TY - JOUR
ID - 36058
TI - $L_{p;r} $ spaces: Cauchy Singular Integral, Hardy Classes and Riemann-Hilbert Problem in this Framework
JO - Sahand Communications in Mathematical Analysis
JA - SCMA
LA - en
SN - 2322-5807
AU - Huseynli, Ali
AU - Mirzabalayeva, Asmar
AD - Department of Mathematics, Khazar University, AZ1096, Baku, Azerbaijan and Department of Non-harmonic analysis, Institute of Mathematics and Mechanics of NAS of Azerbaijan, AZ1141, Baku, Azerbaijan.
AD - Department of Non-harmonic analysis", Institute of Mathematics and Mechanics of NAS of Azerbaijan, AZ1141, Baku, Azerbaijan.
Y1 - 2019
PY - 2019
VL - 16
IS - 1
SP - 83
EP - 91
KW - Function space
KW - Hardy class
KW - singular integral
KW - Riemann-Hilbert problem
DO - 10.22130/scma.2018.81285.391
N2 - In the present work the space $L_{p;r} $ which is continuously embedded into $L_{p} $ 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 results in the classical theory of Hardy spaces are proved for the new spaces. It is shown that the Cauchy singular integral operator is bounded in $L_{p;r} $. The problem of basisness of the system $\left\{A\left(t\right)e^{{\mathop{\rm int}} }; B\left(t\right)e^{-{\mathop{\rm int}} } \right\}_{n\in Z_{+} }, $ is also considered. It is shown that under an additional condition this system forms a basis in $L_{p;r} $ if and only if the Riemann-Hilbert problem has a unique solution in corresponding Hardy class ${ H}_{p;r}^{+} \times { H}_{p;r}^{+} $.
UR - https://scma.maragheh.ac.ir/article_36058.html
L1 - https://scma.maragheh.ac.ir/article_36058_e30acb2ad0eafa93148679627a197562.pdf
ER -