Traces of Besov spaces revisited

TL;DR

This paper revisits the trace properties of Besov spaces on hyperplanes, especially in borderline cases with $s$ near critical values, revealing new dependencies on the sum-exponent $q$ and simplifying classical results.

Contribution

It provides new analysis of Besov space traces at critical smoothness levels, introduces a simple mixed-norm estimate approach, and simplifies proofs of classical borderline surjectivity results.

Findings

New dependence on sum-exponent $q$ in borderline Besov trace cases

Restriction operator for $s$ down to $1/p$ is unsuitable for elliptic problems

Simplified proof of surjectivity at $s=1/p$ for $q\in(0,1]$

Abstract

For the trace of Besov spaces $B^s_{p,q}$ onto a hyperplane, the borderline case with $s=\frac{n}{p}-(n-1)$ and $0<p<1$ is analysed and a new dependence on the sum-exponent $q$ is found. Through examples the restriction operator defined for $s$ down to $1/p$, and valued in $L_p$, is shown to be distinctly different and, moreover, unsuitable for elliptic boundary problems. All boundedness properties (both new and previously known) are found to be easy consequences of a simple mixed-norm estimate, which also yields continuity with respect to the normal coordinate. The surjectivity for the classical borderline $s=\frac1p$ ($1\le p<\infty$) is given a simpler proof for all $q\in\,]0,1]$, using only basic functional analysis. The new borderline results are based on corresponding convergence criteria for series with spectral conditions.

Equations

$${\gamma }_0:f(x_1,\dots ,x_n)\mapsto f(x_1,\dots ,x_{n-1},0).$$

$${\gamma }_0:B_{p,q}^s({\mathbb{R}}^n)\to B_{p,q}^{s-\frac{1}{\text{raise}1.0ptp}}({\mathbb{R}}^{n-1})$$

$$f(0)\text{defined for}f(t)\in C(\mathbb{R},{\mathcal{D}}^{\prime }({\mathbb{R}}^{n-1})).$$

$$T:B_{p,q}^s({\mathbb{R}}^n)\to L_p({\mathbb{R}}^{n-1}),\text{for}s>\frac{1}{\text{raise}1.0ptp},0<p<1.$$

$${\gamma }_0({\delta }_0(x^{\prime })\otimes \phi (x_n))$$

$$\text{whereas}T({\delta }_0(x^{\prime })\otimes \phi (x_n))$$

$$\Vert k^n\eta (k\cdot )|L_p\Vert =\Vert \eta |L_p\Vert k^{n(1-\frac{1}{\text{raise}1.0ptp})}\to 0,\text{for each p<1}.$$

$${\gamma }_0{\psi }_k$$

$$\text{whereas}T{\psi }_k$$

$${\gamma }_0:B_{p,q}^{\frac{n}{\text{raise}1.0ptp}-n+1}({\mathbb{R}}^n)\to B_{p,\infty }^{(n-1)(\frac{1}{\text{raise}1.0ptp}-1)}({\mathbb{R}}^{n-1})\text{if}q\le p<1,$$

$${\gamma }_0:B_{p,q}^{\frac{n}{\text{raise}1.0ptp}-n+1}({\mathbb{R}}^n)\to B_{q,\infty }^{(n-1)(\frac{1}{\text{raise}1.0ptq}-1)}({\mathbb{R}}^{n-1})\text{when}p<q\le 1.$$

$$T:B_{p,q}^{\frac{1}{\text{raise}1.0ptp}}({\mathbb{R}}^n)\to L_p({\mathbb{R}}^{n-1})\text{for}0<q\le p<1.$$

$$B_{p,q}^{\frac{n}{\text{raise}1.0ptp}-n+1}({\mathbb{R}}^n)\hookrightarrow B_{q,q}^{\frac{n}{\text{raise}1.0ptq}-n+1}({\mathbb{R}}^n)\stackrel{{\gamma }_0}{\to }{B}_{q,\infty }^{(n-1)(\frac{1}{\text{raise}1.0ptq}-1)}({\mathbb{R}}^{n-1}),$$

$${\gamma }_0:B_{p,q}^{\frac{1}{\text{raise}1.0ptp}}({\mathbb{R}}^n)\to L_p({\mathbb{R}}^{n-1}).$$

$$B_{p,q}^s({\mathbb{R}}^n)\subset C(\mathbb{R},{\mathcal{D}}^{\prime }({\mathbb{R}}^{n-1})),$$

$$\Vert u|B_{p,q}^s\Vert :=(\sum _{k=0}^{\infty }{2}^{skq}\Vert {\mathcal{F}}^{-1}({\Phi }_k\hat{u})|{\mathcal{L}}_p{\Vert }^q{)}^{\frac{1}{\text{raise}1.0ptq}}<\infty .$$

$${\tilde{\Psi }}_k(\xi ):={\Psi }_k^{(1)}({\xi }_1)\dots {\Psi }_k^{(1)}({\xi }_n)$$

$${\tilde{\Psi }}_k(\xi )={\tilde{\Psi }}_{k-1}(\xi )+\sum _{\varnothing \ne J\subset \{1,\dots ,n\}}{\Theta }_{J,k}(\xi ),$$

$$\text{whereby}{\Theta }_{J,k}(\xi )=\prod _{j\in J}{\Phi }_k^{(1)}({\xi }_j)\prod _{j\notin J}{\Psi }_{k-1}^{(1)}({\xi }_j).$$

$$1=\sum _{k=0}^{\infty }\sum _J{\Theta }_{J,k}(\xi ).$$

$$\mathrm{supp}{\Theta }_{J,k}\subset B_{\infty }(0,\frac{13}{10}{2}^k)\setminus B_{\infty }(0,\frac{11}{10}{2}^{k-1}).$$

$$\Vert u|B_{p,q}^s{\Vert }^{\Theta }:=(\sum _{k=0}^{\infty }\sum _J{2}^{skq}\Vert {\mathcal{F}}^{-1}({\Theta }_{\mathcal{J},k}\hat{u})|{\mathcal{L}}_p{\Vert }^q{)}^{\frac{1}{\text{raise}1.0ptq}}.$$

$$\gamma_{0}u=\sum_{j=0}^{\infty}\cal F^{-1}(\Phi_{j}\cal Fu)\big{|}_{x_{n}=0}$$

$$B:=(\sum _{j=0}^{\infty }{2}^{sjq}\Vert u_j|L_p{\Vert }^q{)}^{\frac{1}{\text{raise}1.0ptq}}$$

$$\mathrm{supp}\mathcal{F}{u}_j\subset \{\xi \mid |\xi |\le \mathcal{A}{2}^j\}$$

$$\Vert u|B_{p,\infty }^{\frac{n}{\text{raise}1.0ptp}-n}\Vert$$

$$\Vert u|B_{q,\infty }^{\frac{n}{\text{raise}1.0ptq}-n}\Vert$$

$$\mathrm{supp}\mathcal{F}{u}_j\subset \{\xi \mid \frac{1}{\mathcal{A}}{2}^j\le |\xi |\le \mathcal{A}{2}^j\}$$

