Anisotropic classes of homogeneous pseudodifferential symbols

Volume 200 / 2010

Árpád Bényi, Marcin Bownik Studia Mathematica 200 (2010), 41-66 MSC: Primary 47G30, 42B20, 43A85; Secondary 42B15, 42B35, 42C40. DOI: 10.4064/sm200-1-3

Abstract

We define homogeneous classes of $x$-dependent anisotropic symbols $\dot{S}^{m}_{\gamma, \delta}(A)$ in the framework determined by an expansive dilation $A$, thus extending the existing theory for diagonal dilations. We revisit anisotropic analogues of Hörmander–Mikhlin multipliers introduced by Rivière [Ark. Mat. 9 (1971)] and provide direct proofs of their boundedness on Lebesgue and Hardy spaces by making use of the well-established Calderón–Zygmund theory on spaces of homogeneous type. We then show that $x$-dependent symbols in $\dot{S}^0_{1,1} (A)$ yield Calderón–Zygmund kernels, yet their $L^2$ boundedness fails. Finally, we prove boundedness results for the class $\dot{S}^m_{1,1} (A)$ on weighted anisotropic Besov and Triebel–Lizorkin spaces extending isotropic results of Grafakos and Torres [Michigan Math. J. 46 (1999)].

Authors

  • Árpád BényiDepartment of Mathematics
    Western Washington University
    516 High Street
    Bellingham, WA 98225-9063, U.S.A.
    e-mail
  • Marcin BownikDepartment of Mathematics
    University of Oregon
    Eugene, OR 97403-1222, U.S.A.
    and
    Institute of Mathematics
    Polish Academy of Sciences
    Abrahama 18
    81-825 Sopot, Poland
    e-mail

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