the space L ∞ ( D ) is a Banach space with the above norm (cf. [7,Page 2]). The Sobolev space W ,p (Ω) . For k ≥ p ≥
1, the Sobolev space W ,p (Ω) isthe Banach space of k -times weak differentiable p -integrable functions. The normin W ,p (Ω) is defined by (cid:107) u (cid:107) W ,p = (cid:90) Ω (cid:88) | α |≤ k | D α u | p d A /p . Date : August 31, 2020.2000
Mathematics Subject Classification.
Primary 30H20, 32A36; secondary 47B38.
Key words and phrases.
Cauchy transform for Dirichlet problem, Poisson equation, Morrey’sinequality, L p norm. a r X i v : . [ m a t h . F A ] A ug J.-F. Zhu and A. Rasila
The space W ,p (Ω) is obtained by taking the closure of C k (Ω) in W k,p (Ω), where C k,p (Ω) is the space of k times continuously differentiable functions with a compactsupport in Ω (cf. [6, Pages 153–154]). The Cauchy transform of a solution to the Dirichlet problem.
The Poissonequation is given as follows:(1.1) u z ¯ z = g ( z ) , z ∈ Ω ,u ∈ W ,p , where ∆ u = 4 u z ¯ z is the Laplacian of u and g ∈ L p (Ω). It is known that if Ω = D and g ∈ L p ( D ), where 1 ≤ p ≤ ∞ , then the weak solution of the Poisson equation is u ( z ) = G [ g ]( z ) = (cid:90) D log (cid:12)(cid:12)(cid:12)(cid:12) z − τ − ¯ zτ (cid:12)(cid:12)(cid:12)(cid:12) g ( τ )d A ( τ ) , where z , τ ∈ D , and G ( z, τ ) = log (cid:12)(cid:12)(cid:12)(cid:12) z − τ − ¯ zτ (cid:12)(cid:12)(cid:12)(cid:12) is the Green function.Suppose u = G [ g ] is a solution to (1.1), where g ∈ L p ( D ) and 1 < p < ∞ . Wemay define the Cauchy transform of the solution to the Dirichlet problem as follows: C [ g ]( z ) = ∂∂z u = (cid:90) D (cid:18) z − τ + ¯ τ − z ¯ τ (cid:19) g ( τ )d A ( τ ) . The operator C is then induced by the complex partial z -derivative of the Green’sfunction (cf. [2, Page 155]).Let C be the Cauchy integral operator ( the Cauchy transform ) which is defined asfollows: C [ g ]( z ) = (cid:90) Ω g ( τ ) τ − z d A ( τ ) . The following integral operator J ∗ was introduced in [3, Page 12] and is given asfollows: J ∗ [ g ]( z ) = (cid:90) D ¯ τ − z ¯ τ g ( τ )d A ( τ ) . Now, it is easy to see that C = J ∗ − C (cf. [2, 3, 10]). Moreover, elementarycalculations show that (cf. [10, (1.4) and (1.5)]) C [ g ]( z ) = ∂∂z u = (cid:90) D − | τ | ( z − τ )(1 − z ¯ τ ) g ( τ )d A ( τ ) , and C [ g ]( z ) = ∂∂ ¯ z u = (cid:90) D − | τ | (¯ z − ¯ τ )(1 − ¯ zτ ) g ( τ )d A ( τ ) . Recall that the standard operator norm of an operator T : X → Y betweennormed spaces X and Y is defined by (cid:107) T (cid:107) X → Y = sup {(cid:107) T x (cid:107) Y : (cid:107) x (cid:107) X = 1 } . p → L q norm estimates of Cauchy transforms on the Dirichlet problem and their applications 3 For simplicity, if X = Y = L p ( D ), then we write (cid:107) T (cid:107) p instead of (cid:107) T (cid:107) L p → L p for the L p norm of the operator T .For p >
2, it was shown in [2, Page 155] that C [ g ]( z ) is a continuous functionon the closed disk D . In 2012, Kalaj proved in [10, Theorem B] that there existsa constant M p depending only on p , such that (cid:107)C(cid:107) L p ( D ) → L ∞ ( D ) = M p . Moreover, heobtained norm estimates for (cid:107)C(cid:107) p , and showed that the results are sharp for p = 1 , ∞ (cf. [10, Theorem A]). The pseudo-hyperbolic distance on D . For each z ∈ D , let ϕ w denote theM¨obius transform of the form ϕ w ( z ) = w − z − ¯ wz , where w ∈ D . The pseudo-hyperbolic distance on D is defined as follows (cf. [5]):(1.2) ρ ( z, w ) = | ϕ w ( z ) | , z, w ∈ D . We note that the pseudo-hyperbolic distance is invariant under M¨obius transforma-tions, that is, ρ ( f ( z ) , f ( w )) = ρ ( z, w ) , for any f ∈ Aut( D ), the M¨obius automorphism of D , where z, w ∈ D (cf. [5]).Moreover, it has the following useful property:(1.3) 1 − ρ ( z, w ) = (1 − | z | )(1 − | w | ) | − ¯ zw | . Definition 1.1. (cf. [2, Page 115] ) The H¨older spaces C µ ( C ), 0 < µ ≤
1, consistof continuous functions f : C → C that satisfy the H¨older condition (cid:107) f (cid:107) C µ ( C ) = sup z (cid:54) = w | f ( z ) − f ( w ) || z − w | µ < ∞ . The space C ,ν ( C ), 0 < ν ≤
1, consists of C functions f : C → C that satisfy thefollowing condition: (cid:107) f (cid:107) C ,ν ( C ) = sup z (cid:54) = w |∇ f ( z ) − ∇ f ( w ) || z − w | ν < ∞ . Main results.