$$\begin{array}{rl}\Vert {\mathcal{F}}^{-1}({\Phi }_j\hat{u})|{\mathcal{L}}_p\Vert & \le (\sum _{k=j-h}^{\infty }\Vert {\check{\Phi }}_j\ast u_k|L_p{\Vert }^p{)}^{\frac{1}{\text{raise}1.0ptp}}\\ & \le c(\sum _{k=j-h}^{\infty }{2}^{k(\frac{n}{\text{raise}1.0ptp}-n)p}\Vert {\check{\Phi }}_j|L_p{\Vert }^p\Vert u_k|L_p{\Vert }^p{)}^{\frac{1}{\text{raise}1.0ptp}}\\ & \le c\max (\Vert {\check{\Phi }}_0|L_p\Vert ,\Vert {\check{\Phi }}_1|L_p\Vert )2^{j(n-\frac{n}{\text{raise}1.0ptp})}B.\end{array}$$

$$\Vert u|B_{r,\infty }^t\Vert \le c(\sum _{j=0}^{\infty }{2}^{j(\frac{n}{\text{raise}1.0ptp}-n)q}\Vert u_j|L_p{\Vert }^q{)}^{\frac{1}{\text{raise}1.0ptq}},$$

$${\omega }_N=\sum _{k=1}^N{\check{\Psi }}_k(x-x_k).$$

$$\Vert {\omega }_N|B_{r,\infty }^t\Vert \ge c(r)\cdot N^{\frac{1}{\text{raise}1.0ptr}};$$

Full text

 

Appeared in Journal of analysis and its applications (Zeischrift für Analysis und ihre Anwendungen), vol. 19 (2000), no. 3, 763-779

1. Introduction

This note concerns the (distributional) trace operator $\gamma_{0}$ that restricts to the hyperplane $\Gamma:=\{x_{n}=0\}$ in ${{\mathbb{R}}}^{n}$ for $n\geq 2$,

[TABLE]

The title should indicate both that there remains unexplored borderlines in the $L_{p}$-theory of $\gamma_{0}$ and that the existing litterature do not reveal the full efficacy of the Fourier analytic proof methods.

The main purpose is to describe the borderline cases for $0<p<1$. See the below Theorem 1.2 concerning s={\textstyle\frac{n}{\raise 1.0pt\hbox{\scriptstyle p}}}-n+1, where it is shown that the smallest Besov space containing $\gamma_{0}(B^{s}_{p,q})$ has its integral-exponent equal to $\max(p,q)$, hence depending on both the integral- and the sum-exponent of the domain. This result seems to be hitherto undescribed.

Secondly Theorem 1.2 is proved in a mere two lines, deriving from the Paley–Wiener–Schwartz theorem and the Nikolskiĭ–Plancherel–Polya inequality a basic mixed-norm, in fact $L_{p}({{\mathbb{R}}}^{n-1};L_{\infty,x_{n}})$, estimate. In addition all the known boundedness results are recovered equally easily from the same calculation. The ensuing unified treatment is in contrast with the existing litterature, which has various page-long arguments both for the generic cases (s>{\textstyle\frac{1}{\raise 1.0pt\hbox{\scriptstyle p}}}+(n-1)({\textstyle\frac{1}{\raise 1.0pt\hbox{\scriptstyle p}}}-1)_{+}) and the classical borderline s={\textstyle\frac{1}{\raise 1.0pt\hbox{\scriptstyle p}}} ($1\leq p\leq\infty$). The present paper should also be interesting for this reason.

Thirdly, another perspective on $\gamma_{0}$ is also gained from the mixed-norm estimate, for this yields (since the value $x_{n}=0$ har no special significance) that all the treated $B^{s}_{p,q}$ are contained in $C({{\mathbb{R}}},\cal D^{\prime}({{\mathbb{R}}}^{n-1}))$ and that $\gamma_{0}$ is a restriction of the natural trace on the latter space. This property has not been given much attention in the Besov space litterature (J. Peetre’s report [Pee75] seems to be the only example), although in practice $\gamma_{0}$ has been defined space by space by means of a limiting procedure. Evidently this raises the question whether $\gamma_{0}u$ is consistently defined when $u$ belongs to both $C({{\mathbb{R}}}^{n})$ and $B^{1}_{1,1}({{\mathbb{R}}}^{n})$ or to another intersection of two spaces. However, the consistency is always assured by the below embedding into $C({{\mathbb{R}}},\cal D^{\prime}({{\mathbb{R}}}^{n-1}))$.

Finally, the surjectiveness of \gamma_{0}\colon B^{{\frac{1}{\raise 1.0pt\hbox{\scriptscriptstyle p}}}}_{p,q}({{\mathbb{R}}}^{n})\to L_{p}({{\mathbb{R}}}^{n-1}) for $1\leq p<\infty$ and $0<q\leq 1$ is given a new proof by an easy extension of the Closed Range Theorem to quasi-Banach spaces.

For precision’s sake it should be mentioned that $\gamma_{0}$ first of all refers to a working definition of the trace as $\gamma_{0}u=\sum(\check{\Phi}_{k}*u)|_{x_{n}=0}$, whereby $u=\sum\cal F^{-1}(\Phi_{k}\hat{u})$ is a Littlewood–Paley decomposition; cf. Section 3 below. Consistency and independence of the $\Phi_{k}$ are obtained post-festum, cf. (1.3) and Theorem 1.4 below. As the point of departure, the generic properties of $\gamma_{0}$ are recalled:

Theorem 1.1** ([Tri78],[Jaw77]).**

When applied to the Besov spaces $B^{s}_{p,q}({{\mathbb{R}}}^{n})$ with $0<p,q\leq\infty$, the trace $\gamma_{0}$ is continuous

[TABLE]

for s>{\textstyle\frac{1}{\raise 1.0pt\hbox{\scriptstyle p}}} if $p\geq 1$, and for s>{\textstyle\frac{n}{\raise 1.0pt\hbox{\scriptstyle p}}}-n+1 if $p<1$. Moreover, $\gamma_{0}$ has a right inverse $K$, which is bounded from B^{s-{\frac{1}{\raise 1.0pt\hbox{\scriptscriptstyle p}}}}_{p,q}({{\mathbb{R}}}^{n-1}) to $B^{s}_{p,q}({{\mathbb{R}}}^{n})$ for every $s\in{{\mathbb{R}}}$.

It is known, but proved explicitly here, that on the one hand $\gamma_{0}$ in (1.2) is a restriction of the distributional trace, that is of

[TABLE]

(This is also denoted by $\gamma_{0}f$ in the rest of the introduction.) On the other hand, the restriction of $\gamma_{0}$ to the Schwartz space $\cal S({{\mathbb{R}}}^{n})$ extends by continuity, cf. [Jaw78, FJ85, FJ90, Tri92], to an operator

[TABLE]

It should be emphasised that $T$ is rather different from $\gamma_{0}$ when s<{\textstyle\frac{n}{\raise 1.0pt\hbox{\scriptstyle p}}}-n+1={\textstyle\frac{1}{\raise 1.0pt\hbox{\scriptstyle p}}}+(n-1)({\textstyle\frac{1}{\raise 1.0pt\hbox{\scriptstyle p}}}-1) (whereby $\gamma_{0}$ acts only on the intersection of $B^{s}_{p,q}$ and $C({{\mathbb{R}}},\cal D^{\prime}({{\mathbb{R}}}^{n-1}))$, cf. (1.3)). Their incompatibility may be exemplified by tensorising some $\varphi\in C^{\infty}_{0}({{\mathbb{R}}})$ equal to $1$ near $x_{n}=0$ with the delta measure $\delta_{0}$ in ${{\mathbb{R}}}^{n-1}$, for

[TABLE]

Here (1.5) is clear by (1.3), since $a\delta_{0}$ depends continuously on the scalar $a$.

