Decompositions of local Morrey-type spaces

Abstract

We develop and apply a decomposition theory for generic local Morrey-type spaces. Our result is nonsmooth decomposition, which follows from the fact that local Morrey-type spaces are isomorphic to Hardy local Morrey-type spaces in the generic case. As an application of our results, we consider the Hardy operator.

Appendices

Appendix

We can relate local Morrey-type spaces with Herz spaces.

Lemma 8.1

Let $1<p<\infty$, $1 \le \theta \le \infty$ and $0<\lambda <\frac{n}{p}$. Then

$$\begin{aligned} \Vert f\Vert _{LM^\lambda _{p\theta }}&\sim \left\{ \sum _{j=-\infty }^\infty \left( 2^{-\lambda j} \left( \int _{2^{j-1}<|y|<2^j}|f(y)|^p\,dy\right) ^{\frac{1}{p}} \right) ^\theta \right\} ^{\frac{1}{\theta }} \end{aligned}$$

for all measurable functions $f:{\mathbb R}^n \rightarrow {\mathbb C}$.

The right-hand side above is called the Herz norm.

Proof

It is clear from (1.14) that

$$\begin{aligned} \Vert f\Vert _{LM^\lambda _{p\theta }} \gtrsim \left\{ \sum _{j=-\infty }^\infty \left( 2^{-\lambda j} \left( \int _{2^{j-1}<|y|<2^j}|f(y)|^p\,dy\right) ^{\frac{1}{p}} \right) ^\theta \right\} ^{\frac{1}{\theta }}. \end{aligned}$$

To prove the reverse estimate, we have

$$\begin{aligned} \Vert f\Vert _{LM^\lambda _{p\theta }}&\sim \left\{ \sum _{j=-\infty }^\infty \left( 2^{-\lambda j} \left( \int _{|y|<2^j}|f(y)|^p\,dy\right) ^{\frac{1}{p}} \right) ^\theta \right\} ^{\frac{1}{\theta }}\\&= \left\{ \sum _{j=-\infty }^\infty \left( \sum _{k=-\infty }^j 2^{-\lambda j} \left( \int _{2^{k-1}<|y|<2^k}|f(y)|^p\,dy\right) ^{\frac{1}{p}} \right) ^\theta \right\} ^{\frac{1}{\theta }}\\&= \left\{ \sum _{j=-\infty }^\infty \left( \sum _{k=-\infty }^\infty \chi _{(-\infty ,j]}(k) 2^{-\lambda j} \left( \int _{2^{k-1}<|y|<2^k}|f(y)|^p\,dy\right) ^{\frac{1}{p}} \right) ^\theta \right\} ^{\frac{1}{\theta }}\\&\le \sum _{k=-\infty }^\infty \left\{ \sum _{j=-\infty }^\infty \left( \chi _{(-\infty ,j]}(k) 2^{-\lambda j} \left( \int _{2^{k-1}<|y|<2^k}|f(y)|^p\,dy\right) ^{\frac{1}{p}} \right) ^\theta \right\} ^{\frac{1}{\theta }}\\&= \sum _{k=-\infty }^\infty \left\{ \left( \frac{1}{1-2^{-\lambda }} \cdot 2^{-\lambda k} \left( \int _{2^{k-1}<|y|<2^k}|f(y)|^p\,dy\right) ^{\frac{1}{p}} \right) ^\theta \right\} ^{\frac{1}{\theta }}, \end{aligned}$$

as was to be shown. $\square$

Comparison of various Morrey spaces and Nikolskii spaces

1.1 Global weighted Morrey type spaces

Theorems 1.3 and 1.4 are translated into the following results on global Morrey spaces.

