Coincidence of Variable Exponent Herz Spaces with Variable Exponent Morrey Type Spaces and Boundedness of Sublinear Operators in these Spaces

Abstract

We introduce generalized local and global Herz spaces where all their characteristics are variable. As one of the main results we show that variable Morrey type spaces and complementary variable Morrey type spaces are included into the scale of these generalized variable Herz spaces. We also prove the boundedness of a class of sublinear operators in generalized variable Herz spaces with application to variable Morrey type spaces and their complementary spaces, based on the mentioned inclusion.

Appendix: Matuszewska–Orlicz Indices

Let $\omega : \mathbb {R}_{+} \to \mathbb {R}_{+}$ be a measurable function satisfying the condition

$$ 0< \inf_{\delta < t< N} \omega(t) \leqslant \sup_{\delta < t < N} \omega(t) < \infty $$

(1)

for all $0< \delta < N < \infty$ and continuous on (0,δ) and on $(N, \infty )$ for some small 0 < δ and large $N< \infty$. The Matuszewska-Orlicz indices of the function ω at the origin and at infinity, introduced in [40, 41], are defined as:

$$ m_{0}(\omega) = \sup_{0<t<1} \frac{ \ln \left (\varlimsup_{h \to 0} \frac{\omega(ht)}{\omega(h)} \right)}{ \ln t} = \lim_{t \to 0} \frac{ \ln \left (\varlimsup_{h \to 0} \frac{\omega(ht)}{\omega(h)} \right)}{ \ln t}, $$

(2)

$$ M_{0}(\omega) = \sup_{t>1} \frac{ \ln \left (\varlimsup_{h \to 0} \frac{\omega(ht)}{\omega(h)} \right)}{ \ln t} = \lim_{t \to \infty } \frac{ \ln \left (\varlimsup_{h \to 0} \frac{\omega(ht)}{\omega(h)} \right)}{ \ln t}, $$

(3)

and

$$ m_{\infty} \left (\omega \right) = \sup_{t>1} \frac{\ln \left (\varliminf_{h \to \infty } \frac{\omega(ht)}{\omega(h)} \right) }{ \ln t}, \quad M_{\infty} \left (\omega \right) = \inf_{t>1} \frac{\ln \left (\varlimsup_{h \to \infty } \frac{\omega(ht)}{\omega(h)} \right) }{ \ln t}. $$

(4)

Various properties of indices, in particular given below, may be found in the Appendix of [52].

It is known that the existence of finite indices is related to almost monotonicity properties of ω:

$$ \frac{\omega(t)}{t^{\alpha}} \textrm{ is almost increasing near the origin for some } \alpha \Leftrightarrow m_{0}(\omega)>- \infty, $$

(5)

$$ \frac{\omega(t)}{t^ \beta} \textrm{ is almost decreasing near the origin for some } \beta \Leftrightarrow M_{0}(\omega)< + \infty, $$

(6)

and similarly for $m_{\infty } (\omega )$ and $M_{\infty } (\omega )$.

It is also known that $m_{0} (\omega ) = \sup \alpha$ and $M_{0} (\omega ) = \inf \beta$ where α and β are all possible values in Eqs. (5) and (6), and similarly for $m_ \infty (\omega )$ and $M_{\infty } (\omega )$.

Definition 1

By ${\Phi } (\mathbb {R}_{+})$ we denote the class of positive functions on $(0, \infty )$ such that Eq. (1) holds and satisfy C₀ and $C_{\infty }$, where

C ₀ :

there exist finite numbers α₀,β₀ such that $\frac {\omega (t)}{t^{\alpha _{0}}}$ is almost increasing on (0, 1) and $\frac {\omega (t)}{t^{\beta _{0}}}$ is almost decreasing on (0, 1), and

${C_{\infty }}$ :

the numbers $\alpha _{\infty } , \beta _{\infty }$ such that $\frac {\omega (t)}{ t^{\alpha _{\infty } }}$ is almost increasing on $(1,\infty )$ and $\frac {\omega (t)}{t^{\beta _{\infty }}}$ is almost decreasing on $(1,\infty )$.

