Covariant functions of characters of compact subgroups

Abstract

This paper presents a systematic study for abstract harmonic analysis on classical Banach spaces of covariant functions of characters of compact subgroups. Let G be a locally compact group and H be a compact subgroup of G. Suppose that $\xi :H\rightarrow \mathbb {T}$ is a character, $1\le p<\infty$ and $L_\xi ^p(G,H)$ is the set of all covariant functions of $\xi$ in $L^p(G)$. It is shown that $L^p_\xi (G,H)$ is isometrically isomorphic to a quotient space of $L^p(G)$. It is also proven that $L^q_\xi (G,H)$ is isometrically isomorphic to the dual space $L^p_\xi (G,H)^*$, where q is the conjugate exponent of p. The paper is concluded by some results for the case that G is compact.

1 Introduction

The Banach spaces consist of covariant functions on locally compact groups associated to characters (one-dimensional continuous irreducible unitary representations) of closed subgroups appear in variant mathematical areas and their applications including calculus of pseudo-differential operators, number theory (automorphic forms), induced representations, homogeneous spaces, complex (hypercomplex) analysis, theoretical aspects of mathematical physics including coherent states and covariant analysis, see [1, 4, 15,16,17,18,19,20,21, 23].

The following paper presents a unified operator theoretic approach to study abstract harmonic analysis of $L^p$-spaces of covariant functions of characters of compact subgroups in locally compact groups. The introduced approach canonically extends classical methods of abstract harmonic analysis and functional analysis on coset spaces (homogeneous spaces) of compact subgroups by assuming the character to be the trivial character of the compact subgroup, see [2, 6, 10,11,12, 22].

This article contains 4 sections and organized as follows. Section 2 is devoted to fix notations and provides a summary of classical harmonic analysis on locally compact groups and covariant functions of characters of closed subgroups. Let G be a locally compact group, H be a compact subgroup of G, and $1\le p<\infty$. Suppose that $\xi :H\rightarrow \mathbb {T}$ is a fixed character of H. In Sect. 3, we study some of the operator theoretic aspects of covariant functions on G associated to the character $\xi$ of the subgroup H. Next section investigates harmonic analysis foundations on the Banach space $L^p_\xi (G,H)$, the space of covariant functions of the character $\xi$ in $L^p(G)$. In this direction, we study fundamental properties of classical Banach spaces of covariant functions of characters of compact subgroups. It is shown that $L^p_\xi (G,H)$ is isometrically isomorphic to a quotient space of $L^p(G)$. We then proved that $L^q_\xi (G,H)$ is isometrically isomorphic to the dual space $L^p_\xi (G,H)^*$, where q is the conjugate exponent of p. Finally, we conclude the paper by some related results for the case that G is compact.

2 Preliminaries and notations

Let X be a locally compact Hausdorff space. Then, $\mathcal {C}_c(X)$ denotes the space of all continuous complex valued functions on X with compact support. If $\uplambda$ is a positive Radon measure on X, for each $1\le p<\infty$ the Banach space of equivalence classes of $\uplambda$-measurable complex valued functions $f:X\rightarrow \mathbb {C}$ such that

$$\begin{aligned} \Vert f\Vert _{L^p(X,\uplambda )}:=\left( \int _X|f(x)|^p\mathrm {d}\uplambda (x)\right) ^{1/p}<\infty , \end{aligned}$$

is denoted by $L^p(X,\uplambda )$ which contains $\mathcal {C}_c(X)$ as a $\Vert \cdot \Vert _{L^p(X,\uplambda )}$-dense subspace.

Let G be a locally compact group with the modular function $\Delta _G$ and a fixed left Haar measure $\uplambda _{G}$. For a function $f:G\rightarrow \mathbb {C}$ and $x\in G$, the functions $L_xf,R_xf:G\rightarrow \mathbb {C}$ are given by $L_xf(y):=f(x^{-1}y)$ and $R_xf(y):=f(yx)$ for $y\in G$. For $1\le p<\infty$, $L^p(G)$ stands for the Banach space $L^p(G,\uplambda _G)$. The convolution for $f,g\in L^1(G)$, is defined via

$$\begin{aligned} f*_G g(x):=\int _Gf(y)g(y^{-1}x)\mathrm {d}\uplambda _G(y)\quad (x\in G). \end{aligned}$$

(2.1)

We then have $f*_G g\in L^p(G)$ with $\Vert f*_G g\Vert _{L^p(G)}\le \Vert f\Vert _{L^1(G)}\Vert g\Vert _{L^p(G)}$, if $f\in L^1(G)$ and $g\in L^p(G)$. It is well known as a classical result in abstract harmonic analysis that the Banach function space $L^1(G)$ is a Banach $*$-algebra with respect to the bilinear product $*_G:L^1(G)\times L^1(G)\rightarrow L^1(G)$ given by $(f,g)\mapsto f*_G g$, with $f*_G g$ defined by (2.1) and involution given by $f\mapsto f^{*_G}$ where $f^{*^G}(x):=\Delta _G(x^{-1})\overline{f(x^{-1})}$ for $x\in G$. In addition, for each $p>1$, the Banach function space $L^p(G)$ is a Banach left $L^1(G)$-module equipped with the left module action $*_G:L^1(G)\times L^p(G)\rightarrow L^p(G)$ given by $(f,g)\mapsto f*_G g$, with $f*g$ defined by (2.1), see [3, 5, 13, 14, 24] and the classical list of references therein.

Suppose that H is a closed subgroup of G. A character $\xi$ of H, is a continuous group homomorphism $\xi :H\rightarrow \mathbb {T}$, where $\mathbb {T}:=\{z\in \mathbb {C}:|z|=1\}$ is the circle group. In terms of group representation theory, each character of H is a 1-dimensional irreducible continuous unitary representation of H. We then denote the set of all characters of H by $\chi (H)$.

