Eigenvalue decay of positive integral operators on the sphere (Q2840013)

The paper deals with the Hilbert-Schmidt integral operator NEWLINE\[NEWLINE {\mathcal K}(f)=\int_{S^m}K(\cdot,y)f(y)\,d\sigma_m(y), \tag{1}NEWLINE\]NEWLINE where \(S^m\) is a unit sphere in \({\mathbb R}^{m+1}\) \((m\geq2)\) endowed with Lebesgue measure \(\sigma_m\).NEWLINENEWLINEThe authors study decay rates of the sequence of eigenvalues (or, alternatively, singular numbers) of the operator (1) depending on additional assumptions on the kernel \(K(\cdot,\cdot).\) These assumptions are defined in terms of its derivatives. In contrast to the existing works in this direction, here instead of standard derivatives for this purpose the so-called Laplace-Beltrami derivatives are used, which are said to be more appropriate for the spherical case.