Density of translates in weighted $L^p$ spaces on locally compact groups

TL;DR

This paper characterizes when translation actions are hypercyclic on weighted $L^p$ spaces over locally compact groups, extending Salas's criterion from integers to general groups and subsets.

Contribution

It provides a general criterion for hypercyclicity of translation actions on weighted $L^p$ spaces for any locally compact group, broadening previous results.

Findings

Extended Salas's hypercyclicity criterion to general locally compact groups.

Established conditions on weights for translation actions to be hypercyclic.

Analyzed hypercyclicity for translations by subsets of the group.

Abstract

Let $G$ be a locally compact group, and let $1\le p < \infty$. Consider the weighted $L^p$-space $L^p(G,\omega)=\{f:\int|f\omega|^p<\infty\}$, where $\omega:G\to \mathbb R$ is a positive measurable function. Under appropriate conditions on $\omega$, $G$ acts on $L^p(G,\omega)$ by translations. When is this action hypercyclic, that is, there is a function in this space such that the set of all its translations is dense in $L^p(G,\omega)$? H. Salas (1995) gave a criterion of hypercyclicity in the case $G=\mathbb Z$ . Under mild assumptions, we present a corresponding characterization for a general locally compact group $G$. Our results are obtained in a more general setting when the translations only by a subset $S\subset G$ are considered.

Equations

$$(T_{s}f)(t)=f(s^{-1}t),t\in G$$

$$\sum_{n,k:n\neq k}\Big{(}\int_{s_{n}s_{k}^{-1}F_{k}}\omega^{p}\Big{)}^{1/p}<\infty.$$

$${\mathrm{esssup}}_{sE\cup s^{-1}E}\omega <\delta .$$

$$\Vert T_s\Vert ={\mathrm{esssup}}_{t\in G}\frac{\omega (st)}{\omega (t)}.$$

$${\int }_{s_j{K}_C}|{\lambda }_j-x(t){|}^{p}dt$$

$$⩽C^p\Vert {\lambda }_j{I}_K-{}^{s_j}x{\Vert }_{p,\omega }^p<C^p\frac{\delta }{4C^p}=\frac{\delta }{4}.$$

$$\frac{\delta }{4}>{\int }_{K_j}|{\lambda }_j-x(s_j^{-1}t){|}^{p}dt⩾\delta \mu (K_j),$$

$$\mu (K_j)<\frac{1}{4}.$$

$$\Vert {}^{s_{\alpha }}x{\Vert }_{p,\omega }^p=\int {\omega }^p|{}^{s_{\alpha }}x{|}^p=\underset{N\to \infty }{\lim }{\int }_{\omega ⩽N}{\omega }^p|{}^{s_{\alpha }}x{|}^p.$$

$${\int }_{\omega ⩽N}|{}^{s_{\alpha }}x-{}^{s_{\beta }}x{|}^p$$

$$\displaystyle\geqslant N^{-p}\Big{(}\|{}^{s_{\alpha}}\mkern-3.0mux-{}^{s_{\beta}}\mkern-3.0mux\|_{p,\omega}^{p}-\int_{\omega\geqslant N_{\alpha}}\omega^{p}|{}^{s_{\alpha}}\mkern-3.0mux|^{p}-\int_{\omega\geqslant N_{\alpha}}\omega^{p}|{}^{s_{\beta}}\mkern-3.0mux|^{p}\Big{)}>N^{-p}.$$

$$\sum _{n,k⩾0:n\ne k}\Vert \omega {\Vert }_{p,s_n{s}_k^{-1}{K}_k}^p<\infty .$$

$${\mathrm{essinf}}_{F_k^{\prime }}\omega ⩾c_k,{\mathrm{esssup}}_{F_k^{\prime }}\omega ⩽c_k^{-1}$$

$$c_k^{2p}{\delta }_k>{\int }_{F_k^{\prime }}|{}^{s_k}y(t)-2{|}^p\omega (t{)}^{p}dt⩾c_k^p{\int }_{F_k^{\prime }}|{}^{s_k}y(t)-2{|}^{p}dt$$

$$c_k^p{\delta }_k>\Vert {}^{s_k}y-2{\Vert }_{p,F_k^{\prime }}^p={\int }_{K_k^{\prime }}|y(s_k^{-1}t)-2{|}^{p}dt+{\int }_{F_k^{\prime }\setminus K_k^{\prime }}|y(s_k^{-1}t)-2{|}^{p}dt,$$

$$|F_k^{\prime }\setminus K_k^{\prime }|⩽{\int }_{F_k^{\prime }\setminus K_k^{\prime }}|y(s_k^{-1}t)-2{|}^{p}dt<c_k^p{\delta }_k<{\delta }_k/4.$$

$$\sum _k{\int }_{s_k^{-1}{K}_k}{\omega }^p⩽\sum _k{\int }_{s_k^{-1}{K}_k}{\omega }^p(t)|y(t){|}^{p}dt⩽\Vert y{\Vert }_{p,\omega }^p<\infty .$$

$${\int }_{G\setminus K_n}{\omega }^p|{}^{s_n}y{|}^p$$

$$⩽|F_n^{\prime }\setminus K_n|c_n^{-p}+\Vert 2I_{F_n^{\prime }}-{}^{s_n}y{\Vert }_{p,\omega }^p<{\delta }_n+c_n^p{\delta }_n⩽2{\delta }_n.$$

$$2{\delta }_n$$

$$a_k=\sum _{n:n\ne k}\Vert \omega {\Vert }_{p,s_n{s}_k^{-1}{K}_k}^p.$$

$$\sum _{n,k:n\ne k}{p}_k\Vert \omega {\Vert }_{p,t_n{t}_k^{-1}{E}_k}^p<\infty .$$

$$|\mathrm{supp}{P}_n\setminus E_n|=|\mathrm{supp}{P}_{l_n}\setminus K_{l_n}|<{\delta }_{l_n}⩽{\delta }_n$$

$$X=\sum _k{T}_{t_k^{-1}}(P_k{I}_{E_k}).$$

$$\Vert X{\Vert }_{p,\omega }^p$$

$$=\sum _k{p}_k\Vert \omega {\Vert }_{p,t_0{t}_k^{-1}{E}_k}^p<\infty .$$

$$\Vert {}^{t_n}X-P_n{\Vert }_{p,\omega }^p=\Vert P_n(1-I_{E_n}){\Vert }_{p,\omega }^p+\sum _{k:k\ne n}\Vert {}^{t_n{t}_k^{-1}}(P_k{I}_{E_k}){\Vert }_{p,\omega }^p,$$