Let G be a locally compact group, H be a closed subgroup of G, and $\xi \in \chi (H)$. A function $\psi :G\rightarrow \mathbb {C}$ satisfies covariant property associated to the character $\xi$, if

$$\begin{aligned} \psi (xs)=\xi (s)\psi (x), \end{aligned}$$

(2.2)

for every $x\in G$ and $s\in H$. In this case, $\psi$ is called a covariant function of $\xi$. The covariant functions appear in abstract harmonic analysis in the construction of induced representations, see [5, 15]. We here employ some of the classical tools in this direction. Suppose that $\uplambda _H$ is a left Haar measure on H. For each character $\xi \in \chi (H)$ and a function $f\in \mathcal {C}_c(G)$, define the function $T_\xi (f):G\rightarrow \mathbb {C}$ via

$$\begin{aligned} T_\xi (f)(x):=\int _Hf(xs)\overline{\xi (s)}\mathrm {d}\uplambda _H(s), \end{aligned}$$

for every $x\in G$. Suppose $\mathcal {C}_\xi (G,H)$ is the linear subspace of $\mathcal {C}(G)$ given by

$$\begin{aligned} \mathcal {C}_\xi (G,H):=\{\psi \in \mathcal {C}_c(G|H):\psi (xh)=\xi (h)\psi (x),\ \mathrm{for\ all}\ x\in G,\ h\in H\}, \end{aligned}$$

where

$$\begin{aligned} \mathcal {C}_c(G|H):=\{\psi \in \mathcal {C}(G):\mathrm {q}(\mathrm{supp}(\psi ))\ \mathrm{is\ compact\ in}\ G/H\}, \end{aligned}$$

and $\mathrm {q}:G\rightarrow G/H$ is the canonical map given by $\mathrm {q}(x):=xH$ for $x\in G$. Using continuity of the canonical map $\mathrm {q}:G\rightarrow G/H$, one can deduce that $\mathcal {C}_c(G)\subseteq \mathcal {C}_c(G|H)$. It is shown that the linear operator $T_\xi$ maps $\mathcal {C}_c(G)$ onto $\mathcal {C}_\xi (G,H)$, see Proposition 6.1 of [5].

3 Covariant functions of characters of compact subgroups

In this section, we study some of the fundamental theoretical aspects of covariant functions of characters of compact subgroups in locally compact groups. Throughout, let G be a locally compact group, H be a compact subgroup of G, and $\xi \in \chi (H)$ be a fixed character.

Proposition 3.1

Let G be a locally compact group and H be a compact subgroup of G. Suppose that $\xi \in \chi (H)$ is a character and $\uplambda _H$ is the probability Haar measure of H. Then,

• 1. $\mathcal {C}_\xi (G,H)\subseteq \mathcal {C}_c(G)$.

• 2. $T_\xi \circ T_\xi =T_\xi$ on $\mathcal {C}_c(G)$.

Proof

(1) Let H be a compact subgroup of G. We then have $\mathcal {C}_c(G)=\mathcal {C}_c(G|H)$. To see this, let $\psi \in \mathcal {C}_c(G|H)$ be given. Then $\mathrm {q}(\mathrm{supp}(\psi ))$ is compact in G/H. Using Lemma 2.46 of [5], there exists a compact subset F of G such that $\mathrm {q}(F)=\mathrm {q}(\mathrm{supp}(\psi ))$. This implies that $\mathrm {supp}(\psi )\subseteq FH$. Since H is compact, we conclude that $\mathrm {supp}(\psi )$ is compact in G and so $\psi \in \mathcal {C}_c(G)$. Therefore, we get $\mathcal {C}_\xi (G,H)\subseteq \mathcal {C}_c(G)$.

(2) Let $f\in \mathcal {C}_c(G)$ be given. Using (1), we have $T_\xi (f)\in \mathcal {C}_c(G)$. Suppose $x\in G$. Since $\uplambda _H$ is a probability measure, we get

$$\begin{aligned} T_\xi (T_\xi (f))(x)&=\int _HT_\xi (f)(xs)\overline{\xi (s)}\mathrm {d}\uplambda _H(s)\\&=\int _HT_\xi (f)(x)\xi (s)\overline{\xi (s)}\mathrm {d}\uplambda _H(s)\\&=T_\xi (f)(x)\int _H|\xi (s)|^2\mathrm {d}\uplambda _H(s)\\&=T_\xi (f)(x)\left( \int _H\mathrm {d}\uplambda _H(s)\right) =T_\xi (f)(x). \end{aligned}$$

$\square$

Invoking Proposition 3.1, if H is compact, one can regard the linear map $T_\xi :\mathcal {C}_c(G)\rightarrow \mathcal {C}_c(G)$ as a projection of the linear space $\mathcal {C}_c(G)$ onto the subspace $\mathcal {C}_\xi (G,H)\subseteq \mathcal {C}_c(G)$.

Remark 3.2

If $\xi =1$ is the trivial character of H, we then have $T_1(f)=T_H(f)$, see formula (3) of [9]. Then, $\mathcal {C}_1(G,H)$ consists of functions on G which are constant on cosets of H. Therefore, $\mathcal {C}_1(G,H)$ can be canonically identified with $\mathcal {C}_c(G/H)$ via the isometric identification $\psi \mapsto \widetilde{\psi }$, where $\widetilde{\psi }:G/H\rightarrow \mathbb {C}$ is given by $\widetilde{\psi }(xH):=\psi (x)$ for every $x\in G$, see Corollary 3.4 of [8]. In this case, harmonic analysis on $\mathcal {C}_1(G,H)$ studied from different perspectives in [6, 7, 9, 12, 22].

We then have the following observations.

Proposition 3.3

Let G be a locally compact group, H be a compact subgroup of G, and $\xi \in \chi (H)$ be a character. Suppose $y\in G$ and $h\in H$. Then

• 1. $T_\xi \circ R_h=\xi (h)T_\xi$ on $\mathcal {C}_c(G)$.

• 2. $T_\xi \circ L_y=L_y\circ T_\xi$ on $\mathcal {C}_c(G)$.