In this paper, we show that if g ∈ L p ( D ), where 3 / < p < C will map L p ( D ) to L q ( D ), where q = p/ ( p −
1) is the conjugateexponent of p . This result partly improves the corresponding results in [10, TheoremA]. As an application, by using Morrey’s inequality (cf. [8, 12]), we have G [ g ] ∈ C µ ( D ) is H¨older continuous with the exponent µ = 1 − /q . Moreover, we provethat if 2 < p < ∞ and g ∈ L p ( D ), then G [ g ] ∈ C ,ν ( D ), where ν = 1 − /p . For thecase p = ∞ , we give Example 1.2 which shows that C is not of C ( D ) space or eventhe space of Lipschitz continuous functions. This implies that when g ∈ L ∞ ( D ), itsGreen potential G [ g ] is not in the space C , ( D ), or even in C , Lip ( D ), the space of thefunctions with Lipschitz continuous derivatives. However, by applying the pseudo-hyperbolic distance, we show that |∇ G [ g ] | is Lipschitz continuious with respect tothe pseudo-hyperbolic metric.More precisely, our results are as follows: J.-F. Zhu and A. Rasila - - - - - - - - - Figure 1.
The images of D under the mappings u ( z ) , g ( z ), and u z ( z )of Example 1.1. Theorem 1.1.
Suppose g ∈ L p ( D ) , where / < p < , and q = p/ ( p − . Then C is an operator of L p ( D ) to L q ( D ) . Moreover, we have (cid:107)C(cid:107) qL p → L q ≤ / (cid:18) − q/p + 1Γ(2 − q/p ) (cid:19) . The following example shows that for the case p = 3 / q = 3, there existsa function u = G [ g ], such that g ∈ L p ( D ), but C [ g ]( z ) = ∂∂z u ( z ) / ∈ L q ( D ). Example 1.1.
Let u ( z ) = z (1 − | z | α ), where z ∈ D and α > . Then u ( ζ ) = 0, forany ζ = e iθ ∈ T , and g ( z ) = ∆ u = − α ( α + ) z | z | α − . Elementary calculationsshow that g ∈ L ( D ) and ∂∂z u ( z ) = z − ( − ( + α ) | z | α ) / ∈ L ( D ). See Figure 1.As an application of Theorem 1.1, we have the following results: Theorem 1.2.
Suppose g ∈ L p ( D ) , where / < p < ∞ , and G [ g ] is the Greenpotential of g . Then :(1) for / < p < , G [ g ] ∈ C µ ( D ) , where µ = 2 /p − ; (2) for < p < ∞ , G [ g ] ∈ C ,ν ( D ) , where ν = 1 − /p . For the case p = ∞ , the following example shows that G [ g ] does not necessarilybelong to the space C , ( D ) or even to C , Lip ( D ). Example 1.2. (cf. [2, Page 116]) Let g ( z ) = z/ ¯ z ∈ L ∞ ( D ). Then C [ g ]( z ) = − z log | z | and J ∗ [ g ]( z ) = z/
2. This shows that C = J ∗ − C does not take L ∞ ( D ) tothe space C ( D ) or even the space of Lipschitz continuous functions. See Figure 2.Next, we show that |∇ G [ g ] | is Lipschitz continuous with respect to the pseudo-hyperbolic metric. Suppose that g ∈ L ∞ ( D ) and that u ( z ) = G [ g ]( z ) is the Greenpotential of g . Then the gradient of u is defined byΛ u ( z ) = | u z ( z ) | + | u ¯ z ( z ) | = (cid:12)(cid:12) C [ g ]( z ) (cid:12)(cid:12) + (cid:12)(cid:12) C [ g ]( z ) (cid:12)(cid:12) . p → L q norm estimates of Cauchy transforms on the Dirichlet problem and their applications 5 - - - - Figure 2.
