NWO sequences, weighted potential operators, and Schrödinger eigenvalues
The author applies the theory of nearly weakly orthonormal (NWO) sequences of functions to give sufficient conditions for boundedness of various potential operators between spaces with weights. He applies these results to give eigenvalue estimates for the Schrödinger operator. Let \({\mathbf Q}\) denote the set of all dyadic cubes in \(\mathbb{R}^n\). For each \(Q\in{\mathbf Q}\) let \(\zeta_Q\) denote the center of \(Q\), \(\eta_Q\) denote the sidelength of \(Q\), \(|Q|\) the volume of \(Q\), and denote by \(KQ\) the cube with the same center as \(Q\) but \(K\) times the side length. A set of functions \(\{e_Q(x)\}_{Q\in{\mathbf Q}}\) is an NWO sequence if the associated maximal function \[ f\to f^*(x)= \sup_{|\zeta_Q- x|\leq \eta_Q} (|Q|^{- 1/q}\langle f, e_Q\rangle) \] is bounded on \(L^q\), where \(1/q+ 1/p= 1\). Sufficient conditions for this are that the \(\{e_Q\}\) have support in \(KQ\) and that there exists a \(p< r\leq \infty\) and \(c> 0\) so that for all \(Q\), \(|e_Q|_r\leq c|Q|^{1/r- 1/p}\). Another preliminary result worth mentioning is that it is easy to give estimates and finite rank approximations for operators written as tensor products of NWO sequences of functions. Given a sequence \(\{a_Q\}_{Q\in{\mathbf Q}}\), define a new sequence \(\{a^*_Q\}_{Q\in{\mathbf Q}}\) by \(a^*_Q= \sum_{R\in{\mathbf Q}, R\subset Q} {|R|\over |Q|} |a_R|\), where one should note the typo in the article which writes \(a_Q\) instead of \(a_R\). If the sequence \(\{a^*_Q\}\) is bounded, the discrete Carleson measure norm of the sequence is \(|\{a_Q\}|_{CM_d}= \sup\{a^*_Q\}\). If an operator acting on \(L^p\) can be written as \(T= \sum_{Q\in{\mathbf Q}} a_Q e_Q\otimes f_Q\), where \(\{e_Q\}_{Q\in{\mathbf Q}}\) is NWO for \(L^q\) and \(\{f_Q\}_{Q\in{\mathbf Q}}\) is NWO for \(L^p\), then \(T\) is bounded from \(L^p\) into \(L^p\) and \(|T|\leq C|\{a_Q\}|_{CM_d}\). If \(\{a^*_Q\}\) tends to zero, one can define the nonincreasing rearrangement of \(\{a^*_Q\}\), which is denoted \(\{a^*(n)\}\). In this case \(T\) will be a compact operator and if one constructs the finite rank approximation with \(a^*(j)= a^*_{Q_j}\), \(j= 1, 2,\dots, N\), \(F_N= \sum^N_{j= 1} a_{Q_j} e_{Q_j}\otimes f_{Q_j}\), then \(|T- F_N|\leq Ca^*(N+ 1)\). The author studies, e.g., the Riesz potential \(I_\alpha\) as a map from \(L^p_v\) into \(L^q_u\), where \[ I_\alpha f(x)= \int_{\mathbb{R}^n} {f(y)\over |x- y|^{d- \alpha}} dy \] by writing it as an integral operator with kernel \[ K(x, y)= u(x)^{1/q} v(y)^{- 1/p}/|x- y|^{d- \alpha}, \] considered as a map of \(L^p\to L^q\). He uses a Whitney decomposition to find a tensor product decomposition which dominates this operator and applies the above estimates for tensor products to estimate \(I_\alpha\). In the tensor product \(e_Q(y)= |v^{- 1/p} \chi_Q|^{- 1}_{p'} v(y)^{- 1/p} \chi_Q(y)\), where \(Q\in{\mathbf Q}\), and \(\chi_Q\) denotes the characteristic function of the cube \(Q\). The expression for \(f\) is similar but involves \(u^{1/q}\) and the \(L^q\) norm. When \(w^p\in A_\infty\), he has shown that \(|w_{\chi_Q}|^{- 1}_p w_{\chi_Q}\) is NWO for \(L^p\). Now \(I_\alpha\) maps \(L^p_v\) into \(L^q_u\) if the tensor product maps \(L^p\) into \(L^q\), and under the assumption that \(u[= (u^{1/q})^q]\) and \(v^{- p'/p}[= (v^{- 1/p})^{p'}]\) belong to \(A_\infty\), the boundedness condition for the tensor product becomes \[ A= \sup_{Q\in{\mathbf Q}} \eta^{- d+ \alpha}_Q \Biggl( \int_Q v(x)^{- p'/p} dx\Biggr)^{1/p'} \Biggl( \int_Q u(x)dx\Biggr)^{1/q}< + \infty, \] and the norm of the tensor product estimator is bounded by \(cA\) and thus so is the norm of \(T\). Results are also given for the Bessel potential operator, local versions (for a fixed \(R\in{\mathbf Q}\)), and results on a finite rank approximation for the compact case. He concludes with estimates for the singular values of the compact operators leading to sufficient conditions on the kernel for an operator to belong to a Schatten-von Neumann class. He concludes by showing that if \(\{|w_{\chi_Q}|^{- 1}_p w_{\chi_Q}\}_{Q\in{\mathbf Q}}\) (the operators making up the tensor product decomposition of \(I_\alpha\)) are NWO, then \(w^p\) must be dyadic \(A_\infty\).
scientific article; zbMATH DE number 558999
scientific article