1 Introduction

The well-known Herz spaces introduced in [15] attracted a lot of attention last decade as an example of decomposition spaces widely used in applications ([7]), see for instance studies of Herz spaces in [8, 10, 14, 18, 21, 23], including results in variable exponent setting in [2, 16, 20, 26,27,28,29,30, 33], see also references therein.

Let \(\alpha \in {\mathbb {R}}\), \(1\leqslant p<\infty , 1\leqslant q<\infty \). Homogeneous Herz spaces are defined by the norm

$$\begin{aligned} \Vert f\Vert _{\dot{{\varvec{\mathcal {K}}}}_\alpha ^{p,q}({\mathbb {R}}^n )}:=\left\{ \sum \limits _{k\in {\mathbb {Z}}}2^{k\alpha q}\left( \int \limits _{2^{k-1}<|x|<2^k}|f(x)|^p \mathrm{d}x\right) ^\frac{q}{p}\right\} ^\frac{1}{q}. \end{aligned}$$
(1)

It is known that the norm (1) is equivalent to the continual form:

$$\begin{aligned} \Vert f\Vert _{{\varvec{K}}_\alpha ^{p,q}({\mathbb {R}}^n )}:= \left\{ \int _{0}^\infty t^{\alpha q} \left( \int \limits _{t<|x|<2t }|f(x)|^p \mathrm{d}x\right) ^\frac{q}{p} \frac{\mathrm{d}t }{t}\right\} ^\frac{1}{q}, \end{aligned}$$
(2)

see [9, 17].

Besides Herz spaces, in analysis there are known the spaces, which we denote by , and , defined by the norms

(3)
(4)

and

(5)

where \( \omega \) is a non-negative function in \( {\mathbb {R}} _+ \) and stands for the Haar measure .

The local Morrey type space \({\varvec{M}} ^{p,q } _{\alpha ,0} ({\mathbb {R}}^n ) \equiv {\varvec{M}}^{p,q }_{t^\alpha , 0} \left( {\mathbb {R}}^n \right) \) first appeared in 1981 in D.R. Adams [1, p. 44]. The local Morrey type space \( {\varvec{M}} ^{p,q }_ {\omega ,0} \left( {\mathbb {R}}^n \right) \) and the complementary Morrey type space \( ^\complement \! {\varvec{M}} ^{p,q } _{\omega ,0} \left( {\mathbb {R}}^n \right) \) were introduced in V. Guliyev [11], cf. [12, 13], together with global Morrey type space \({\varvec{M}} ^{p,q } _{\omega } \left( {\mathbb {R}}^n \right) \), see additionally [5, 6]. We refer also to surveying papers [3, 4] on mapping properties of operators of harmonic analysis in such spaces.

The spaces \({\varvec{M}}^{p,q}_{\alpha ,0} ({\mathbb {R}}^n )\) and \({\varvec{M}}^{p,q}_{\alpha } ({\mathbb {R}}^n )\) may be called local and global Morrey–Adams spaces, respectively, taking into account [1] and the spaces \( ^\complement {\varvec{M}}^{p,q}_{\alpha ,0} ({\mathbb {R}}^n) \) complementary Morrey–Adams spaces. Though from the point of view of the role in the development of these spaces the spaces \( {\varvec{M}} ^{p,q } _{\omega ,0} \left( {\mathbb {R}}^n \right) \) and \({\varvec{M}} ^{p,q }_{\omega } \left( {\mathbb {R}}^n \right) \) may be called local and global Morrey–Guliyev spaces, respectively, taken into account [11] and the space \(^\complement {\varvec{M}} ^{p,q } _{\omega ,0} \left( {\mathbb {R}}^n\right) \) complementary Morrey–Guliyev space. Note that, the local and global Morrey–Guliyev spaces may also be called local and global Morrey–Adams–Guliyev spaces. Everywhere in the sequel we use the name “Morrey type spaces” for all these spaces, following the tradition adopted in the literature.

In this paper, we consider generalization of Herz spaces with \( t^\alpha \) replaced by \( \omega (t) \) both global and local versions, like in  (3) and (4). We use Matuszewska–Orlicz indices of \(\omega \) to describe appropriate properties of \( \omega \) at the origin and at infinity.

