Estimates for continuity envelopes and approximation numbers of Bessel potentials. (English) Zbl 1293.46020
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 \(k\geq2\).
In 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 \[ \|u\|_{C(\mathbb R^n)}\leq c\int_0^T\tau^{\frac\alpha n-1}f^\ast(\tau)\,d\tau \] and, for all \(t\in(0,T)\), \[ \omega_k(u;t^\frac1n)\leq \widetilde{c}\int_0^T\frac{\tau^{\frac{\alpha-k}{n}-1}} {\tau^{-\frac kn}+t^{-\frac kn}}f^\ast(\tau)\,d\tau. \] Conversely, the authors find some \(T_0\in(0,\infty)\) and construct an extremal function \(f_0\) corresponding to \(T_0\) such that \[ \omega_k(u_0;t^\frac1n)\geq c_0\int_0^{T_0}\frac{\tau^{\frac{\alpha-k}{n}-1}} {\tau^{-\frac kn}+t^{-\frac kn}}f_0^\ast(\tau)\,d\tau \] 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.
In 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 \[ \|u\|_{C(\mathbb R^n)}\leq c\int_0^T\tau^{\frac\alpha n-1}f^\ast(\tau)\,d\tau \] and, for all \(t\in(0,T)\), \[ \omega_k(u;t^\frac1n)\leq \widetilde{c}\int_0^T\frac{\tau^{\frac{\alpha-k}{n}-1}} {\tau^{-\frac kn}+t^{-\frac kn}}f^\ast(\tau)\,d\tau. \] Conversely, the authors find some \(T_0\in(0,\infty)\) and construct an extremal function \(f_0\) corresponding to \(T_0\) such that \[ \omega_k(u_0;t^\frac1n)\geq c_0\int_0^{T_0}\frac{\tau^{\frac{\alpha-k}{n}-1}} {\tau^{-\frac kn}+t^{-\frac kn}}f_0^\ast(\tau)\,d\tau \] 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.
Reviewer: Dachun Yang (Beijing)
MSC:
| 46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |
| 47B06 | Riesz operators; eigenvalue distributions; approximation numbers, \(s\)-numbers, Kolmogorov numbers, entropy numbers, etc. of operators |
| 26A15 | Continuity and related questions (modulus of continuity, semicontinuity, discontinuities, etc.) for real functions in one variable |
| 31B15 | Potentials and capacities, extremal length and related notions in higher dimensions |
| 46E30 | Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.) |
Keywords:
convolution; rearrangement invariant space; space of potentials; continuity envelope; compact embedding; approximation numberReferences:
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