Let α≥ 0 and 0 〈 ρ ≤ n/2, the boundedness of hypersingular parameterized Marcinkiewicz integrals μΩ,α^ρ with variable kernels on Sobolev spaces Lα^ρ and HardySobolev spaces Hα^ρ is established.
The behavior on the space L^∞（R^n） for the multilinear singular integral operator defined by TAf（x）=∫RnΩ（x-y）/｜x-y｜^n＋1（A（x）-A（y）△A（y）（x-y））f（y）dy is considered, where 12 is homogeneous of deg...The behavior on the space L^∞（R^n） for the multilinear singular integral operator defined by TAf（x）=∫RnΩ（x-y）/｜x-y｜^n＋1（A（x）-A（y）△A（y）（x-y））f（y）dy is considered, where 12 is homogeneous of degree zero, integrable on the unit sphere and has vanishing is considered, where Ω is homogeneous of degree zero, integrable on the unit sphere and has vanishingmoment of order one, A has derivatives of order one in BMO（R^n）. It is proved that if Ω satisfies some minimum size condition and an L1-Dini type regularity condition, then for f ∈ L^∞（R^n）, TAf is either infinite almost everywhere or finite almost everywhere, and in the latter case, TAf ∈ BMO（R^n）.展开更多
基金Supported by the National Natural Science Foundation of China(1057115610871173)
文摘Let α≥ 0 and 0 〈 ρ ≤ n/2, the boundedness of hypersingular parameterized Marcinkiewicz integrals μΩ,α^ρ with variable kernels on Sobolev spaces Lα^ρ and HardySobolev spaces Hα^ρ is established.
文摘The behavior on the space L^∞（R^n） for the multilinear singular integral operator defined by TAf（x）=∫RnΩ（x-y）/｜x-y｜^n＋1（A（x）-A（y）△A（y）（x-y））f（y）dy is considered, where 12 is homogeneous of degree zero, integrable on the unit sphere and has vanishing is considered, where Ω is homogeneous of degree zero, integrable on the unit sphere and has vanishingmoment of order one, A has derivatives of order one in BMO（R^n）. It is proved that if Ω satisfies some minimum size condition and an L1-Dini type regularity condition, then for f ∈ L^∞（R^n）, TAf is either infinite almost everywhere or finite almost everywhere, and in the latter case, TAf ∈ BMO（R^n）.
