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Characterizations of Hardy-type, Bergman-type and Dirichlet-type spaces on certain classes of complex-valued functions

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Characterizations of Hardy-type, Bergman-type and Dirichlet-type spaces on certain classes of complex-valued functions
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    a  r   X   i  v  :   1   4   1   0 .   8   2   8   3  v   1   [  m  a   t   h .   C   V   ]   3   0   O  c   t   2   0   1   4 CHARACTERIZATIONS OF HARDY-TYPE, BERGMAN-TYPEAND DIRICHLET-TYPE SPACES ON CERTAIN CLASSES OFCOMPLEX-VALUED FUNCTIONS SHAOLIN CHEN, ANTTI RASILA, AND MATTI VUORINEN Abstract.  In this paper, we continue our investigation of function spaces on cer-tain classes of complex-valued functions. In particular, we give characterizationson Hardy-type, Bergman-type and Dirichlet-type spaces. Furthermore, we presentapplications of our results to certain nonlinear PDEs. 1.  Introduction and main results For a positive integer  n  ≥  1, let  C n denote the complex  Euclidean   n -space  . For z   := ( z  1 ,...,z  n ) and  w  = ( w 1 ,...,w n ) in  C n , we let  z   = ( z  1 ,...,z  n ) ,  and   z,w   :=  nk =1  z  k w k  with the  Euclidean norm    z    :=   z,z   1 / 2 which makes  C n into an  n -dimensional complex  Hilbert   space. For  a  ∈  C n and  r >  0,  B n ( a,r ) denotes the(open) ball of radius  r  with center  a . Also, we let  B n ( r ) :=  B n (0 ,r ) and denotethe unit ball by  B n :=  B n (1). In particular, let  B 1 ( r ) =  D ( r ) and  D  =  B 1 . Fora domain Ω  ⊂  C n with non-empty boundary, let  d Ω ( z  ) be the Euclidean distancefrom  z   to the boundary  ∂  Ω of Ω. Moreover, we always use  d ( z  ) to denote theEuclidean distance from  z   to the boundary of   B n . We denote by  C m ( B n ) the setof all  m -time continuously differentiable complex-valued functions  f   of   B n into  C ,where  m ∈{ 0 , 1 ,... } .For  k  ∈ { 1 ,...,n } , z   = ( z  1 ,...,z  n )  ∈  C n and  f   ∈ C 1 ( B n ), we introduce thefollowing notations: ∇ f   = ( f  z 1 ,...,f  z n ) ,  ∇ f   = ( f  z 1 ,...,f  z n ) and  D f   = ( ∇ f,  ∇ f  ) , where  f  z k  =  ∂f/∂z  k  = 1 / 2  ∂f/∂x k  − i∂f/∂y k  ,  f  z k  =  ∂f/∂z  k  = 1 / 2  ∂f/∂x k  + i∂f/∂y k   and  z  k  =  x k  +  iy k , with  x k  and  y k  real. Let   D f    be the  Hilbert-Schmidt semi-norm   given by  D f   = ( ∇ f   2 + ∇ f   2 ) 1 / 2 . Let  f   =  u  +  iv  ∈ C 1 ( B n ), where  u  and  v  are real-valued functions. Then for z   = ( z  1 ,...,z  n ) = ( x 1  +  iy 1 ,...,x n  +  iy n ) ∈ B n ,(1.1)  ∇ f  ( z  )  + ∇ f  ( z  ) ≤∇ u ( z  )  + ∇ v ( z  )  , where ∇ u  =   ∂u∂x 1 , ∂u∂y 1 ,..., ∂u∂x n , ∂u∂y n   and  ∇ v  =   ∂v∂x 1 , ∂v∂y 1 ,..., ∂v∂x n , ∂v∂y n  . File: crv1.tex, printed: 31-10-2014, 11.46 2000  Mathematics Subject Classification.  Primary: 32A10, 30D55; Secondary: 30C65, 58J10. Key words and phrases.  Hardy-type space, Bergman-type space, Dirichlet-type space. 