|
ВИДЕОТЕКА |
27th International Conference on Finite and Infinite Dimensional Complex Analysis and Applications
|
|||
|
Oscillatory integrals and Weierstrass polynomials A. S. Sadullaev National University of Uzbekistan named after M. Ulugbek, Tashkent |
|||
Аннотация: The well-known Weierstrass theorem states that if \begin{equation} f\left( z,w \right)=\left[ {{\left( w-{{w}^{0}} \right)}^{m}}+{{c}_{m-1}}\left( z \right){{\left( w-{{w}^{0}} \right)}^{m-1}}+...+{{c}_{0}}\left( z \right) \right]\varphi \left( z,w \right),\,\,\,\,(1) \end{equation} where In recent years, the Weierstrass representation (1) has found a number of applications in the theory of oscillatory integrals. Using a version of Weierstrass representation the first author (see [ikr]) obtained a solution of famous Sogge-Stein problem (see [SS]). He obtained also close to a sharp bound for maximal operators associated to analytic hypersurfaces. In the obtained results the phase function is an analytic function at a fixed critical point without requiring the condition \begin{equation} f\left( z,w \right)=\left[ {{c}_{m}}\left( z \right){{\left( w-{{w}^{0}} \right)}^{m}}+{{c}_{m-1}}\left( z \right){{\left( w-{{w}^{0}} \right)}^{m-1}}+...+{{c}_{0}}\left( z \right) \right]\varphi \left( z,w \right). (2) \end{equation} Such kind of results may be useful to studying of the oscillatory integrals and in estimates for maximal operators on a Lebesgue spaces. However, the well-known Osgood counterexample [O], p.90 (see also [F], p. 68) shows that when In the talk we will discuss, that there is a global option (see [S1], [S2]), also a global multidimensional (in This is a joint work with I. Ikramov (Samarkand State University, Samarkand, Uzbekistan). Язык доклада: английский Список литературы
|