Herz Spaces Meet Morrey Type Spaces and Complementary Morrey Type Spaces

Abstract

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

Appendix on Matuszewska–Orlicz (M.O.) Indices

The indices called Matuszewska–Orlicz indices were introduced in [24, 25]. They are widely used in the literature, for instance, [19, 31], see also a good presentation of them in the Appendix of [32]. These indices are closely related with the property of a function to be almost increasing or almost decreasing after multiplication (division) by a power function.

For a positive function $\omega$ on ${\mathbb {R}} _+$ these indices are defined as follows:

$$\begin{aligned} m_0(\omega )= & {} \sup _{0<t<1} \frac{ \ln \left( \varlimsup _{h \rightarrow 0} \frac{\omega (ht)}{\omega (h)} \right) }{ \ln t} = \lim _{t \rightarrow 0} \frac{ \ln \left( \varlimsup _{h \rightarrow 0} \frac{\omega (ht)}{\omega (h)} \right) }{ \ln t}, \end{aligned}$$

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$$\begin{aligned} M_0(\omega )= & {} \sup _{t>1} \frac{ \ln \left( \varlimsup _{h \rightarrow 0} \frac{\omega (ht)}{\omega (h)} \right) }{ \ln t} = \lim _{t \rightarrow \infty } \frac{ \ln \left( \varlimsup _{h \rightarrow 0} \frac{\omega (ht)}{\omega (h)} \right) }{ \ln t}, \end{aligned}$$

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$$\begin{aligned} m_\infty \left( \omega \right)= & {} \sup _{t>1} \frac{\ln \left( \varliminf _{h \rightarrow \infty } \frac{\omega (ht)}{\omega (h)} \right) }{ \ln t}, \quad M_\infty \left( \omega \right) = \inf _{t>1} \frac{\ln \left( \varlimsup _{h \rightarrow \infty } \frac{\omega (ht)}{\omega (h)} \right) }{ \ln t}. \end{aligned}$$

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The following estimates are known (cf. [32, (6.22)–(6.23)])

$$\begin{aligned} c_1 t ^{M_0(\omega )+ \varepsilon } \leqslant \inf _{0<\tau<1} \frac{\omega (ty)}{\omega (\tau )}, \quad \sup _{0<\tau<1} \frac{\omega (t\tau )}{\omega (\tau )} \leqslant c_2 t ^{m_0(\omega )- \varepsilon }, \quad 0<x<1, \end{aligned}$$

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and

$$\begin{aligned} c_1 t ^{m_ \infty (\omega )- \varepsilon } \leqslant \inf _{\tau>1} \frac{\omega (t\tau )}{\omega (\tau )}, \quad \sup _{\tau>1} \frac{\omega (t\tau )}{\omega (\tau )} \leqslant c_2 t ^{M_\infty (\omega )+ \varepsilon }, \quad x>1. \end{aligned}$$

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