Estimates of Kolmogorov, Gelfand and linear \(n\)-widths on compact Riemannian manifolds (Q2802117)

The author determines lower and exact estimates of Kolomogorov, Gelfand and linear \(n\)-widths of unit balls in Sobolev norms in spaces \(L_p(M)\) on a compact manifold \(M\).NEWLINENEWLINERecall the definition of the Kolmogorov \(n\)-width \(d_n\): for a normed vector space \(Y\) and a subset \(H\) of \(Y\), NEWLINE\[NEWLINEd_n(H,Y)=\inf _{Z_n}\sup _{x\in H}\inf _{z\in Z_n}\| x-y\|,NEWLINE\]NEWLINE where \(Z_n\) runs over all the \(n\)-dimensional subspaces of \(Y\).NEWLINENEWLINERecall also, that, for \(1\leq q\leq p \leq \infty\), the unit ball \(B_p^r(M)\) of the Sobolev space \(W_p^r(M)\) is a subset of \(L_q(M)\) by the Sobolev embedding theorem.NEWLINENEWLINEOne of the main results is the following asymptotic estimate: NEWLINE\[NEWLINEd_n\bigl(B_p^r(M),L_q(M)\bigr)\asymp n^{-\frac{r}{s}},NEWLINE\]NEWLINE where \(s=\dim M\). Similar results are obtained for the Gelfand \(n\)-width and the linear \(n\)-width. The proofs use spectral analysis for a Laplace operator \(L\) on \(M\). One introduces the operator \(F(t^2L)\) (\(t\in\mathbb R\)), whose kernel is given by NEWLINE\[NEWLINEK_t^F(x,y)=\sum _\ell F(t^2\lambda _\ell) u_\ell(x)u_\ell(y),NEWLINE\]NEWLINE where \(0\leq \lambda _0<\lambda_1\leq \lambda _2\leq \cdots \) are the eigenvalues of \(L\), and the functions \(u_\ell\) are the corresponding orthogonal eigenfunctions. The author establishes, for a suitable function \(F\), estimates for the \(L_p\)-norm of the function \(y\mapsto K_t^F(x,y)\), uniform in \(x\). Then, for a positive integer \(N\), one considers a collection of disjoint balls \(B_i^N=B\bigl(x_i^N,N^{-{1\over s}}\bigr)\), and for each \(i\), the function NEWLINE\[NEWLINE\varphi _i^N(x)={1\over N}K_t^F(x_i^N,x),NEWLINE\]NEWLINE which is supported in \(B_i^N\) for \(t\) small enough. It is proven that NEWLINE\[NEWLINE\|\varphi _i ^N\|\asymp N^{-{1\over p}}.NEWLINE\]NEWLINE Then, the problem is reduced to a problem in the finite dimensional space generated by the functions \(\varphi _i^N\).