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27th International Conference on Finite and Infinite Dimensional Complex Analysis and Applications
16 августа 2019 г. 11:30, г. Красноярск, Сибирский федеральный университет


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 $f\left( z,w \right)$ is holomorphic at a point $\left( {{z}^{0}},{{w}^{0}} \right)\in \mathbb{C}_{z}^{n}\times {{\mathbb{C}}_{w}}$ and $f\left( {{z}^{0}},{{w}^{0}} \right)=0,$ but $f\left( {{z}^{0}},w \right) \not \equiv 0,$ then in some neighborhood $U=V\times W$of this point $f$ is represented as
\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 ${c}_{k}\left( z \right) $ are holomorphic in $V$ and $\varphi(z,w)$ is holomorphic in $U,$ $\varphi(z,w)\neq 0,\, (z,w) \in U$.
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 $f\left( {{z}^{0}},w \right)\not\equiv 0.$ It is natural to expect the validity of Weierstrass theorem without requiring a condition $f\left( {{z}^{0}},w \right)\not\equiv 0,$ in form
\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 $n>1$ it is not always possible.
In the talk we will discuss, that there is a global option (see [S1][S2]), also a global multidimensional (in $w$) analogue of (2) is true without requiring the condution $f\left( {{z}^{0}},w \right)\not\equiv 0$. In addition, for an arbitrary germ of a holomorphic function, we will prove one representation, that is useful in the study of oscilatory integrals.
This is a joint work with I. Ikramov (Samarkand State University, Samarkand, Uzbekistan).

Язык доклада: английский

Список литературы
  1. B. A. Fuks, Vvedenie v teoriyu analiticheskikh funktsiy mnogikh kompleksnykh peremennykh, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1962 , 419 pp.  mathscinet
  2. I. A. Ikromov, “Damped oscillatory integrals and maximal operators”, Mat. Zametki, 78:6 (2005), 833–852  mathnet  crossref  mathscinet
  3. W. F. Osgood, Lehrbuch der Funktionentheorie. Erster Band, Chelsea Publishing Co., New York, 1965 , xiv+818 pp.  mathscinet
  4. A. Sadullaev, “A criterion for the algebraicity of analytic sets”, Funckional. Anal. i Prilozen., 6:1 (1972), 85–86  mathscinet
  5. A. Sadullaev, “Criteria algebraicity of analytic sets”, Institute of Physics named after L.V. Kirensky, Krasnoyarsk, 1976, (Russian), 1976, 107–122
  6. Christopher D. Sogge, Elias M. Stein, “Averages of functions over hypersurfaces in $R^n$”, Invent. Math., 82:3 (1985), 543–556  crossref  mathscinet


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