The result in (1.6) is connected to the fact that the co-domain $L_{p}$ is not continuously embedded into $\cal D^{\prime}$ when $p<1$; this fact is elementary, for when $\eta\in\cal S({{\mathbb{R}}}^{n})$ with $\int\eta=1$, then $k^{n}\eta(k\cdot)$ tends to $\delta_{0}$ in $\cal D^{\prime}$ and to [math] in $L_{p}$ for $k\to\infty$ because

[TABLE]

With a similar $\eta\in\cal S({{\mathbb{R}}}^{n-1})$ and $\psi_{k}(x)=k^{n-1}\eta(kx^{\prime})\varphi(x_{n})$,

[TABLE]

so the sequence $(\psi_{k})$ is treated rather differently by $\gamma_{0}$ and $T$ (in fact (1.6) can be proved thus, cf. Remark 8.1 below).

These phenomena also depend on the domain chosen in (1.4). Indeed, $\gamma_{0}$ in (1.3) is for $p<1$ continuous B^{{\frac{n}{\raise 1.0pt\hbox{\scriptscriptstyle p}}}-n+1}_{p,q}\to\cal D^{\prime} only if $q\leq 1$ (and a fortiori not at all for s<{\textstyle\frac{n}{\raise 1.0pt\hbox{\scriptstyle p}}}-n+1) by [Joh96, Lem. 2.8], or [Joh93, Lem. 2.5.2]; however, the counterexample there does not contradict (1.4), cf. Remark 8.2. (Similarly, for s={\textstyle\frac{1}{\raise 1.0pt\hbox{\scriptstyle p}}} and $q>1$, hence for s<{\textstyle\frac{1}{\raise 1.0pt\hbox{\scriptstyle p}}}, it was shown too that $\gamma_{0}$ is never continuous from $B^{s}_{p,q}$, regardless of the co-domain.)

Moreover, the severe shortcomings of $T$ in connection with elliptic boundary problems for s\leq{\textstyle\frac{n}{\raise 1.0pt\hbox{\scriptstyle p}}}-n+1 are reviewed in Remark 8.3 below.

Altogether $T$ discards so much information that it is inconsistent with the distribution trace $\gamma_{0}$, seemingly to the extent that it is inappropriate, for the usual applications, to maintain s={\textstyle\frac{1}{\raise 1.0pt\hbox{\scriptstyle p}}} as the borderline when $p<1$.

In view of the above, it is natural to analyse s={\textstyle\frac{n}{\raise 1.0pt\hbox{\scriptstyle p}}}-n+1 when $p<1$. The main point is that $q\leq p\leq 1$ and $p<q\leq 1$ constitute two rather different cases:

Theorem 1.2**.**

For $0<p<1$ the operator $\gamma_{0}$ is continuous

[TABLE]

whereas it is bounded

[TABLE]

Furthermore, $q$ is the smallest possible integral-exponent for the co-domain in (1.11), for even $B^{t}_{r,\infty}$ can only receive when $r\geq q$.

This shows that the smallest Besov space one may use as a co-domain of $\gamma_{0}$ is B^{(n-1)({\frac{1}{\raise 1.0pt\hbox{\scriptscriptstyle r}}}-1)}_{r,\infty} with $r=\max(p,q)$ when s={\textstyle\frac{n}{\raise 1.0pt\hbox{\scriptstyle p}}}-n+1 and $0<p<1$; in addition neither (1.10) nor (1.11) is a surjection (hence the range is not a Besov space, cf. Remark 1.5 below). Altogether this makes a noteworthy contrast with Theorem 1.1.

To elucidate Theorem 1.2, one can observe that the above-mentioned operator $T$ is a continuous surjection, see [FJ85, Th. 5.1], [Tri92, 4.4.3],

[TABLE]

Here the condition $q\leq p$ is known to be necessary, and formally a distinction between the same cases appear in Theorem 1.2 too. This seems surprising and unnoticed hitherto, and a fortiori the theorem is a novelty; cf. Remark 1.5 below.

As an interpretation of (1.11), note that it follows from (1.10) when combined with a Sobolev embedding. In fact, given (1.10) then

[TABLE]

and since $q$ is the optimal integral-exponent on the right hand side of (1.11), cf. Section 7 below, this is the only way to apply $\gamma_{0}$ when $p<q\leq 1$.

Moreover, in both (1.10) and (1.11) one can take $L_{1}({{\mathbb{R}}}^{n-1})$ as the receiving space, for by a Sobolev embedding into $B^{1}_{1,1}({{\mathbb{R}}}^{n})$ the question is reduced to a case (viz. $p=1$) of the following

Theorem 1.3**.**

Let $1\leq p\leq\infty$ and $0<q\leq 1$. Then $\gamma_{0}$ in (1.3) is bounded

[TABLE]

Moreover, (1.14) is a surjection if $1\leq p<\infty$ and $0<q\leq 1$.

Earlier Burenkov, Gol’dman and Peetre [Pee75, BG79, Gol79] proved surjectiveness for $q=1$ (the latter two even for anisotropic spaces), but the first to consider this borderline were seemingly Agmon and Hörmander [AH76] (cf. their note), who covered $p=2$. However, the borderline itself was found in 1951 by Nikolskiĭ [Nik51]. Using atomic decompositions, Frazier and Jawerth [FJ85] proved the surjectivity for $0<q\leq 1$. An alternative argument is given below by means of a short application of the Closed Range Theorem (extended to quasi-Banach spaces); it should be interesting because of the simplicity.

Theorems 1.1, 1.2 and 1.3 are proved and re-proved here, for they may actually all be obtained by combining general principles with a single, mixed-norm estimate; in its turn, this estimate follows straightforwardly from the Paley–Wiener–Schwartz theorem and the Nikolskiĭ–Plancherel–Polya inequality; see Section 4 below. Besides being a unified proof, it is also simple compared to those in e.g. [BL76, Tri83, FJ85].

The mixed-norm estimate actually shows $\cal S^{\prime}$-convergence of the series used as the working definition of $\gamma_{0}u$ in (3.1) below. In Theorem 1.3 this is a consequence of $L_{p}$’s completeness, and for the generic cases in Theorem 1.1 it follows from the known convergence criteria for series with spectral conditions, summed up in (ii) of Theorem 3.1 below.

Furthermore, a small reflection about this estimate yields

Theorem 1.4**.**

Let s\geq{\textstyle\frac{1}{\raise 1.0pt\hbox{\scriptstyle p}}}+(n-1)({\textstyle\frac{1}{\raise 1.0pt\hbox{\scriptstyle p}}}-1)_{+}, and suppose $q\leq 1$ holds in the case of equality. Then there is an inclusion

[TABLE]

and the working definition of $\gamma_{0}$ amounts to a restriction of the natural trace on $C({{\mathbb{R}}},\cal D^{\prime}({{\mathbb{R}}}^{n-1}))$.