Proof

(1) Suppose $f\in \mathcal {C}_c(G)$ and $h\in H$. Let $x\in G$ be given. We then have

$$\begin{aligned} T_\xi (R_hf)(x)&=\int _HR_hf(xs)\overline{\xi (s)}\mathrm {d}\uplambda _H(s)\\&=\int _Hf(xsh)\overline{\xi (s)}\mathrm {d}\uplambda _H(s)\\&=\int _Hf(xs)\overline{\xi (sh^{-1})}\mathrm {d}\uplambda _H(sh^{-1})\\&=\xi (h)\int _Hf(xs)\overline{\xi (s)}\mathrm {d}\uplambda _H(s)=\xi (h)T_\xi (f)(x), \end{aligned}$$

which implies that $T_\xi (R_hf)=\xi (h)T_\xi (f)$.

(2) Suppose $f\in \mathcal {C}_c(G)$ and $y\in G$. Let $x\in G$ be given. We then have

$$\begin{aligned} T_\xi (L_yf)(x)&=\int _HL_yf(xs)\overline{\xi (s)}\mathrm {d}\uplambda _H(s)\\&=\int _Hf(y^{-1}xs)\overline{\xi (s)}\mathrm {d}\uplambda _H(s)=L_y(T_\xi (f))(x), \end{aligned}$$

implying that $T_\xi (L_yf)=L_y(T_\xi (f))$. $\square$

For functions $f,g\in \mathcal {C}_c(G)$, the function $f\overline{g}$ is continuous with compact support. Therefore, it is integrable over G with respect to every left Haar measure $\uplambda _G$ on G. We denote the later integral by $\langle f,g\rangle$.

Theorem 3.4

Let G be a locally compact group and H be a compact subgroup of G. Suppose $\xi \in \chi (H)$ is a character and $f,g\in \mathcal {C}_c(G)$. Then

$$\begin{aligned} \langle T_\xi (f),g\rangle =\langle f,T_\xi (g)\rangle . \end{aligned}$$

(3.1)

Proof

Let $f,g\in \mathcal {C}_c(G)$ be given. We then have

$$\begin{aligned} \langle T_\xi (f),g\rangle&=\int _GT_\xi (f)(x)\overline{g(x)}\mathrm {d}\uplambda _G(x)\\&=\int _G\left( \int _Hf(xs)\overline{\xi (s)}\mathrm {d}\uplambda _H(s)\right) \overline{g(x)}\mathrm {d}\uplambda _G(x)\\&=\int _H\left( \int _Gf(xs)\overline{g(x)}\mathrm {d}\uplambda _G(x)\right) \overline{\xi (s)}\mathrm {d}\uplambda _H(s)\\&=\int _H\left( \int _Gf(x)\overline{g(xs^{-1})}\mathrm {d}\uplambda _G(xs^{-1})\right) \overline{\xi (s)}\mathrm {d}\uplambda _H(s)\\&=\int _H\Delta _G(s^{-1})\left( \int _Gf(x)\overline{g(xs^{-1})}\mathrm {d}\uplambda _G(x)\right) \overline{\xi (s)}\mathrm {d}\uplambda _H(s). \end{aligned}$$

Since H is compact in G, we have $\Delta _G(s)=1$ for all $s\in H$. Therefore, we get

$$\begin{aligned} \langle T_\xi (f),g\rangle&=\int _H\left( \int _Gf(x)\overline{g(xs^{-1})}\mathrm {d}\uplambda _G(x)\right) \overline{\xi (s)}\mathrm {d}\uplambda _H(s)\\&=\int _Gf(x)\left( \int _H\overline{g(xs^{-1})}\overline{\xi (s)}\mathrm {d}\uplambda _H(s)\right) \mathrm {d}\uplambda _G(x)\\&=\int _Gf(x)\left( \int _H\overline{g(xs)}\xi (s)\mathrm {d}\uplambda _H(s^{-1})\right) \mathrm {d}\uplambda _G(x)\\&=\int _Gf(x)\left( \int _H\overline{g(xs)}\xi (s)\mathrm {d}\uplambda _H(s)\right) \mathrm {d}\uplambda _G(x)=\langle f,T_\xi (g)\rangle . \end{aligned}$$

$\square$

We then conclude this section by the following convolution property of the linear map $T_\xi$.

Theorem 3.5

Let G be a locally compact group, H be a compact subgroup of G, and $\xi \in \chi (H)$ be a character. Suppose $f,g\in \mathcal {C}_c(G)$ are given. Then

$$\begin{aligned} T_\xi (f*_G g)=f*_G T_\xi (g). \end{aligned}$$

Proof

Suppose that $f,g\in \mathcal {C}_c(G)$ and $x\in G$ are given. We then have

$$\begin{aligned} T_\xi (f*_G g)(x)&=\int _H f*_G g(xs)\overline{\xi (s)}\mathrm {d}\uplambda _H(s)\\&=\int _H \left( \int _Gf(y)g(y^{-1}xs)\mathrm {d}\uplambda _G(y)\right) \overline{\xi (s)}\mathrm {d}\uplambda _H(s)\\&=\int _Gf(y)\left( \int _H g(y^{-1}xs)\overline{\xi (s)}\mathrm {d}\uplambda _H(s)\right) \mathrm {d}\uplambda _G(y)\\&=\int _Gf(y)T_\xi (g)(y^{-1}x)\mathrm {d}\uplambda _G(y)=f*_G T_\xi (g)(x). \end{aligned}$$

$\square$

4 $L^p$-spaces of covariant functions of characters of compact subgroups

In this section, we study $L^p$-spaces of covariant functions of characters of compact subgroups. We then investigate some of the basic properties of these classical Banach spaces of covariant functions of characters of compact subgroups on locally compact groups. Throughout, suppose that G is a locally compact group, H is a compact subgroup of G, and $\xi \in \chi (H)$. Let $\uplambda _{G}$ be a left Haar measure on G and $\uplambda _H$ be a Haar measure on H.

The following theorem proves the boundedness property of the linear map $T_\xi$ in the sense of $L^p(G)$.