As one of the main results, we reveal the situation when generalized Herz spaces coincide with Morrey type spaces local and global, respectively, and when local Herz spaces coincide with complementary Morrey type spaces. In other words, we show that Morrey type spaces and complementary Morrey type spaces are included in the scale of generalized Herz spaces. To our surprise, it was never observed before, up to our knowledge. In particular, for the classical Herz spaces we will see that \( {\varvec{K}}^{p,q}_{\alpha } ({\mathbb {R}}^n )\) coincides with Morrey type space \( {\varvec{M}}^{p,q}_{\alpha ,0}({\mathbb {R}}^n) \) when \( \alpha <0 \) and it coincides with complementary Morrey type space \(^\complement {\varvec{M}}^{p,q}_{\alpha ,0} ({\mathbb {R}}^n ) \) when \( \alpha >0 \) (in the case \( \alpha = 0 \) both Morrey type and complementary Morrey type spaces are trivial), see Corollary 1. We also prove the boundedness of a class of sublinear operators in generalized global Herz spaces. As a by-product, we obtain for free some results on the boundedness of a class of sublinear operators with application to maximal, singular, and Hardy operators.

The paper is organized as follows. In Sect. 2 we define global and local generalized Herz spaces and prove the equivalence between discrete and continual norms for these spaces. In Sect. 3 we prove the coincidence of generalized Herz spaces with Morrey type or complementary Morrey type, depending on the sign of the indices of the function \( \omega \). In Sect. 4 we prove a general theorem on boundedness of a class of sublinear operators, with application to Morrey type and complementary Morrey type spaces, based on the coincidence proved in Sect. 3.

Notation:

B(x, r) is the ball of radius r centered at the point x;

\(^\complement B(x,r)\) is the set \( {\mathbb {R}}^n \setminus B(x,r) \) ;

\({\mathbf {1}}_E\) is the characteristic function of the set E;

denotes the Haar measure on \({\mathbb {R}}_+\);

\(A \simeq B\) for non-negative expressions A and B means that \(C_1 A\leqslant B \leqslant C_2 A\) where \(C_1>0\) and \(C_2>0\) do not depend on A and B.

2 Global Generalized Herz Spaces

Everywhere in the sequel we assume that \( \omega \in M({\mathbb {R}} _+) \) where the class \( M( {\mathbb {R}} _+) \) is defined as follows.

Definition 2.1

By \( M ({\mathbb {R}} _+) \) we denote the class of positive functions in \( (0, \infty ) \) such that

$$\begin{aligned} 0< \inf _{\delta< t< N} \omega (t) \leqslant \sup _{\delta< t< N} \omega (t) < \infty \end{aligned}$$
(6)

for all \( 0< \delta< N < \infty \) and additionally there exist finite numbers \( \alpha _0, \beta _0, \alpha _\infty , \beta _\infty \) such that \( \omega (t) t^{\alpha _0} \) is almost increasing on \( (0,t_0) \), is almost decreasing on \( (0,t_0) \), \( \omega (t) t^{\alpha _\infty } \) is almost increasing on \( (t_0,\infty ) \) and is almost decreasing on \( (t_0,\infty ) \).

Under the condition (6), in Definition 2.1 we can take any other positive point, in particular \( t_0=1 \). It is known that a positive and continuous function on \( (0,\infty ) \) is in \( M({\mathbb {R}} _+)\) if and only if it has finite Matuszewska–Orlicz indices \( m_0(\omega )\), \(m_\infty (\omega )\), \(M_0 (\omega ),\) and \( M_\infty (\omega ) \), see definition of such indices in Appendix.

Note that

$$\begin{aligned} \omega (kt ) \leqslant C_{k} \omega (t), \end{aligned}$$
(7)

for \( \omega \in M({\mathbb {R}} _+) \).

