Anisotropic classes of homogeneous pseudodifferential symbols

Árpád Bényi¹ ; Marcin Bownik²

¹ Department of Mathematics Western Washington University 516 High Street Bellingham, WA 98225-9063, U.S.A.
² Department 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

Studia Mathematica, Tome 200 (2010) no. 1, pp. 41-66

Résumé

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)].

Détail

DOI : 10.4064/sm200-1-3

Keywords: define homogeneous classes x dependent anisotropic symbols dot gamma delta framework determined expansive dilation extending existing theory diagonal dilations revisit anisotropic analogues rmander mikhlin multipliers introduced rivi ark mat provide direct proofs their boundedness lebesgue hardy spaces making well established calder zygmund theory spaces homogeneous type x dependent symbols dot yield calder zygmund kernels yet their boundedness fails finally prove boundedness results class dot weighted anisotropic besov triebel lizorkin spaces extending isotropic results grafakos torres michigan math