In this paper, the boundedness of an oscillating multiplier mγ,β for different β on the Herz type spaces is obtained. This operator was initially studied by Wainger and FeffermanStein. 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.
We study the windowed Fourier transform in the framework of Clifford analysis, which we call the Clifford windowed Fourier transform （CWFT）. Based on the spectral representation of the Clifford Fourier transform （CFT）, we derive several important properties such as shift, modulation, reconstruction formula, orthogonality relation, isometry, and reproducing kernel. We also present an example to show the differences between the classical windowed Fourier transform （WFT） and the CWFT. Finally, as an application we establish a Heisenberg type uncertainty principle for the CWFT.
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 Lp（Rn） with ｜1/p-1/2｜〈（1＋2α）/（2n）.
We study the maximal super-singular integral operator T＊Ω,α,β（f）（x,y）=sup ∈1〉0,∈2〉0｜∫｜u｜〉ε1,｜v｜〉ε2 b1（｜u｜）b2（｜u｜）Ω（u＇,u＇）/｜u｜^n＋α｜u｜^m＋β-f（x-u,y-u）dudu｜defined on all f ∈ S（R^n ×R^m）, where 0 ≤ α,β〈∞, b1 b2 ∈ L∞（R＋1 ）,Ω satisfies certain cancellation conditions and Ω∈L1（S^n-1×S^m-1）in the case α,β〉0;Ω∈L（log＋L）（S^n-1×S^m-1）in the case αβ=0 and α＋β 〉0. It is proved that, for 1〈p〈∞.T＊Ω,α,βis a bounded operator from the homogeneous Sobolev space Lα,β^p（R^n×R^m）to the Lebesgue space L^p（R^n×R^m）.
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 Hard^Littlewood-Pdlya operators on L^q （｜x｜apdx）. Also, we prove that the commutators generated by the p-adic Hardy operators （Hardy-Littlewood-Pdlya operators） and the central BMO functions are bounded on L^q （｜x｜apdx）, more generally, on Herz spaces.
The authors establish A-central BMO estimates for commutators of maximal multilinear Calderon-Zygmund operators TIIb and multilinear fractional operators Ia,b on central Morrey spaces respectively. Similar results still hold for Tb, Tb and Ia,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.