Theorem 4.1

Let G be a locally compact group and $1\le p<\infty$. Suppose H is a compact subgroup of G and $\xi \in \chi (H)$. The linear operator

$$\begin{aligned} T_\xi :(\mathcal {C}_c(G),\Vert \cdot \Vert _{L^p(G)})\rightarrow (\mathcal {C}_\xi (G,H),\Vert \cdot \Vert _{L^p(G)}) \end{aligned}$$

is bounded. In particular, if $\uplambda _H$ is the probability Haar measure of H then the linear operator $T_\xi :(\mathcal {C}_c(G),\Vert \cdot \Vert _{L^p(G)})\rightarrow (\mathcal {C}_\xi (G,H),\Vert \cdot \Vert _{L^p(G)})$ is a contraction.

Proof

Let $f\in \mathcal {C}_c(G)$ be given. Since H is compact and $\uplambda _H(H)<\infty$, we have

$$\begin{aligned} \left( \int _H|f(ys)|\mathrm {d}\uplambda _H(s)\right) ^p\le \uplambda _H(H)^{p-1}\int _H|f(ys)|^p\mathrm {d}\uplambda _H(s), \end{aligned}$$

(4.1)

for every $y\in G$. Using compactness of H, we get $\Delta _G(s)=1$ for all $s\in H$. Thus, using (4.1), we obtain

$$\begin{aligned} \Vert T_\xi (f)\Vert _{L^p(G)}^p&=\int _G\left| \int _Hf(xs)\overline{\xi (s)}\mathrm {d}\uplambda _H(s)\right| ^p\mathrm {d}\uplambda _G(x)\\&\le \int _G\left( \int _H|f(xs)|\mathrm {d}\uplambda _H(s)\right) ^p\mathrm {d}\uplambda _G(x)\\&\le \uplambda _H(H)^{p-1}\int _G\int _H|f(xs)|^p\mathrm {d}\uplambda _H(s)\mathrm {d}\uplambda _G(x)\\&=\uplambda _H(H)^{p-1}\int _H\int _G|f(xs)|^p\mathrm {d}\uplambda _G(x)\mathrm {d}\uplambda _H(s)\\&=\uplambda _H(H)^{p-1}\int _H\int _G|f(x)|^p\mathrm {d}\uplambda _G(xs^{-1})\mathrm {d}\uplambda _H(s)\\&=\uplambda _H(H)^{p-1}\Vert f\Vert _{L^p(G)}^p\int _H\Delta _G(s^{-1})\mathrm {d}\uplambda _H(s) =\uplambda _H(H)^{p}\Vert f\Vert _{L^p(G)}^p, \end{aligned}$$

which guarantees that $\Vert T_\xi (f)\Vert _{L^p(G)}\le \uplambda _H(H)\Vert f\Vert _{L^p(G)}$. Since $f\in \mathcal {C}_c(G)$ was arbitrary, we conclude that the linear map $T_\xi :(\mathcal {C}_c(G),\Vert \cdot \Vert _{L^p(G)})\rightarrow (\mathcal {C}_\xi (G,H),\Vert \cdot \Vert _{L^p(G)})$ is bounded with $\Vert T_\xi \Vert \le \uplambda _H(H)$. In particular, if $\uplambda _H$ is the probability Haar measure of H then the operator norm $\Vert T_\xi \Vert \le 1$. So the linear map $T_\xi :(\mathcal {C}_c(G),\Vert \cdot \Vert _{L^p(G)})\rightarrow (\mathcal {C}_\xi (G,H),\Vert \cdot \Vert _{L^p(G)})$ is a contraction. $\square$

Corollary 4.2

Let G be a compact group and $1\le p<\infty$. Suppose H is a closed subgroup of G and $\xi \in \chi (H)$. The linear operator

$$\begin{aligned} T_\xi :(\mathcal {C}_c(G),\Vert \cdot \Vert _{L^p(G)})\rightarrow (\mathcal {C}_\xi (G,H),\Vert \cdot \Vert _{L^p(G)}), \end{aligned}$$

is bounded. In particular, if $\uplambda _H$ is the probability Haar measure on H then the linear operator $T_\xi :(\mathcal {C}_c(G),\Vert \cdot \Vert _{L^p(G)})\rightarrow (\mathcal {C}_\xi (G,H),\Vert \cdot \Vert _{L^p(G)})$ is a contraction.

Remark 4.3

If $\xi =1$ is the trivial character then Theorem 4.1 coincides with Proposition 3.4 of [8], if $\mathcal {C}_1(G,H)$ is identified with $\mathcal {C}_c(G/H)$.

From now on, we assume that $\uplambda _H$ is the probability Haar measure on H and hence the linear operator $T_\xi :(\mathcal {C}_c(G),\Vert \cdot \Vert _{L^p(G)})\rightarrow (\mathcal {C}_\xi (G,H),\Vert \cdot \Vert _{L^p(G)})$ is a contraction.

We then conclude the following property of $\Vert \cdot \Vert _{L^p(G)}$.

Proposition 4.4

Let G be a locally compact group and H be a compact subgroup of G. Suppose $1\le p<\infty$, $\xi \in \chi (H)$ and $\psi \in \mathcal {C}_\xi (G,H)$. Then,

$$\begin{aligned} \Vert \psi \Vert _{L^p(G)}=\inf \left\{ \Vert f\Vert _{L^p(G)}:f\in \mathcal {C}_c(G), \ T_\xi (f)=\psi \right\} . \end{aligned}$$

Proof

Let $\psi \in \mathcal {C}_\xi (G,H)$ be given. Suppose $\mathcal {B}_\psi :=\{f\in \mathcal {C}_c(G): T_\xi (f)=\psi \}$ and $\gamma _\psi :=\inf \left\{ \Vert f\Vert _{L^p(G)}:f\in \mathcal {B}_\psi \right\}$. Using Theorem 4.1, for each $f\in \mathcal {B}_\psi$, we have $\Vert \psi \Vert _{L^p(G)}\le \Vert f\Vert _{L^p(G)}$. Hence, we get $\Vert \psi \Vert _{L^p(G)}\le \gamma _\psi$. Now, we claim that $\Vert \psi \Vert _{L^p(G)}\ge \gamma _\psi$ as well. To this end, since H is compact, using Proposition 3.1, we have $\psi \in \mathcal {C}_c(G)$ and $T_\xi (\psi )=\psi$. Therefore, $\psi \in \mathcal {B}_\psi$ and hence we get $\Vert \psi \Vert _{L^1(G)}\ge \gamma _\psi$, which completes the proof. $\square$

Remark 4.5

Let $\xi =1$ be the trivial character of H. Then Proposition 4.4 is a consequence of Proposition 3.4 and Corollary 3.4 of [8], if $\mathcal {C}_1(G,H)$ is identified with $\mathcal {C}_c(G/H)$.