Definition 2.2

We define the global generalized Herz space \({\varvec{H}}^{p, q}_{\omega } ({\mathbb {R}}^n ) \) as the space of functions such that

$$\begin{aligned} \Vert f \Vert _{{\varvec{H}}^{p, q}_{\omega } ({\mathbb {R}}^n ) } := \sup _{\xi \in {\mathbb {R}}^n } \left( \int _{0}^\infty \omega (t)^q \left( \; \int _{t<|\xi -y|< 2t} |f(y)|^p \mathrm{d}y \right) ^{\frac{q}{p}}\frac{\mathrm{d}t}{t} \right) ^{\frac{1}{q}} < \infty , \end{aligned}$$
(8)

where \( \omega \in M({\mathbb {R}} _+) \). We also define the local generalized Herz space \( {\varvec{H}}^{p,q}_{\omega ,0} ({\mathbb {R}}^n )\) taking \( \xi =0 \) in (8).

We also introduce the discrete version of (8), namely

$$\begin{aligned} \Vert f \Vert _{\varvec{\dot{\mathcal {K}}}^{p, q}_{\omega } ({\mathbb {R}}^n )} := \sup _{\xi \in {\mathbb {R}}^n } \left( \sum _{k=- \infty }^{\infty } \omega (2^k)^q \big \Vert f {\mathbf {1}}_{ B(\xi , 2^k) \setminus B(\xi ,2^{k-1}) } \big \Vert ^q_{L^{p}({\mathbb {R}}^n)} \right) ^{\frac{1}{q}}. \end{aligned}$$
(9)

Lemma 2.3

Let \( \omega \in M({\mathbb {R}} _+) \). The norm (8) remains equivalent if the integration over the layer \( \{ y : t< | \xi -y| < 2 t\} \) is replaced by integration over the layer \( \{ y : \delta t< | \xi -y| < \gamma t\} \) with \( 0< \delta< \gamma < \infty \). Moreover, the norm (8) is equivalent to the norm (9).

Proof

In this proof we follow the arguments used in [33].

Equivalence between the norms (8) for different positive \(\gamma \) and \(\delta \).

For simplicity we take

By the dilation change of the variable t, it is easy to see that

$$\begin{aligned} A_{f,\xi }(\gamma ,\delta )\simeq A_{f,\xi }\left( 1,\frac{\delta }{\gamma }\right) , \ \ \ \ 0<\gamma <\delta \end{aligned}$$

(where the constants in the equivalence relation depend only on the constant from the inequality (7), valid for \( \omega \in M ({\mathbb {R}} _+) \). Consequently, it suffices to deal only with the case \(\gamma =1\) and \(\delta >1\). Let \(\delta <\lambda .\) Then

$$\begin{aligned} A_{f,\xi }(1,\delta )\leqslant A_{f,\xi }(1,\lambda ) \leqslant C_{\omega ,\delta } \left[ A_{f,\xi }(1,\delta )+A_{f,\xi }\left( 1,\frac{\lambda }{\delta }\right) \right] , \end{aligned}$$

with \(C_{\omega ,\delta }\) depending only on \(\delta ,\) but not depending on f,  where the left-hand side inequality is obvious, and the right-hand side one is easily obtained by splitting \({\mathbf {1}}_{(t,\lambda t)}={\mathbf {1}}_{(t,\delta t)}+{\mathbf {1}}_{(\delta t,\lambda t)}\).

If \(\lambda \leqslant \delta ^2,\) then and the proof of the equivalence \(A(1,\lambda )\simeq A(1,\delta )\) is over. If \(\delta ^2<\lambda \leqslant \delta ^3,\) we similarly proceed and have . Iterating this N times, we obtain that \(A_{f,\xi }(1,\lambda )\leqslant CA_{f,\xi }(1,\delta )\) with C depending on \(\lambda \), \( \omega \) and \(\delta \), but not depending on f.

Equivalence of the norms (8) and (9).

We have

Using the monotonicity of \( \omega \), we have

which, after taking appropriate supremum in both sides, yields the inequality \( \Vert f \Vert _{ {\varvec{H}}^{p,q}_\omega ({\mathbb {R}}^n )} \geqslant C \Vert f\Vert _{\dot{{\varvec{\mathcal {K}}}}^{p,q}_{\omega } ({\mathbb {R}}^n )} \).

