대한수학회지 (Journal of the Korean Mathematical Society)

  • 제45권1호
  • /
  • Pages.63-78
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  • 2008
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  • 0304-9914(pISSN)
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  • 2234-3008(eISSN)
대한수학회 (Korean Mathematical Society)
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HOLOMORPHIC FUNCTIONS ON THE MIXED NORM SPACES ON THE POLYDISC

  • Stevic, Stevo (Mathematical Institute of the Serbian Academy of Science)
  • 발행 : 2008.01.31
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초록

We generalize several integral inequalities for analytic functions on the open unit polydisc $U^n={\{}z{\in}C^n||zj|<1,\;j=1,...,n{\}}$. It is shown that if a holomorphic function on $U^n$ belongs to the mixed norm space $A_{\vec{\omega}}^{p,q}(U^n)$, where ${\omega}_j(\cdot)$,j=1,...,n, are admissible weights, then all weighted derivations of order $|k|$ (with positive orders of derivations) belong to a related mixed norm space. The converse of the result is proved when, p, q ${\in}\;[1,\;{\infty})$ and when the order is equal to one. The equivalence of these conditions is given for all p, q ${\in}\;(0,\;{\infty})$ if ${\omega}_j(z_j)=(1-|z_j|^2)^{{\alpha}j},\;{\alpha}_j>-1$, j=1,...,n (the classical weights.) The main results here improve our results in Z. Anal. Anwendungen 23 (3) (2004), no. 3, 577-587 and Z. Anal. Anwendungen 23 (2004), no. 4, 775-782.

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  4. Mixed norm variable exponent Bergman space on the unit disc vol.61, pp.8, 2016, https://doi.org/10.1080/17476933.2016.1140750
  5. On a New Integral-Type Operator from the Weighted Bergman Space to the Bloch-Type Space on the Unit Ball vol.2008, 2008, https://doi.org/10.1155/2008/154263
  6. Weighted Integrals and Bloch Spaces of n-Harmonic Functions on the Polydisc vol.29, pp.1, 2008, https://doi.org/10.1007/s11118-008-9087-3
  7. The Numerical Range of Toeplitz Operator on the Polydisk vol.2009, 2009, https://doi.org/10.1155/2009/757964