25 февраля 2026 года в 14:00 пройдет совместный семинар Института математики имени В.И. Романовского Академии наук Республики Узбекистан, Института прикладной математики и автоматизации КБНЦ Российской академии наук и Analysis & PDE Center of Ghent University (Belgium) «Современные проблемы математической физики».
Докладчик: Professor Roland Duduchava, V. Kupradze Institute of mathematics, University of Georgia & A. Razmadze Institute of Mathematics TSU, Tbilisi, Georgia.
Название доклада: Spectral properties of convolution integro-differential equations on the submonoid M = [0,1).
Краткая аннотация: The interval G = (-1,1) becomes a Lie group under the binary operation x∘y∶= (x + y)/(1 + xy), x,y∈G, equipped with the Fourier transform F_G and the invariant Haar measure dμ_G ∶=dt/(1-t^2 ) . The subinterval M = [0,1) forms a submonoid of G with the same operation x∘y, which naturally induces the Haar measure dμ_M and the Fourier transform F_M. The principal object of study is the Fourier convolution operator
W_(M,a) ∶= r^+ W_(G,a)^0 l^+, W_(G,a)^0 ∶= F_(G,a)^(-1),F_G,
representing the restriction of a convolution operator on G to the submonoid M. A typical convolution integro-differential equation on M =[0,1) has the form
∑_(j=0)^n( a_j D^(m_j ) u(t)- b_j D^(n_j ) ∫_0^1k_j ((t -τ)/(1 — tτ)) D^(l_j ) u(τ) dτ/(1 -τ^2 )〗 ) = f(t), t∈ M,
where a_j,b_j∈ C , m_j,n_j,l_j∈ N, and k_j∈ L^1 (G,dμ_G). Here the differential operator
Du(x)∶= -(1 — x^2)d/dx u(x) acts as a convolution operator, and under the Fourier transform its symbol satisfies (F_G D)(ξ)= -iξ,ξ∈ R. The theory of convolution operators W_(M,a) on the submonoid M is substantially more intricate than on Lie groups, yet richer in applications. For instance, Wiener–Hopf equations on the submonoid [0,∞) of the Lie group R form a classical example. In the scale of generic Bessel potential spaces
f∈ GH_p^(s-r) (M,dμ_M ), φ∈ GH_p^s (M,dμ_M ), 1 < p <∞,s,r∈ R,
the convolution equation W_(M,a) φ = f has a nontrivial Fredholm index. Fredholm properties and solvability depend delicately on the symbol a(ξ) and the parameters of the underlying function spaces. A complete theory is presented, including Fredholm criteria, solvability conditions, index formulas, and explicit solution representations via symbol factorization. Notably, classical equations of Prandtl, Tricomi, and Lavrentiev–Bitsadze belong to this class of convolution equations on the Lie group G = (-1,1). The results are based on [1], [2] and related recent work.
- D. Cardona, R. Duduchava, A. Hendrix, M. Ruzhansky. Generic Bessel potential spaces on Lie groups, Tbilisi Analysis & PDE Seminar — Extended Abstracts, Trends in Mathematics, Birkhäuser/Springer, 2024.
- R. Duduchava, Convolution equations on the submonoid $M=[0,1)$, submitted to Boletín de la Sociedad Matemática Mexicana, 37 pp.
Подключение к конференции Zoom:
https://us06web.zoom.us/j/88273567781?pwd=6TMH3UOjALVQmLsLlm6vTMPo5YpCYr.1
Идентификатор конференции: 882 7356 7781
Код доступа: 151325
Руководители семинара: академик Ш. А. Алимов, профессор А. В. Псху, профессор Р. Р. Ашуров, профессор Э. Каримов.
Секретарь семинара: Р. Т. Зуннунов.