Similarly,

$$\begin{aligned} A_{f,\xi }(1,2)&\leqslant C\left( \left( \sum \limits _{k\leqslant 0} \omega (2^k)^q + \sum \limits _{k\geqslant 0} \omega (2^{k-1})^q \right) \left\| f{\mathbf {1}}_{ B(\xi ,2^{k+2}) \setminus B(\xi ,2^k) }\right\| ^q_{L^{p}({\mathbb {R}}^n)} \right) ^\frac{1}{q}\\&\leqslant C \left( \sum \limits _{k\in {\mathbb {Z}}} \left( \omega (2^k)^q + \omega (2^{k-1})^q \right) \left\| f{\mathbf {1}}_{ B(\xi ,2^{k+1}) \setminus B(\xi ,2^k) }\right\| ^q_{L^{p}({\mathbb {R}}^n)}\right) ^\frac{1}{q}\\&\leqslant C \left( \sum \limits _{k\in {\mathbb {Z}}} \omega (2^k)^q \left\| f{\mathbf {1}}_{ B(\xi ,2^{k+1}) \setminus B(\xi ,2^k) }\right\| ^q_{L^{p}({\mathbb {R}}^n)}\right) ^\frac{1}{q}\\&= C \Vert f\Vert _{\dot{\varvec{{\mathcal {K}}}}^{p,q}_{\omega } ({\mathbb {R}}^n )}. \end{aligned}$$

\(\square \)

3 Herz Spaces Meet Morrey Type Spaces and Complementary Morrey Type Spaces

To prove our main result, we will need the following auxiliary estimates for functions \( \omega \in M({\mathbb {R}}_+ ) \).

Lemma 3.1

Let \( \omega \in M({\mathbb {R}} _+) \). Then for every \( \varepsilon >0 \) there exists \( c_\varepsilon >0 \) such that

$$\begin{aligned} \frac{\omega (t)}{\omega (\tau )} \leqslant c_\varepsilon \left( \frac{t}{\tau } \right) ^{\min (m_0(\omega ), m_\infty (\omega ) ) - \varepsilon }, \quad 0<t<\tau <\infty , \end{aligned}$$
(10)

and

$$\begin{aligned} \frac{\omega (t)}{\omega (\tau )} \leqslant c_\varepsilon \left( \frac{t}{\tau } \right) ^{\max (M_0(\omega ), M_\infty (\omega ) ) + \varepsilon }, \quad 0<\tau<t<\infty . \end{aligned}$$
(11)

Proof

We prove (10), since (11) is similar. In fact, a more precise estimate holds

$$\begin{aligned} \frac{\omega (t)}{\omega (\tau )} \leqslant c_\varepsilon {\left\{ \begin{array}{ll} \left( \frac{t}{\tau } \right) ^{m_0(\omega )-\varepsilon }, \quad &{}0<t<\tau<1;\\ \frac{t^{m_0(\omega )-\varepsilon }}{\tau ^{m_\infty (\omega ) -\varepsilon }}, \quad &{}t<1<\tau ;\\ \left( \frac{t}{\tau } \right) ^{m_\infty (\omega ) -\varepsilon }, \quad &{}1<t<\tau . \end{array}\right. } \end{aligned}$$
(12)

from which (10) will follow (it suffices to observe that ).

In (12) the first and third estimates are known, see [19, eqs. (2.26)–(2.27)] or [32]. It remains to check the estimate in the second line, which follows from the known estimates

$$\begin{aligned} \omega (t) \leqslant c_\varepsilon t^{m_0(\omega )-\varepsilon }, 0< t<1, \quad \omega (\tau ) \geqslant c_\varepsilon \tau ^{m_\infty (\omega ) -\varepsilon }, \tau >1, \end{aligned}$$

cf. [19, eqs. (2.26)-(2.27)]. For (11) we can, in the same vein, obtain the more precise estimate

$$\begin{aligned} \frac{\omega (t)}{\omega (\tau )} \leqslant c_\varepsilon {\left\{ \begin{array}{ll} \left( \frac{t}{\tau } \right) ^{M_0(\omega )+\varepsilon }, \quad &{}0<\tau<t<1;\\ \frac{t^{M_\infty (\omega ) +\varepsilon }}{\tau ^{M_0(\omega ) +\varepsilon }}, \quad &{}\tau<1<t;\\ \left( \frac{t}{\tau } \right) ^{M_\infty (\omega ) +\varepsilon }, \quad &{}1<\tau <t, \end{array}\right. } \end{aligned}$$
(13)

which ends the proof. \(\square \)