1  2 Sh. Chen, A. Rasila and M. Vuorinen Note that the converse of (1.1) is not always true (see [4]). Generalized Hardy spaces.  For  p  ∈  (0 , ∞ ], the  generalized Hardy space   H  pg ( B n )consists of measurable functions  f   :  B n →  C  such that  M   p ( r,f  ) exists for all r  ∈ (0 , 1) and   f    p  < ∞ , where  f    p  =  sup 0 <r< 1 M   p ( r,f  ) ,  if   p ∈ (0 , ∞ ) , sup z ∈ B n | f  ( z  ) | ,  if   p  = ∞ , M   p ( r,f  ) =   ∂  B n | f  ( rζ  ) |  p dσ ( ζ  )  1 /p and  dσ  denotes the normalized Lebesgue surface measure in  ∂  B n .There are numerous characterizations of the classical analytic Hardy spaces in theliterature, see for example [14, 17, 18, 21, 22, 28]. But, to our knowledge, there are few analogous results for general complex-valued functions. In this paper, we givethe following characterization of a class of complex-valued functions  f   in Hardy-typespaces. Theorem 1.  For   p ≥ 2 , let   f   ∈C 2 ( B n )  with   Re( f  ∆ f  ) ≥ 0 .  Then,   B n d ( z  )∆  | f  ( z  ) |  p  dV  N  ( z  )  < ∞ if and only if   f   ∈H  pg ( B n ) , where   ∆  is the usual complex Laplacian operator  ∆ := 4 n  k =1 ∂  2 ∂z  k ∂z  k = n  k =1   ∂  2 ∂x 2 k +  ∂  2 ∂y 2 k   for   z   = ( z  1 ,...,z  n ) = ( x 1  +  iy 1 ,...,x n  +  iy n ) ∈ B n . Yukawa PDE.  Let  τ, η  :  B n →  [0 , ∞ ) be continuous and  f   =  u  +  iv  ∈ C 2 ( B n ),where  u  and  v  are real-valued functions in  B n . The nonlinear elliptic partial differ-ential equation (PDE) of the form(1.2) ∆ f  ( z  ) =  τ  ( z  ) f  ( z  ) +  η ( z  )Re  f  ( z  )  is called the  non-homogeneous Yukawa PDE  , where  z   ∈ B n . If   τ   in (1.2) is a positiveconstant function and  η  ≡  0, then we have the usual Yukawa PDE. This equationarose from the work of the Japanese Nobel physicist Hideki Yukawa, who used it todescribe the nuclear potential of a point charge as  e −√  τr /r  (cf. [2, 3, 7, 11, 12, 13, 16, 30, 34]). As an application of Theorem 1, we obtain the following result. Corollary 1.1.  For   p ≥ 2 , let   f   ∈C 2 ( B n )  satisfying   (1.2) . Then,   B n d ( z  )∆  | f  ( z  ) |  p  dV  N  ( z  )  < ∞ if and only if   f   ∈H  pg ( B n ) . A continuous increasing function  ω  : [0 , ∞ )  →  [0 , ∞ ) with  ω (0) = 0 is called a majorant   if   ω ( t ) /t  is non-increasing for  t >  0 (cf. [9, 10, 26, 27]). Given a subset Ω  Characterizations of Hardy-type, Bergman-type and Dirichlet-type spaces 3 of   C , a function  f   : Ω → C  is said to belong to the  Lipschitz space   L ω (Ω) if there isa positive constant  C   such that | f  ( z  ) − f  ( w ) |≤ Cω ( | z  − w | ) for all  z,w  ∈ Ω . A classical result of Hardy and Littlewood asserts that if   p  ∈  (0 , ∞ ],  α  ∈  (1 , ∞ )and  f   is an analytic function in  D , then (cf. [14, 21, 22]) M   p ( r,f  ′ ) =  O   11 − r  α   as  r  → 1 , if and only if  M   p ( r,f  ) =  O  log 11 − r  α − 1   as  r  → 1 . In [18], via the closed graph theorem, Girela, Pavlovi´c and Pel´aez refined theabove result for the case  α  = 1 as follows. Theorem A.  ([18, Theorem 1.1])  Let   p ∈ (2 , ∞ ) . For   r  ∈ (0 , 1) , if   f   is analytic in  D  such that  M   p ( r,f  ′ ) =  O   11 − r   as   r  → 1 , then  M   p ( r,f  ) =  O  log 11 − r  12   as   r  → 1 and the exponent   1 / 2  is sharp. Theorem A gives an affirmative answer to the open problem in [17, p. 464,Equation (26)]. For related investigations on this topic, we refer to [3, 5, 7]. Next we study the relationship between the integral means of solutions to theequation (1.2) and those of their two order partial derivative. Our result is given asfollows. Theorem 2.  Let   ω  be a majorant and   f   ∈C 2 ( B n )  satisfying   (1.2)  with   η + τ <  4 n/p ,where   τ   and   η  are nonnegative constant functions. For   p ≥ 2  and   r  ∈ (0 , 1) , if  M   p ( r,D ∗ f  ) ≤ M  ∗ ω   11 − r  , then  M   p ( r,D f  ) ≤   M  ∗ 2   D f  (0)  2 +  M  ∗ 1    10 ω   11 − rt  dt  12 , and   f   ∈H  pg ( B n ) , where   M  ∗  is a positive constant, D ∗ f   =   n   j =1 n  k =1  | f  z k z j | 2 + | f  z k z j | 2 + | f  z k z j | 2 + | f  z k z j | 2  12 ,M  ∗ 1  = 2  p (2  p − 3)( M  ∗ ) 2 ω (1)  and   M  ∗ 2  = 1 / [1 −  p ( η  +  τ  ) / (4 n )] . In particular, by taking  ω ( t ) =  t  in Theorem 2, we obtain the following result.  4 Sh. Chen, A. Rasila and M. Vuorinen Corollary 1.2.  Let   p ≥ 2  and   f   ∈C 2 ( B n )  satisfying   (1.2)  with   η + τ <  4 n/p , where  τ   and   η  are nonnegative constant functions. For   r  ∈ (0 , 1) , if  M   p ( r,D ∗ f  ) =  O   11 − r   as   r  → 1 , then  M   p ( r,D f  ) =  O  log 11 − r  12   as   r  → 1 , and   f   ∈H  pg ( B n ) . Dirichlet-type spaces and Bergman-type spaces.  For  ν, µ, t ∈ R , D f  ( ν,µ,t ) =   B n d ν  ( z  ) | f  ( z  ) | µ  D f  ( z  )  t dV  N  ( z  )  < ∞ is called  Dirichlet-type energy integral   of the complex-valued function  f  , where  dV  N  denotes the normalized Lebesgue volume measure in  B n (cf. [1, 2, 7, 16, 18, 19, 31, 32, 33, 34]). In particular, for  ν   ≥  0,  µ  = 0 and 0  < t <  ∞ , we use  D ν,t ( B n ) todenote the  Dirichlet-type space   consisting of all  f   ∈C 1 ( B n ) with the norm  f   D ν,t  = | f  (0) | +  D f  ( ν, 0 ,t )  1 /t < ∞ . Moreover, for  ν >  − 1, 0  < µ <  ∞  and  t  = 0, we denote by  b ν,µ ( B n ) the  Bergman-type space   consisting of all  f   ∈C 0 ( B n ) with the norm  f   b ν,µ  = | f  (0) | +  D f  ( ν,µ, 0)  1 /µ < ∞ . We refer to [15, 18, 19, 20, 25, 27, 35] for basic characterizations of analytic (or harmonic) Bergman-type spaces and Dirichlet-type spaces. Again, for generalcomplex-valued functions, very little related research can be found from the litera-ture. The following is a characterization of a class of complex-valued functions  f   inBergman-type spaces. Theorem 3.  Let   f   ∈C 2 ( B n )  with   Re( f  ∆ f  ) ≥ 0 .  Then, for   p ≥ 2  and   α ≥ 2 ,   B n (1 −| z  | 2 ) α ∆  | f  ( z  ) |  p  dV  N  ( z  )  < ∞ , if and only if   f   ∈ b α − 2 ,p ( B n ) . The following result easily follows from Theorem 3. Corollary 1.3.  