For the two cases in Theorem 1.2 it is also noteworthy that they stem from an analogous destinction in (iii) of Theorem 3.1 below. However, part (iii) of the latter theorem is actually a generalisation of the criteria to the borderline s={\textstyle\frac{n}{\raise 1.0pt\hbox{\scriptstyle p}}}-n, and the necessity of the splitting into two cases is shown in Proposition 3.2. Hence this paper also contributes to the convergence criteria in general Besov spaces.

Remark 1.5*.*

In a subsequent joint work [FJS], inspired by the present article, especially Theorems 1.2 and 1.4, the traces of all admissible Besov and Triebel–Lizorkin spaces were determined. In particular the exact ranges in (1.10) and (1.11) were found to be the approximation space A^{(n-1)({\frac{1}{\raise 1.0pt\hbox{\scriptscriptstyle p}}}-1)}_{p,q} in both cases. So although $r=\max(p,q)$ is the smallest possible integral-exponent when the co-domain is stipulated to be a Besov space (as in Theorem 1.2 ff. and throughout this paper), the situation is different if the scale of $A^{s}_{p,q}$ spaces is adopted.

Acknowledgement

In the early stages I benefitted from discussions with prof. H. Triebel, who also kindly provided [Pee75].

2. Preliminaries

For the general notions in distribution theory standard notation is used, similarly to [Hör85]; $C({{\mathbb{R}}},X)$ denotes the vector space of continuous functions from ${{\mathbb{R}}}$ to $X$, and if $X$ is a Banach space, $C_{\operatorname{b}}({{\mathbb{R}}},X)$ stands for the sup-normed space of continuous bounded functions.

For the Besov spaces $B^{s}_{p,q}$ the conventions of [Yam86] are adopted, so the norm is defined from a Littlewood–Paley decomposition $1=\sum_{j=0}^{\infty}\Phi_{j}(\xi)$, where the $\Phi_{j}(\xi)$ vanish unless $\frac{11}{20}2^{j}\leq|\xi|\leq\frac{13}{10}2^{j}$ when $j>0$. This may, moreover, be obtained by letting $\Phi_{0}=\Psi_{0}$ and $\Phi_{j}=\Psi_{j}-\Psi_{j-1}$ when $\Psi_{j}(\xi)=\Psi(2^{-j}|\xi|)$ for some real $C^{\infty}$ function $\Psi(t)$ on ${{\mathbb{R}}}$ vanishing for $t>13/10$ and equalling $1$ for $t<11/10$; in this case $\Psi_{j}=\Phi_{0}+\dots+\Phi_{j}$.

Then $B^{s}_{p,q}$ is defined to consist of the $u\in\cal S^{\prime}({{\mathbb{R}}}^{n})$ for which

[TABLE]

On ${{\mathbb{R}}}^{n-1}$ a partition of unity $1=\sum\Phi_{j}^{\prime}$ with $\Phi_{j}^{\prime}(\xi^{\prime})=\Phi_{j}(\xi^{\prime},0)$ is used.

Equivalently a partition may be used in which each function is a product of $n$ factors, each depending on a single coordinate $\xi_{j}$ of $\xi$. This is folklore, but for precision the following easy construction and Lemma 2.1 below are given. Let $\Phi^{(1)}_{k}$ and $\Psi^{(1)}_{k}$ denote the functions obtained in the manner above for $n=1$. Then

[TABLE]

equals $1$ in $B_{\infty}(0,\tfrac{11}{10}2^{k})$, the max-norm ball of radius $\tfrac{11}{10}2^{k}$, centred at the origin; $\operatorname{supp}\tilde{\Psi}_{k}$ lies in $B_{\infty}(0,\tfrac{13}{10}2^{k})$. Now insertion of $\Psi^{(1)}_{k}=\Psi^{(1)}_{k-1}+\Phi^{(1)}_{k}$ gives, for $k\geq 1$,

[TABLE]

Letting $\Theta_{J,0}=\tilde{\Psi}_{0}$, this yields a smooth partition of unity since for $\xi\in{{\mathbb{R}}}^{n}$,

[TABLE]

When $k\geq 1$, then evidently

[TABLE]

Observe also the tensor product structure of the function $\Theta_{J,k}$ and that $\Theta_{J,k}(\xi)=\Theta_{J,1}(2^{-(k-1)}\xi)$ for $k\geq 1$.

Finally, the next lemma may be proved in the usual way by means of (iv) in Theorem 3.1 below, using also that independently of $k$ there are (1 or) $2^{n}-1$ terms in the sum over $J$.

Lemma 2.1**.**

For every $s\in{{\mathbb{R}}}$ and $p$, $q\in\,]0,\infty]$ the Besov space $B^{s}_{p,q}({{\mathbb{R}}}^{n})$ coincides with the set of $u\in\cal S^{\prime}({{\mathbb{R}}}^{n})$ for which the following quasi-norm is finite:

[TABLE]

Moreover, $\mathinner{\|}\cdot\,|B^{s}_{p,q}\|^{\Theta}$ is an equivalent quasi-norm for $B^{s}_{p,q}$.

3. Definition of the trace

3.1. The working definition

When dealing with $\gamma_{0}u$ it is convenient to take a Littlewood–Paley partition of unity, say $1=\sum_{j=0}^{\infty}\Phi_{j}$, and let

[TABLE]

for those $u\in\cal S^{\prime}({{\mathbb{R}}}^{n})$ for which the sum converges in $\cal D^{\prime}({{\mathbb{R}}}^{n-1})$: by the Paley–Wiener–Schwartz theorem each summand $\cal F^{-1}(\Phi_{j}\cal Fu)$ is an entire analytic function for which restriction to $x_{n}=0$ makes sense.

However, the limit in (3.1) might depend on the $\Phi_{j}$, but in Proposition 5.1 below, this is shown not to be the case for the spaces treated here. (The procedure in (3.1) was used to define the trace in [Jaw77], but without justification or relation to other trace notions.)

The usefulness of (3.1) depends on the availability of easy-to-apply results for the convergence of a series $\sum_{j=0}^{\infty}u_{j}$. While for a general Banach space $X$ a finite norm series, $\sum_{j=0}^{\infty}\mathinner{\|}u_{j}\,|X\|<\infty$, is such a criterion, $B^{s}_{p,q}$ has a variant with $\ell^{s}_{q}(L_{p})$-norms without the troublesome $\cal F^{-1}\Phi_{j}\cal F$ acting on $u_{j}$.

For the reader’s sake, these criteria for series with spectral conditions are recalled with [Yam86, Thms. 3.6, 3.7] in (ii) and (iv) below, together with supplements on the borderline cases for s=\max(0,{\textstyle\frac{n}{\raise 1.0pt\hbox{\scriptstyle p}}}-n) in (i) and (iii).

Theorem 3.1**.**

Let a series $\sum_{j=0}^{\infty}u_{j}$ of distributions $u_{j}$ in $\cal S^{\prime}({{\mathbb{R}}}^{n})$ be given together with numbers $s\in{{\mathbb{R}}}$ and $p$ and $q$ in $\,]0,\infty]$, and consider then

[TABLE]

as a constant in $[0,\infty]$ (with sup-norm over $j$ if $q=\infty$).

