## The Bergman kernel on the intersection of two balls in $$\mathbb{C}^2$$.(English)Zbl 1051.32004

The authors study the Bergman kernel $$K_\Omega(z,\zeta)$$ of the domain $$\Omega= B_1\cap B_2$$, where $$B_1= \mathbb{B}^2\subset \mathbb{C}^2$$, is the unit ball and $$B_2= B(a, r)\subset\mathbb{C}^2$$ is the ball of center $$a= (a_1, a_2)$$ and radius $$r$$, under the assumption that the boundaries $$\partial B_i$$ $$(i\in\{1, 2\})$$ intersect transversally. If this is the case, it is shown that $$\partial B_1\cap\partial B_2$$ has precisely two complex tangent points $$p,q\in\partial E\cap\partial B_2$$. The main result is that i) $$K_\Omega(\cdot,\zeta)$$ is holomorphic in a neighborhood of $$\overline\Omega\setminus \{p,q\}$$, for any $$\zeta\in\Omega$$, and ii) for $$z$$ near $$q$$ the Bergman kernel of $$\Omega$$ admits an asymptotic expansion $K_\Omega(z,\zeta)\sim \sum_j\langle z- q,a^t\rangle^{n_j}\langle z-q,a\rangle^{\gamma_j} P_j(\log\langle z- q,a\rangle, \zeta),$ where $$a^t= (\overline a_2,-\overline a_1)$$, $$\langle\,,\,\rangle$$ is the Hermitian inner product in $$\mathbb{C}^2$$, $$n_j\in\mathbb{Z}$$, $$n_j\geq 0$$ and $$\gamma\in \{w\in\mathbb{C}: \text{Re}(w)> -1-n_j/2\}$$. Also $$P(\xi,\zeta)$$ is a polynomial in $$\xi$$ with antiholomorphic coefficients (as functions of $$\zeta$$).

### MSC:

 32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.) 32W05 $$\overline\partial$$ and $$\overline\partial$$-Neumann operators
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