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L p;r spaces: Cauchy Singular Integral, Hardy Classes and Riemann-Hilbert Problem in this Framework

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In the present work the space Lp;r which is continuously embedded into Lp is introduced. The corresponding Hardy spaces of analytic functions are defined as well. Some properties of the functions from these spaces are studied. The analogs of some
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  Sahand Communications in Mathematical Analysis (SCMA) Vol. 16 No. 1 (2019), 83-91 http://scma.maragheh.ac.ir DOI: 10.22130/scma.2018.81285.391 L  p ; r  spaces: Cauchy Singular Integral, Hardy Classes andRiemann-Hilbert Problem in this Framework Ali Huseynli 1 ∗ and Asmar Mirzabalayeva 2 Abstract.  In the present work the space  L p ; r  which is continu-ously embedded into  L p  is introduced. The corresponding Hardyspaces of analytic functions are defined as well. Some properties of the functions from these spaces are studied. The analogs of someresults in the classical theory of Hardy spaces are proved for thenew spaces. It is shown that the Cauchy singular integral oper-ator is bounded in  L p ; r . The problem of basisness of the system 󰁻 A ( t ) e int ; B ( t ) e − int 󰁽 n ∈ Z + ,  is also considered. It is shown thatunder an additional condition this system forms a basis in  L p ; r  if and only if the Riemann-Hilbert problem has a unique solution incorresponding Hardy class  H  + p ; r × H  + p ; r . 1.  Introduction During the last two decades, non-standard function spaces becamean extremely popular subject because of their appearance in modernproblems of analysis and qualitative theory of PDEs. Introduction of Lebesgue spaces with variable exponents at the end of last century andvariety of extraordinary results obtained therein were the main moti-vation and the inception of this new tendency in analysis. For origi-nal results in the theory of Lebesgue spaces with variable exponent thereader may consult the books [9, 10] and references therein. Anotherkind of non-standard function spaces-small and grand Lebesgue spaceswere defined motivated by C.B. Morrey’s seminal work, in which Morrey 2010  Mathematics Subject Classification.  32A55, 30H10, 42B35. Key words and phrases.  Function space, Hardy class, singular integral, Riemann-Hilbert problem.Received: 10 February 2018, Accepted: 18 July 2018. ∗ Corresponding author. 83  84 A. HUSEYNLI AND A. MIRZABALAYEVA space has been defined and proved to be extremely useful tool in qual-itative theory of elliptic equations(for further discussions, see, [1, 14]and references therein). We only mention few recent monographs witha comprehensive bibliography, where in-depth treatment of these issuescan be found: [1, 9, 14]. We mention the works [3–7, 12] because of theircloseness to the spirit of the present paper.In [13] Y. Katznelson considered the class of functions, whose Fouriercoefficients are  p -th power summable and proved a uniqueness theo-rem for the Fourier series of the functions of this class. In the presentwork equipping this class of Katznelson by a norm, the  L  p ; r  functionspace which is continuously embedded into  L  p  is introduced. The corre-sponding Hardy spaces of analytic functions are defined as well. Someproperties of the functions from these spaces are studied. The analogsof some results in the classical theory of Hardy spaces are proved for thenew spaces. It is shown that the Cauchy singular integral operator isbounded in  L  p ; r .2.  