Let $\mathcal {N}_\xi =\mathcal {N}_\xi (G,H)$ be the kernel of the linear map $T_\xi$ in $\mathcal {C}_c(G)$, that is the linear subspace given by

$$\begin{aligned} \mathcal {N}_\xi (G,H):=\{f\in \mathcal {C}_c(G):T_\xi (f)=0\}. \end{aligned}$$

Then, $\mathcal {N}_\xi (G,H)$ is a closed linear subspace of $(\mathcal {C}_c(G),\Vert \cdot \Vert _{L^p(G)})$ for every $1\le p<\infty$. Using Proposition 3.3(1), we also have

$$\begin{aligned} \mathrm {span}\{R_hf-\xi (h)f:h\in H,\ f\in \mathcal {C}_c(G)\}\subseteq \mathcal {N}_\xi (G,H). \end{aligned}$$

(4.2)

For every $1\le p<\infty$, suppose that $\mathfrak {X}_\xi (G,H):=\mathcal {C}_c(G)/\mathcal {N}_\xi (G,H)$ is the quotient normed space of $\mathcal {N}_\xi (G,H)$ in $\mathcal {C}_c(G)$, that is

$$\begin{aligned} \mathcal {C}_c(G)/\mathcal {N}_\xi (G,H)=\left\{ f+\mathcal {N}_\xi :f\in \mathcal {C}_c(G)\right\} , \end{aligned}$$

with the quotient norm given by

$$\begin{aligned} \Vert f+\mathcal {N}_\xi \Vert _{[p]}:=\inf \left\{ \Vert f+g\Vert _{L^p(G)}:g\in \mathcal {N}_\xi \right\} . \end{aligned}$$

(4.3)

Also, let $\mathfrak {X}_\xi ^p(G,H)$ be the Banach completion of the normed linear space $\mathfrak {X}_\xi (G,H)$ with respect to the quotient norm $\Vert \cdot \Vert _{[p]}$. We may denote $\mathfrak {X}_\xi (G,H)$ by $\mathfrak {X}_\xi$ and $\mathfrak {X}^p_\xi (G,H)$ by $\mathfrak {X}_\xi ^p$ at times.

We also denote by $L^p_\xi (G,H)$ the Banach completion of the normed linear space $\mathcal {C}_\xi (G,H)$ with respect to $\Vert \cdot \Vert _{L^p(G)}$. We shall use the completion norm by $\Vert \cdot \Vert _{L^p(G)}$ or just $\Vert \cdot \Vert _{p}$ as well.

It is then clear that $L^p_\xi (G,H)\subseteq L^p(G)$. In addition, one can see that

$$\begin{aligned} L^p_\xi (G,H)=\{\psi \in L^p(G):R_h\psi =\xi (h)\psi ,\ \mathrm{for}\ h\in H\}. \end{aligned}$$

Next we conclude the following characterization of $L^p_\xi (G,H)$.

Theorem 4.6

Let G be a locally compact group and H be a compact subgroup of G. Suppose $1\le p<\infty$ and $\xi \in \chi (H)$. Then, $\mathfrak {X}^p_\xi (G,H)$ is isometrically isomorphic to the Banach space $L^p_\xi (G,H)$.

Proof

Invoking the structure of the spaces $\mathfrak {X}^p_\xi (G,H)$ and $L^p_\xi (G,H)$, it is enough to show that $(\mathcal {C}_\xi (G,H),\Vert \cdot \Vert _{L^p(G)})$ is isometrically isomorphic with the quotient normed space $(\mathfrak {X}_\xi (G,H),\Vert \cdot \Vert _{[p]})$. Since $T_\xi :\mathcal {C}_c(G)\rightarrow \mathcal {C}_\xi (G,H)$ is a surjective linear map, we conclude that the linear space $\mathcal {C}_\xi (G,H)$ is isomorphic to the quotient linear space $\mathfrak {X}_\xi (G,H)$ via the canonical linear map $U_\xi :\mathfrak {X}_\xi (G,H)\rightarrow \mathcal {C}_\xi (G,H)$ given by $U_\xi (f+\mathcal {N}_\xi ):=T_\xi (f)$ for every $f\in \mathcal {C}_c(G)$. Further, the isomorphism $U_\xi$ is not only algebraic, but also isometric, if the quotient linear space $\mathfrak {X}_\xi (G,H)$ is equipped with the classical quotient norm (4.3). Using Proposition 4.4, for $f\in \mathcal {C}_c(G)$, we have

$$\begin{aligned} \Vert U_\xi (f+\mathcal {N}_\xi )\Vert _{L^p(G)}&=\Vert T_\xi (f)\Vert _{L^p(G)}\\&=\inf \left\{ \Vert h\Vert _{L^p(G)}:T_\xi (h)=T_\xi (f)\right\} \\&=\inf \left\{ \Vert h\Vert _{L^p(G)}:h-f\in \mathcal {N}_\xi \right\} \\&=\inf \left\{ \Vert f+g\Vert _{L^p(G)}:g\in \mathcal {N}_\xi \right\} =\Vert f+\mathcal {N}_\xi \Vert _{[p]}. \end{aligned}$$