Note that the conditions \( \max (M_0(\omega ), M_\infty (\omega ))<0 \) and \( \min (m_0(\omega ),m_\infty (\omega ))>0 \) used in the following theorem guarantee that Morrey type spaces and its complementary spaces are not trivial. Note that, in general, even for a positive on \( (0, \infty ) \) function \( \omega \notin M({\mathbb {R}}_+) \) the conditions

$$\begin{aligned} \int _{1}^\infty \frac{\omega (t)^q}{t}\mathrm{d}t< \infty \end{aligned}$$

and

$$\begin{aligned} \int _{0}^1 \frac{\omega (t)^q}{t} \mathrm{d}t < \infty \end{aligned}$$

are necessary and sufficient for the non-triviality of Morrey type and complementary Morrey type spaces, respectively, see [5, Lemma 1].

Theorem 3.2

Let \( 1 \leqslant p < \infty \), \( 1 \leqslant q < \infty \), and \( \omega \in M ({\mathbb {R}}^n )\).

  1. (a)

    If \( \max (M_0(\omega ), M _\infty (\omega )) <0\) then \( {\varvec{H}}^{p ,q}_{\omega } ({\mathbb {R}}^n ) \) and \( {\varvec{M}}^{p ,q}_{\omega } ({\mathbb {R}}^n ) \) coincide up to equivalence of norms.

  2. (b)

    If \( \min (m_0(\omega ), m _\infty (\omega )) >0 \) then \( {\varvec{H}}^{p ,q}_{\omega ,0 } ({\mathbb {R}}^n )\) and \(\, ^\complement {\varvec{M}}^{p ,q}_{\omega ,0 } ({\mathbb {R}}^n ) \) coincide up to equivalence of norms.

Proof

The embedding \( {\varvec{M}}^{p,q}_{\omega } ({\mathbb {R}}^n)\hookrightarrow {\varvec{H}}^{p,q}_{\omega } ({\mathbb {R}}^n ) \) is obvious if we take into account that for \( \omega \in M({\mathbb {R}} _+) \).

To prove the inverse embedding, via dyadic decomposition we obtain

(14)

where we used (11) and the fact that \( \max (M_0(\omega ), M_ \infty (\omega ) )< 0 \). Since the constant \( C_\varepsilon >0 \) does not depend on \( \xi \), it remains to pass to the supremum.

The proof for (b) is similar via dyadic decomposition of the exterior of the ball and the use of (10).\(\square \)

Remark 3.3

The statement of Theorem 3.2.(a) is valid also for local Morrey type spaces; \( {\varvec{H}}^{p,q}_{\omega ,0} ({\mathbb {R}}^n ) = {\varvec{M}}^{p,q}_{\omega ,0} ({\mathbb {R}}^n ) \) under the assumptions of Theorem 3.2 on p, q, and \(\omega \), since its proof was made for a fixed \( \xi \).

Corollary 1

Let \( 1 \leqslant p < \infty \), \( 1 \leqslant q < \infty \), and \( \omega (t) = t^\alpha \) with \( \alpha \in {\mathbb {R}} \). Then

$$\begin{aligned} {\varvec{H}}^{p,q}_{\omega ,0} ({\mathbb {R}}^n ) = {\varvec{M}}^{p,q}_{\omega ,0} ({\mathbb {R}}^n ), \quad {\mathrm {if}}\; \alpha <0, \end{aligned}$$

and

$$\begin{aligned} {\varvec{H}}^{p,q}_{\omega ,0} ({\mathbb {R}}^n ) = {} ^\complement {\varvec{M}}^{p,q}_{\omega ,0} ({\mathbb {R}}^n ), \quad {\mathrm {if}}\; \alpha >0, \end{aligned}$$