$$\mathrm{supp}{}^{t_n{t}_k^{-1}}{P}_k\subset t_n{t}_k^{-1}{E}_k.$$

$$\Vert P_n(1-I_{E_n}){\Vert }_{p,\omega }^p={\int }_{\mathrm{supp}{P}_n\setminus E_n}|P_n{|}^p⩽\Vert P_n{\Vert }_{\infty }^p|\mathrm{supp}{P}_n\setminus E_n|<p_n{\delta }_n<2^{-n}.$$

$$\Vert {}^{t_n{t}_k^{-1}}(P_k{I}_{E_k}){\Vert }_{p,\omega }^p⩽\Vert P_k{\Vert }_{\infty }^p\Vert \omega {\Vert }_{p,t_n{t}_k^{-1}{E}_k}^p=p_k\Vert \omega {\Vert }_{p,t_n{t}_k^{-1}{E}_k}^p.$$

$${\epsilon }_n=\sum _{n,k:k\ne n}{p}_k\Vert \omega {\Vert }_{p,t_n{t}_k^{-1}{E}_k}^p.$$

$$\Vert {}^{t_n}X-P_n{\Vert }_{p,\omega }^p⩽{\epsilon }_n+2^{-n}$$

Full text

1. Introduction

Let $G$ be a locally compact group with identity $e$ and a left Haar measure $\mu$. Fix $p,\,1\leqslant p<\infty$, and a weight $\omega$, that is, a measurable strictly positive function on $G$ supposed to be locally $p$-summable.

Consider the weighted space $L^{p}(G,\omega)=\{f:\int_{G}|f\omega|^{p}d\mu<\infty\}$ of complex-valued functions on $G$. In this paper the following question is addressed. Can all functions in $L^{p}(G,\omega)$ be approximated arbitrarily well by the left translates of some single function $f\in L^{p}(G,\omega)$? The same question can be asked if we allow to translate $f$ only by elements of a fixed subset $S\subset G$.

To be more precise, let $S$ be a subset of $G$, and suppose that, for any $s\in S$, the (left) translation operator

[TABLE]

is continuous from $L^{p}(G,\omega)$ into itself. A function $f\in L^{p}(G,\omega)$ is called $S$-dense if its $S$-orbit $Orb_{S}(f)=\{T_{s}f:s\in S\}$ is dense in $L^{p}(G,\omega)$. So, we are interested in the existence of $S$-dense vectors in the space $L^{p}(G,\omega)$.

$S$-density is a particular case of a more general situation, known as the universality phenomenon, see a survey of K.-G. Grosse-Erdmann [8]. One of the first examples in this area was given in 1929 by G. D. Birkhoff (see [8]): there exists an entire function $f$ whose translates are dense in the space ${\mathcal{H}}(\mathbb{C})$ of entire functions. In the case when $S$ is the semigroup generated by one element $s\in G$, $S$-density is equivalent to the hypercyclicity of the operator $T_{s}$, which means that there exists a single vector $x$ such that the sequence $(T_{s}^{n}x)_{n\in\mathbb{N}}$ is dense.

If $S$ is a sub-semigroup or a subgroup of $G$, $S$-density means in other terms that the action of $S$ by translations on the space $L^{p}(G,\omega)$ admits a hypercyclic vector. On a non-weighted $L^{p}$-space, the translation operators are isometric and therefore cannot have hypercyclic vectors. In the weighted case hypercyclicity can occur, depending on the weight; moreover, the conditions on weight are as a rule explicit and not difficult to calculate. This problem was considered by many authors. The characterisation in the discrete case $G=\mathbb{Z}$, $S=\mathbb{Z}_{+}$ was obtained by H. Salas in 1995 [10]. In 1997, W. Desch, W. Schappacher, and G. Webb [5] characterized the hypercyclicity of the $C_{0}$-semigroup of translations by $S=\mathbb{R}_{+}$, with $G=\mathbb{R}$. The dynamics of the translation $C_{0}$–semigroup indexed by a sector of the complex plane was considered by J. A. Conejero and A. Peris [2]. Ch.-Ch. Chen [1] characterized the hypercyclicity of a single translation operator (when $S$ is the semigroup generated by an element of $G$) for locally compact groups.

In the present paper, we do not suppose that the set $S$ has any algebraic structure: it can be an arbitrary subset of $G$. We denote by $\mathbf{\Sigma}$ the class of non-compact second countable locally compact groups. For a locally compact group $G$, we show that $G\in\mathbf{\Sigma}$ if and only if for some weight $\omega$ on $G$ and for some $S\subset G$ there exist $S$-dense functions.

The main result of the paper is Theorem 5, which provides a necessary and sufficient condition for the existence of $S$-dense functions in $L^{p}(G,\omega)$. To avoid technical details, we formulate here this condition in the case when $G$ is discrete:

Corollary 6. Let $G$ be an infinite countable discrete group, and let $S\subset G$. Let $\omega$ be a weight on $G$. There is an $S$-dense vector in $L^{p}(G,\omega)$ if and only if for every increasing sequence $(F_{n})_{n\geqslant 1}$ of finite subsets of $G$, there exists a sequence $(s_{n})_{n\geqslant 1}\subset S$ such that the sets $s_{n}^{-1}F_{n}$ are pairwise disjoint and, setting $s_{0}=e$, $F_{0}=\emptyset$,

[TABLE]

If we suppose that $S$ generates a commutative subgroup of $G$ (which is not necessarily discrete), the criterion is straightforward:

Theorem 10. Let $G\in\mathbf{\Sigma}$, and let $S\subset G$ generate an abelian subgroup in $G$. Let $\omega$ be a weight on $G$. There is an $S$-dense vector in $L^{p}(G,\omega)$ if and only if for every compact set $F\subset G$ and any given $\delta>0$, there exist $s\in S$ and a compact $E\subset F$ such that $\mu(F\setminus E)<\delta$ and

[TABLE]

Note that if, in addition, $G$ is discrete, compact subsets of $G$ are just finite, and $E=F$ in this condition.

Our theorems generalize the results [5], [10], [1] mentioned above.

2. Notations and assumptions on the group

Recall that we consider the weighted space $L^{p}(G,\omega)=\{f:f\omega\in L^{p}(G)\}$ with the norm \|f\|_{p,\omega}=\big{(}\int|f\omega|^{p}\big{)}^{1/p}. The assumption that $\omega$ is locally $p$-summable (that is $\omega\in L^{p}(K)$ for any compact set $K\subset G$) implies that $L^{p}(G,\omega)$ contains all measurable compactly supported bounded functions. For a set $K\subset G$, denote by $I_{K}$ the characteristic function of $K$. For a measurable function $f$, denote $\|f\|_{p,K}=(\int_{K}|f|^{p})^{1/p}$, with $\|f\|_{p}=\|f\|_{p,G}$.