Theorem 9.1

Let $1<p<\infty$, $1< \theta \le \infty$ and $w\in \Omega _{\theta }$. Assume that $w$ satisfies the doubling condition; $C^{-1}w(r) \le w(2r) \le Cw(r)$ for all $r>0$. We define $\hat{v}_1,\hat{v}_2$ by (1.8). Assume that $H^{*}$ is bounded on $L_{\theta ,\hat{v}_1}(0,\infty )$ to $L_{\theta ,\hat{v}_2}(0,\infty )$. Suppose that a real parameter s satisfies (1.9) for all $r>0$. Assume that we are given $\{Q_j\}_{j=1}^\infty \subset {\mathcal Q}({\mathbb R}^n)$, $\{a_j\}_{j=1}^\infty \subset L^s({\mathbb R}^n)$, $\{\lambda _j\}_{j=1}^\infty \subset [0,\infty )$ satisfying

$$\begin{aligned} \Vert a_j\Vert _{L^s} \le \Vert \chi _{Q_j}\Vert _{L^s}=|Q_j|^{1/s}, \quad \mathrm{supp}(a_j) \subset Q_j, \quad \left\| \sum _{j=1}^\infty \lambda _j\chi _{Q_j} \right\| _{GM_{p\theta ,w(\cdot )}} <\infty . \end{aligned}$$

(9.1)

Then the series $f \equiv \sum _{j=1}^\infty \lambda _j a_j$ converges in $L^1_\mathrm{loc}({\mathbb R}^n)$ and in the Schwartz space ${\mathcal S}'({\mathbb R}^n)$ of tempered distributions and satisfies the estimate

$$\begin{aligned} \Vert f\Vert _{GM_{p\theta ,w(\cdot )}} \le C\left\| \sum _{j=1}^\infty \lambda _j\chi _{Q_j} \right\| _{GM_{p\theta ,w(\cdot )}}, \end{aligned}$$

(9.2)

where $C>0$ depends only on $n,p,q,w$ and s.

Proof

Just use (1.7) and reexamine the proof of Theorem 1.4. We omit the further detail. $\square$

Theorem 9.2

Let $L \in {\mathbb N}_0={\mathbb N} \cup \{0\}$, $1<p<\infty$, $1<\theta \le \infty$ and $w \in \Omega _{\theta }$. Assume that $w$ satisfies the doubling condition; $C^{-1}w(r) \le w(2r) \le Cw(r)$ for all $r>0$. We define $\hat{v}_1,\hat{v}_2$ by (1.8). Assume that $H^*$ is bounded from $L_{\theta ,\hat{v}_1}(0,\infty )$ to $L_{\theta ,\hat{v}_2}(0,\infty )$. Suppose that a real parameter s satisfies (1.9) for all $r>0$. Let $f \in LM_{p\theta ,w(\cdot )}({\mathbb R}^n)$. Then there exist $\{\lambda _j\}_{j=1}^\infty \subset [0,\infty )$, $\{Q_j\}_{j=1}^\infty \subset {\mathcal Q}({\mathbb R}^n)$ and $\{a_j\}_{j=1}^\infty \subset L^\infty ({\mathbb R}^n)$ such that $f\equiv \sum _{j=1}^\infty \lambda _j a_j$ converges in ${\mathcal S}'({\mathbb R}^n) \cap L^1_\mathrm{loc}({\mathbb R}^n)$, that $a_j$ satisfies (1.12) for all multi-indices $\alpha =(\alpha _1,\alpha _2,\ldots ,\alpha _n)$ with $|\alpha | \equiv \alpha _1+\alpha _2+\cdots +\alpha _n \le L$ and, that for all $v>0$

$$\begin{aligned} \left\| \left( \sum _{j=1}^\infty (\lambda _j\chi _{Q_j})^{v} \right) ^{1/v}\right\| _{GM_{p\theta ,w(\cdot )}} \le C_v \Vert f\Vert _{GM_{p\theta ,w(\cdot )}}. \end{aligned}$$

(9.3)

Here the constant $C_v>0$ is independent of f.

Proof

Use the method of Theorem 9.1 to obtain $\{\lambda _j\}_{j=1}^\infty \subset [0,\infty )$, $\{Q_j\}_{j=1}^\infty \subset {\mathcal Q}({\mathbb R}^n)$ and $\{a_j\}_{j=1}^\infty \subset L^\infty ({\mathbb R}^n)$ such that $f\equiv \sum _{j=1}^\infty \lambda _j a_j$ converges in ${\mathcal S}'({\mathbb R}^n) \cap L^1_\mathrm{loc}({\mathbb R}^n)$, that $a_j$ satisfies (1.12) and (1.13) for all multi-indices $\alpha =(\alpha _1,\alpha _2,\ldots ,\alpha _n)$ with $|\alpha | \equiv \alpha _1+\alpha _2+\cdots +\alpha _n \le L$ and for all $v>0$. We need to check (9.3). To this end, we have only to show