Note that if ω(t)≢0 and $\omega (t) \not \equiv \infty$ in $(0,\infty )$ then $C_{0}, C_{\infty }$ imply (1).

The following estimates hold:

$$ c_{1} t^{M_{0}(\omega)+ \varepsilon} \leqslant \inf_{0<\tau<1} \frac{\omega(t \tau)}{\omega(\tau)}, \quad \sup_{0<\tau<1} \frac{\omega(t \tau)}{\omega(\tau)} \leqslant c_{2} t^{m_{0}(\omega)- \varepsilon} , \quad 0<t<1, $$

(7)

and

$$ c_{1} t^{m_ \infty(\omega)- \varepsilon} \leqslant \inf_{\tau>1} \frac{\omega(t \tau)}{\omega(\tau)}, \quad \sup_{\tau>1} \frac{\omega(t \tau)}{\omega(\tau)} \leqslant c_{2} t^{M_{\infty}(\omega)+ \varepsilon} , \quad t>1. $$

(8)

We need the following result.

Lemma 1

Let $\omega \in {\Phi }(\mathbb {R}_{+})$. Then for every ε > 0 there exists c_ε > 0 such that

$$ \frac{\omega(t)}{\omega(\tau)} \leqslant c_{\varepsilon} \left (\frac{t}{\tau} \right)^{\min(m_{0}, m_{\infty} ) - \varepsilon}, \quad 0<t<\tau<\infty, $$

(9)

and

$$ \frac{\omega(t)}{\omega(\tau)} \leqslant c_{\varepsilon} \left (\frac{t}{\tau} \right)^{\max(M_{0}, M_{\infty} ) + \varepsilon}, \quad 0<\tau<t<\infty. $$

(10)

Proof

We prove (9), since (10) is similar. In fact, a more precise estimate holds

$$ \frac{\omega(t)}{\omega(\tau)} \leqslant c_{\varepsilon} \begin{cases} \left (\frac{t}{\tau} \right)^{m_{0}-\varepsilon}, \quad &0<t<\tau<1;\\ \frac{t^{m_{0}-\varepsilon}}{\tau^{m_{\infty} -\varepsilon}}, \quad &t<1<\tau;\\ \left (\frac{t}{\tau} \right)^{m_{\infty} -\varepsilon}, \quad &0<t<\tau<1. \end{cases} $$

(11)

from which Eq. (9) will follow (it suffices to observe that $\frac {t^{m_{0}}}{\tau ^{m_{\infty } }} \leqslant \left (\nicefrac {t}{\tau }\right )^{\min \limits (m_{0}, m_{\infty } )}$.

In Eq. (11) the first and third estimates are known, see [46, Eqs. (2.26)-(2.27)] or [51]. It remains to check the estimate in the second line, which follows from the known estimates

$$ \omega(t) \leqslant c_{\varepsilon} t^{m_{0}-\varepsilon}, 0< t<1, \quad \omega(\tau) \geqslant c_{\varepsilon} \tau^{m_{\infty} -\varepsilon }, \tau >1, $$

cf. [46, Eqs. (2.26)-(2.27)]. For Eq. (10) we can, in the same way, obtain the more precise estimate

$$ \frac{\omega(x)}{\omega(y)} \leqslant c_{\varepsilon} \begin{cases} \left (\frac{t}{\tau} \right)^{M_{0}+\varepsilon}, \quad &0<\tau<t<1;\\ \frac{t^{M_{\infty} +\varepsilon}}{\tau^{M_{0} +\varepsilon}}, \quad &\tau<1<t;\\ \left (\frac{t}{\tau} \right)^{M_{\infty} +\varepsilon}, \quad &1<\tau<t, \end{cases} $$

(12)

which ends the proof. □