The Haar measure is denoted by $\mu$, but we will also use the notations $|A|\equiv\mu(A)$ for $A\subset G$ and $\int_{G}fd\mu\equiv\int_{G}f(t)\,dt$. The translation $T_{s}f$ of a function $f\in L^{p}(G,\omega)$ will also be denoted by ${}^{s}\mkern-3.0muf$ ($s\in G$).

For $s\in G$, one has the following expression for the norm of the translation operator:

[TABLE]

We say that a weight $\omega$ is $S$-admissible if $\|T_{s}\|<+\infty$ for all $s\in S$.

Let $\mathbf{\Sigma}$ be the class of non-compact second countable locally compact groups. Note that every group in $\mathbf{\Sigma}$ is $\sigma$-compact, that is, represented as a countable union of compact sets (to see this, pick an open compactly generated subgroup $H\subset G$, which is a fortiori $\sigma$-compact; $G$ being second countable, $[G:H]$ is necessarily countable, so finally $G$ is $\sigma$-compact). In this section we show that $G$ must be in $\mathbf{\Sigma}$ to admit $S$-dense vectors for some $S\subset G$ and some weight $\omega$.

In the statements 1-3 it is supposed that $\S\subset G$ and the weight $\omega$ is $S$-admissible.

Our first claim is that $G$ can’t be compact. In the following lemma we prove a bit more, and it is exactly this result that we use below in the proof of Theorem 5.

Lemma 1**.**

Let $x$ be an $S$-dense vector in $L^{p}(G,\omega)$ for some $S,p,\omega$, and let $K\subset G$ be a compact set. Then for every $\varepsilon>0,\lambda\in\mathbb{R}$ and every compact set $L\subset G$ there is $s\in S\setminus L$ such that $\|\lambda I_{K}-{}^{s}\mkern-3.0mux\|_{p,\omega}<\varepsilon$.

Proof.

We can scale the Haar measure if necessary to have $\mu(K)=1$. Increasing $L$ if necessary, we can suppose that $L=L^{-1}$. Suppose the contrary: that is, $\|\lambda I_{K}-{}^{s}\mkern-3.0mux\|_{p,\omega}\geqslant\varepsilon$ for every $s\in S\setminus L$. There is clearly $C>0$ such that for $K_{C}=\{t\in K:\omega(t)>1/C\}$ we have $\mu(K_{C})>3/4$.

Since $LK$ is also compact, its measure is finite. Pick $N\in\mathbb{N}$ with $N>2\mu(LK)$. Set $\lambda_{j}=\lambda+j\,\varepsilon/(2N)$, $j=0,\dots,N$ and pick $\delta>0$ such that $\delta<\varepsilon^{p}/(4N)^{p}$ and $\delta/C^{p}<\varepsilon$. For $j=0,\dots,N-1$, there is $s_{j}\in S$ such that $\|\lambda_{j}I_{K}-{}^{s_{j}}\mkern-3.0mux\|_{p,\omega}^{p}<\delta/(4C^{p})$. Since $\|\lambda I_{K}-{}^{s_{j}}\mkern-3.0mux\|_{p,\omega}<\delta/(4C^{p})+|\lambda-\lambda_{j}|<\varepsilon/4+\varepsilon/2<\varepsilon$, by assumption we have $s_{j}\in L$.

It follows that

[TABLE]

Set $K_{j}=\{t\in K_{C}:|\lambda_{j}-x(s_{j}^{-1}t)|^{p}\geqslant\delta\}$. By trivial estimates,

[TABLE]

so that

[TABLE]

For $t\in K^{\prime}_{j}=s_{j}^{-1}(K_{C}\setminus K_{j})$ we have the estimate $|x(t)-\lambda_{j}|<\delta^{1/p}<\varepsilon/(4N)$. As $|\lambda_{j}-\lambda_{k}|\geqslant\varepsilon/(2N)$ for $j\neq k$, it follows that the sets $K^{\prime}_{j}$ are pairwise disjoint. Moreover, $\mu(K^{\prime}_{j})\geqslant\mu(K_{C})-\mu(K_{j})>1/2$ for every $j$, hence $\mu(\cup K^{\prime}_{j})>N/2>\mu(LK)$ and in particular, $\cup K^{\prime}_{j}\not\subset LK$. But by assumption, we have $K^{\prime}_{j}\subset s_{j}^{-1}K\subset LK$ for every $j$. This contradiction proves the lemma. ∎

Corollary 2**.**

If for some $p$, $S$, $\omega$ there exists an $S$-dense vector in $L^{p}(G,\omega)$, then $G$ is not compact.

Proposition 3**.**

If for some $p$, $S$, $\omega$ there exists an $S$-dense vector in $L^{p}(G,\omega)$, then $G$ is second countable.

Proof.

We use then the fact that $G$ is second countable if and only if $L^{p}(G)$ is separable [4, Theorem 2], and the same is valid for $L^{p}(G,\omega)$ since this space is isometrically isomorphic to $L^{p}(G)$.

Suppose that $L^{p}(G,\omega)$ is not separable but $x\in L^{p}(G,\omega)$ is an $S$-dense vector. There exists then an uncountable set $\{f_{\alpha}\}\subset L^{p}(G,\omega)$ such that $\|f_{\alpha}-f_{\beta}\|_{p}>3$ (this is easy to show usign the Zorn’s lemma). Approximating $f_{\alpha}$ by ${}^{s_{\alpha}}\mkern-3.0mux$ so that $\|f_{\alpha}-{}^{s_{\alpha}}\mkern-3.0mux\|_{p}<1/2$, we get a family $s_{\alpha}\in S$ such that $\|{}^{s_{\alpha}}\mkern-3.0mux-{}^{s_{\beta}}\mkern-3.0mux\|_{p,\omega}^{p}>2$. In particular, $s_{\alpha}$ are all different.

For every $\alpha$,

[TABLE]

There exists $N_{\alpha}\in\mathbb{N}$ such that $\int_{\omega\geqslant N_{\alpha}}\omega^{p}|{}^{s_{\alpha}}\mkern-3.0mux|^{p}<1/2$.

