JournalsrmiVol. 29, No. 1pp. 53–74

Regularizations of general singular integral operators

  • Constanze Liaw

    Baylor University, Waco, USA
  • Sergei Treil

    Brown University, Providence, USA
Regularizations of general singular integral operators cover
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Abstract

In the theory of singular integral operators significant effort is often required to rigorously define such an operator. This is due to the fact that the kernels of such operators are not locally integrable on the diagonal s=ts=t, so the integral formally defining the operator TT or its bilinear form Tf,g\langle Tf, g \rangle is not well defined (the integrand in not in L1L^1) even for “nice” ff and gg. However, since the kernel only has singularities on the “diagonal” s=ts=t, the bilinear form Tf,g\langle Tf, g \rangle is well defined, say, for bounded compactly supported functions with separated supports.

One of the standard ways to interpret the boundedness of a singular integral operators is to consider the regularized kernel

Kε(s,t)=K(s,t)m((st)/ε),K_\varepsilon(s, t) = K(s, t)\, m((s-t)/\varepsilon),

where the cut-off function mm is 00 in a neighborhood of the origin, so the integral operators TεT_\varepsilon with kernel KεK_\varepsilon are well defined (at least on a dense set). Then instead of asking about the boundedness of the operator TT, which is not well defined, one can ask about uniform boundedness (in ε\varepsilon) of the regularized operators TεT_\varepsilon.

For the standard regularizations one usually considers truncated operators TεT_\varepsilon with m(s)=1[1,)(s)m(s) =\mathbf{1}_{[1, \infty)} (|s|), although smooth cut-off functions were also considered in the literature.

The main result of the paper is that for a wide class of singular integral operators (including the classical Calderón–Zygmund operators in nonhomogeneous two weight settings), the so called restricted LpL^p boundedness, i.e., the uniform estimate

Tf,gCfpgp|\langle Tf, g \rangle| \le C \,\|f\|_p \,\|g\|_{p'}

for bounded compactly supported ff and gg with separated supports implies the uniform LpL^p-boundedness of regularized operators TεT_\varepsilon for any reasonable choice of smooth cut-off function mm. For example, any mC(RN)m \in C^\infty(\mathbb{R}^N)}, m0m\equiv 0 in a neighborhood of 00, and such that 1m1-m is compactly supported would work.

If the kernel KK satisfies some additional assumptions (which are satisfied for classical singular integral operators like the Hilbert transform, Cauchy transform, Ahlfors–Beurling transform, and generalized Riesz transforms), then the restricted LpL^p boundedness also implies the uniform LpL^p boundedness of the classical truncated operators TεT_\varepsilon (m(s)=1[1,)(s)m(s) =\mathbf{1}_{[1, \infty)} (|s|)).

Cite this article

Constanze Liaw, Sergei Treil, Regularizations of general singular integral operators. Rev. Mat. Iberoam. 29 (2013), no. 1, pp. 53–74

DOI 10.4171/RMI/712