Estimates for continuity envelopes and approximation numbers of Bessel potentials (Q390527)

For \(\alpha\in(0,\infty)\), let \(G_\alpha\) be the Bessel-MacDonald kernel. For a measurable function \(f\) on \(\mathbb R^n\), denote by \(f^\ast\) the decreasing rearrangement of \(f\) and let \(u(x)=G_\alpha\ast f(x)\) for all \(x\in\mathbb R^n\). Let \(C(\mathbb R^n)\) be the space of all complex-valued bounded uniformly continuous functions \(f\) on \(\mathbb R^n\), equipped with the sup-norm \(\|f\|_{C(\mathbb R^n)}:=\sup_{x\in \mathbb R^n}|f(x)|\). The \(k\)-th modulus of smoothness of a function \(f\in C(\mathbb R^n)\) is defined by setting, for all \(t\in(0,\infty)\) and \(k\in\mathbb N:=\{1,2,\dots\}\), \(\omega_k(f;t):=\sup_{|h|\leq t}\|\Delta_h^kf(x)\|_{C(\mathbb R^n)}\), where \(\Delta_h^1f(x):=f(x+h)-f(x)\) and \(\Delta_h^{k}f(x):=\Delta_h^1(\Delta_h^{k-1}f)(x)\) for all \(k\geq2\).NEWLINENEWLINEIn this paper, the authors prove the following conclusion. Let \(k\in\mathbb N\), \(\alpha\in(0,n)\) and \(f:\;\mathbb R^n\to\mathbb R\) be a function such that, for fixed \(T\in(0,\infty)\), \(\int_0^T\tau^{\frac\alpha n-1}f^\ast(\tau)\,d\tau<\infty\). Then there exist positive constants \(c\) and \( \widetilde c\) such that NEWLINE\[NEWLINE\|u\|_{C(\mathbb R^n)}\leq c\int_0^T\tau^{\frac\alpha n-1}f^\ast(\tau)\,d\tauNEWLINE\]NEWLINE and, for all \(t\in(0,T)\), NEWLINE\[NEWLINE\omega_k(u;t^\frac1n)\leq \widetilde{c}\int_0^T\frac{\tau^{\frac{\alpha-k}{n}-1}} {\tau^{-\frac kn}+t^{-\frac kn}}f^\ast(\tau)\,d\tau.NEWLINE\]NEWLINE Conversely, the authors find some \(T_0\in(0,\infty)\) and construct an extremal function \(f_0\) corresponding to \(T_0\) such that NEWLINE\[NEWLINE\omega_k(u_0;t^\frac1n)\geq c_0\int_0^{T_0}\frac{\tau^{\frac{\alpha-k}{n}-1}} {\tau^{-\frac kn}+t^{-\frac kn}}f_0^\ast(\tau)\,d\tauNEWLINE\]NEWLINE for all \(t\in(0,T_0]\), where \(u_0(x):=G_\alpha\ast f_0(x)\) for all \(x\in\mathbb R^n\) and \(c_0\) is a positive constant only depending on \(\alpha\), \(n\), \(k\). As applications of these results, the authors determine the continuity envelope functions for spaces of Bessel potential in \(n\)-dimensional Euclidean spaces and estimate the approximation numbers of some embeddings as well.