Multilinear theory of strongly singular Calderón-Zygmund operators and applications (Q1985822)

The main purpose of this paper is to study the multilinear strongly singular Calderón-Zygmund operators whose kernels are more singular near the diagonal than that of the standard multilinear Calderón-Zygmund operator. The sharp maximal pointwise estimate for the multilinear strongly singular Calderón-Zygmund operator is established. As applications, the boundedness of the multilinear strongly singular Calderón-Zygmund operator on the product of weighted Lebesgue spaces and on the product of variable exponent Lebesgue spaces is obtained respectively. Moreover, three kind of endpoint estimates of the types $L^{\infty} \times \cdots \times L^{\infty} \rightarrow BMO$, $BMO \times \cdots \times BMO \rightarrow BMO$, and $LMO \times \cdots \times LMO \rightarrow LMO$ are established for the multilinear strongly singular Calderón-Zygmund operator. These results improve the corresponding known ones for the standard multilinear Calderón-Zygmund operators.