$\square$

Invoking Theorem 4.1, one can conclude that if H is a compact subgroup of G then the bounded linear operator

$$\begin{aligned} T_\xi :(\mathcal {C}_c(G),\Vert \cdot \Vert _{L^p(G)})\rightarrow (\mathcal {C}_\xi (G,H),\Vert \cdot \Vert _{L^p(G)}) \end{aligned}$$

has a unique extension to a bounded linear map from $L^p(G)$ onto $L^p_\xi (G,H)$, which we still denote it by $T_\xi$. Then $T_\xi \circ R_h=\xi (h)T_\xi$ and $T_\xi \circ L_y=L_y\circ T_\xi$, for every $y\in G$ and $h\in H$. It is easy to see that the extended map $T_\xi :L^p(G)\rightarrow L^p_\xi (G,H)$ is given by $f\mapsto T_\xi (f)$, where

$$\begin{aligned} T_\xi (f)(x)=\int _Hf(xs)\overline{\xi (s)}\mathrm {d}\uplambda _H(s),\ \ \mathrm{for\ a.e.}\ x\in G. \end{aligned}$$

In particular, if $\uplambda _H$ is the probability Haar measure of H, the extended linear operator $T_\xi :L^p(G)\rightarrow L^p_\xi (G,H)$ is a contraction.

Invoking the structure of $L^p_\xi (G,H)$ and since $L^p(G)$ is a Banach $L^1(G)$-module, we get that $L^p_\xi (G,H)$ is a Banach $L^1(G)$-submodule of $L^p(G)$ as well. In particular, we conclude that $L^1_\xi (G,H)$ is a closed left ideal in $L^1(G)$.

We then prove the following multiplier property concerning convolution structure of $L^p_\xi (G,H)$ in terms of the linear operator $T_\xi$.

Proposition 4.7

Let G be a locally compact group and H be a compact subgroup of G. Suppose $\xi \in \chi (H)$ and $1\le p<\infty$. Then, $T_\xi :L^p(G)\rightarrow L^p(G)$ is a $L^1(G)$-multiplier.

Proof

Let $f\in \mathcal {C}_c(G)$ and $g\in L^p(G)$. Suppose $(g_n)\subset \mathcal {C}_c(G)$ with $g=\lim _n g_n$ in $L^p(G)$. Then $f*_G g=\lim _nf*_Gg_n$ in $L^p(G)$. Using boundedness of the linear operator $T_\xi :L^p(G)\rightarrow L^p(G)$, continuity of the module action in each argument, and Theorem 3.5, we get

$$\begin{aligned} T_\xi (f*_G g)&=T_\xi (\lim _nf*_Gg_n)\\&=\lim _nT_\xi (f*_Gg_n)\\&=\lim _nf*_GT_\xi (g_n)=f*_GT_\xi (g). \end{aligned}$$

Using a similar method, we get $T_\xi (f*_G g)=f*_GT_\xi (g)$, if $f\in L^1(G)$. $\square$

For $1\le p<\infty$, let $\mathcal {N}_\xi ^p(G,H)$ be the kernel of extension of the linear map $T_\xi$ in $L^p(G)$, that is the linear subspace given by

$$\begin{aligned} \mathcal {N}_\xi ^p(G,H):=\left\{ f\in L^p(G):T_\xi (f)=0\ \mathrm{in}\ L^p_\xi (G,H)\right\} . \end{aligned}$$

Proposition 4.8

Let G be a locally compact group and H be a compact subgroup of G. Suppose $1\le p<\infty$ and $\xi \in \chi (H)$. Then, $\mathcal {N}_\xi ^p(G,H)$ is the closure of $\mathcal {N}_\xi (G,H)$ in $L^p(G)$.

Proof

Let $\mathcal {X}$ be the closure of $\mathcal {N}_\xi (G,H)$ in $L^p(G)$. Suppose $f\in \mathcal {X}$ is given. Then, we have $f\in L^p(G)$ and $\lim _nf_n=f$ for some sequence $(f_n)\subset \mathcal {N}_\xi (G,H)$. Continuity of the extended $T_\xi :L^p(G)\rightarrow L^p_\xi (G,H)$ implies that $T_\xi (f)=0$ in $L^p_\xi (G,H)$. Therefore, we get $f\in \mathcal {N}_\xi ^p(G,H)$. Since f was arbitrary, we deduce that $\mathcal {X}\subseteq \mathcal {N}_\xi ^p(G,H)$. Conversely, let $f\in \mathcal {N}^p(G,H)$ be given. Then, $f\in L^p(G)$ with $T_\xi (f)=0$ in $L^p_\xi (G,H)$. Suppose that $\varepsilon >0$ is given. Pick $h\in \mathcal {C}_c(G)$ with $\Vert f-h\Vert _{L^p(G)}<\varepsilon /2$. Let $\phi :=T_\xi (h)$. We then define the function $g:G\rightarrow \mathbb {C}$ by $g(x):=h(x)-\phi (x)$ for every $x\in G$. Then, we have $g\in \mathcal {C}_c(G)$ and $T_\xi (g)=0$. Indeed, for $x\in G$, we have

$$\begin{aligned} T_\xi (g)(x)=\phi (x)-\int _H\phi (xs)\overline{\xi (s)}\mathrm{d}\uplambda _H(s)=\phi (x)-\phi (x)=0. \end{aligned}$$

Hence, we get $g\in \mathcal {N}_\xi (G,H)$. In addition, we have

$$\begin{aligned} \Vert h-g\Vert _{L^p(G)}&=\Vert \phi \Vert _{L^p(G)}=\Vert T_\xi (h)\Vert _{L^p(G)}\\&\le \Vert T_\xi (h-f)\Vert _{L^p(G)}+\Vert T_\xi (f)\Vert _{L^p(G)}\le \Vert h-f\Vert _{L^p(G)}<\frac{\varepsilon }{2}. \end{aligned}$$

Therefore, we achieve

$$\begin{aligned} \Vert f-g\Vert _{L^p(G)}\le \Vert f-h\Vert _{L^p(G)}+\Vert h-g\Vert _{L^p(G)}<\varepsilon , \end{aligned}$$

which implies that f is in the closure of $\mathcal {N}_\xi (G,H)$ in $L^p(G)$, that is $f\in \mathcal {X}$. Since $f\in \mathcal {N}_\xi ^p(G,H)$ was given, we conclude that $\mathcal {N}_\xi ^p(G,H)\subseteq \mathcal {X}$ as well. $\square$

We then have the following interesting characterization of the Banach space $L_\xi ^p(G,H)$ as a quotient space of $L^p(G)$.