Then the following assertion is valid:

• (i)

If $s=0$, $1\leq p\leq\infty$ and $q\leq 1$, then $B<\infty$ implies that $\sum u_{j}$ converges in $L_{p}({{\mathbb{R}}}^{n})$ to a sum $u(x)$ for which $\mathinner{\|}u\,|L_{p}\|\leq B$ holds.

In addition, suppose that for some $A>0$ the spectral condition

[TABLE]

is satisfied by each $u_{j}$, $j\geq 0$. Then one has:

• (ii)

If s>\max(0,{\textstyle\frac{n}{\raise 1.0pt\hbox{\scriptstyle p}}}-n), then $B<\infty$ implies convergence of $\sum u_{j}$ in $\cal S^{\prime}({{\mathbb{R}}}^{n})$ to a limit $u(x)$ in $B^{s}_{p,q}({{\mathbb{R}}}^{n})$ for which $\mathinner{\|}u\,|B^{s}_{p,q}\|\leq cB$ holds for some constant $c=c(n,s,p,q)$.

• (iii)

If s={\textstyle\frac{n}{\raise 1.0pt\hbox{\scriptstyle p}}}-n, $p\in\,]0,1[$ and $q\in\,]0,1]$, then $B<\infty$ implies convergence of $\sum u_{j}$ in $L_{1}({{\mathbb{R}}}^{n})$ to a limit $u(x)$ in $L_{1}$ for which $\mathinner{\|}u\,|L_{1}\|\leq cB$ holds for some constant $c=c(n,p,q)$.

Moreover, there is then a constant $c=c(n,p,q)$ such that $u(x)$ belongs to B^{{\frac{n}{\raise 1.0pt\hbox{\scriptscriptstyle p}}}-n}_{p,\infty} or B^{{\frac{n}{\raise 1.0pt\hbox{\scriptscriptstyle q}}}-n}_{q,\infty} and satisfies the estimate

[TABLE]

respectively.

• (iv)

Furthermore, if the stronger condition

[TABLE]

holds for $j>0$, then the assertion (ii) holds even for all $s\in{{\mathbb{R}}}$.

Proof.

The completeness of $L_{p}$ easily gives (i); cf. [Joh95, Prop. 2.5]. The $L_{1}$-part of (iii) may be reduced to (i) by means of the Nikolskiĭ–Plancherel–Polya inequality, cf. [Joh95, Prop. 2.6] (modulo typos there: $L^{p}$ should have been $L_{1}$ and the corresponding estimate ‘$\mathinner{\|}u\,|L_{1}\|\leq cB$’).

This gives the existence of $u$, and since $\cal F^{-1}(\Phi_{j}\hat{u})=\sum_{k=j-h}^{\infty}\cal F^{-1}(\Phi_{j}\hat{u}_{k})$ for some fixed $h\in{\mathbb{Z}}$, we may for $q\leq p$ use $\ell_{q}\hookrightarrow\ell_{p}$ to get that

[TABLE]

Therefore $u$ is in $B^{s}_{p,\infty}$ for s={\textstyle\frac{n}{\raise 1.0pt\hbox{\scriptstyle p}}}-n with the required estimate. For $p<q$ the Nikolskiĭ–Plancherel–Polya inequality applied to $B$ reduces the question to the case with $p=q$. ∎

It was also shown in [Joh95, Ex. 2.4] that in both (i) and (iii) the restriction $q\leq 1$ is optimal; for $q>1$ there exists series diverging in $\cal D^{\prime}({{\mathbb{R}}}^{n})$ for which the associated $B$ is finite.

In addition to this, the receiving spaces in (iii) must have sum-exponents equal to infinity (see [FJS, Th. 6], where this is derived from trace estimates) and the integral-exponents cannot be smaller than $p$ and $q$, respectively:

Proposition 3.2**.**

If for some $t\in{{\mathbb{R}}}$ and $r>0$ there exists $c\in\,]0,\infty[$ such that every $u\in\cal S({{\mathbb{R}}}^{n})$ satisfies

[TABLE]

whenever $u=\sum u_{j}$ is a decomposition satisfying (3.3), then $r\geq q$.

Consequently, for $p<q\leq 1$ in Theorem 3.1 (iii), the receiving space B^{{\frac{n}{\raise 1.0pt\hbox{\scriptscriptstyle q}}}-n}_{q,\infty} is optimal with respect to the integral-exponent.

Proof.

The latter statement follows from the former, for on the one hand B^{{\frac{n}{\raise 1.0pt\hbox{\scriptscriptstyle q}}}-n}_{q,\infty}\hookrightarrow B^{{\frac{n}{\raise 1.0pt\hbox{\scriptscriptstyle r}}}-n}_{r,\infty} for $r\geq q$, and if, on the other hand, B^{{\frac{n}{\raise 1.0pt\hbox{\scriptscriptstyle r}}}-n}_{r,\infty} receives with an estimate for some $r<q$, then (3.8) holds. In particular it does so when $u=\sum u_{j}$ is a decomposition of a Schwartz function, so the contradicting conclusion $r\geq q$ follows.

When (3.8) holds, one may for arbitrary fixed points $x_{j}\in{{\mathbb{R}}}^{n}$ define

[TABLE]

Independently of the choice of the points $x_{j}$, the right hand side of (3.8) equals cN^{{\frac{1}{\raise 1.0pt\hbox{\scriptscriptstyle q}}}}\mathinner{\|}\check{\Psi}_{0}\,|L_{p}\|, and it is well known that $x_{1}$, $x_{2}$, … may be chosen such that

[TABLE]

so in view of (3.8) the inequality $r\geq q$ must hold.

For completeness’ sake it is remarked that (3.10) may be seen thus: clearly the fact that $\Psi_{k}\equiv 1$ on $\operatorname{supp}\Phi_{0}$ yields that

[TABLE]

Moreover, $\Phi_{0}(\xi)=\Phi_{0}(-\xi)\geq 0$, so $\check{\Phi}_{0}$ is real-valued with $\check{\Phi}_{0}(0)>0$, hence some $\delta>0$ fulfills that $\check{\Phi}_{0}(x)>\check{\Phi}_{0}(0)/2>0$ for $|x|<\delta$.

There is also $R>\delta$ such that $|\check{\Phi}_{0}(x)|<\check{\Phi}_{0}(0)/(2N)$ for $|x|>R$, so if $x_{j}=3jR(1,0,\dots,0)$,

[TABLE]

Indeed, $|\check{\Phi}_{0}*\omega_{N}|\geq\check{\Phi}_{0}(\cdot-x_{j})-\tfrac{N-1}{2N}\check{\Phi}_{0}(0)\geq\check{\Phi}_{0}(\cdot-x_{j})/2$ holds on the ball $B(x_{j},\delta)$ because $|x_{k}-x|>R$ does so for $k\neq j$. This shows (3.10). ∎

Remark 3.3*.*

In (ii) and (iv) of Theorem 3.1, the series $u=\sum u_{j}$ converges in $B^{s}_{p,q}$ if $q<\infty$ and in $B^{s-\varepsilon}_{p,1}$ for $\varepsilon>0$ if $q=\infty$. This is a well-known easy consequence of the completeness and the norm estimate in the theorem.