Since the set of $\alpha$ is uncountable, there is a single $N\in\mathbb{N}$ such that the set $A=\{\alpha:N_{\alpha}=N\}$ is uncountable. For every $\alpha,\beta\in A$, we have

[TABLE]

But this contradicts the fact that the $G$-orbit of any function in the non-weighted $L^{p}(G)$ is separable [11]: pick $\gamma\in A$ and set $y={}^{s_{\gamma}}\mkern-3.0mux\cdot I_{\omega\geqslant N}$, then $y\in L^{p}(G)$ (with norm $\leqslant N/2^{1/p}$), and $\int_{\omega\leqslant N}|{}^{s_{\alpha}}\mkern-3.0mux-{}^{s_{\beta}}\mkern-3.0mux|^{p}=\|{}^{(s_{\alpha}s_{\gamma}^{-1})}\mkern-3.0muy-{}^{(s_{\beta}s_{\gamma}^{-1})}\mkern-3.0muy\|_{p}^{p}$ for all $\alpha\neq\beta$ in $A$. ∎

Corollary 2 and Proposition 3 prove the implication $(2)\Rightarrow(1)$ of the following

Theorem 4**.**

For a locally compact group $G$, the following assumptions are equivalent:

1. (1)

$G$* is in the class $\mathbf{\Sigma}$ (non-compact second countable);* 2. (2)

for every subset $S\subset G$ with non-compact closure there is an $S$-admissible weight $\omega$ such that $L^{p}(G,\omega)$ contains $S$-dense vectors, for all (or some) $p\geqslant 1$; 3. (3)

there is a weight $\omega$ for which all left and right translations are bounded and such that $L^{p}(G,\omega)$ contains $G$-dense functions for all (or some) $p\geqslant 1$.

The implication $(1)\Rightarrow(3)$ is proved in the beginning of Section 5 (and $(3)\Rightarrow(2)$ is obvious).

3. Density criterion

In the most general case, the criterion has the following form. Below we give similar conditions in several particular cases. Recall that in every statement we assume that $T_{s}$ is a bounded operator on $L^{p}(G,\omega)$ for all $s\in S$.

Theorem 5**.**

Let $G\in\mathbf{\Sigma}$, and let $S\subset G$. Let $\omega$ be an $S$-admissible weight on $G$. There is an $S$-dense vector in $L^{p}(G,\omega)$ if and only if for every increasing sequence $(F_{n})_{n\geqslant 1}$ of compact subsets of $G$ and any given $\delta_{n}>0$, there exists a sequence $(s_{n})_{n\geqslant 1}\subset S$ and compact sets $K_{n}\subset F_{n}$ such that the sets $s_{n}^{-1}F_{n}$ are pairwise disjoint, $|F_{n}\setminus K_{n}|<\delta_{n}$ and, setting $s_{0}=e$, $K_{0}=\emptyset$,

[TABLE]

Proof.

$\kern 4.0pt\raise 2.5pt\hbox{\circle{17.0}}\kern-5.0pt{\Rightarrow}$:

We set $s_{0}=e$ and $K_{0}=\emptyset$.

Decreasing $\delta_{k}$ if necessary, we can assume that $\sum\delta_{k}<\infty$. Let $y\in L^{p}(G,\omega)$ be $S$-dense. For every $k$, there is $F^{\prime}_{k}\subset F_{k}$ such that $|F_{k}\setminus F^{\prime}_{k}|<\delta_{k}/2$ and

[TABLE]

with some $0<c_{k}<1/4$. By Lemma 1, there exists $s_{k}\in S$ such that $\|2I_{F^{\prime}_{k}}-{}^{s_{k}}y\|_{p,\omega}^{p}<c_{k}^{2p}\,\delta_{k}$ and $s_{k}^{-1}F_{k}$ does not intersect $s_{j}^{-1}F_{j}$ for $j<k$ (this condition is equivalent to $s_{k}\notin F_{k}F_{n}^{-1}s_{n}$). This implies that

[TABLE]

and so $\|{}^{s_{k}}y-2\|_{p,F^{\prime}_{k}}^{p}<c_{k}^{p}\delta_{k}<\delta_{k}$.

Set $K^{\prime}_{k}=\{t\in F^{\prime}_{k}:|y(s_{k}^{-1}t)|>1\}$. Then

[TABLE]

which implies

[TABLE]

By regularity of the Haar measure, there is a compact set $K_{k}\subset K^{\prime}_{k}$ such that $|K^{\prime}_{k}\setminus K_{k}|<\delta_{k}/4$. By estimates above, we have $|F_{k}\setminus K_{k}|<\delta_{k}$.

Since the norm of $y$ is finite, we have:

[TABLE]

By the choice of $s_{n}$, we have

[TABLE]

At the same time, $|{}^{s_{n}}y|^{p}>1$ on $s_{n}s_{k}^{-1}K_{k}$. For $k\neq n$, the condition $s_{n}s_{k}^{-1}F_{k}\cap F_{n}=\emptyset$ implies that $s_{n}s_{k}^{-1}K_{k}\subset G\setminus K_{n}$; moreover, the sets $s_{n}s_{k}^{-1}F_{k}$, $k\neq n$, are pairwise disjoint. It follows that

[TABLE]

By the choice of $\delta_{n}$ we have $\sum_{n}\delta_{n}<\infty$, so we get immediately (2).

$\kern 4.0pt\raise 2.5pt\hbox{\circle{17.0}}\kern-5.0pt{\Leftarrow}$:

From the assumptions it follows that $L^{p}(G,\omega)$ is separable. Choose a dense sequence $(Q_{n})_{n=1}^{\infty}\subset L^{p}(G,\omega)$ such that every $Q_{n}$ is compactly supported and essentially bounded. Arrange them with repetitions in a sequence $(P_{n})_{n=1}^{\infty}$ so that every $Q_{n}$ appears infinitely many times in this sequence. We will seek for $X$ “almost” of the form $\sum_{k}T_{s_{k}^{-1}}P_{k}$, with $s_{n}\in S$ chosen so that the series converges and $\|{}^{s_{n}}X-P_{n}\|_{p,\omega}<\varepsilon_{n}$ for every $n$, with some $\varepsilon_{n}\to 0$. This guarantees that $X$ is $S$-dense.

