Asymptotics of singular numbers of smooth kernels via trigonometric transforms
For a kernel K in \(L^ 2[0,2\pi]^ 2\) define \(\psi (h,p,s)\doteq \| \Delta_ hK(s,\cdot)\|_{L^ p}\) and \(w(h;p;K)\doteq h^{1-1/p} \sup \{\| \psi (h';p;\cdot)\|_{L^ 2}:\quad 0<h'\leq h\}.\) Suppose \(\{\ell_ n\}\) is a slowly varying sequence. Then, if \(1\leq p\leq 2\), \(0<r<2\), and \(0<\beta <\infty\), \[ (*)\quad \{\sum^{\infty}_{N}\ell_ ns_ n^{\beta}(K)n^{-1+\beta /r}\}^{1/\beta}\leq A\{\sum^{\infty}_{N}w^{\beta}(n^{- 1};p;K)n^{-1+\beta /r}\}^{1/\beta}; \] and \(\sup \{\ell_ nn^{1/r}s_ n(K):\) \(n\geq N\}\leq \sup \{\ell_ nn^{1/r}w(n^{- 1};p;K):\) \(n\geq N\}\). A depends only upon p, r, \(\beta\) and the sequence \(\{\ell_ n\}\). With suitable restrictions on \(\{\ell_ n\}\) (e.g., if limsup \(\ell_ n=\infty\), liminf \(\ell_ n>0\), and \(\exists \lambda \geq 0\) such that \(\sum \ell_ n^{-\alpha}n^{-1}<\infty\) iff \(\alpha >\lambda)\) then (*) is, in a sense, unimprovable. The Hardy-Littlewood method of ``blending'' Lipschitz spaces extends to prove, via (*), that when \(\mu >0\), \(\lambda >-1\), \(1\leq p<2<r\leq \infty\), \(K\in BV_ p\) (in the sense that the Wiener p-variation \(V_ p(K(s,\cdot))\leq M(s)\) with \(M\in L^ 2[0,2\pi])\), and that \(\sum^{\infty}_{n=1}w^{\mu}(n^{-1};r;K)n^{\lambda}<\infty\), then \(\sum s_ n^{\beta}(K)n^{\gamma}<\infty\), where \(\delta =(1/2- 1/n)/(1/p-1/2),\) \(\beta =\mu +\delta \mu\), and \(\gamma =\lambda +\delta \mu\). In particular for any \(\beta >0\), \(\sum s_ n^{\beta}(K)<\infty\) provided \(K\in BV_ 1\) and \(\sum n^{-\beta}w^{\beta /2}(n^{- 1};\infty;K)<\infty.\) Another consequence of blending is that (with a suitable definition of lip(\(\alpha\),q)) if \(1\leq p<2<q\leq \infty\) and \(K\in BV_ p\cap lip(q^{-1},q)\) then \(s_ n(L)=o(n^{-1})\). A corollary is that a \(2\pi\)-periodic function \(f\in L'\) which has no removable discontinuities and belongs to \(BV_ p\), with \(p<2\), is continuous if and only if the nonincreasing rearrangement \(\hat f^*(n)=o(n^{-1}).\) Several corrections are required: In (2.10.2) read \(``c<1''\). Formulas (2.12.5) and (2.12.6) require that \(\gamma >0\). Theorem 4.3.1 requires that \(\lambda >-1\). Cor. 4.3.4(a) requires \(0<\kappa <1/\delta\). In Cor. 4.3.4(d) read \(\kappa =\beta /2\). Prop. 4.2.3 requires that \(p\leq 2\).
scientific article; zbMATH DE number 4146787
scientific article