Remark 3.4*.*

The spectral conditions in (3.3) are robust under restriction: when $x=(x^{\prime},x^{\prime\prime})$ is a splitting of the variables and $x^{\prime\prime}$ is kept fixed, then

[TABLE]

holds by the Paley–Wiener–Schwartz theorem, for $u_{j}(\cdot,x^{\prime\prime})$ is still an analytic function satisfying the relevant estimates in $\operatorname{Re}z^{\prime}$ and $\operatorname{Im}z^{\prime}$.

By the same argument, (3.6) goes over into (3.13) for $u_{j}(\cdot,x^{\prime\prime})$.

3.2. The distribution trace

A rather general definition of the trace is obtained as $r_{0}f:=f(0)$ on the subspace

[TABLE]

For the spaces considered in this note, the working definition in (3.1) actually amounts to a restriction of $r_{0}$. This is proved in Proposition 5.1 below by means of the injection in (3.14), so this folklore is explicated (in lack of a reference):

Proposition 3.5**.**

Let $f(t)$ belong to $C({{\mathbb{R}}},\cal D^{\prime}({{\mathbb{R}}}^{n-1}))$, whereby $\cal D^{\prime}({{\mathbb{R}}}^{n-1})$ has the w∗-topology. Then

[TABLE]

defines an injection of $C({{\mathbb{R}}},\cal D^{\prime}({{\mathbb{R}}}^{n-1}))$ into $\cal D^{\prime}({{\mathbb{R}}}^{n})$.

Proof.

When $\varphi\in C^{\infty}_{0}$ is supported by the rectangle $K:=[-k,k]^{n}$, bilinearity and the Banach–Steinhaus theorem for $C^{\infty}_{0}([-k,k]^{n-1})$ give continuity of the map $t\mapsto\langle\,f(t),\,\varphi(\cdot,t)\,\rangle$ and the bound

[TABLE]

while $\varphi$ of the form $\psi(x^{\prime})\chi(t)$ yields the injectivity of $f\mapsto\Lambda_{f}$. ∎

While it is meaningful, for every subspace $X$ of $\cal D^{\prime}({{\mathbb{R}}}^{n})$, to ask whether

[TABLE]

it is for arbitrary $u\in\cal D^{\prime}({{\mathbb{R}}}^{n})$ meaningless to ask whether the dependence on $x_{n}$ is continuous. Despite this peculiarity, the estimates yielding boundedness of $\gamma_{0}$ in (3.1) do also give inclusions like (3.17) for the domains of $\gamma_{0}$; cf. Proposition 5.1.

Remark 3.6*.*

On $X=C_{\operatorname{b}}({{\mathbb{R}}}^{n})$, where the inclusion in (3.17) is clear, it follows that (3.1) converges to the continuous function obtained from the operation in (1.1) as expected. Indeed, since $\Psi_{k}=\Phi_{0}+\dots+\Phi_{k}$ gives an approximative identity, viz. $\cal F^{-1}\Psi_{k}$, for the convolution algebra $X$,

[TABLE]

Remark 3.7*.*

Considering $\rho_{0}\colon H^{1}(\overline{{{\mathbb{R}}}}_{+})\to{{\mathbb{C}}}$ given by $\rho_{0}u=u(0)$, the restriction \rho_{0}\big{|}_{C^{\infty}_{0}} extends by continuity to the zero-operator $L_{2}\to{{\mathbb{C}}}$. This exemplifies that when a restriction of an operator is extended by continuity between another pair of spaces, the resulting map may be very different from the original one.

A less obvious example is \gamma_{0}\big{|}_{\cal S} extended as $T$ in (1.4); cf. (1.6)–(1.8).

Remark 3.8*.*

To avoid phenomena as those in Remark 3.7, the approach of this paper is first of all to define $r_{0}$ as the distributional trace on $C({{\mathbb{R}}},\cal D^{\prime}({{\mathbb{R}}}^{n-1}))$; for this reason Proposition 3.5 is included. Secondly, boundedness of $\gamma_{0}\colon X\to Y$ is obtained together with the identity \gamma_{0}=r_{0}\big{|}_{X} without extension by continuity.

4. Boundedness

To obtain the continuity properties, observe that since $\cal F^{-1}(\Phi_{j}\hat{u})$ has spectrum in the ball $B(0,R2^{j})$ for $R=\tfrac{13}{10}$, it follows from Remark 3.4 by freezing $x^{\prime}$ that $\cal F^{-1}(\Phi_{j}\hat{u})(x^{\prime},\cdot)$ has spectrum in $[-R2^{j},R2^{j}]$, hence by the Nikol’skiĭ–Plancherel–Polya inequality that

[TABLE]

when the latter is applied in the $x_{n}$-variable only.

Integration with respect to $x^{\prime}$ then gives the basic $L_{p}$-$L_{\infty}$ estimate

[TABLE]

and taking in particular $x_{n}=0$,

[TABLE]

The boundedness in Theorems 1.1, 1.2 and 1.3 now follows by Theorem 3.1 and Remark 3.4.

For example, that u\in B^{{\frac{1}{\raise 1.0pt\hbox{\scriptscriptstyle p}}}}_{p,1}({{\mathbb{R}}}^{n}) means that the right hand side of (4.3) is in $\ell_{1}$, so \sum_{j=0}^{\infty}\cal F^{-1}(\Phi_{j}\cal Fu)\big{|}_{x_{n}=0} converges in $L_{p}$ (because of its convergent norm series); hence also in $\cal D^{\prime}({{\mathbb{R}}}^{n-1})$ when $1\leq p\leq\infty$. So, with the limit denoted $\gamma_{0}u$ according to the working definition of $\gamma_{0}$,

[TABLE]

For B^{{\frac{1}{\raise 1.0pt\hbox{\scriptscriptstyle p}}}}_{p,q} with $0<q<1$ part (i) of Theorem 3.1 applies.

When s={\textstyle\frac{n}{\raise 1.0pt\hbox{\scriptstyle p}}}-n+1 for $p<1$, then (4.3) may be multiplied by 2^{j(s-{\frac{1}{\raise 1.0pt\hbox{\scriptscriptstyle p}}})} and the $\ell_{q}$-norm of both sides calculated. By Remark 3.4 — this time applied with the freezing $x_{n}=0$ — and (iii) of Theorem 3.1, the properties in (1.10)–(1.11) are obtained. Observe here that the assumption on $s$ is equivalent to

[TABLE]

which is required when (iii) is applied to the co-domain B^{s-{\frac{1}{\raise 1.0pt\hbox{\scriptscriptstyle p}}}}_{p,q}({{\mathbb{R}}}^{n-1}).

In the same way (4.3) and (ii) of Theorem 3.1 show the boundedness in Theorem 1.1.

Following [Tri83, 2.7.2], the right inverse $K$ of $\gamma_{0}$ may be taken as

[TABLE]

when $\psi\in\cal S({{\mathbb{R}}})$ has $\operatorname{supp}\cal F\psi\subset[-1,1]$ and $\psi(0)=1$.

Indeed, letting $v_{j}=\cal F^{-1}\Phi_{j}^{\prime}\cal Fv$,

[TABLE]

so part (iv) of Theorem 3.1 gives that $Kv$ is well defined with

[TABLE]

for $s\in{{\mathbb{R}}}$. Moreover, for s>{\textstyle\frac{1}{\raise 1.0pt\hbox{\scriptstyle p}}}+(n-1)({\textstyle\frac{1}{\raise 1.0pt\hbox{\scriptstyle p}}}-1)_{+} the already shown continuity of $\gamma_{0}$ gives

[TABLE]

This reproves the claims on $K$ in Theorem 1.1.