Denote $F_{n}=\operatorname{supp}P_{n}$ and set $p_{n}=\|P_{n}\|_{\infty}^{p}$. Pick a decreasing sequence $(\delta_{n})$ such that $0<\delta_{n}<p_{n}^{-1}\,2^{-n}$. Choose $s_{n}$, $K_{n}$ according to (2). Set

[TABLE]

By (2), $\sum_{k}a_{k}<\infty$. Choose a subsequence $(a_{l_{k}})$, with $l_{0}=0$, such that $\sum p_{k}a_{l_{k}}<\infty$ and $P_{l_{k}}=P_{k}$ for every $k$. By the choice of $(P_{k})$, this is always possible. Set $t_{n}=s_{l_{n}}$ and $E_{k}=K_{l_{k}}$, then the following series converges:

[TABLE]

In particular, the sets $t_{n}^{-1}E_{n}$ are pairwise disjoint, and

[TABLE]

for $n\geqslant 1$. We can still assume that $t_{0}=e$, $E_{0}=\emptyset$. Set

[TABLE]

This series converges, since (note that the support of $T_{t_{k}^{-1}}P_{k}$ is $t_{k}^{-1}\operatorname{supp}P_{k}$)

[TABLE]

Now, for every $n$

[TABLE]

since again the supports of the summands in (5) are pairwise disjoint:

[TABLE]

One has

[TABLE]

and for $k\neq n$,

[TABLE]

Denote

[TABLE]

By (4), $\sum_{n}\varepsilon_{n}<\infty$, so $\varepsilon_{n}\to 0$. Thus, we have

[TABLE]

with $\varepsilon_{n}\to 0$, which proves that $X$ is $S$-dense. ∎

Corollary 6**.**

Let $G$ be an infinite countable discrete group, and let $S\subset G$. Let $\omega$ be an $S$-admissible weight on $G$. There is an $S$-dense vector in $L^{p}(G,\omega)$ if and only if for every increasing sequence $(F_{n})_{n\geqslant 1}$ of finite subsets of $G$, there exists a sequence $(s_{n})_{n\geqslant 1}\subset S$ such that the sets $s_{n}^{-1}F_{n}$ are pairwise disjoint and, setting $s_{0}=e$, $F_{0}=\emptyset$,

[TABLE]

Remark 7**.**

It is enough to check the conditions of Theorem 5 for an increasing sequence $(F_{n})$ of compact sets such that $G=\cup_{n}F_{n}$. Indeed, for another sequence $(\tilde{F}_{k})$ one can choose $n_{k}$ so that $\tilde{F}_{k}\subset F_{n_{k}}$, and the rest is obvious.

4. Simplifications of conditions

Theorem 8**.**

Let $G\in\mathbf{\Sigma}$, and let $S\subset G$. Let $\omega$ be an $S$-admissible weight on $G$. If for every compact set $K\subset G$ of positive measure and every $\varepsilon,\delta>0$ there exist $s\in S$ and compact $E\subset K$ such that $|K\setminus E|<\delta$ and

[TABLE]

then $L^{p}(G,\omega)$ contains an $S$-dense vector.

Proof.

Let us show first that under the assumptions above, $s$ can be chosen outside of a given compact set $L$. Suppose the contrary: that (8) does not hold for any $s\notin L$. We can assume that $\delta_{1}=|K|-\delta>0$. Choose a compact set $E_{0}$ and $s_{0}\in S$ so that $|K\setminus E_{0}|<\delta$ and (8) holds for $E=E_{0}$, $s=s_{0}$. In particular, $|E_{0}|>|K|-\delta=\delta_{1}$. There is a compact subset $K_{1}\subset E_{0}$ such that $|E_{0}\setminus K_{1}|<\delta_{1}/9$ and $\varepsilon_{1}=\mathop{\rm ess\,inf}\{\omega(t):t\in s_{0}K_{1}\cup s_{0}^{-1}K_{1}\}>0$.

Next choose by induction compact sets $E_{n}$, $K_{n}$ and $s_{n}\in S$, $n\geqslant 1$, so that $|K_{n}\setminus E_{n}|<\delta_{1}/9^{n}$, $\mathop{\rm ess\,sup}_{\!s_{n}E_{n}\,\cup\,s_{n}^{-\!1}\!E_{n}}\;\omega<\varepsilon_{n}$, $|E_{n}\setminus K_{n+1}|<\delta_{1}/9^{n}$ and $\varepsilon_{n+1}=\mathop{\rm ess\,inf}\{\omega(t):t\in s_{n}K_{n+1}\cup s_{n}^{-1}K_{n+1}\}>0$. For every $n\geqslant 1$, we have $|E_{n}|>|K_{n}|-\delta_{1}/9^{n}>|E_{n-1}|-2\delta_{1}/9^{n}$, which implies

[TABLE]

Denote $D_{n}=s_{n}E_{n}\,\cup\,s_{n}^{-\!1}\!E_{n}$, $D_{n}^{-}=\cup_{k<n}D_{k}\cap D_{n}$ and $D_{n}^{+}=\cup_{k>n}D_{k}\cap D_{n}$. Since $D_{k}\cap D_{n}$ is contained (except probably for a zero measure subset) in $(\{s_{n}\}\cup\{s_{n}^{-1}\})(E_{n}\setminus K_{n+1})$ for all $k>n$, we have $|D_{n}^{+}|\leqslant 2|E_{n}\setminus K_{n+1}|<2\delta_{1}/9^{n}$. As a consequence,

[TABLE]

and for every $n$,

[TABLE]

whence it follows that

[TABLE]

which is impossible since $LE\cup L^{-1}E$ is compact. This contradiction shows that $s$ can be chosen outside $L$.

Now let us show that the assumptions of the theorem imply the assumptions of Theorem 5.

Let $(F_{n})$ be an increasing sequence of compact sets and $(\delta_{n})$ a sequence of positive numbers. We can assume that $\delta_{1}<1/2$ and $\delta_{n+1}<\delta_{n}/2$ for all $n$, in particular $\delta_{n}<2^{-n}$. Choose firstly sequences $E^{\prime}_{n}\subset G$, $s_{n}\in S$ by induction. Set $E^{\prime}_{0}=\emptyset$, $s_{0}=e$; for $n\geqslant 1$, set

[TABLE]

Set $C_{n}=\max_{k<n}\|T_{s_{k}}\|$ (automatically $C_{n}\geqslant 1$) and choose $s_{n}\in S$, $E^{\prime}_{n}\subset K_{n}$ so that $E^{\prime}_{n}$ is compact, $|K_{n}\setminus E^{\prime}_{n}|<\delta_{n}/2$ and $\|\omega\|_{\infty,s_{n}E^{\prime}_{n}\cup s_{n}^{-1}E^{\prime}_{n}}<(2^{n}C_{n}|E^{\prime}_{n}|^{1/p})^{-1}$. We can choose $s_{n}$ so that $s_{n}^{-1}F_{n}$ does not intersect $s_{k}^{-1}F_{k}$ for $k<n$ (that is, $s_{n}\notin F_{n}F_{k}^{-1}s_{k}$).

