By Kopachevsky N.D., Orlov I.V. (eds.)
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Extra resources for Spectral and evolution problems. Proceedings 14th Crimean Autumn Mathematical School-Symposium (KROMSH-2003). v. 14
M, π) можно перейти к ориентирующей двулистной накрывающей (M , π) с ориентированной проекцией π : U −→ Y . 1. Spectral Theory of Not Self-Adjoint Operators 41 Через D обозначим поднятие оператора D на пару M , π . В силу локальности интеграла имеем 2indt D = indt D. Аналогичным образом, аналитический индекс также мультипликативен относительно поднятия оператора на накрытие (ср. ]). Поэтому индексы операторов D и D связаны равенством 2inda D = inda D. Последние два равенства доказывают первую часть леммы.
3) Эту формулу можно получить, если сначала заметить, что индекс оператора на M равен индексу соответствующей задачи сопряжения на M \U ⊔ U с граничным условием непрерывности: (DM \U ⊗ 1E )u = f, π ! ) как два условия в дополнительных подпространствах: Π+ (u − v)|∂U = h+ ∈ Im Π+ , (1 − Π+ ) (u − v)|∂U = h− ∈ Im (1 − Π+ ) . Краевая задача π ! (DY ⊗ 1E0 ) , 1 − Π+ эквивалентна обычной задаче Атьи–Патоди– Зингера для оператора DY ⊗ 1E0 на базе Y накрытия. ) indDM ⊗ 1E = L(T M ) ∧ chE + L (T Y ) ∧ chE0 .
When considering this question in Lp (I) for p = 2, additional diﬃculties arise and the answer is not as complete yet. ]). In the present paper, this question will be considered in Lp spaces with weights. , Theorem 8]). Recall that a weight Φ(x) on R satisﬁes the Muckenhoupt’s condition (Ap ) (we write Φ ∈ (Ap )) if p−1 1 1 sup Φ(x)dx Φ(x)−1/(p−1) dx < +∞ (Ap ) |I| |I| I⊂R I I where I is an interval in R and 1 < p < ∞. We now introduce the basic notation and deﬁnitions use throughout this paper.