Oscillatory singular integrals on and Hardy spaces

We consider boundedness properties of oscillatory singular integrals on and Hardy spaces. By constructing a phase function, we prove that boundedness may fail while boundedness holds for all . This shows that the theory and theory for such operators are fundamentally different.

     

The critical exponent of degenerate parabolic systems

The Cauchy problemu _t=u +v ^p ,v _t =v +u ^q is studied, wherex R ^N , 0 <t < and ,,p andq, are positive exponents. It is proved that ifp,q 1 and 1 <pq < 1 + 2 max(p + ,q + )/n then every nontrivial non-negative solution is not global in time; whereaspq > 1 + 2 max(p + , q + )/n then there exist both positive global solutions and non-global solutions. In addition, the decaying in time of solutions tou _t,=u inR ⁿ × (0, ), an equation which occurs naturally in our study of above systems, is studied and solutions with the fastest decaying in time are constructed.     

Regularity of minimizers of non-isotropic integrals of the calculus of variations

The regularity of the minimizers of a special type of non-isotropic variational minimization problem is studied. The particularity of the potential of energy is that it has different growth rate with respect to different parts of the derivatives of the function. In particular, the model treated in this paper can be described as $$\Phi (Du) = |\partial _1 u|^2 + |\partial _2 u|^2 + |\partial _3 u|^2 + |\partial _3 u - |^p .$$ By using a result of P.Marcellini (cf. [4]) and perturbation method, it is proved that the minimizer of the Dirichlet boundary value problem is a function of W _loc ^{1, ∞} .     

Regio- and stereocontrolled synthesis and conformational analysis of benzimidazole nucleosides

Regio- and stereocontrolled synthesis and conformational analysis of a series of benzimidazole nucleosides were achieved. A simple method by ¹H NMR 1D NOE experiment was developed for estimation of syn or anti conformation of benzimidazole nucleosides. Substituents at C2 of benzimidazole demonstrated to play a key role both in the unexpected regioselectivity of the glycosidic reaction and in the conformation distributions of the final products.