In this paper, the boundedness of an oscillating multiplier m γ,β for different β on the Herz type spaces is obtained. This operator was initially studied by Wainger and Fefferman-Stein. Our results extend one of the main results in a paper by Xiaochun Li and Shanzhen Lu for the non-weighted case, if β is close to 1 or α is suitably large. For β≥ 1, the results with no weights on the Herz type spaces are also new.
In this paper, we study central BMO estimates for commutators of n-dimensional rough Hardy operators. Furthermore, λ-central BMO estimates for commutators on central Morrey spaces are discussed.
In this article, we prove the boundedness of commutators generated by BochnerRiesz operators below the critical index and BMO functions on the class of radial functions in L p (R n ) with | 1/p -1/2 | < (1 + 2α)/(2n).
We study the maximal super-singular integral operator TΩ* ,α,β(f )(x1 y) = sup e1>0, e2 >0 |∫|u|>ε1,|v|>ε2 b1(|u|)b2(|v|)Ω(u', v')/ |u|n+α|v|m+β f (x-u, y-v)dudv |defined on all f ∈ S(Rn×Rm), where 0≦α, β < ∞, b1, b2 ∈ L∞(R1+), Ω satisfies certain cancellation conditions and Ω∈ L1(Sn-1 × Sm-1) in the case α, β > 0; Ω∈ L(log+L)(Sn-1 × Sm -1) in the case αβ = 0 and α + β > 0. It is proved that, for 1 < p < ∞, TΩ* ,α,β is a bounded operator from the homogeneous Sobolev space L p α,β (Rn × Rm) to the Lebesgue space Lp(Rn × Rm).
WANG Hui1 & CHEN JieCheng2,3, 1Department of Mathematics, Xidian University, Xi’an 710071, China
In this paper we get the sharp estimates of the p-adic Hardy and Hardy-Littlewood-Pólya operators on Lq(|x|αpdx). Also, we prove that the commutators generated by the p-adic Hardy operators(Hardy-Littlewood-Pólya operators) and the central BMO functions are bounded on Lq(|x|αpdx), more generally, on Herz spaces.
The authors establish λ-central BMO estimates for commutators of maximal multilinear Calderón-Zygmund operators TΠb* and multilinear fractional operators Iα,b on central Morrey spaces respectively. Similar results still hold for Tb,Tb* and Iα,b*.
For a compact Riemannian manifold NRK without boundary, we establish the existence of strong solutions to the heat flow for harmonic maps from Rn to N, and the regularizing rate estimate of the strong solutions. Moreover, we obtain the analyticity in spatial variables of the solutions. The uniqueness of the mild solutions in C([0,T]; W1,n) is also considered in this paper.