and if \( \alpha = 0\) then \({\varvec{M}}^{p,q}_{\omega , 0} ({\mathbb {R}}^n ) \), \({\varvec{M}}^{p,q}_{\omega } ({\mathbb {R}}^n ) \) and \( ^\complement {\varvec{M}}^{p,q}_{\omega ,0} ({\mathbb {R}}^n ) \) are trivial; \( {\varvec{M}}^{p,q}_{\omega , 0} ({\mathbb {R}}^n ) = {\varvec{M}}^{p,q}_{\omega } ({\mathbb {R}}^n ) = \; ^\complement {\varvec{M}}^{p,q}_{\omega } ({\mathbb {R}}^n ) = \{ 0 \}\), which is not the case for the Herz spaces \({\varvec{H}}^{p,q}_{\omega ,0} ({\mathbb {R}}^n )\) and \( {\varvec{H}}^{p,q}_{\omega } ({\mathbb {R}}^n )\).

4 Boundedness of Sublinear Operators

We now deal with the boundedness of a sublinear operator T satisfying the well-known size condition

$$\begin{aligned} |Tf(x)|\leqslant C \int \limits _{{\mathbb {R}}^n}\frac{|f(y)| }{|x-y|^n} \mathrm{d}y, \ \ \ \ x \notin \, {\text {spt}}\,f, \end{aligned}$$
(15)

in the global generalized Herz spaces \({\varvec{H}}^{p, q}_{\omega } ({\mathbb {R}}^n )\).

To prove the boundedness result we will need the following technical auxiliary lemma, cf. [19, Lemma 13.61].

Lemma 4.1

The following relations

$$\begin{aligned} \int \limits _{2a<|y|<t} |\phi (y)| \mathrm{d}y= & {} \frac{1}{\ln \,2}\int \limits _a^t \frac{\mathrm{d}\tau }{\tau }\int \limits _{\max (2a,\tau )<|y|<\min (t,2\tau )}|\phi (y)| \mathrm{d}y, \quad t>2a>0, \nonumber \\\end{aligned}$$
(16)
$$\begin{aligned} \int _{|y|>2t}|\phi (y)| \mathrm{d}y= & {} \frac{1}{\ln \,2}\int _{t}^\infty \frac{\mathrm{d}\tau }{\tau }\int _{\max (\tau ,2t)<|y|<2\tau } |\phi (y)| \mathrm{d}y, \quad t>0. \end{aligned}$$
(17)

hold for every measurable function \(\phi \) for which the integrals on the left-hand side exist.

In the proof of Theorem 4.3 we use the following Hardy operators

$$\begin{aligned} H^\beta f(t) = \frac{1}{t} \int _{0}^t \left( \frac{t}{\tau } \right) ^\beta f(\tau ) \mathrm{d}\tau , \quad {\mathcal {H}}^\gamma f(t) = \int _{t}^\infty \left( \frac{t}{\tau } \right) ^\gamma f(\tau ) \frac{\mathrm{d}\tau }{\tau }, \quad \beta , \gamma \in {\mathbb {R}}. \end{aligned}$$

As is known, these operators are bounded in \( L^q ({\mathbb {R}}_+) \) if and only if and , respectively, see [22]. Via direct recalculation we formulate the following lemma with respect to the Haar measure for its use in Theorem 4.3.

Lemma 4.2

Let \( 1< q< \infty \). Then \( H^\beta \) and \( {\mathcal {H}}^{\gamma } \) are bounded in if and only if \( \beta <1 \) and \( \gamma > 0 \).

The following theorem was proved in [33] in a particular case of local Herz spaces and \( \omega (t) = t^{\alpha } \), but in a more general setting of variable exponent p, q and \( \alpha \).

Theorem 4.3

Let \(1< p<\infty \), \(1< q <\infty \) and \( \omega \in M({\mathbb {R}}_+ ) \). Then every sublinear operator T satisfying the size condition (15), bounded in \(L^{p}({\mathbb {R}}^n),\) is also bounded in \( {\varvec{H}}^{p, q} _{\omega } \left( {\mathbb {R}}^n \right) \)

if

$$\begin{aligned} - \frac{n}{p}< m_0(\omega ) \leqslant M_0 (\omega ) < \frac{n}{p^\prime }, \end{aligned}$$
(18)

and

$$\begin{aligned} -\frac{n}{p}< m_\infty (\omega ) \leqslant M_\infty (\omega ) < \frac{n}{p^\prime }. \end{aligned}$$
(19)

Proof

For a fixed \( \xi \in {\mathbb {R}}^n \), we evaluate .

