Abstract
In the present paper, we consider a kind of singular integralTf(x)=p.v.∫RnΩ(y)|y|n−βf(x−y)dy which can be viewed as an extension of the classical Calderón-Zygmund type singular integral. This kind of singular integral appears in the approximation of the surface quasi-geostrophic (SQG) equation from the generalized SQG equation. We establish an estimate of the singular integral in the Lq space for 1<q<∞ and a weak (1,1) type of the singular integral when 0<β<(q−1)nq without any smoothness assumed on Ω. Moreover, the bounds do not depend on β and the strong (q,q) type estimate and weak (1,1) type estimate of the Calderón-Zygmund type singular integral can be recovered when β→0 from our obtained estimates.
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