Set now $E_{0}=\emptyset$ and

[TABLE]

for $n\geqslant 1$. Then, since $F_{n}\subset K_{n}$ and for $k>n$  $s_{n}^{-1}F_{n}\subset K_{k}$, that is $F_{n}\subset s_{n}K_{k}$, we have

[TABLE]

The sets $s_{n}^{-1}F_{n}$ are pairwise disjoint; for $k<n$ we have $E_{k}\subset s_{k}E^{\prime}_{n}$ by (9), which implies $s_{k}^{-1}E_{k}\subset s_{k}^{-1}F_{k}\subset E^{\prime}_{n}$. Now

[TABLE]

Finally, the assumptions of Theorem 5 are satisfied with $E_{n}$ in place of $K_{n}$. ∎

Corollary 9**.**

Let $G\in\mathbf{\Sigma}$, and let $S\subset G$. Let $\omega$ be an $S$-admissible weight on $G$. If for every compact set $K\subset G$ of positive measure

[TABLE]

then $L^{p}(G,\omega)$ contains an $S$-dense vector.

In general, this condition is not necessary, see Example 15. But if $S$ generates an abelian subgroup, this is the case:

Theorem 10**.**

Suppose that $G\in\mathbf{\Sigma}$, and let $S\subset G$ generate an abelian subgroup. Let $\omega$ be an $S$-admissible weight on $G$. The following conditions are equivalent:

1. (1)

There is an $S$-dense vector in $L^{p}(G,\omega)$ for every $p$, $1\leqslant p<\infty$; 2. (2)

There is an $S$-dense vector in $L^{p}(G,\omega)$ for some $p$, $1\leqslant p<\infty$; 3. (3)

For every compact set $F\subset G$ and any given $\delta>0$, there exist $s\in S$ and a compact $E\subset F$ such that $|F\setminus E|<\delta$ and

[TABLE] 4. (4)

For some $p$, $1\leqslant p<\infty$, the following condition holds: For every compact set $F\subset G$ and any given $\delta>0$, there exist $s\in S$ and a compact $E\subset F$ such that $|F\setminus E|<\delta$ and

[TABLE] 5. (5)

For every $1\leqslant p<\infty$, the assumptions of (4) above hold.

Proof.

(4)$\Rightarrow$(3) is an easy exercise, (3)$\Rightarrow$(1) was shown in Theorem 8 without the commutativity assumption; (1)$\Rightarrow$(2) and $(5)\Rightarrow(4)$ are obvious.

It remains to prove (2)$\Rightarrow$(5).

If there is an $S$-dense vector in $L^{p}(G,\omega)$, then for $F_{n}=F$ and $\delta_{n}=2^{-n}$, $n\geqslant 1$, there exist $s_{n}\in S$, $E_{n}\subset F$ with $s_{0}=e$, $E_{0}=\emptyset$ such that $|F\setminus E_{n}|<\delta_{n}$ and

[TABLE]

It follows (by considering the subseries with $n=0$) that $\|\omega\|^{p}_{p,s_{k}^{-1}E_{k}}\to 0$.

Let $k$ be such that $\delta_{k}<\delta/2$. Set $C=\|T_{s_{k}}\|$. For $n>k$, we have

[TABLE]

Pick now $n>k$ such that $\|\omega\|_{p,s_{n}^{-1}E_{n}}<\delta/2$ and $\|\omega\|_{p,s_{n}E_{k}}<\delta/2$. Set $E=E_{k}\cap E_{n}$, $s=s_{n}$, then $|F\setminus E|<\delta_{k}+\delta_{n}<\delta$ and $\|\omega\|_{p,sE\cup s^{-1}E}\leqslant\|\omega\|^{p}_{p,s_{n}E_{k}}+\|\omega\|^{p}_{p,s_{n}^{-1}E_{n}}<\delta$. ∎

The following case is quite particular and is meaningful especially in the abelian case, see Corollary 13.

Remark 11**.**

If $\omega$ is locally summable and

[TABLE]

for all $g\in G$, then $\omega$ is equivalent to a continuous weight [7, Satz 2.7]. Since (13) holds or does not hold simlutaneously for $\omega^{p}$ for all $1\leqslant p<\infty$, we have the same conclusion if $\omega$ is locally $p$-summable and satisfies (13). Hence in the following statement we can assume, without loss of generality, that $\omega$ is continuous.

Proposition 12**.**

Let $G\in\mathbf{\Sigma}$, and suppose that the weight $\omega$ is continuous and such that for all $g\in G$,

[TABLE]

Then for any $S\subset G$, the following conditions are equivalent:

1. (1)

There are $S$-dense vectors in $L^{p}(G,\omega)$. 2. (2)

[TABLE] 3. (3)

[TABLE]

Proof.

Obviously (15)$\Rightarrow$(16). Let us prove the inverse implication. Let $K$ be a compact set; we can assume that $K=K^{-1}$. It is proved in a rather general situation in [6, Proposition 1.16] that $L$ and $R$ are locally bounded (provided they are finite). There exists thus a constant $C$ such that $L|_{K}\leqslant C$ and $R|_{K}\leqslant C$.

Now for every $s\in G$,

[TABLE]

In the same way we get the inequality $\sup_{s^{-1}K}\omega\leqslant C\omega(s^{-1})$, thus (15) and (16) are equivalent.

Since they both imply (10), they imply also the assumption (2) of Theorem 5 and the existence of $S$-dense vectors in $L^{p}(G,\omega)$.

Suppose now that there exist $S$-dense vectors in $L^{p}(G,\omega)$. Pick a compact set $F$ of positive measure and set $F_{n}=F$ for all $n\in\mathbb{N}$. Choose $s_{n}\in S$ and $K_{n}\subset F$ so that (2) holds. By setting $\delta_{n}=|F|/4$, we can guarantee that $|K_{n}\cap K_{1}|>|F|/2$ for all $n\geqslant 1$. From (2) it follows, in particular, that

[TABLE]

Let $C_{1}>0$ be such that $L(s_{1}^{-1})\leqslant C_{1}$, $R|_{K_{1}^{-1}}\leqslant C_{1}$ and $R|_{K_{1}^{-1}s_{1}}\leqslant C_{1}$. We have

[TABLE]

which implies $\omega(s_{n})\to 0$, $n\to\infty$. At the same time,

[TABLE]

By (2), $\omega(s_{n}^{-1})\to 0$, $n\to\infty$, and this implies (16). ∎

In the case when $G$ is abelian our conditions take an especially simple form:

Corollary 13**.**

Let $G\in\mathbf{\Sigma}$ be abelian, $1\leqslant p<\infty$, and let $\omega$ be a continuous $G$-admissible weight. For any given $S\subset G$, there are $S$-dense vectors in $L^{p}(G,\omega)$ if and only if

[TABLE]

5. Examples and counter-examples

First we show that every group $G\in\mathbf{\Sigma}$ admits a weight with $G$-dense vectors; this completes the proof of Theorem 4.

Proof.