Remark 4.1*.*

The spaces B^{{\frac{1}{\raise 1.0pt\hbox{\scriptscriptstyle p}}}}_{p,1}({{\mathbb{R}}}^{n}) with $1\leq p\leq\infty$ are maximal among those under consideration, for when s>{\textstyle\frac{1}{\raise 1.0pt\hbox{\scriptstyle p}}}+(n-1)({\textstyle\frac{1}{\raise 1.0pt\hbox{\scriptstyle p}}}-1)_{+},

[TABLE]

and this also holds when s={\textstyle\frac{1}{\raise 1.0pt\hbox{\scriptstyle p}}}+(n-1)({\textstyle\frac{1}{\raise 1.0pt\hbox{\scriptstyle p}}}-1)_{+} and $q\leq 1$.

5. Continuity in the $x_{n}$-variable

In view of Remark 4.1, the proof of Theorem 1.4 need only be conducted for the B^{{\frac{1}{\raise 1.0pt\hbox{\scriptscriptstyle p}}}}_{p,1}({{\mathbb{R}}}^{n}) spaces with $1\leq p\leq\infty$. Clearly $x_{n}=0$ does not play a special role, for the mixed norm estimate in (4.2) ‘absorbs’ any value equally well: obviously

[TABLE]

follows in the same way as (4.3). This means that the function series

[TABLE]

converges in the Banach space $C_{\operatorname{b}}({{\mathbb{R}}},L_{p}({{\mathbb{R}}}^{n-1}))$, say, with the limit denoted by $f_{u}(t)$. So for every u\in B^{{\frac{1}{\raise 1.0pt\hbox{\scriptscriptstyle p}}}}_{p,1}({{\mathbb{R}}}^{n}) with $1\leq p\leq\infty$,

[TABLE]

and $f_{u}(0)=\gamma_{0}u$ by the working definition of $\gamma_{0}$. By (3.15), the injection in (5.3) is well defined and continuous; in fact

[TABLE]

for every test function $\varphi$, when $p+p^{\prime}=pp^{\prime}$. However, since the series of $C^{\infty}$ functions in (5.2) converges to the given $u$ in $\cal S^{\prime}({{\mathbb{R}}}^{n})$, hence in $\cal D^{\prime}({{\mathbb{R}}}^{n})$, it follows from (5.3)–(5.4) that $u=f_{u}$.

This proves

Proposition 5.1**.**

Let u\in B^{{\frac{1}{\raise 1.0pt\hbox{\scriptscriptstyle p}}}}_{p,q}({{\mathbb{R}}}^{n}) for some $p\in[1,\infty]$ and $q\leq 1$. Then the function $f_{u}(t)$ given by (5.2)–(5.3) defines a distribution $\Lambda_{f_{u}(t)}$, by Proposition 3.5, that coincides with $u$; that is, $\Lambda_{f_{u}(t)}=u$.

Thereby (3.17) has been verified for the result in Theorem 1.3, so the distribution trace $u(0)$ is defined for every u\in B^{{\frac{1}{\raise 1.0pt\hbox{\scriptscriptstyle p}}}}_{p,1}; viewing $u$ as an element of $C({{\mathbb{R}}},\cal D^{\prime}({{\mathbb{R}}}^{n}))$ gives $u(0)=f_{u}(0)=\gamma_{0}u$ as desired. In particular $\gamma_{0}u$ in (3.1) is independent of the choice of partition of unity.

6. Surjectiveness

Since $\gamma_{0}$ in (1.14) has dense range, it is for $q=1$ surjective precisely when its adjoint $\gamma_{0}^{*}$ has a bounded inverse from $\operatorname{ran}(\gamma_{0}^{*})$ to $L_{p}^{*}$ (see e.g. [Rud73, Th. 4.15]).

For $1\leq p<\infty$ and $q=1$ the adjoint is bounded, when $p+p^{\prime}=pp^{\prime}$,

[TABLE]

cf. [Tri83] for the dual space; and $\gamma_{0}^{*}u=u\otimes\delta_{0}$ for $u\in L_{p^{\prime}}$ since for $\varphi\in\cal S$

[TABLE]

It remains to be shown, with primes omitted for simplicity, that

[TABLE]

for all $u\in L_{p}({{\mathbb{R}}}^{n-1})$ whenever $p\in\,]1,\infty]$.

Using Lemma 2.1 we have a partition of unity $1=\sum_{k=0}^{\infty}\sum_{J\neq\emptyset}\Theta_{J,k}$, where each $\Theta_{J,k}$ is a product:

[TABLE]

By (6.1), the corresponding B^{{\frac{1}{\raise 1.0pt\hbox{\scriptscriptstyle p}}}-1}_{p,\infty}-norm with supremum over $(J,k)$ gives

[TABLE]

Since $\eta_{J}(0)\neq 0$ for some $J$, we can take $J$ such that

[TABLE]

if $a:=\int\check{\eta}_{J}\neq 0$. The w∗-compactness of the balls in $L_{p}$ together with (6.6)–(6.5) show that (6.3) holds with $c$ equal to $(a\mathinner{\|}\check{\theta}_{J}\,|L_{p}\|)^{-1}$.

From the Besov spaces’ point of view the surjectiveness is proved in a natural way above; essentially it is known from the technical report [Pee75].

For $q\leq 1$ the dual of B^{{\frac{1}{\raise 1.0pt\hbox{\scriptscriptstyle p}}}}_{p,q} is independent of $q$, because (B^{{\frac{1}{\raise 1.0pt\hbox{\scriptscriptstyle p}}}}_{p,q})^{*}=B^{-{\frac{1}{\raise 1.0pt\hbox{\scriptscriptstyle p}}}}_{p^{\prime},\infty} then; cf. [Tri83, 2.11.2]. Therefore the adjoint remains equal to (6.1) for $q<1$, so it suffices to show that the Closed Range Theorem is valid when the domain is a quasi-Banach space.

Observe first, for precision, that $B^{s}_{p,q}$ is an F-space in Rudin’s terminology [Rud73] when $d(u,v):=\mathinner{\|}u-v\,|B^{s}_{p,q}\|^{\lambda}$ and $\lambda=\min(1,p,q)$. Hence continuity and boundedness are equivalent for operators between these quasi-Banach spaces [Rud73, 1.32].

Moreover, defining the operator norm in the usual way, ${\mathbb{B}}(X,Y)$ becomes a quasi-Banach space; $\|S+T\|\leq c(\|S\|+\|T\|)$ holds with the same constant as it does for $\mathinner{\|}\cdot\,|Y\|$. In particular, $X^{*}$ is always a Banach space. As usual each $T\in{\mathbb{B}}(X,Y)$ has an adjoint $T^{*}\in{\mathbb{B}}(Y^{*},X^{*})$.

Proposition 6.1**.**

Let $X$ be a quasi-Banach space such that $\mathinner{\|}\cdot\,|X\|^{\lambda}$ is subadditive for some $\lambda\in\,]0,1]$, let $Y$ be a Banach space and $T\colon X\to Y$ be a bounded linear operator. When $\overline{T(X)}=Y$, then boundedness of $T^{*-1}$ from $T^{*}(Y^{*})$ to $Y^{*}$ implies that $T$ is surjective, i.e. $T(X)=Y$.