Theorem 4.9

Let G be a locally compact group and H be a compact subgroup of G. Suppose $1\le p<\infty$ and $\xi \in \chi (H)$. The Banach space $L_\xi ^p(G,H)$ is isometrically isomorphic to the quotient Banach space $L^p(G)/\mathcal {N}_\xi ^p(G,H)$.

Proof

Applying Proposition 4.8 and Lemma 3.4.4 of [24] we deduce that the Banach space $\mathfrak {X}_\xi ^p(G,H)$ is isometrically isomorphic to the quotient Banach space $L^p(G)/\mathcal {N}_\xi ^p(G,H)$. Then, Theorem 4.6 implies that the Banach space $L_\xi ^p(G,H)$ is isometrically isomorphic to the quotient Banach space $L^p(G)/\mathcal {N}_\xi ^p(G,H)$. $\square$

We continue by proving results concerning adjoint of the linear map $T_\xi :L^p(G)\rightarrow L^p(G)$, when H is compact.

Proposition 4.10

Let G be a locally compact group and H be a compact subgroup of G. Suppose $\xi \in \chi (H)$, and $1<p,q<\infty$ with $p^{-1}+q^{-1}=1$. The adjoint of the bounded linear map $T_\xi :L^p(G)\rightarrow L^p(G)$ can be identified by $T_\xi :L^q(G)\rightarrow L^q(G)$.

Proof

Let $g\mapsto \Lambda _g$ be the canonical antilinear isometric isomorphism identification of $L^q(G)$ as $L^p(G)^*$, where the bounded linear functional $\Lambda _g:L^p(G)\rightarrow \mathbb {C}$ is given by $\Lambda _g(f):=\langle f,g\rangle$ for every $f\in L^p(G)$. Suppose that $f,g\in \mathcal {C}_c(G)$ are given. Invoking the abstract structure of the adjoint linear map $T_\xi ^*:L^p(G)^*\rightarrow L^p(G)^*$, and using (3.1), we get

$$\begin{aligned} T_\xi ^*(\Lambda _g)(f)=\Lambda _g(T_\xi (f))=\langle T_\xi (f),g\rangle =\langle f,T_\xi (g)\rangle =\Lambda _{T_\xi (g)}(f). \end{aligned}$$

Since $f\in \mathcal {C}_c(G)$ was arbitrary, and using density of $\mathcal {C}_c(G)$ in $L^p(G)$, we get $T_\xi ^*(\Lambda _g)=\Lambda _{T_\xi (g)}$. Then density of $\mathcal {C}_c(G)$ in $L^q(G)$ implies that $T_\xi ^*(\Lambda _g)=\Lambda _{T_\xi (g)}$ for every $g\in L^q(G)$. So, $T_\xi$ identifies $T_\xi ^*$. $\square$

Corollary 4.11

Let G be a locally compact group and H be a compact subgroup of G. Suppose $\xi \in \chi (H)$, $1\le p<\infty$, and $\uplambda _{H}$ be the probability Haar measure of H. Then, $T_\xi :L^p(G)\rightarrow L^p(G)$ is the projection onto $L^p_\xi (G,H)$. In particular, $T_\xi :L^2(G)\rightarrow L^2(G)$ is the orthogonal projection onto $L_\xi ^2(G,H)$.

We then obtain the following unified characterization for the dual space $L^p_\xi (G,H)^*$, if $p>1$.

Theorem 4.12

Let G be a locally compact group and H be a compact subgroup of G. Suppose $\xi \in \chi (H)$, and $1<p,q<\infty$ with $p^{-1}+q^{-1}=1$. Then $L^q_\xi (G,H)$ is isometrically isomorphic to $L^p_\xi (G,H)^*$.

Proof

Let $g\mapsto \Lambda _g$ be the canonical antilinear isometric isomorphism identification of $L^q(G)$ as $L^p(G)^*$, where $\Lambda _g:L^p(G)\rightarrow \mathbb {C}$ is given by $\Lambda _g(f):=\langle f,g\rangle$ for every $f\in L^p(G)$. Invoking Theorem 4.9, we conclude that the dual space $L^p_\xi (G,H)^*$ is isometrically isomorphic to $\mathcal {N}_\xi ^p(G,H)^\perp$, where

$$\begin{aligned} \mathcal {N}_\xi ^p(G,H)^\perp =\left\{ g\in L^q(G):\Lambda _g(f)=0,\ \ \mathrm{for\ all}\ f\in \mathcal {N}_\xi ^p(G,H)\right\} . \end{aligned}$$

We then claim that $\mathcal {N}_\xi ^p(G,H)^\perp =L^q_\xi (G,H)$. To show this, let $g\in \mathcal {N}_\xi ^p(G,H)^\perp$ be given. Then, $g\in L^q(G)$ and $\Lambda _g(f)=0$ for all $f\in \mathcal {N}_\xi ^p(G,H)$. Suppose that $h\in H$ is arbitrary. Then, using (4.2) and Proposition 4.8, for every $f\in \mathcal {C}_c(G)$ we have $R_{h^{-1}}f-\overline{\xi (h)}f\in \mathcal {N}_\xi ^p(G,H)$. This implies that $\Lambda _g(R_{h^{-1}}f)=\overline{\xi (h)}\Lambda _g(f)$. Therefore, using compactness of H, we get

$$\begin{aligned} \Lambda _{R_hg}(f)&=\int _Gf(x)\overline{g(xh)}\mathrm {d}\uplambda _G(x)\\&=\int _Gf(xh^{-1})\overline{g(x)}\mathrm {d}\uplambda _G(xh^{-1})\\&=\Delta _G(h^{-1})\int _Gf(xh^{-1})\overline{g(x)}\mathrm {d}\uplambda _G(x)\\&=\int _Gf(xh^{-1})\overline{g(x)}\mathrm {d}\uplambda _G(x)\\&=\Lambda _g(R_{h^{-1}}f)=\overline{\xi (h)}\Lambda _g(f)=\Lambda _{\xi (h)g}(f). \end{aligned}$$