For \( x \in B(\xi , 2t) \setminus B(\xi , t) \) we split the function f(x) as

$$\begin{aligned} f(x)= f_t(x)+g_t(x)+h_t(x), \end{aligned}$$

where

depending on the parameter \(t\in (0,\infty )\) and \( \xi \in {\mathbb {R}}^n \).

Then

$$\begin{aligned} |Tf(x)|\leqslant |Tf_t(x)|+ |Tg_t(x)| + |Th_t(x)|. \end{aligned}$$

Estimation of \(Tf_t\). We have

Since , so that

Using the relation (16) we obtain

and evaluating the norm of the characteristic function we obtain

whence

Consequently,

This may be rewritten in the form

where .

Via splittings \(1 ={\mathbf {1}}_{(0,1]} (t) + {\mathbf {1}}_{(1, \infty )} (t) \) and \( 1 ={\mathbf {1}}_{(0,1]} (\tau ) + {\mathbf {1}}_{(1, \infty )} (\tau ) \) and using the relations (12) we have

$$\begin{aligned} \omega (t) \Vert {\mathbf {1}}_{ B(\xi , 2t) \setminus B(\xi , t) }Tf_t\Vert _{L^{p}({\mathbb {R}}^n)} \leqslant c_\varepsilon \left( H^{\beta _1} \varphi + H^{\beta _2} \varphi + A \varphi \right) , \end{aligned}$$
(20)

where \( c_\varepsilon \) does not depend on \( \xi \), and and

where for the estimate of A we used the more precise estimate (13).

By Lemma 4.2 the Hardy operators \( H^{\beta _1} \) and \( H^{\beta _2} \) are bounded in under the conditions (18)–(19) taken into account that \( \varepsilon \) may be taken arbitrarily small.

The boundedness of the operator A in obviously follows from the same conditions and Hölder’s inequality.

Estimation of \(Tg_t\). By the boundedness of the operator T in the space \(L^{p}({\mathbb {R}}^{n})\) we obtain

which yields the estimate of \(T g_t\) in Herz space taking into account Lemma 2.3.

Estimation of \(Th_t\). We take \(x\in B(\xi , 2t) \setminus B(\xi ,t)\) and by the size condition it follows

$$\begin{aligned} |Th_t(x)|\leqslant C \int _{|y - \xi |> 8 t}\frac{|f(y)|}{|x-y|^n} \mathrm{d}y. \end{aligned}$$

Now from (17) we have

Since . Using Hölder’s inequality we arrive at

where \( \varphi (\tau ) = \omega (\tau ) \Vert f{\mathbf {1}}_{ B(\xi , 2 \tau ) \setminus B(\xi , \tau ) }\Vert _{L^{p}({\mathbb {R}}^n)} \).

Proceeding as in the estimation of \( T f_t \) we see that

$$\begin{aligned} \omega (t) \Vert Th_t \, {\mathbf {1}}_{B(\xi , 2 t ) \setminus B(\xi ,t)}\Vert _{L^{p}({\mathbb {R}}^n)} \leqslant C \left( {\mathcal {H}}^{\gamma _1} \varphi + {\mathcal {H}}^{\gamma _2} \varphi + {\mathcal {A}} \varphi \right) , \end{aligned}$$
(21)

where \( \gamma _1 = m_0 (\omega ) - \varepsilon + \frac{n}{p } \), \( \gamma _2 = m_\infty (\omega ) - \varepsilon + \frac{n}{p } \) and

By Lemma 4.2 the Hardy operators \( {\mathcal {H}}^{\gamma _1} \) and \( {\mathcal {H}}^{\gamma _2} \) are bounded under the conditions (18)–(19) taken into account that \( \varepsilon \) may be taken arbitrarily small.