L  p ; r  spaces Let  L  p  ≡  L  p  ( − π,π )  ,  1  ≤  p  ≤  + ∞  and  l r ,  1  ≤  r  ≤  + ∞  be theusual spaces of   p -th power summable functions and  r -th power summablesequences of scalars, respectively; ˆ f   denotes the sequence of Fouriercoefficients of the function f  :ˆ f   ≡{ f  n } n ∈ Z  , f  n  ≡  1 √  2 π 󲈫   π − π f   ( t ) e − int dt, n ∈ Z . Denote the set 􀁻 f   ∈ L  p  : ˆ f   ∈ l r 􀁽  by  L  p ; r . It is evident that  L  p ; r  is alinear space with respect to pointwise operations and  ∥ f  ∥  p ; r  =  ∥ f  ∥  p  +  ˆ f   l r defines a norm in  L  p ; r  here and thereafter ∥ f  ∥  p  = ∥ f  ∥ L p . We showthat the space  L  p ; r  is a Banach space. Indeed, if  􀁻 ˆ f  m 􀁽 m ∈ N ⊂  L  p ; r  isany fundamental sequence, then the sequences  { f  m } m ∈ N  and 􀁻 ˆ f  m 􀁽 m ∈ N are fundamental in the spaces  L  p  and  l r , respectively. Hence there exist f   ∈  L  p  and ˆ a  ∈  l r  such that 􀁻 ˆ f  m 􀁽 m ∈ N → { f  m } m ∈ N  and  ˆ f  m  →  ˆ a , as m →∞ . It is easy to observe that ˆ f   = ˆ a .Take any  g  ∈ L q 󰀨 1  p  +  1 q  = 1 󰀩  and consider the linear functional l g  ( f  ) = 1 √  2 π 󲈫   π − π f   ( t ) g ( t ) dt defined in  L  p ; r . As  l g  is bounded,  L q  ⊂  ( L  p ; r ) ∗ . We shall identifythe function from  L q  with the linear functional generated by itself. In  L p ; r  SPACES: CAUCHY SINGULAR INTEGRAL, HARDY CLASSES AND ... 85 that sense the system  E   ≡ 󐁻 e int 󐁽 n ∈ Z  is biorthogonal to the system 12 π E   = 󐁻  12 π e int 󐁽 n ∈ Z  ⊂ L  p ; r .Consider the problem of basicity of   E   in  L  p ; r . First, let  p ∈ (1 , + ∞ ).It is known that in that case  E   forms a basis in  L  p . Take any  f   ∈ L  p ; r .We can write ∥ f   − T  m f  ∥ L p ; r = ∥ f   − T  m f  ∥  p  +  + ∞ 󲈑 | n |≥ m +1 | f  n | r  1 r , where  T  m f   =  ∑ | n |≤ m  f  n e int . Since  E   is a basis in  L  p ,  ∥ f   − T  m f  ∥→ 0,as  m →∞ . Also, since ˆ f   ∈ l r , the second term of above sum also has azero limit. The uniqueness of expansion is obvious.Now consider the case when  p  = 1 and  r ∈ [1 ,  2]. Take any  f   ∈ L 1; r .Then ˆ f   ∈ l 2 , and therefore, it implies that  f   ∈ L 2 . We can write ∥ f   − T  m f  ∥ L p ; r =  ˆ f   −  ˆ T  m f   l r + ∥ f   − T  m f  ∥ 1 ≤  + ∞ 󲈑 | n |≥ m +1 | f  n | r  1 r +  c ·∥ f   − T  m f  ∥ 2  → 0as  m →∞ .It proves the following Proposition 2.1.  The system   E   forms a basis in   L  p ; r  for any   p  ∈ (1 , + ∞ )  and   r  ∈  [1 , + ∞ ] ; the same property holds in   L 1; r  for any   r  ∈ [1 ,  2] . 3.  