The image of D under the mapping C ( z ) of Example 1.2.We have the following result: Theorem 1.3.
Let u = G [ g ] be the Green potential of g , where g ∈ L ∞ ( D ) . Then (1.4) (cid:12)(cid:12) (1 − | z | )Λ u ( z ) − (1 − | w | )Λ u ( w ) (cid:12)(cid:12) ≤ (cid:107) g (cid:107) ∞ ρ ( z, w ) , holds for all z, w ∈ D . The rest of this paper is organized as follows: In Section 2, we recall some knownresults and prove two lemmas which will be used in proving Theorem 1.1. In Section2, we prove Theorems 1.1 to 1.3. In Section 3, we provide some additional insightsrelated to the constant C q , which occurs in Theorem 1.2.2. Auxiliary results
In this section, we recall some known results and prove two lemmas which will beused in proving our main theorems. We start with the following Morrey’s inequality(cf. [8, 12]).
Theorem A. ( Morrey’s inequality ) Assume that n < p < ∞ , and let u ∈ D ,p ( R n )( i.e., the derivative of u exists and of L p ( R n ) space ) . Then there exists a constant C > such that sup x (cid:54) = y (cid:26) | u ( x ) − u ( y ) || x − y | − n/p (cid:27) ≤ C (cid:18)(cid:90) R n | Du | p d x (cid:19) /p . The following equality (2.1) and Lemma B will be applied in the proofs of Lemmas2.1, 2.2, and Theorem 1.1.For β > z ∈ D and ζ = e iθ ∈ T , we have1(1 − zζ ) β = ∞ (cid:88) n =0 Γ( n + β ) n !Γ( β ) z n ζ n . J.-F. Zhu and A. Rasila
By using
Parseval’s theorem , one gets the identity:(2.1) 12 π (cid:90) π d θ | − ze iθ | β = ∞ (cid:88) n =0 (cid:18) Γ( n + β ) n !Γ( β ) (cid:19) | z | n . Recall the following estimate:
Lemma B. (cf. [4, (7)])
For ≤ a ≤ and n = 1 , , . . . , n + 1) − a ≤ Γ( n + a ) n ! ≤ n − a . Next, we prove two lemmas which will be used in proving our Theorem 1.1.
Lemma 2.1.
For ≤ β < , let (2.2) I β = (cid:90) D (cid:18) − | w | | z − w || − ¯ zw | (cid:19) β d A ( w ) , where z ∈ D . Then (2.3) I β ≤ β/ Γ(1 + β )Γ(2 − β ) . Proof.
By using the M¨obius transformation η = z − w − ¯ zw , we obtain I β = (1 − | z | ) − β (cid:90) D (cid:18) − | η | | η | (cid:19) β | − ¯ zη | d A ( η ) . Suppose that η = re it ∈ D . By applying (2.1) and the following equality: (cid:90) r n +1 − β (1 − r ) β d r = Γ(1 + β )Γ( n + 1 − β/ n + 2 + β/ , n = 0 , , . . . , we have I β = (1 − | z | ) − β π (cid:90) r − β (1 − r ) β d r (cid:90) π | − ¯ zre it | d θ (2.4) = (1 − | z | ) − β Γ(1 + β ) ∞ (cid:88) n =0 ( n + 1) Γ( n + 1 − β/ n + 2 + β/ | z | n . Note that for 0 < a <
1, the formula(2.5) 1(1 − z ) a = ∞ (cid:88) n =0 Γ ( n + a ) n !Γ ( a ) z n holds for every z ∈ D . Moreover, according to Lemma B, one obtains the followinginequality:(2.6) ( n + 1) Γ( n + 1 − β/ n + 2 + β/ ≤ β/ Γ( n + 2 − β ) n ! , n = 1 , , . . . . It is easy to see that when n = 0, the above inequality (2.6) still holds. Combining(2.4), (2.5) and (2.6), we see that (2.3) holds, which completing the proof. (cid:3) p → L q norm estimates of Cauchy transforms on the Dirichlet problem and their applications 7 Lemma 2.2.