Proof.

Since $\ker T^{*}\subset T(X)^{\perp}=\{0\}$ the inverse is well defined; by assumption there is a constant $c<\infty$ such that

[TABLE]

This inequality implies that $T$ is open. Indeed, if $X$ is a Banach space this is the content of [Rud73, Lem. 4.13]. When only $Y$ is assumed to be a Banach space, the reduction from part (b) to (a) there carries over verbatim (since the Hahn–Banach theorem is only used for $Y$), and in the proof of (a) the sequence $(\varepsilon_{n})$ should be picked in $\ell_{\lambda}$ such that $\sum_{n=1}^{\infty}\varepsilon_{n}^{\lambda}<1-\mathinner{\|}y_{1}\,|Y\|^{\lambda}$. Then the sequences $(x_{n})$ and $(y_{n})$ defined there satisfy

[TABLE]

hence $x=\sum x_{n}$ converges in $X$ and has $\mathinner{\|}x\,|X\|<1$ as desired.

Thus (6.7) implies that $T$ is an open mapping, but as such it’s necessarily surjective. ∎

Altogether this shows that $L_{p}({{\mathbb{R}}}^{n-1})$ is the image of B^{{\frac{1}{\raise 1.0pt\hbox{\scriptscriptstyle p}}}}_{p,q}({{\mathbb{R}}}^{n}) under $\gamma_{0}$ for every $q\leq 1$ when $1\leq p<\infty$.

Remark 6.2*.*

It is known that every quasi-Banach space $X$ has an equivalent quasi-norm such that $\mathinner{\|}\cdot\,|X\|^{\lambda}$ is sub-additive for some $\lambda\in\,]0,1]$. In view of this, the proposition holds for all quasi-Banach spaces.

7. The borderline for $0<p<1$

Since the boundedness in Theorem 1.2 is proved in Section 4 above, it remains to show the claim on the integral-exponents.

That it is necessary for $p<q\leq 1$ in Theorem 1.2 to let B^{(n-1)({\frac{1}{\raise 1.0pt\hbox{\scriptscriptstyle q}}}-1)}_{q,\infty}({{\mathbb{R}}}^{n-1}) receive $\gamma_{0}u$ follows because the inequality $r\geq q$ is implied by the estimate

[TABLE]

To show this implication, it suffices to extend the $\omega_{N}$ in the proof of Proposition 3.2 by taking some $\eta\in\cal S({{\mathbb{R}}})$ satisfying $\operatorname{supp}\eta\subset\,]1,2[\,$ and $\check{\eta}(0)=1$ and set

[TABLE]

Using (7.1) and part (ii) of Theorem 3.1 to estimate the Besov norm of $E\omega_{N}$, it is easily seen that

[TABLE]

Because of (3.10) the inequality $r\geq q$ holds.

8. Final remarks

Remark 8.1*.*

In addition to (1.9), $\psi_{k}:=2^{k(n-1)}\eta(2^{k}\cdot)\varphi\to\delta\otimes\varphi$ in $B^{s}_{p,1}$ for $k\to\infty$ when {\textstyle\frac{1}{\raise 1.0pt\hbox{\scriptstyle p}}}<s<{\textstyle\frac{n}{\raise 1.0pt\hbox{\scriptstyle p}}}-n (which entails p<1-{\textstyle\frac{1}{\raise 1.0pt\hbox{\scriptstyle n}}}), at least if $\hat{\eta}=1$ in a ball around $\xi^{\prime}=0$. For by Remark 3.3, $\delta_{0}=\eta+\sum_{k=1}^{\infty}(2^{k(n-1)}\eta(2^{k}\cdot)-2^{(k-1)(n-1)}\eta(2^{k-1}\cdot))$ converges in $B^{s}_{p,1}({{\mathbb{R}}}^{n-1})$ while $\cdot\otimes\varphi$ maps continuously into $B^{s}_{p,1}({{\mathbb{R}}}^{n})$ by [Fra86]. Hence $\psi_{k}\to\delta_{0}\otimes\varphi$ there, and $T\psi_{k}\to 0$ as shown in (1.9); i.e. (1.6) holds.

Remark 8.2*.*

For s={\textstyle\frac{n}{\raise 1.0pt\hbox{\scriptstyle p}}}-n+1 it is useful to consider

[TABLE]

for Schwartz functions $f$ and $g$ with their spectra in balls of radius $1/2$ such that $\int f=1$ and $g(0)=1$. As shown in [Joh96, Lem. 2.8], $\gamma_{0}v_{k}\to\delta_{0}$ in $\cal D^{\prime}$ while $v_{k}\to 0$ in B^{{\frac{n}{\raise 1.0pt\hbox{\scriptscriptstyle p}}}-n+1}_{p,q} if $q>1$, so that $\gamma_{0}$ is only continuous from B^{{\frac{n}{\raise 1.0pt\hbox{\scriptscriptstyle p}}}-n+1}_{p,q} if $q\leq 1$.

However, $Tv_{k}=\frac{1}{k}\sum_{l=k+1}^{2k}2^{l(n-1)}f(2^{l}\cdot)$ since $v_{k}\in\cal S$, and its norm $\mathinner{\|}Tv_{k}\,|L_{p}\|$ is \cal O(k^{(n-1)(1-{\frac{1}{\raise 1.0pt\hbox{\scriptscriptstyle p}}})}) and so tends to [math] for $k\to\infty$; that is, already at the borderline $\gamma_{0}$ and $T$ behave differently.

Remark 8.3*.*

For $\Omega$ equal to the unit ball in ${{\mathbb{R}}}^{n}$, $n\geq 3$, Franke and Runst [FR95, Sect. 6.5] proved that B^{{\frac{n}{\raise 1.0pt\hbox{\scriptscriptstyle p}}}-n+1}_{p,\infty}(\overline{\Omega}) contains an infinite-dimensional solution space for the problem

[TABLE]

In fact, for each boundary point $z\in S^{n-1}$ they showed that $\Phi(x-z)-\tfrac{1}{n-2}z\cdot\operatorname{grad}\Phi(x-z)$, where $\Phi(x)=c|x|^{2-n}$ is the fundamental solution of $-\!\operatorname{\Delta}$, belongs to this space and solves (8.2).

Moreover, in [Joh93, Joh96] it was proved that the Boutet de Monvel calculus of pseudo-differential boundary operators (for elliptic problems) extends nicely to spaces with $p<1$. However for trace operators and $P_{\Omega}+G$ that precisely have class $r\in{\mathbb{Z}}$, it was proved that s\geq{\textstyle\frac{n}{\raise 1.0pt\hbox{\scriptstyle p}}}-n+r is necessary for continuity from $B^{s}_{p,q}$ to $\cal D^{\prime}$ when $p<1$.

Taken together, these facts show that not only the usual Fredholm properties but also the continuity of solution operators for elliptic problems break down for $p<1$ unless s={\textstyle\frac{n}{\raise 1.0pt\hbox{\scriptstyle p}}}-n+r is taken as the borderline for operators of class $r$. (For the Dirichlét realisation of $-\!\operatorname{\Delta}$, the latter fact was also shown by Chang, Krantz and Stein [CKS93].)