Let \(\Omega \subset\mathbb{R}^{n}\) be a measurable subset, and let \(p(\cdot) \) be a measurable function \(p(\cdot) :\Omega \rightarrow (0,\infty ]\) with \(0<\) essinf\(_{x\in \Omega }p( x) \). We denote by \(L^{p(\cdot) }(\Omega ) \) the variable exponent Lebesgue space and equip it with the Luxemburg norm \[ \| f\| _{L^{p(\cdot) }( \Omega ) }=\inf\{ \lambda >0:\rho _{p}( \frac{f}{\lambda }) \leq 1\} ,\text{ where }\rho _{p}( f) =\int_{\{ p(x) <\infty \} }| f( x) | ^{p(x) }dx+\| f\| _{L^{\infty }}. \] We also denote by \(\wp _{0}(\mathbb{R}^{n}) \) and \(\wp (\mathbb{R}^{n}) \) the sets of all measurable functions \(p(\cdot) : \mathbb{R}^{n}\rightarrow (0,\infty )\) such that \(0<p_{-}\leq p_{+}<\infty \) and \(1<p_{-}\leq p_{+}<\infty \), respectively, where \(p_{-}=\mathrm{ess\, inf}_{x\in\mathbb{R}^{n}}p( x) \), \(p_{+}=\mathrm{ess\, sup}_{x\in\mathbb{R}^{n}}p( x)\). Let \(p(\cdot) ,q(\cdot) \in \wp_{0}(\mathbb{R}^{n}) \). The mixed Lebesgue sequence space \(\ell ^{q(\cdot)}( L^{p(\cdot) }) \) is the collection of all sequences \(\{ f_{j}\} _{j=0}^{\infty }\) of measurable functions on \(\mathbb{R}^{n}\) such that \[ \| \{ f_{j}\} _{j=0}^{\infty }\| _{\ell^{q(\cdot) }( L^{p(\cdot) }) }=\inf \{ \mu:\rho _{\ell ^{q(\cdot) }( L^{p(\cdot) }) }(\{ \frac{f_{j}}{\mu }\} _{j=0}^{\infty }) \leq 1\}<\infty, \] where \[ \rho _{\ell ^{q(\cdot) }( L^{p(\cdot) }) }(\{ f_{j}\} _{j=0}^{\infty }) =\sum_{j=0}^{\infty }\inf\{ \lambda _{j}:\int_{\mathbb{R}^{n}}( \frac{| f_{j}( x) | }{\lambda_{j}^{\frac{1}{q( x) }}}) ^{p( x) }dx\leq 1\} . \] Also, the mixed space \(L^{p(\cdot) }( \ell ^{q(\cdot) }) \) is the collection of all sequences \(\{ g_{j}\} _{j=0}^{\infty }\) of measurable functions on \(\mathbb{R}^{n}\) such that \[ \| \{ g_{j}\} _{j=0}^{\infty }\| _{L^{p(\cdot) }( \ell ^{q(\cdot) }) }=\| (\sum_{j=0}^{\infty }| g_{j}(\cdot) | ^{q(\cdot)}) ^{\frac{1}{q(\cdot) }}\| _{L^{p(\cdot) }}<\infty . \] Let \(\theta =\{ \theta _{j}\} _{j=1}^{\infty }\subset S(\mathbb{R}^{n}) \) be a system such that

1.

\(\begin{cases}\text{supp}\mathcal F \theta _{0}\subset \{ \xi :| \xi | \leq 2\}, \\ \text{supp}\mathcal F \theta _{j}\subset \{ \xi :2^{j-1}\leq| \xi | \leq 2^{j+1}\} \text{ for }j\geq 1. \end{cases}\)

2.

For every multi-index \(\alpha \), there exists a positive number \(C_{\alpha }\) such that \(\;2^{j| \alpha | }|D^{\alpha }\mathcal F ( \theta _{j}) ( \xi )| \leq c_{\alpha }\) holds for all \(j\in\mathbb{N}_{0}\) and all \(\xi \in\mathbb{R}^{n}\).

3.

\(\sum_{j=0}^{\infty }\mathcal F ( \theta _{j}) ( \xi) =1\) for all \(\xi \in\mathbb{R}^{n}\), where \(\mathcal F \) is the Fourier transform.