$$\begin{aligned} \left\| \left( \sum _{j=1}^\infty (\lambda _j\chi _{Q_j}(\cdot +x))^{v} \right) ^{1/v}\right\| _{LM_{p\theta ,w(\cdot )}} \le C_v \Vert f(\cdot +x)\Vert _{LM_{p\theta ,w(\cdot )}}. \end{aligned}$$

However, since

$$\begin{aligned} f=\sum _{j=1}^\infty \lambda _j a_j \end{aligned}$$

is the atomic decomposition of $f \in LM_{p\theta ,w(\cdot )}$, we can say that

$$\begin{aligned} f(\cdot +x)=\sum _{j=1}^\infty \lambda _j a_j(\cdot +x) \end{aligned}$$

is the atomic decomposition of f. Thus, we obtain

$$\begin{aligned} \left\| \left( \sum _{j=1}^\infty (\lambda _j\chi _{Q_j}(\cdot +x))^{v} \right) ^{1/v}\right\| _{LM_{p\theta ,w(\cdot )}}&\le C_v \Vert {\mathcal M}[f(\cdot +x)]\Vert _{LM_{p\theta ,w(\cdot )}}\\&=C_v \Vert {\mathcal M}f(\cdot +x)\Vert _{LM_{p\theta ,w(\cdot )}}\\&\le C_v \Vert f(\cdot +x)\Vert _{LM_{p\theta ,w(\cdot )}}, \end{aligned}$$

as was to be shown. $\square$

1.2 Classical Morrey spaces

The classical Morrey spaces $M^{\lambda }_{p}({\mathbb R}^n)$ were first introduced by Morrey in [34] to study the local behavior of solutions to second order elliptic partial differential equations. For the boundedness of the Hardy–Littlewood maximal operator, the fractional integral operator and the Calderón-Zygmund singular integral operator on these spaces, we refer the readers to [1, 20, 38]. For the properties and applications of classical Morrey spaces, see [21, 23] and references therein.

Morrey spaces $M^{\lambda }_{p}({\mathbb R}^n)$, named after Morrey, were based on his study of elliptic differential operators in 1938 [34] and they are defined as follows: For $\lambda \in \mathbb {R}, 0<p\le \infty , f\in M^{\lambda }_{p}({\mathbb R}^n)$ if $f\in L^p_\mathrm{loc}({\mathbb R}^n)$ and

$$\begin{aligned} \Vert f\Vert _{M_p^\lambda }\equiv \Vert f\Vert _{M_p^\lambda ({\mathbb R}^n)}=\sup _{x\in \mathbb {R}^n, r>0} r^{-\lambda }\Vert f\Vert _{L^p(B(x,r))}<\infty , \end{aligned}$$

where B(x, r) is the open ball in $\mathbb {R}^n$ centered at the point $x\in \mathbb {R}^n$ of radius $r>0$.

In other words $f\in M^{\lambda }_{p}({\mathbb R}^n)$ if $f\in L^p_\mathrm{loc}(\mathbb {R}^n)$ and there exists $c>0$ (depending on f) such that for all $x\in \mathbb {R}^n$ and for all $r>0$

$$\begin{aligned} \Vert f\Vert _{L^p(B(x,r))}\le cr^\lambda . \end{aligned}$$

The minimal value of c in this inequality is $\Vert f\Vert _{M_p^\lambda }$.

If $\lambda =0$, then

$$\begin{aligned} M_p^0({\mathbb R}^n)=L^p({\mathbb R}^n). \end{aligned}$$

If $\lambda =\frac{n}{p}$, then

$$\begin{aligned} M_p^{\frac{n}{p}}({\mathbb R}^n)=L^\infty ({\mathbb R}^n). \end{aligned}$$

If $\lambda >\frac{n}{p}$ or $\lambda <0$, then

$$\begin{aligned} M_p^\lambda ({\mathbb R}^n)=\Theta , \end{aligned}$$

where $\Theta \equiv \Theta (\mathbb {R}^n)$ is the set of all functions equivalent to 0 on $\mathbb {R}^n$.