Let $U=U^{-1}$ be a neighbourhood of identity with compact closure and let $G_{1}$ be the subgroup generated by $U$. Pick $g_{n}\in G$, $n\in\mathbb{N}$, such that $G=\cup_{n}(g_{n}G_{1})$; this is possible since $G$ is $\sigma$-compact. Set $U_{1}=U$ and for $n>1$ set by induction

[TABLE]

In particular, we have $U^{n}\subset U_{n}$ and $G=\cup_{n}U_{n}$. By construction, $U_{j}U_{k}\subset U_{j+k}$ for all $j,k$.

Every $U_{n}$ has compact closure, so we can choose $s_{n}\in S$, $n\in\mathbb{N}$, so that $E_{n}=U_{n+1}s_{n}U_{n+1}\cup U_{n+1}s_{n}^{-1}U_{n+1}$ are pairwise disjoint and $U_{n+1}s_{n}U_{n+1}\cap U_{n+1}s_{n}^{-1}U_{n+1}=\emptyset$. Denote $V_{n,k}=U_{k}s_{n}U_{k}\cup U_{k}s_{n}^{-1}U_{k}$ for $1\leqslant k\leqslant n$, $V_{n,k}=G$ for $k>n$ and $V_{n,0}=\{s_{n}\}\cup\{s_{n}^{-1}\}$. We have $U_{j}\,(V_{n,k}\setminus V_{n,k-1})U_{j}\subset V_{n,k+j}\setminus V_{n,k-j-1}$ for all $n,k$. Set

[TABLE]

We have thus $\omega(s_{n})=\omega(s_{n}^{-1})=2^{-n}$ and $\omega(t)\leqslant 1$ for all $t\in G$.

Let $t$ be in $U_{j}$. If $s\notin\cup_{n}V_{n,n}$, then $st\notin\cup_{n;\,k\leqslant n-j-1}V_{n,k}$ and $\omega(st)\geqslant 2^{-j}$. This implies

[TABLE]

Similarly $2^{-j}\leqslant\omega(ts)/\omega(s)\leqslant 1$. If $s\in V_{n,k}\setminus V_{n,k-1}$ with $k\leqslant n$, then $\omega(s)=2^{k-n}$ and $st\in V_{n,k+j}\setminus V_{n,k-j-1}$, thus

[TABLE]

Similarly $2^{-j}\leqslant\omega(ts)/\omega(s)\leqslant 2^{j}$, and we see that these inequalities hold for all $s\in G$, $t\in U$.

It follows that in the notations of (14), $L(t)<+\infty$, $R(t)<+\infty$ for all $t\in U_{j}$, and since $G=\cup_{j}U_{j}$, this is true for all $t\in G$. By construction, the condition (16) holds, so we can conclude that $L^{p}(G,\omega)$ contains an $S$-dense vector. ∎

Let ${\mathbb{F}}_{2}$ be the free discrete group of two generators $a$ and $b$.

Example 14**.**

There exists an ${\mathbb{F}}_{2}$-admissible weight on ${\mathbb{F}}_{2}$ which satisfies the condition (16) but does not admit ${\mathbb{F}}_{2}$-dense vectors.

Set $U=\{e,a,b,a^{-1},b^{-1}\}$, then $G=\cup_{n}U^{n}$. Set $n_{k}=2^{2^{k}}$, $k=1,2,\dots$. Define the weight $\omega$ as follows: $\omega(a^{\pm n_{k}})=2^{-k}$, $\omega(t)=2^{j-k}$ if $t\in(U^{j}\setminus U^{j-1})\,a^{\pm n_{k}}$, $1\leqslant j\leqslant k$, and 1 elsewhere. In particular, $\omega|_{U}=1$.

For $\alpha\in\{a,b,a^{-1},b^{-1}\}$, denote by $F_{\alpha}$ the set of reduced words in ${\mathbb{F}}_{2}$ which end by $\alpha$. Note that $U^{k}a^{\pm n_{k}}\subset F_{a}\cup F_{a^{-1}}$ for all $k$ (since every word in $U^{k}$ has length at most $k<n_{k}$). It follows that $\omega\equiv 1$ on $F_{b}\cup F_{b^{-1}}$.

Obviously $\omega(sg)\leqslant 2\omega(g)$ for $s\in U$ and for all $g\in{\mathbb{F}}_{2}$. It follows that $\omega$ is of moderate growth in the sense of Edwards [6], that is, $\|T_{g}\|<\infty$ for all $g\in{\mathbb{F}}_{2}$. Also, $\inf_{g}\max(\omega(g),\omega(g^{-1}))\leqslant\inf_{k}\max(\omega(a^{n_{k}}),\omega(a^{-n_{k}}))=0$.

Suppose now that there is a ${\mathbb{F}}_{2}$-dense vector $x\in L^{p}(G,\omega)$. For every $0<\varepsilon<1/2$, there exists $g\in{\mathbb{F}}_{2}$ such that $\|I_{U}-{}^{g}\mkern-3.0mux\|_{p,\omega}<\varepsilon$. This implies that

[TABLE]

for $t=e,b$. It follows that $|x(t)|>1/2$ for $t=g^{-1}$ and $t=g^{-1}b$.

By Lemma 1, for the same $\varepsilon$ there is $h\in{\mathbb{F}}_{2}\setminus(Ug\cup Ub^{-1}g)$ such that ${\|{}^{h}x-I_{U}\|_{p,\omega}<\varepsilon}$. This implies that $|x(h^{-1}t)|\,\omega(t)<\varepsilon$ for $t\notin U$. In particular, setting $t=hg^{-1}$ we get: $|x(h^{-1}hg^{-1})|\,\omega(hg^{-1})=|x(g^{-1})|\,\omega(hg^{-1})<\varepsilon$ which implies $\omega(hg^{-1})<2\varepsilon<1$; setting $t=hg^{-1}b$, we infer that $\omega(hg^{-1}b)<1$. By construction of $\omega$, this implies that $hg^{-1},hg^{-1}b\notin F_{b}\cup F_{b^{-1}}$, which is impossible. This contradiction proves that there are no ${\mathbb{F}}_{2}$-dense vectors.

Example 15**.**

There exists an $S$-admissible weight on $G={\mathbb{F}}_{2}$ which admits $S$-dense vectors in $L^{p}(G,\omega)$ for some semigroup $S$ but does not satisfy the sufficient condition (10).

Set $U=\{e,a,b,a^{-1},b^{-1}\}$, then $G=\cup_{n}U^{n}$. Set $n_{k}=2^{2^{k}}$, $k=1,2,\dots$.

