Approximation property and nuclearity on mixed-norm \(L^{p}\), modulation and Wiener amalgam spaces (Q2835326)

Let \(B\) be a Banach space and let \(F(B)\) denote the finite rank operators on \(B\). If for every compact \(K\) and every \(\epsilon>0\), there is a finite rank \(F\) with \(\|F\|\leq 1\) such that for all \(x\in K\), \(\|x-Fx\|<\epsilon\), then one says that \(B\) has the metric approximation property. This is equivalent to the statement that, given a finite subset of \(B\) and \(\epsilon\), there is an \(F\) with \(\|F\|\leq 1\) and \(\|x-Fx\|<\epsilon\) for all \(x\) in the set (Lemma 2.1). Recall that a linear operator \(T\) is \(r\)-nuclear if there exist sequences \(\{x_n'\}\subset B'\) and \(\{y_n\}\subset B\) such that \(Tx=\sum \langle x, x_n'\rangle y_n\) with \(\sum \|x_n'\|_{B'}^r\|y_n\|_B^r<\infty\).NEWLINENEWLINELet \((\Omega_i,S_i,\mu_i)\), \(i=1,\dots ,m\), be \(\sigma\)-finite measure spaces with product \(\Omega=\Omega_1\times\cdots\times \Omega_m\). For \(P=(p_1,\dots,p_m)\), one defines the weighted mixed-norm space \(L^P_w\) with norm NEWLINE\[NEWLINE\|f\|_{L^P_w}=\Bigl(\int\Bigl(\cdots \Bigl(\int |f(x_1,\dots, x_m)|^{p_1}w(x)\, d\mu_1\Bigr)^{\frac{p_2}{p_1}} \, d\mu_2\Bigr)^{\frac{p_3}{p_2}}\cdots d\mu_n\Bigr)^{\frac{1}{p_m}}. NEWLINE\]NEWLINE Assuming that \(w(x)\leq w_1(x_1)\cdots w_m(x_m)\) on \(\Omega\), the first main result (Theorem 2.3) states that \(L^P_w\) has the metric approximation property.NEWLINENEWLINEThe Feichtinger-Gröchenig modulation spaces and Wiener amalgam spaces are closely related to mixed norm spaces. Define the short-time Fourier transform NEWLINE\[NEWLINEV_gf(x,\xi)=\int_{\mathbb{R}^d} f(y) \overline{g(y-x)}\, e^{-i y\cdot\xi}\, dy.NEWLINE\]NEWLINE Define the modulation space \(M^{p,q}_w(\mathbb{R}^d)\) to consist of those \(f\) such that \(V_gf w\) (for suitable fixed \(g\)) belongs to \(L^{(p,q)}(\mathbb{R}^d\times \mathbb{R}^d)\). For \(F\in L^1_{\text{loc}}(\mathbb{R}^d\times \mathbb{R}^d)\), set \(\mathcal{R}F(x,\xi)=F(\xi,x)\). Define the amalgam space \(W^{p,q}_w(\mathbb{R}^d)\) to consist of those functions such that \(\mathcal{R}(V_g f w)\in L^{(q,p)}(\mathbb{R}^d\times \mathbb{R}^d)\).NEWLINENEWLINEAs a corollary of Theorem 2.3, it is proved that if \(w\) is a submultiplicative (\(w(x+y)\leq w(x)w(y)\)), polynomially moderate weight (growth bounded by a polynomial) and \(1\leq p,q<\infty\), then \(M^{p,q}_w(\mathbb{R}^d)\) satisfies the metric approximation property and, as a further consequence, \(W^{p,q}_w(\mathbb{R}^d)\) also satisfies the metric approximation property. The authors then go on to characterize nuclear operators on \(L^P_w\) (Theorem 4.3) and \(r\)-nuclear operators (Theorem 4.7) with a corresponding trace formula, and then to characterize when functions \(F\) of the harmonic oscillator \(-\Delta+|x|^2\) are \(r\)-nuclear on \(M^{p,q}_s\) (with weight \(w_s(\xi)=(1+|\xi|)^s\)).