For ≤ β < , let J β = (cid:90) D (cid:18) − | w | | z − w || − ¯ zw | (cid:19) β d A ( z ) , where w ∈ D . Then (2.7) J β ≤ − β + 1Γ(2 − β ) . Proof.
Following the proof of Lemma 2.1, by letting z = w − η − ¯ wη and applying (2.1),one has J β = (1 − | w | ) − β (cid:90) D | η | β | − ¯ zη | − β d A ( η )= 2(1 − | w | ) − β ∞ (cid:88) n =0 (cid:18) Γ( n + 2 − β ) n !Γ(2 − β ) (cid:19) | w | n n + 2 − β = 2(1 − | w | ) − β (cid:32) − β + ∞ (cid:88) n =1 (cid:18) Γ( n + 2 − β ) n !Γ(2 − β ) (cid:19) | w | n n + 2 − β (cid:33) . Since 1 ≤ β <
2, by using Lemma B we see thatΓ( n + 2 − β ) n !(2 n + 2 − β ) < n β ≤ , n = 1 , , . . . . Thus J β ≤ − | w | ) − β (cid:32) − β + 12Γ(2 − β ) ∞ (cid:88) n =1 Γ( n + 2 − β ) n !Γ(2 − β ) | w | n (cid:33) . Again, by (2.5), we have J β ≤ − | w | ) − β (cid:20) − β + 12Γ(2 − β ) (cid:18) − | w | ) − β − (cid:19)(cid:21) ≤ − β + 1Γ(2 − β ) . Therefore, the desired inequality (2.7) follows. (cid:3) Proofs of the main results
Proof of Theorem 1.1.
Recall that C [ g ]( z ) = (cid:90) D − | τ | ( z − τ )(1 − z ¯ τ ) g ( τ )d A ( τ ) . By using Lemma 2.1 and the H¨older’s inequality for integrals, we have |C [ g ]( z ) | ≤ I /q (cid:18)(cid:90) D − | τ | | z − τ || − ¯ zτ | | g ( τ ) | p d A ( τ ) (cid:19) /p , J.-F. Zhu and A. Rasila where q = pp − . The assumption g ∈ L p ( D ) ensures that (cid:90) D | g ( τ ) | p d A ( τ ) = (cid:107) g (cid:107) pp < ∞ , and thus, | g ( τ ) | p d A ( τ ) (cid:107) g (cid:107) pp is a probability measure in D .Observe that under the assumption of 3 / < p <
2, we have 1 < q/p < |C [ g ]( z ) | q ≤ I (cid:107) g (cid:107) qp (cid:18)(cid:90) D − | τ | | z − τ || − ¯ zτ | | g ( τ ) | p d A ( τ ) (cid:107) g (cid:107) pp (cid:19) q/p (3.1) ≤ I (cid:107) g (cid:107) q − p (cid:90) D (cid:18) − | τ | | z − τ || − ¯ zτ | (cid:19) q/p | g ( τ ) | p d A ( τ ) . It follows from Lemma 2.1 and the assumption g ∈ L p ( D ) that (cid:18) − | τ | | z − τ || − ¯ zτ | (cid:19) q/p | g ( τ ) | p ∈ L ( D × D ) . By using Lemma 2.2 and (3.1), one obtains (cid:90) D |C [ g ]( z ) | q d A ( z ) ≤ I (cid:107) g (cid:107) q − p (cid:18)(cid:90) D | g ( τ ) | p d A ( τ ) (cid:19) (cid:90) D (cid:18) − | τ | | z − τ || − ¯ zτ | (cid:19) q/p d A ( z ) ≤ (cid:107) g (cid:107) q − pp I J q/p < ∞ . This shows that C [ g ] ∈ L q ( D ), where q = p/ ( p − (cid:107)C(cid:107) qL p → L q = sup {(cid:107)C [ g ] (cid:107) qq : (cid:107) g (cid:107) p = 1 }≤ / (cid:18) − q/p + 1Γ(2 − q/p ) (cid:19) . The proof is complete. (cid:3)
