In this paper, a weak type (1,1) bound criterion is established for singular integral operator with rough kernel. As some applications of this criterion, we show some important operators with rough kernel in harmonic analysis, such as Calderón commutator, higher order Calderón commutator, general Calderón commutator, Calderón commutator of Bajsanski-Coifman type and general singular integral of Muckenhoupt type, are all of weak type (1,1).
In this paper, we investigate the behavior of the bounds of the composition for rough singular integral operators on the weighted space. More precisely, we obtain the quantitative weighted bounds of the composition operator for two singular integral operators with rough homogeneous kernels on L p (R d , w), p ∈ (1, ∞), which is smaller than the product of the quantitative weighted bounds for these two rough singular integral operators. Moreover, at the endpoint p = 1, the L log L weighted weak type bound is also obtained, which may have its own interest in the theory of rough singular integral even in the unweighted case. A slightly better bilinear sparse domination is established for obtaining the quantitative weighted bounds.2010 Mathematics Subject Classification. Primary 42B20, Secondary 47B33. Key words and phrases. Rough singular integral operator, composite operator, weighted bound, bilinear sparse operator.
In this paper, we establish a general discrete Fourier restriction theorem. As an application, we make some progress on the discrete Fourier restriction associated with KdV equation.2010 Mathematics Subject Classification. 42B05,11L07.
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