Cauchy Singular Integral Throughout the paper  ω  will denote the open unit disc ω  =  { z  ∈ C   : | z | <  1 }  and  γ   =  ∂ω  will denote the unit circle  ω  = { z  ∈ C   : | z | = 1 } . Let  f   ∈ L 1  ( γ  ). Consider the Cauchy-type integral F   ( z ) = 12 πi 󲈫  γ  f   ( τ  ) τ   − zdτ, z / ∈ γ, and the singular integral( Sf  )( τ  ) = 12 πi 󲈫  γ  f   ( ξ  ) dξ ξ  − τ  , τ   ∈ γ  corresponding to it.It is well known that  Sf   exists a.e. on  γ   (see, e.g. [8, 11]). In thesequel, we will use the following space of analytic functions generalized  86 A. HUSEYNLI AND A. MIRZABALAYEVA by  L  p ; ν  . Denote H  +  p ; ν   = 󰁻 f   :  f   is analytic on  ω  and  ∥ f  ∥ H  + p ; ν = sup 0 <r< 1 ∥ f  r  ( · ) ∥ L p ; ν <  + ∞ 󰁽 , where  f  r  ( t ) =  f  󐀨 re it 󐀩 .Let  f   ∈ H  +  p ; ν  ,  1  < p <  + ∞ ,  1 ≤ ν   ≤ + ∞ , and f  󐀨 re it 󐀩  = ∞ 󲈑 n =0 f  n r n e int . Then we have ∥ f  ∥ H  + p ; ν = sup 0 <r< 1 􀀨  ∞ 󲈑 n =0 | f  n | ν  r νn 􀀩 1 ν + ∥ f  r  ( · ) ∥ L p  . Denote by  f  + ( τ  ) the nontangential boundary values of   f   ( τ  ) on  γ  .By a classical theorem  f  + ( τ  ) exists a.e. on  γ   and sup 0 <r< 1 ∥ f  r  ( · ) ∥ L p = ∥ f  + ( · ) ∥ L p (see, e.g. [15]). As each summand on the right-hand side of the above equality is monotonic increasing function of   r , we get ∥ f  ∥ H  + p ; ν = 􀀨  ∞ 󲈑 n =0 | f  n | ν  􀀩 1 ν +  f  +  L p =  f  +  L p ; ν . Now define the set  m H  −  p ; ν   for fixed integer  m . Let  f   ( z ) be a function,analytic outside  ω  and(3.1)  f   ( z ) = k 󲈑 n = −∞ f  n z n with some  k ≤ m  . Write  f   as f   ( z ) =  P  k  ( z ) +  f  1  ( z ) , where  P  k  ( · ) and  f  1  ( · ) are the analytic, if any, and principal parts of theexpansion (3.1), respectively. We will say that  f   ( · ) belongs to  m H  −  p ; ν   if  f  1 󐀨 1¯ z 󐀩 ∈ H  +  p ; ν  .Let  f   ∈ H  +  p ; ν  , 1 ≤  p <  + ∞ , 1 ≤ ν   ≤ + ∞ . It immediately follows that f   ∈ H  +  p  , and therefore, we have the Cauchy formula(3.2)  f   ( z ) = 12 πi 󲈫  γ  f  + ( τ  ) τ   − z dz,  ∀ z  ∈ ω. As shown above,  f  + ( · ) ∈ L  p ; ν  . It is clear that,  f   ( · ) ∈ H  +1  .  L p ; r  SPACES: CAUCHY SINGULAR INTEGRAL, HARDY CLASSES AND ... 87 Conversely, suppose  f   ( · ) ∈ H  +1  and  f  + ( · ) ∈ L  p ; ν  . From here we have f  + ∈  L  p , and as a result the representation (3.2) is true. Then, fromthe equality(3.3)  ∥ f  ∥ H  + p ; ν =  f  +  L p ; ν , it follows that  f   ∈ H  +  p ; ν  .Hence, the following theorem was proved Theorem 3.1.  The function   f   ( · )  belongs to  H  +  p ; ν  ,  1  ≤  p <  + ∞ ,  1  ≤ ν   ≤ + ∞ iff   f  + ( · ) ∈ L  p ; ν  ; in that case the Cauchy formula   (3.2)  is valid. By (3.3), we deduce the following theorem. Theorem 3.2.  H  +  p ; ν   and   m H  −  p ; ν  ,  1 ≤  p <  + ∞ ,  1 ≤ ν   ≤∞  are Banach spaces. Now consider the singular integral  S   in  L  p ; ν  , for 1  < p <  + ∞  and1  ≤  ν   ≤ ∞ . Let  F   ( z ) be the corresponding Cauchy-type integral with f   ( · ) as its density. Therefore, the following Sokhotski-Plemely formulais true F  ± ( τ  ) = ± 12 f   ( τ  ) + ( Sf  )( τ  ) , τ   ∈ γ, where  F  + ( · ) (( F  − ( · )) is the interior(exterior) non-tangential boundaryvalues of   F   ( z ) along  γ  . Hence f   ( τ  ) =  F  + ( τ  ) − F  − ( τ  ) , τ   ∈ γ. Let f   ( τ  ) = 󲈑 n ∈ Z f  n τ  n be the Fourier expansion of   f  ( · ) ∈ L  p  ( γ  ). Then F  + ( τ  ) = ∞ 󲈑 n =0 f  n τ  n , F  − ( τ  ) = − ∞ 󲈑 n =1 f  − n τ  − n . We have( Sf  )( τ  ) =  F  + ( τ  ) −  12 f   ( τ  )= 󲈑 n ∈ Z g n τ  n , where g n  = 12 󰁻  f  n , − f  n ,n ≥ 0 ,n <  0 . Then(3.4)  ∥ Sf  ∥ L p ; ν = 􀀨󲈑 n ∈ Z | g n | ν  􀀩 1 ν + ∥ Sf  ∥ L p .
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