Proof of Theorem 1.2. (1) The assumption 3 / < p < < q = p/ ( p − < p . According to Theorem 1.1, we see that C [ g ] ∈ L q ( D ). Similarly, wemay prove that C [ g ] ∈ L q ( D ). Then, by Theorem A, we see that G [ g ] ∈ C µ ( D ) isH¨older continuous with the exponent µ = 1 − /q = 2 /p − g ∈ L p ( D ), where 2 < p < ∞ . Then, for z , w ∈ D satisfying z (cid:54) = w ,we have C [ g ]( z ) − C [ g ]( w ) = J ∗ [ g ]( z ) − J ∗ [ g ]( w ) − (cid:2) C [ g ]( z ) − C [ g ]( w ) (cid:3) = (cid:90) D (cid:20) ( z − w )¯ τ (1 − z ¯ τ )(1 − w ¯ τ ) + w − z ( z − τ )( w − τ ) (cid:21) g ( τ )d A ( τ ) . p → L q norm estimates of Cauchy transforms on the Dirichlet problem and their applications 9 We first estimate J ∗ [ g ]( z ) − J ∗ [ g ]( w ) as follows: | J ∗ [ g ]( z ) − J ∗ [ g ]( w ) | ≤ | z − w | (cid:90) D | g ( τ ) || ¯ z − /τ || ¯ w − /τ | d A ( τ ) ≤ | z − w | (cid:90) C \ D | g (1 /η ) || ¯ z − η || ¯ w − η | d A ( η ) , where the last inequality holds because 1 / | η | ≤
1, for any η ∈ C \ D . Because g ∈ L p ( D ) (2 < p < ∞ ), by using H¨older inequality for integrals, we have | J ∗ [ g ]( z ) − J ∗ [ g ]( w ) | ≤ (cid:107) g (cid:107) p | z − w | (cid:18)(cid:90) C \ D d A ( η ) | ¯ z − η | q | ¯ w − η | q (cid:19) /q ≤ | z − w | /q − (cid:107) g (cid:107) p (cid:18)(cid:90) C d A ( ξ ) | ξ | q | − ξ | q (cid:19) /q = C q (cid:107) g (cid:107) p | z − w | − /p , where q = p/ ( p − ∈ (1 , C q = (cid:18)(cid:90) C d A ( ξ ) | ξ | q | − ξ | q (cid:19) /q is a constant depending only on q (see Section 4 for more details).Next, we estimate C [ g ]( z ) − C [ g ]( w ) as follows (cf. [2, Theorem 4.3.13]): | C [ g ]( z ) − C [ g ]( w ) | ≤ (cid:107) g (cid:107) p | z − w | (cid:18)(cid:90) D | z − τ | q | w − τ | q d A ( τ ) (cid:19) /q ≤ (cid:107) g (cid:107) p | z − w | /q − (cid:18)(cid:90) C | η | q | − η | q d A ( η ) (cid:19) /q = C q (cid:107) g (cid:107) p | z − w | − /p . Then |C [ g ]( z ) − C [ g ]( w ) | ≤ C q (cid:107) g (cid:107) p | z − w | − /p . Similarly, we obtain the following estimate: |C [ g ]( z ) − C [ g ]( w ) | ≤ C q (cid:107) g (cid:107) p | z − w | − /p . Because C [ g ]( z ) = ∂∂z G [ g ]( z ) and C [ g ]( z ) = ∂∂ ¯ z G [ g ]( z ), we see that G [ g ] ∈ C ,ν ( D ),where ν = 1 − /p . The proof of Theorem 1.2 is complete. (cid:3) Proof of Theorem 1.3.
