JournalsrmiVol. 15, No. 2pp. 233–265

Singular integral operators with non-smooth kernels on irregular domains

  • Xuan Thinh Duong

    Macquarie University, Sydney, Australia
  • Alan G.R. McIntosh

    Australian National University, Canberra, Australia
Singular integral operators with non-smooth kernels on irregular domains cover
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Abstract

Let χ\chi be a space of homogeneous type. The aims of this paper are as follows: i) Assuming that TT is a bounded linear operator on L2(χ)L_2(\chi) we give a sufficient condition on the kernel of TT so that TT is of weak type (1,1), hence bounded on Lp(χ)L_p(\chi) for 1<p21 < p ≤ 2; our condition is weaker than the usual Hörmander integral condition. ii) Assuming that TT is a bounded linear operator on L2(Ω)L_2(\Omega)  where Ω\Omega is a measurable subset of χ\chi, we give a sufficient condition on the kernel of TT so that TT is of weak type (1,1), hence bounded on Lp(Ω)L_p(\Omega) for 1<p21 < p ≤2. iii) We establish sufficient conditions for the maximal truncated operator TT_*, which is defi ned by Tu(x)T_*u(x) = supϵ>0Tϵu(x)_{\epsilon>0} | T_\epsilon u(x) |, to be LpL_p bounded, 1<p<1 < p < \infty. Applications include weak (1,1) estimates of certain Riesz transforms and LpL_p boundedness of holomorphic functional calculi of linear elliptic operators on irregular domains.

Cite this article

Xuan Thinh Duong, Alan G.R. McIntosh, Singular integral operators with non-smooth kernels on irregular domains. Rev. Mat. Iberoam. 15 (1999), no. 2, pp. 233–265

DOI 10.4171/RMI/255