For $k\in\mathbb{N}$, set $s_{k}=a^{n_{k}}b$ and let $S$ be the semigroup generated by $\{s_{k}:k\in\mathbb{N}\}$. Set $V_{l,k}=s_{l}s_{k}^{-1}U^{k}$. Every element of $V_{l,k}$, $l\neq k$, has the form $s_{l}s_{k}^{-1}t=a^{n_{l}-n_{k}}t$ with some $t\in U^{k}$; in the reduced form it equals $a^{p}r$ with $|p|\geqslant|n_{l}-n_{k}|-k>0$ and $r\in U^{2k}$ not starting with $a^{\pm 1}$.

Let us show that the sets $SV_{l,k}$, $k\neq l$, are pairwise disjoint. For $s=a^{n_{j_{1}}}b\dots a^{n_{j_{m}}}b$, $s^{\prime}=a^{n_{j^{\prime}_{1}}}b\dots a^{n_{j^{\prime}_{m^{\prime}}}}b$ and $a^{p}r\in V_{l,k}$, $a^{p^{\prime}}r^{\prime}\in V_{l^{\prime},k^{\prime}}$ (the latter two reduced as above) the equality $sa^{p}r=s^{\prime}a^{p^{\prime}}r^{\prime}$ would mean

[TABLE]

and clearly both sides are in their reduced form. We can suppose that $m\leqslant m^{\prime}$. It follows that $j_{i}=j^{\prime}_{i}$ for $i=1,\dots,m$. The equality $p=n_{j_{m+1}}$ is in fact impossible: it would imply $p>0$, then $l>k$; but in this case $p\geqslant n_{l}-n_{k}-k>n_{l}/2>n_{l-1}$ and $p\leqslant n_{l}-n_{k}+k<n_{l}$, so $p$ cannot be equal to $n_{\nu}$ for any $\nu\in\mathbb{N}$. We conclude that $m=m^{\prime}$, $s=s^{\prime}$ and $p=p^{\prime}$.

As discussed above, $p=n_{l}-n_{k}+\xi$ with $|\xi|\leqslant k$, and $p^{\prime}=n_{l^{\prime}}-n_{k^{\prime}}+\xi^{\prime}$ with $|\xi^{\prime}|\leqslant k^{\prime}$. Either $p=p^{\prime}>0$ and then $k>l$, $k^{\prime}>l^{\prime}$, or $p=p^{\prime}<0$ and $k<l$, $k^{\prime}<l^{\prime}$. In the first case we can suppose that $l\geqslant l^{\prime}$. We have $n_{l}=n_{l^{\prime}}+n_{k}-n_{k^{\prime}}+\xi^{\prime}-\xi$; by the choice of $(n_{k})$, an equality is possible only if $l=l^{\prime}$. But then $n_{k}=n_{k^{\prime}}+\xi-\xi^{\prime}$, and for the same reason it follows that $k=k^{\prime}$. In the second case one arrives similarly at the same conclusion. Finally, $l=l^{\prime}$ and $k=k^{\prime}$, which shows that $V_{l,k}$ are pairwise disjoint indeed.

Set now

[TABLE]

By the reasoning above, $\omega$ is well defined. We have

[TABLE]

and this implies that the assumptions of Theorem 5 hold.

It remains to show that $\|T_{s_{n}}\|<\infty$ for all $n$. If $t\notin\cup_{l,k}SV_{l,k}$, then $\omega(t)=1$ and $\omega(s_{n}t)/\omega(t)\leqslant 1$. If $t=ss_{l}s_{k}^{-1}u$ with $s\in S$ and $u\in U^{k}$, then $\omega(s_{n}t)=\omega(s_{n}ss_{l}s_{k}^{-1}u)=\omega(t)$. It follows that $\|T_{s_{n}}\|\leqslant 1$.

Example 16**.**

In general, one cannot have $G_{n}=F_{n}$ in Theorem 5, so that

[TABLE]

This is shown below: there exists a weight $\omega$ on $G=\mathbb{R}^{2}$ which is $S$-admissible and such that the condition (11) is satisfied for $S=\mathbb{R}\times\{0\}$, i.e. $L^{p}(G,\omega)$ contains an $S$-dense vector, but the strengthened condition (18) does not hold.

Denote $\ell_{0}=0$ and $\ell_{n}=\sum_{k=1}^{n}k=n(n+1)/2$ for $n\geqslant 1$. Define $\omega$ as follows: $\omega(-x,y)=\omega(x,y)$, and for $x\geqslant 0$

[TABLE]

This is illustrated as follows:

[TABLE]

For all $t\in[0,1)$,

[TABLE]

it is clear from the picture above that this ratio is always bounded by 4. It follows that $\|T_{(t,0)}\|=4$ for all $t\in[0,1)$, so that $S=\mathbb{R}\times\{0\}$ acts continuously on $L^{p}(\mathbb{R}^{2})$.

Setting $F_{n}=[n,n+1]\times[0,1]$, we have $\int_{F_{n}}\omega^{p}\geqslant 1$ for all $n$, so clearly the condition (18) does not hold.

To check (11), it is sufficient to consider $F=[-n,n]\times[-n,n]$. Choose $t\in\mathbb{N}$ so that

[TABLE]

Set $E=F\setminus([-n,n]\times(1-2^{2-t},1))$. Then $|F\setminus E|=n2^{3-t}<\delta$.

Set $s=(2\ell_{t},0)$. Since $\ell_{t}>t$,

[TABLE]

and

[TABLE]

This implies

[TABLE]

thus (11) holds.

If $S$ generates $G$ and $G$ is abelian, then such an example is impossible, see Proposition 12.

In [2], it was shown on an example that a semigroup indexed by a complex sector can be hypercyclic without any single operator being hypercyclic. We provide a simple example illustrating the same situation:

Example 17**.**

There is a $\mathbb{Z}^{2}$-admissible weight on $\mathbb{Z}^{2}$ with $\mathbb{Z}^{2}$-dense vectors such that no single operator $T_{(m,n)}$ is hypercyclic. The weight is defined as follows: set $\omega(l,m)=2^{k-n}$ if $\max(|l-n|,|m-2^{n}|)=k$ or $\max(|l+n|,|m+2^{n}|)=k$ for some $n\in\mathbb{N}$ and $0\leqslant k\leqslant n$, and $\omega(l,m)=1$ otherwise.

It is easy to check that all the translations $T_{(l,m)}$, $(l,m)\in\mathbb{Z}^{2}$, are bounded. By Corollary 13, the action of $\mathbb{Z}^{2}$ on $L^{p}(\mathbb{Z}^{2})$ for every $1\leqslant p<\infty$ is hypercyclic. But for fixed $l,m$, there are only finitely many points on the line $(l,m)\mathbb{Z}$ in which $\omega$ is different from 1; thus, the action of $S=\{T_{(l,m)}^{n}\}_{n\in\mathbb{N}}$ is not hypercyclic, by Proposition 12.