So the admissible range of the parameters is

$$\begin{aligned} 0<p\le \infty \quad and\quad 0\le \lambda \le \frac{n}{p}. \end{aligned}$$

(9.4)

The cases $p=\infty$ forces $\lambda$ to be 0 and $M_\infty ^0({\mathbb R}^n)=L^\infty ({\mathbb R}^n)$. Under these assumptions, which will always be assumed in the sequel, the space $M^{\lambda }_{p}({\mathbb R}^n)$ is a Banach space for $1\le p\le \infty$ and a quasi-Banach space for $0<p<1$.

Also the space $M_p^\lambda ({\mathbb R}^n)$ does not coincide with a Lebesgue space, if and only if

$$\begin{aligned} 0<p<\infty \quad and\quad 0<\lambda <\frac{n}{p}. \end{aligned}$$

(9.5)

Furthermore,

$$\begin{aligned} L^{\infty }({\mathbb R}^n)\cap L^p({\mathbb R}^n)\subset M_p^\lambda ({\mathbb R}^n). \end{aligned}$$

If $f\in L^p$, then $f\in M_p^\lambda ({\mathbb R}^n)$ if and only if $\sup _{x\in \mathbb {R}^n, 0<r\le 1}r^{-\lambda }\Vert f\Vert _{L^p(B(x,r))}<\infty$, hence under this assumption only local properties of f are of importance.

Consider the Nikolskii space $H_p^\lambda \equiv H_p^\lambda (\mathbb {R}^n)$ of functions possessing common smoothness of order $\lambda$ measured in the $L^p$ metrics. For $\lambda >0, 1\le p\le \infty$ they are defined in the following way: fix an integer $\sigma >\lambda$. We say that $f\in H_p^\lambda ({\mathbb R}^n)$ if $f\in L^p({\mathbb R}^n)$ and

$$\begin{aligned} \Vert f\Vert _{H_p^\lambda }=\Vert f\Vert _{L^p}+\sup _{h\in \mathbb {R}^n, h\ne 0}\left| h\right| ^{-\lambda }\left\| \Delta _h^\sigma f\right\| _{L^p}<\infty , \end{aligned}$$

where $\Delta _h^\sigma f$ is the difference of f of order $\sigma \in \mathbb {N}$ with step h. For different $\sigma >\lambda$ the definitions are equivalent.) One can prove that if $0<\lambda <\frac{n}{p}$, then

$$\begin{aligned} H_p^\lambda ({\mathbb R}^n)\subset M_p^\lambda ({\mathbb R}^n). \end{aligned}$$

We refer to [31] for $n = 1$ and [35, 36] $n>1$. Clearly the converse inclusion does not hold, because if $f\in M_p^\lambda ({\mathbb R}^n)$, then clearly $fg\in M_p^\lambda ({\mathbb R}^n)$ for any bounded measurable function g, which is not true for the case of the spaces $H_p^\lambda ({\mathbb R}^n)$. So, $M_p^\lambda ({\mathbb R}^n)$ is not a space of functions possessing any kind of common smoothness of order $\lambda$, but the expressions $\Vert f\Vert _{L^p(B(x,r))}$ behave like the ones for functions f possessing certain smoothness of order $\lambda$. Detailed exposition of properties of these spaces can be found in [9, 36]. Note that the expression for $\Vert f\Vert _{LM_{p\theta }^\lambda }$ is very similar to the semi-norms $\Vert f\Vert _{b_{p\theta }^\lambda }$ of the Nikol’skii-Besov spaces $B_{{p\theta }}^\lambda$. In the latter case, we suppose $\lambda >0, 1\le p,\theta \le \infty$ and $\Vert f\Vert _{L^p(B(r))}$ should be replaced by the $L^p$ modulus of continuity: $w^\sigma (f,r)=\sup _{\left| h\right| \le r}\left\| \Delta _h^\sigma f\right\| _{L^p(\mathbb {R}^n)}$ with $\sigma >\lambda$. Recall that $\Vert f\Vert _{B_{p\theta }^\lambda }=\Vert f\Vert _{L^p}+\Vert f\Vert _{b_{p\theta }^\lambda }$. If $\theta =\infty$ then $B_{p\infty }^\lambda ({\mathbb R}^n)\equiv H_p^\lambda ({\mathbb R}^n)$. There are several definitions, equivalent for these values of the parameters, of the spaces $B_{p\theta }^\lambda ({\mathbb R}^n)$. The definition mentioned above makes sense for a wider range of the parameters, namely for $\lambda >0, 0<p,\theta \le \infty$. For this range of the parameters the equivalence of the quasi-norms $\left\| \cdot \right\| _{B_{p\theta }^\lambda }$ for different $\sigma >\lambda$ was proved in [20]. If $\theta =p$ then

