论文标题
奇异集的整体操作员的光谱估计和渐近学
Spectral estimates and asymptotics for integral operators on singular sets
论文作者
论文摘要
For singular numbers of integral operators of the form $u(x)\mapsto \int F_1(X)K(X,Y,X-Y)F_2(Y)u(Y)μ(dY),$ with measure $μ$ singular with respect to the Lebesgue measure in $\mathbb{R}^\mathbf{N}$, order sharp estimates for the counting function are established.内核$ k（x，y，z）$应该在$ x，y $和$ z \ ne 0 $中平稳，并承认在$ z $变量中以$ z \ z \ z \ 0的均匀功能的渐近扩展为0。 $ f_1，f_2 $。对于该度量$μ$的情况，是某些正编码$ \ mathfrak {d}的Lipschitz表面的表面度量，在自动化案例中$，可以找到该积分操作员特征值的渐近学。
For singular numbers of integral operators of the form $u(x)\mapsto \int F_1(X)K(X,Y,X-Y)F_2(Y)u(Y)μ(dY),$ with measure $μ$ singular with respect to the Lebesgue measure in $\mathbb{R}^\mathbf{N}$, order sharp estimates for the counting function are established. The kernel $K(X,Y,Z)$ is supposed to be smooth in $X,Y$ and in $Z\ne 0$ and to admit an asymptotic expansion in homogeneous functions in $Z$ variable as $Z\to 0.$ The order in estimates is determined by the leading homogeneity order in the kernel and geometric properties of the measure $μ$ and involves integral norms of the weight functions $F_1,F_2$. For the case of the measure $μ$ being the surface measure for a Lipschitz surface of some positive codimension $\mathfrak{d},$ in the self-adjoint case, the asymptotics of eigenvalues of this integral operator is found.