Since $f\in \mathcal {C}_c(G)$ was arbitrary, we get $R_hg=\xi (h)g$ in $L^q(G)$. Since $h\in H$ was arbitrary, we conclude that $g\in L^q_\xi (G,H)$. Conversely, suppose that $g\in L^q_\xi (G,H)$. We shall show that $\Lambda _g(f)=0$ for every $f\in \mathcal {N}_\xi ^p(G,H)$. According to Theorem 2.49 of [5], let $\mu _{G/H}$ be the G-invariant Radon measure on the left coset space G/H normalized with respect to the Weil’s formula (2.50) of [5]. Then, for every $f\in \mathcal {N}_\xi ^p(G,H)$, we have

$$\begin{aligned} \Lambda _g(f)&=\int _G f(x)\overline{g(x)}\mathrm {d}\uplambda _G(x)\\&=\int _{G/H}\left( \int _Hf(xh)\overline{g(xh)}\mathrm {d}\uplambda _H(h)\right) \mathrm {d}\mu _{G/H}(xH)\\&=\int _{G/H}\left( \int _Hf(xh)\overline{\xi (h)}\mathrm {d}\uplambda _H(h)\right) \overline{g(x})\mathrm {d}\mu _{G/H}(xH)\\&=\int _{G/H}T_\xi (f)(x)\overline{g(x})\mathrm {d}\mu _{G/H}(xH)=0, \end{aligned}$$

which implies that $g\in \mathcal {N}_\xi ^p(G,H)^\perp$. $\square$

Suppose that $L^\infty (G)$ is the Banach space of all locally $\uplambda _G$-measurable functions $f:G\rightarrow \mathbb {C}$ that are bounded except on a locally $\uplambda _G$-null set, modulo functions which are zero locally a.e. on G given the norm

$$\begin{aligned} \Vert f\Vert _\infty :=\inf \{t: |f(x)|\le t\ \ \mathrm{l.a.e.}\ \ x\in G\}. \end{aligned}$$

We then have $\mathcal {C}_\xi (G,H)\subset L^\infty (G)$. It is also routine to check that the linear operator $T_\xi :(\mathcal {C}_c(G),\Vert \cdot \Vert _{L^\infty (G)})\rightarrow (\mathcal {C}_\xi (G,H),\Vert \cdot \Vert _{L^\infty (G)})$, is bounded with the operator norm $\Vert T_\xi \Vert \le \uplambda _H(H)$. In particular, if $\uplambda _H$ is the probability Haar measure of H then the linear operator $T_\xi :(\mathcal {C}_c(G),\Vert \cdot \Vert _{L^\infty (G)})\rightarrow (\mathcal {C}_\xi (G,H),\Vert \cdot \Vert _{L^\infty (G)})$ is a contraction.

Let $L^\infty _\xi (G,H)$ be the closed subspace of $L^\infty (G)$ given by

$$\begin{aligned} L^\infty _\xi (G,H):=\left\{ \psi \in L^\infty (G):R_h\psi =\xi (h)\psi ,\ \mathrm{for}\ h\in H\right\} . \end{aligned}$$

We then also obtain the following characterization for the dual space $L^1_\xi (G,H)^*$.

Theorem 4.13

Let G be a locally compact group and H be a compact subgroup of G. Suppose $\xi \in \chi (H)$. Then $L^1_\xi (G,H)^*$ is isometrically isomorphic to $L^\infty _\xi (G,H)$.

Proof

Let $g\mapsto \Lambda _g$ be the canonical antilinear isometric isomorphism identification of $L^\infty (G)$ as $L^1(G)^*$, where $\Lambda _g:L^1(G)\rightarrow \mathbb {C}$ is given by $\Lambda _g(f):=\langle f,g\rangle$ for $f\in L^1(G)$. Invoking Theorem 4.9, the dual space $L^1_\xi (G,H)^*$ is isometrically isomorphic to $\mathcal {N}_\xi ^1(G,H)^\perp$, where

$$\begin{aligned} \mathcal {N}_\xi ^1(G,H)^\perp =\left\{ g\in L^\infty (G):\Lambda _g(f)=0,\ \ \mathrm{for\ all}\ f\in \mathcal {N}_\xi ^1(G,H)\right\} . \end{aligned}$$

Then using a similar method used in Theorem 4.12, we get $\mathcal {N}_\xi ^1(G,H)^\perp =L^\infty _\xi (G,H)$. $\square$

We then conclude the paper by the following inclusion property when G is compact.

Proposition 4.14

Let G be a compact group and H be a closed subgroup of G. Suppose $\xi \in \chi (H)$ and $1\le p<\infty$. Then, $L^p_\xi (G,H)\subseteq L_\xi ^1(G,H)$.

Proof

Since G is compact, we have $\Vert \psi \Vert _{L^1(G)}\le \uplambda _G(G)^{\frac{p-1}{p}}\Vert \psi \Vert _{L^p(G)}$, for $\psi \in \mathcal {C}_c(G)$. Therefore, $\Vert \psi \Vert _{L^1(G)}\le \uplambda _G(G)^{\frac{p-1}{p}}\Vert \psi \Vert _{L^p(G)}$ for every $\psi \in \mathcal {C}_\xi (G,H)$. This implies that $L^p_\xi (G,H)\subseteq L^1_\xi (G,H)$. $\square$

Corollary 4.15

Let G be a compact group and H be a closed subgroup of G. Suppose $\xi \in \chi (H)$ and $1\le p<\infty$. Then

• 1. $T_\xi :L^p(G)\rightarrow L^p(G)$ is a $L^p(G)$-multiplier.

• 2. $L^p_\xi (G,H)$ is a closed left ideal in $L^p(G)$.