$$\begin{aligned} \Vert f\Vert _{LM_{pp}^\lambda }=\left( \lambda p\right) ^{-\frac{1}{p}}\left( \int _{\mathbb {R}^n}\frac{\left| f(x)\right| ^p}{\left| x\right| ^{\lambda p}}dx\right) ^{\frac{1}{p}}. \end{aligned}$$

(9.6)

For $n=1, 1\le p,\theta<\infty , 0<\lambda <\frac{1}{p}$ the inclusion

$$\begin{aligned} B_{p\theta }^\lambda ({\mathbb R}^n)\subset GM_{p\theta }^\lambda ({\mathbb R}^n) \end{aligned}$$

(9.7)

was proved by Kuznetsov [32]. In the diagonal case $p=\theta$ (9.7) follows by equality (9.6) and the estimate of the right-hand side of (4.2) via $\Vert f\Vert _{b_{pp}^\lambda }$ for functions $f\in B_{pp}^\lambda$, proved by Yakovlev [42, 43].

Let us recall some results on local Morrey-type spaces. In 1994 Guliyev initially introdused and studied the local Morrey-type spaces in his doctoral thesis [24]; see also [25]. The main purpose of [24, 25] is to give some sufficient conditions for the boundedness of fractional integral operators and singular integral operators defined on homogeneous Lie groups in the local Morrey-type spaces.

In a series of papers [2, 3, 10,11,12,13,14,15,16] by Burenkov, Husein Guliyev and Vagif Guliyev etc. some necessary and sufficient conditions for the boundedness of fractional maximal operators, fractional integral operators and singular integral operators in local Morrey-type spaces $LM_{p\theta ,w(\cdot )}({\mathbb R}^n)$ were given. The fractional maximal operator, the Hardy–Littlewood maximal operator, the fractional integral operator and the Marcinkiewicz operator are considered in [10, 13, 14, 16, 28], respectively. We refer to [40] for the two-weight estimates for the Hardy–Littlewood maximal operators.

1.3 Local Morrey-type spaces and interpolation

Investigating local Morrey-type spaces is not a mere quest to generality; it appears naturally in the context of real interpolation. In [18], Burenkov and Nursultanov established that

$$\begin{aligned} ( L^p(\Omega ,w^{\lambda _0},\mu ), L^p(\Omega ,w^{\lambda _1},\mu ) )_{\theta ,q} \end{aligned}$$

is a generalized local Morrey-type space, when we are given weights. More precisely, we can state the result as follows: We start with generalizing the space $LM_{p\theta ,w(\cdot )}({\mathbb R}^n)$. Let $0<p,q \le \infty$, $0<\lambda <\infty$ if $q<\infty$, $0 \le \lambda <\infty$ if $q=\infty$. Let $\Omega \subset {\mathbb R}^n$ be a measurable set and $\mu$ be a $\sigma$-finite Borel measure on $\Omega$. Moreover, let $G=(G_t)_{t>0}$ where all the $G_t$’s are $\mu$-measurable subsets for which $G_t \ne \Omega$ for some $t>0$, $G_{t_1} \subset G_{t_2}$ if $t_1<t_2$, and $\bigcup _{t>0}G_t=\Omega$. We define

$$\begin{aligned} \Vert f\Vert _{LM^\lambda _{p,q}(G,\mu )} \equiv \left( \int _0^\infty (t^{-\lambda }\Vert f\Vert _{L^p(G_t,\mu )})^q\frac{dt}{t} \right) ^{\frac{1}{q}}. \end{aligned}$$

