Multilinear theory of strongly singular Calderón–Zygmund operators and applications

Abstract

The main purpose of this paper is to study the multilinear strongly singular Calderón–Zygmund operator whose kernel is more singular near the diagonal than that of the standard multilinear Calderón–Zygmund operator.The sharp maximal estimate for this class of multilinear singular integrals is established, and as applications, its boundedness on product of weighted Lebesgue spaces and product of variable exponent Lebesgue spaces is obtained, respectively. Moreover, the endpoint estimates of the types L∞×⋯×L∞→BMO, BMO×⋯×BMO→BMO, and LMO×⋯×LMO→LMO are established for the multilinear strongly singular Calderón–Zygmund operator.These results improve the corresponding known ones for the standard multilinear Calderón–Zygmund operator. Furthermore, we remove the size condition assumption for the kernel of the multilinear strongly singular Calderón–Zygmund operator which has been imposed in the literature for the case of the standard multilinear Calderón–Zygmund operator to get the sharp maximal estimates (see Remarks 1.1-1.4). Extra care is needed to deal with the mean oscillation over balls with small radius to overcome the stronger singularity in establishing such sharp maximal estimates and endpoint estimates.