For every z, w ∈ D , set λ = ϕ w ( z ) = ( w − z ) / (1 − wz )and let ψ = u ◦ ϕ w . Then(3.2) (1 − | w | )Λ u ( w ) = | ψ z (0) | + | ψ ¯ z (0) | = Λ ψ (0) . On the other hand, as z = ( w − λ ) / (1 − ¯ wλ ) = ϕ w ( λ ), it follows from (1.3) that(1 − | z | )Λ u ( z ) = (1 − | ϕ w ( λ ) | )Λ u ( ϕ w ( λ ))= (1 − | λ | ) | ϕ (cid:48) w ( λ ) | Λ u ( ϕ w ( λ ))= (1 − | λ | )Λ ψ ( λ ) . Therefore, (cid:12)(cid:12) (1 − | z | )Λ u ( z ) − (1 − | w | )Λ u ( w ) (cid:12)(cid:12) = (cid:12)(cid:12) (1 − | λ | )Λ ψ ( λ ) − Λ ψ (0) (cid:12)(cid:12) ≤ | λ | Λ ψ (0) + (1 − | λ | ) | Λ ψ ( λ ) − Λ ψ (0) | . (3.3)It follows from [9, Lemma 2.7] that for any g ∈ L ∞ ( D ), we have Λ u ( z ) ≤ / (cid:107) g (cid:107) ∞ . By using (3.2), we see that(3.4) Λ ψ (0) ≤ (cid:107) g (cid:107) ∞ . Next, we estimate | Λ ψ ( λ ) − Λ ψ (0) | as follows: Let Γ be the line segment joining0 and λ , i.e., Γ has the parametric equation z ( t ) = tλ , where 0 ≤ t ≤
1. Then | Λ ψ ( λ ) − Λ ψ (0) | ≤ (cid:90) Γ Λ u ( ϕ w ( tλ )) | ϕ (cid:48) w ( tλ ) || d( tλ ) | (3.5) ≤ (cid:107) g (cid:107) ∞ | λ | (cid:90) − | tλ | (1 − | tλ | ) d t = 23 (cid:107) g (cid:107) ∞ | λ | (cid:18) | λ | log 11 − | λ | − (cid:19) . By combining (3.3), (3.4) and (3.5), we obtain (cid:12)(cid:12) (1 − | z | )Λ u ( z ) − (1 − | w | )Λ u ( w ) (cid:12)(cid:12) ≤ (cid:107) g (cid:107) ∞ | λ | , because (1 − | λ | ) (cid:18) | λ | log 11 − | λ | − (cid:19) < | λ | <
1. This completes the proof. (cid:3) Appendiex
In this section, we calculate the precise value of C q , C q = (cid:18)(cid:90) C d A ( ξ ) | ξ | q | − ξ | q (cid:19) /q , where 1 < q <
2. Note that in the proof of Theorem 1.2, it was only required thatthis quantity is bounded.Recall that the hypergeometric function p F q is defined for | z | < p F q [ a , a , . . . , a p ; b , b , . . . , b q ; z ] = ∞ (cid:88) n =0 ( a ) n · · · ( a p ) n ( b ) n · · · ( b q ) n z n n ! . Here ( a ) n is the Pochhammer symbol and given as follows ( a ) n = Γ( n + a )Γ( a ) . Lemma C. (cf. [11, Theorem 2.1.2])
The series q +1 F q ( a , . . . , a q +1 ; b , . . . , b q ; x ) converges absolutely for | x | = 1 and Re( (cid:80) qm =1 b m − (cid:80) q +1 n =1 a n ) > . This seriesconverges conditionally for x = e iθ (cid:54) = 1 and ≥ Re( (cid:80) qm =1 b m − (cid:80) q +1 n =1 a n ) > − .This series diverges for Re( (cid:80) qm =1 b m − (cid:80) q +1 n =1 a n ) ≤ − . p → L q norm estimates of Cauchy transforms on the Dirichlet problem and their applications 11 Figure 3.
The constant C q as a function of q ∈ (1 , η = re it ∈ D , by using (2.1) and Lemma C, we have (cid:90) D d A ( η ) | η | q | − η | q = 2 ∞ (cid:88) n =0 (cid:18) Γ( n + q/ n !Γ( q/ (cid:19) (cid:90) r n +1 − q d r = F [1 − q , q , q ; 1 , − q ; 1]1 − q < ∞ . For η = re it ∈ C \ D , again by (2.1) and Lemma C, we have (cid:90) C \ D d A ( η ) | η | q | − η | q = (cid:90) D | η | q − | − η | q d A ( η )= F [ q − , q , q ; 1 , q ; 1] q − < ∞ . By combining the above two identities, we obtain the constant C q . Numerical valuesof C q for q ∈ (1 ,
2) are illustrated in Figure 3.
Acknowledgments . The research of the authors were supported by NSFs of China(No. 11501220, 11971124, 11971182), NSFs of Fujian Province (No. 2016J01020,2019J0101), Subsidized Project for Postgraduates’ Innovative Fund in Science Re-search of Huaqiao University and the Promotion Program for Young and Middle-aged Teachers in Science and Technology Research of Huaqiao University (ZQN-PY402).
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E-mail address : [email protected] Antti Rasila, Technion – Israel Institute of Technology, Guangdong Technion,Shantou, Guangdong 515063, China.
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