To formulate the interpolation result, we consider a special case of G. Let $w_0,w_1$ be positive $\mu$-measurable functions on $\Omega \subset {\mathbb R}^n$ and $0<\lambda _0,\lambda _1<\infty$. Let the family $G_{\lambda _0,\lambda _1}=(G_{t,\lambda _0,\lambda _1})_{t>0}$ be defined by:

$$\begin{aligned}&\displaystyle G_{t,\lambda _0,\lambda _1} = \{x \in \Omega \,:\, w_0(x)^{\alpha _0}w_1(x)^{\alpha _1}<t\} \quad (t>0).\\&\displaystyle d\nu _{\lambda _0,\lambda _1}(x) = (w_0(x)^{\beta _0}w_1(x)^{\beta _1})^p\,d\mu (x), \end{aligned}$$

where

$$\begin{aligned} \alpha _0=\frac{1}{\lambda _1-\lambda _0}, \quad \alpha _1=\frac{1}{\lambda _0-\lambda _1}, \quad \beta _0=\frac{\lambda _1}{\lambda _1-\lambda _0}, \quad \beta _1=\frac{\lambda _0}{\lambda _0-\lambda _1}. \end{aligned}$$

Observe that

$$\begin{aligned} LM^{\lambda _0}_{p,p}(G_{\lambda _0,\lambda _1},\nu _{\lambda _0,\lambda _1}) = L^p(\Omega ,w_0,\mu ), \quad LM^{\lambda _1}_{p,p}(G_{\lambda _0,\lambda _1},\nu _{\lambda _0,\lambda _1}) = L^p(\Omega ,w_1,\mu ). \end{aligned}$$

In words of the book [8], Burenkov and Nursultanov obtained the following interpolation result (9.8) in [18], which compliments the Stein-Weiss type interpolation (9.9):

Theorem 9.3

Suppose that the parameters $p,q,\lambda _0,\lambda _1,\theta \in (0,\infty ]$ satisfy

$$\begin{aligned} \lambda _0,\lambda _1<\infty , \quad \lambda _0 \ne \lambda _1, \quad \theta <1, \end{aligned}$$

and $\lambda =(1-\theta )\lambda _0+\theta \lambda _1$. Then

$$\begin{aligned} (L^p(\Omega ,w_0,\mu ),L^p(\Omega ,w_1,\mu ))_{\theta ,q} = LM^{\lambda }_{p,q}(G_{\lambda _0,\lambda _1},\nu _{\lambda _0,\lambda _1}). \end{aligned}$$

(9.8)

If $q=p$,

$$\begin{aligned} (L^p(\Omega ,w_0,\mu ),L^p(\Omega ,w_1,\mu ))_{\theta ,p} = L^p(\Omega ,w_0{}^{1-\theta }w_1{}^{\theta },\mu ). \end{aligned}$$

(9.9)

Our method seems to be applicable to the anisotropic local Morrey-type spaces defined in [2]. Based on the definition above, Akbulut, Guliyev and Muradova discussed the boundedness property of the anisotropic Riesz potential in the anisotropic local Morrey-type spaces in [3]. We feel that the method employed in [29] seems to be applicable once we obtain a counterpart of Theorems 1.1 and 1.2. In [5], Aykol, Guliyev and Serbetci defined the local Lorentz Morrey spaces as the set of all measurable functions f for which the quasi-norm

$$\begin{aligned} \Vert f\Vert _{M^\mathrm{loc}_{p,q;1}} \equiv \sup _{t>0}t^{-\lambda /q}\Vert s^{1/p-1/q}f^*(s)\Vert _{L_q(0,t)}<\infty . \end{aligned}$$

In [5, Theorem 4.1], Aykol, Guliyev and Serbetci obtained the boundedness of the Hardy–Littlewood maximal operator in the local Lorentz Morrey spaces. In [6, Theorem 3.1], Aykol, Guliyev, Kucukaslan and Serbetci obtained the boundedness of the Hilbert transform in the local Lorentz Morrey spaces. The modification of the argument to the anisotropic setting or to the local Lorentz Morrey spaces will be left as a future work.