The boundedness of the operator \( {\mathcal {A}}\) in obviously follows from the same conditions and Hölder’s inequality. \(\square \)

Remark 4.4

The Theorem 4.3 covers in particular the maximal operator

$$\begin{aligned} M f(x)=\sup \limits _{r>0} \frac{1}{|B(x,r)|}\int \limits _{B(x,r)}|f(y)|\mathrm{d}y, \end{aligned}$$

the Hardy operators

$$\begin{aligned} H f(x) = \frac{1}{|x|^n} \int _{|y|<|x|} f(y) \mathrm{d}y, \quad {\mathcal {H}} f(x) = \int _{|y|>|x|} \frac{f(y)}{|y|^n}\mathrm{d}y, \end{aligned}$$

and the Calderón–Zygmund type singular operators with standard kernels

$$\begin{aligned} T f(x)=\lim \limits _{\varepsilon \rightarrow 0}\int _{{\mathbb {R}}^n\backslash B(x,\varepsilon )}K(x,y) f(y) \mathrm{d}y, \end{aligned}$$

which are bounded in \(L^2({\mathbb {R}}^n)\) and have a standard singular kernel K(x, y), i.e. K(x, y) is continuous on \( \{(x,y)\in {\mathbb {R}}^n\times {\mathbb {R}}^n: x\ne y\}\) and

$$\begin{aligned}&|K(x,y)|\leqslant C|x-y|^{-n} \;\;\text{ for } \text{ all } \;\;x\ne y,\\&|K(x,y)-K(x,z)|\leqslant C \frac{|y-z|^\sigma }{|x-y|^{n+\sigma }},\;\; \sigma>0,\;\; \text{ if }\;\;|x-y|>2|y-z|,\\&|K(x,y)-K(\xi ,y)|\leqslant C \frac{|x-\xi |^\sigma }{|x-y|^{n+\sigma }},\;\; \sigma>0,\;\; \text{ if }\;\;|x-y|>2|x-\xi |. \end{aligned}$$

Example 4.5

Let

$$\begin{aligned} \omega (t) = {\left\{ \begin{array}{ll} t^{\alpha _0} \left( \ln \left( \frac{\mathrm {e}}{t} \right) \right) ^{\beta _0}, &{} t < 1 \\ t^{\alpha _\infty } \left( \ln (\mathrm {e}t) \right) ^{\beta _\infty }, &{} t \geqslant 1. \end{array}\right. } \end{aligned}$$

Then any sublinear operator satisfying the size condition (15) and bounded in \( L^p({\mathbb {R}}^n ) \) is bounded in the generalized Herz spaces \( {\varvec{H}}^{p,q}_{\omega } ({\mathbb {R}}^n ) \) and \( {\varvec{H}}^{p,q}_{\omega ,0} ({\mathbb {R}}^n ) \) under the conditions:

$$\begin{aligned} -\frac{n}{p}< \alpha _0< \frac{n}{p^\prime }, \quad -\frac{n}{p}< \alpha _\infty < \frac{n}{p^\prime }. \end{aligned}$$

Corollary 2

Let \( 1< p < \infty \), \( 1< q < \infty \), and \( \omega \in M({\mathbb {R}}_+) \). Then the maximal operator M, the Hardy operators H and \({\mathcal {H}} \), and Calderón-Zygmund singular operator with standard kernel T are bounded in:

  1. (a)

    \( {\varvec{M}}^{p,q}_{\omega } ({\mathbb {R}}^n) \) if \( -\frac{n}{p}< m_0(\omega ) \leqslant M_0 (\omega )< 0 \) and \( -\frac{n}{p}< m_\infty (\omega ) \leqslant M_\infty (\omega ) <0 \),

  2. (b)

    \( ^\complement {\varvec{M}}^{p,q}_{\omega ,0} (\mathbb R^n) \) if \( 0< m_0 (\omega ) \leqslant M_0(\omega )< \frac{n}{p^\prime } \) and \( 0< m_\infty (\omega ) \leqslant M_\infty (\omega ) < \frac{n}{p^\prime } \).

Proof

Apply Theorem 4.3 and Theorem 3.2. \(\square \)

An overview of results on the boundedness of various operators of harmonic analysis in Morrey type spaces may be found in [3, 4], where conditions on the boundedness are given in some integral type assumptions on the function \( \omega \), not in terms of its indices.