If there exist positive constants \(C_{\log }( p) ,A\) and a real number \(p_{\infty }\) such that \[ | p( x) -p( y) | \leq \frac{C^{\log }( p) }{\log ( e+| x-y| ^{-1}) } \quad ( x,y\in\mathbb{R}^{n},\;x\neq y) \tag{1} \] and \[ | p( x) -p_{\infty }| \leq \frac{A}{\log ( e+| x| ) }\quad ( x\in\mathbb{R}^{n}), \tag{2} \] we write \(C^{\log }(\mathbb{R}^{n}) \) for the set of functions \(p(\cdot) :\mathbb{R}^{n}\rightarrow\mathbb{R}\) satisfying (1) and (2).
Let \(p(\cdot) ,q(\cdot) \in C^{\log }(\mathbb{R}^{n}) \cap \wp _{0}(\mathbb{R}^{n}) \) and \(s(\cdot) \in C^{\log }(\mathbb{R}^{n})\). Furthermore, let \(\theta =\{ \theta _{j}\}_{j=1}^{\infty }\) be the system as above. The Besov space with variable exponents \(B_{p(\cdot) ,q(\cdot) }^{s(\cdot) }(\mathbb{R}^{n}) \) is the collection of \(f\in S' (\mathbb{R}^{n}) \) such that \[ \| f\| _{B_{p(\cdot) ,q(\cdot) }^{s(\cdot)}}=\| \{ 2^{js(\cdot) }\theta _{j}\ast f\} _{j=0}^{\infty }\| _{\ell ^{q(\cdot) }(L^{p(\cdot) }) }<\infty . \] The Triebel-Lizorkin space with variable exponents \(F_{p(\cdot),q(\cdot) }^{s(\cdot) }(\mathbb R^{n}) \) is the collection of \(f\in S'(\mathbb R^{n}) \) such that \[ \| f\| _{F_{p(\cdot) ,q(\cdot) }^{s(\cdot)}}=\| \{ 2^{js(\cdot) }\theta _{j}\ast f\} _{j=0}^{\infty }\| _{L^{p(\cdot) }( \ell^{q(\cdot) }) }<\infty . \] Let \(p(\cdot) ,q(\cdot) \in \wp _{0}(\mathbb R^{n}) \) and \(\alpha (\cdot) \in L^{\infty }(\mathbb R^{n}) \). The non-homogeneous Herz space \(K_{p(\cdot) }^{\alpha (\cdot) ,q(\cdot) }(\mathbb R^{n}) \) consists of all \(f\in L_{\mathrm{loc}}^{p(\cdot) }(\mathbb R^{n}) \) such that \[ \| f\| _{K_{p(\cdot) }^{\alpha (\cdot),q(\cdot) }}=\| \{ 2^{k\alpha (\cdot)}| f\chi _{k}| \} _{k=0}^{\infty }\|_{\ell ^{q(\cdot) }( L^{p(\cdot) }) }<\infty . \] The homogeneous Herz space \(\dot{K}_{p(\cdot)}^{\alpha (\cdot) ,q(\cdot) }(\mathbb R^{n}) \) consists of all \(f\in L_{\mathrm{loc}}^{p(\cdot) }(\mathbb R^{n}\backslash \{ 0\} ) \) such that \[ \| f\| _{\dot{K}_{p(\cdot) }^{\alpha (\cdot) ,q(\cdot) }}=\inf \{ \lambda >0:\sum_{k=-\infty}^{\infty }\| ( \frac{2^{k\alpha (\cdot) }|f\chi _{R_{k}}| }{\lambda }) ^{q(\cdot) }\|_{L^{\frac{p(\cdot) }{q(\cdot) }}}\leq 1\} <\infty . \] In the present paper, the authors are concerned with duality and reflexivity of Triebel-Lizorkin spaces \(F_{p(\cdot) ,q(\cdot) }^{s(\cdot)}(\mathbb R^{n}) \), Besov spaces \(B_{p(\cdot) ,q(\cdot)}^{s(\cdot) }(\mathbb R^{n})\), and Herz spaces \(K_{p(\cdot) }^{\alpha (\cdot) ,q(\cdot) }(\mathbb R^{n}) \) and \(\dot{K}_{p(\cdot) }^{\alpha (\cdot) ,q(\cdot) }(\mathbb R^{n}) \). They prove the following main theorems.
Theorem 1. Let \(p(\cdot) ,q(\cdot) \in C^{\log }(\mathbb R^{n}) \cap \wp (\mathbb R^{n}) \) and \(s(\cdot) \in C^{\log }(\mathbb R^{n}) \). Then we have \[ ( F_{p(\cdot) ,q(\cdot) }^{s(\cdot) }(\mathbb R^{n}) ) `=F_{p'(\cdot),q'(\cdot) }^{-s(\cdot) }(\mathbb R^{n}) , \] where \(p',q'\) are the conjugate exponents.
Theorem 2. Let \(p(\cdot) ,q(\cdot) \in C^{\log }(\mathbb R^{n}) \cap \wp (\mathbb R^{n}) \) and \(s(\cdot) \in C^{\log }(\mathbb R^{n}) \). Then we have \[ ( B_{p(\cdot) ,q(\cdot) }^{s(\cdot) }(\mathbb R^{n}) ) `=B_{p'(\cdot),q'(\cdot) }^{-s(\cdot) }(\mathbb R^{n}) . \]
Theorem 3. Let \(p(\cdot) ,q(\cdot) \in \wp (\mathbb R^{n}) \) and \(\alpha (\cdot) \in L^{\infty }(\mathbb R^{n}) \). Then we have \[ ( K_{p(\cdot) }^{\alpha (\cdot) ,q(\cdot)}(\mathbb R^{n}) ) `=K_{p'(\cdot)}^{-\alpha (\cdot) ,q'(\cdot) }(\mathbb R^{n}) \text{ and }( \dot{K}_{p(\cdot) }^{\alpha(\cdot) ,q(\cdot) }(\mathbb R^{n}) ) `=\dot{K}_{p'(\cdot) }^{-\alpha (\cdot) ,q'(\cdot) }(\mathbb R^{n}) . \]