Let   f   ∈C 2 ( B n )  satisfying   (1.2) . Then, for   p ≥ 2  and   α ≥ 2 ,   B n (1 −| z  | 2 ) α ∆  | f  ( z  ) |  p  dV  N  ( z  )  < ∞ if and only if   f   ∈ b α − 2 ,p ( B n ) . Definition 1.  For  m ∈{ 2 , 3 ,... } , we denote by  HZ  m ( B n ) the class of all functions f   ∈C m ( B n ) satisfying  Heinz’s   nonlinear differential inequality (cf. [23]) | ∆ f  ( z  ) |≤ a ( z  )  D f  ( z  )  +  b ( z  ) | f  ( z  ) | +  c ( z  ) , where  a ( z  ),  b ( z  ) and  c ( z  ) are real-valued nonnegative continuous functions in  B n .  Characterizations of Hardy-type, Bergman-type and Dirichlet-type spaces 5 Theorem 4.  Let   M   be a nonnegative constant and   f   ∈ HZ  3 ( B n ) ∩D γ,α ( B n )  with  Re( f  ∆ f  )  ≥  0  and   Re  nk =1  f  z k (∆ f  ) z k  +  f  z k (∆ f  ) z k   ≥  0 ,  where   2  ≤  α  ≤  2 n , γ >  0 ,  sup z ∈ B n  a ( z  )  <  ∞ ,  sup z ∈ B n  b ( z  )  <  ∞  and   c ( z  )  ≤  M   d ( z  )  − q . Then for   p ≥ 2 ,   B n  d ( z  )   pq ∆  | f  ( z  ) |  p  dV  N  ( z  )  < ∞ , where   q   = (2 n  +  γ  ) /α − 1 . The result given below is a consequence of Theorem 4. Corollary 1.4.  For   2 ≤ α  ≤ 2 n  and   γ >  0 , let   f   ∈HZ  3 ( B n ) ∩D γ,α ( B n )  satisfying  (1.2) , where   τ   and   η  are nonnegative constant functions. Then for   p ≥ 2 ,   B n  d ( z  )   pq ∆  | f  ( z  ) |  p  dV  N  ( z  )  < ∞ , where   q   = (2 n  +  γ  ) /α − 1 . Proof.  By elementary calculations, we see that if   f   is a solution to (1.2), then  f  satisfies Heinz’s nonlinear differential inequality. Hence Corollary 1.4 follows from(2.8), (2.9) and Theorem 4.   By Corollaries 1.1, 1.3 and 1.4, we get Corollary 1.5.  For   2  ≤  α  ≤  2 n  and   γ >  0 , let   q   = (2 n  +  γ  ) /α  −  1  and   f   ∈HZ  3 ( B n ) ∩D γ,α ( B n )  satisfying   (1.2) , where   τ   and   η  are nonnegative constant func-tions. (1)  If   p  =  1 q  ≥ 2 , then   f   ∈H  pg ( B n ) ; (2)  If   p ≥ 2  and   pq   ≥ 2 , then   f   ∈ b  pq − 2 ,p ( B n ) . Definition 2.  For  p  ≥  2,  t 1  >  0,  t 2  >  0 and  m  ∈ { 2 , 3 ,... } , we denote by  IHZ  t 1 ,t 2 m  ( B n ) the class of all functions  f   ∈ C m ( B n ) satisfying the inverse  Heinz’s  nonlinear differential inequality∆( | f  ( z  ) |  p ) ≥ a 1 ( z  )  D f  ( z  )  t 1 +  b 1 ( z  ) | f  ( z  ) | t 2 +  c 1 ( z  ) , where  a 1 ( z  ),  b 1 ( z  ) and  c 1 ( z  ) are real-valued nonnegative continuous functions in  B n . Theorem 5.  Let   f   ∈IHZ  t 1 ,t 2 2  ( B n ) ∩H  pg ( B n ) , where   inf  z ∈ B n  a 1 ( z  )+inf  z ∈ B n  b 1 ( z  )  >  0 and   inf  z ∈ B n  c 1 ( z  ) ≥ 0 . (1)  If   inf  z ∈ B n  a 1 ( z  )  >  0 , then   f   ∈D 1 ,t 1 ( B n ) ; (2)  If   inf  z ∈ B n  b 1 ( z  )  >  0 , then   f   ∈ b 1 ,t 2 ( B n ) . For  k  ∈ { 1 ,...,n } , let  λ k  ∈  R  be a constant and  f   ∈ C 1 ( B n ) satisfying thefollowing nonlinear PDE,(1.3)  ∂f ∂z  k =  λ k | f  | α ,
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