Xiangrong Zhu

SMALL ENERGY COMPACTNESS FOR APPROXIMATE HARMONIC MAPPINGS

In this paper, we consider the elliptic systems

BOUNDEDNESS OF MAXIMAL SINGULAR INTEGRALS

\triangle u=-\Omega \cdot \nabla u+f,

A note on BMO and its application

where u ∈ W1, 2(R2, RK) and f ∈ L ln+ L, and Ω belongs to L2(R2, MK(R)⊗R2) which is antisymmetric. In the first part we prove a compactness theorem for this system. As a corollary, we obtain the compactness theorem for a sequence of mappings from a Riemannian surface to a compact Riemannian manifold with tension fields bounded in L ln+ L. In the second part we prove the energy identity for a sequence of mappings from a surface to a sphere with tension fields bounded in L ln+ L. In the last section we construct a blow-up sequence of mappings from B1 to S2 with tension fields bounded in L ln+ L but there exists a neck with positive length during blowing up.

Boundedness of multidimensional Hausdorff operators on H1(Rn)

We study the oscillatory hyper-Hilbert transform (1) H n,α,β f(x) = ∫ 1 0 f(x-Γ(t))e ιt- t -1-α dt along the curve F(t) = (t p1 , t p2 , ···, t pn ), where p 1 , p 2 , ···, p n,α,β are some real positive numbers. We prove that if β > (n+ 1)α, then H n,α,β is bounded on L p whenever p ∈ (2β 2β-(n+1)α, (2β (n+1)α). Furthermore, we also prove that H n,α,β is bounded on L 2 when β = (n + 1)α. Our work improves and extends some known results by Chandarana in 1996 and in a preprint. As an application, we obtain an L p boundedness result for some strongly parabolic singular integrals with rough kernels.

No neck for approximate harmonic maps to the sphere

The authors study the singular integrals under the Hormander condition and the measure not satisfying the doubling condition. At first, if the corresponding singular integral is bounded from L2 to itself, it is proved that the maximal singular integral is bounded from L∞ to RBMO except that it is infinite μ-a.e. on Rd. A sufficient condition and a necessary condition such that the maximal singular integral is bounded from L2 to itself are also obtained. There is a small gap between the two conditions.