# Singular and fractional integral operators on weighted local Morrey spaces

Javier Duoandikoetxea and Marcel Rosenthal  Sopela, Basque Country, Spain
<EMAIL_ADDRESS>Stuttgart, Germany<EMAIL_ADDRESS>
###### Abstract.

We obtain a characterization of the weighted inequalities for the Riesz
transforms on weighted local Morrey spaces. The condition is sufficient for
the boundedness on the same spaces of all Calderón-Zygmund operators suitably
defined on the functions of the space. In the case of the fractional maximal
operator and the fractional integral we obtain a characterization valid for
exponents satisfying the Sobolev relation. For power weights we get sharp
results in a larger range of exponents for the usual versions of weighted
Morrey spaces.

###### Key words and phrases:

Local Morrey spaces, weights, Calderón-Zygmund operators, fractional maximal
operator, fractional integral

###### 2010 Mathematics Subject Classification:

42B20, 42B35, 46E30, 47G40

##  1\. Introduction

We define for $1\leq p<\infty$ the weighted local (also known as central)
Morrey space $\mathcal{LM}^{p}(\varphi,w)$ as the class of measurable
functions in $\mathbb{R}^{n}$ for which

(1.1)
$\|f\|_{\mathcal{LM}^{p}(\varphi,w)}:=\sup_{B}\left(\frac{1}{\varphi(B)}\int_{B}|f|^{p}w\right)^{1/p}<\infty,$

where the supremum is taken over all the Euclidean balls in $\mathbb{R}^{n}$
centered at $0$. Here $w$ is a weight, that is, a nonnegative locally
integrable function, and $\varphi$ is a function defined on the set of balls
in $\mathbb{R}^{n}$ centered at $0$, with values in $(0,\infty)$. For the
global Morrey spaces all the Euclidean balls in $\mathbb{R}^{n}$ are taken.

The most studied cases of $\varphi$ correspond to the weighted Morrey spaces
introduced by N. Samko in [17], and by Y. Komori and S. Shirai in [9]. In
general, we need to impose some conditions on $\varphi$ to avoid trivial
spaces. We fix some conditions in Section 2.

The characterization of the weights $w$ for which the Hardy-Littlewood maximal
operator is bounded on global Morrey spaces is an open problem even for the
most familiar cases of $\varphi$ mentioned in the preceding paragraph.
Nevertheless, a remarkable result of S. Nakamura, Y. Sawano, and H. Tanaka in
[16] gives necessary and sufficient conditions for the boundedness of the
Hardy-Littlewood maximal operator $M$ on weighted local Morrey spaces. In [4]
the authors of this paper obtained a simplified characterization of the same
result in the form of a Muckenhoupt-type condition adapted to the Morrey
setting. More precisely, we showed that the condition

(1.2)
$\sup_{B}\frac{\|\chi_{B}\|_{\mathcal{LM}^{p}(\varphi,w)}\|\chi_{B}\|_{\mathcal{LM}^{p}(\varphi,w)^{\prime}}}{|B|}<\infty$

is necessary and sufficient for the boundedness of $M$ on
$\mathcal{LM}^{p}(\varphi,w)$. (The notation $X^{\prime}$ stands for the Köthe
dual of the Banach lattice $X$. See Subsection 2.2.) In this paper we deal
first with singular integral operators and show that the slightly stronger
condition

(1.3)
$\sup_{B}\frac{\|\chi_{B}\|_{\mathcal{LM}^{p}(\varphi,w)}\|M\chi_{B}\|_{\mathcal{LM}^{p}(\varphi,w)^{\prime}}}{|B|}<\infty$

is necessary and sufficient for the boundedness of the Riesz transforms on
$\mathcal{LM}^{p}(\varphi,w)$. This is valid for $1<p<\infty$, and for $p=1$
the result holds if we consider weak-type inequalities. The sufficiency is
extended to all Calderón-Zygmund operators. We remark that due to the lack of
density of smooth functions in the Morrey-type spaces the operators need to be
defined in a convenient way. Condition (1.3) is the limiting case of the
sufficient condition obtained in [4, Theorem 7.3] by using extrapolation of
Lebesgue-weighted inequalities.

Next we work with the fractional maximal operator and the fractional
integrals. In this case our results are limited to a particular form of the
spaces. In the classical paper [14], B. Muckenhoupt and R. Wheeden
characterized the weights $w$ such that the fractional integral $I_{\alpha}$
is bounded from $L^{p}(w^{p})$ to $L^{q}(w^{q})$, when $p$ and $q$ are related
by the Sobolev condition $1/p-1/q=\alpha/n$. The characterizations we obtain
for the boundedness of the fractional operators on weighted local Morrey
spaces are also limited by the same condition on the exponents and similar
conditions on the weights. More precisely, we obtain necessary and sufficent
conditions for the boundedness of $M_{\alpha}$ and $I_{\alpha}$ from
$\mathcal{LM}^{p}(\varphi^{p},w^{p})$ to
$\mathcal{LM}^{q}(\varphi^{q},w^{q})$, with $p$ and $q$ as above. They are
given by expressions which are similar to (1.2) and (1.3), respectively. In
the case of $M_{\alpha}$ a necessary and sufficient condition was already
obtained in [16]. As we did for the Hardy-Littlewood maximal operator, we
simplify the characterization and give it by means of a single condition,
namely,

(1.4)
$\sup_{B}\frac{\|\chi_{B}\|_{L\mathcal{M}^{q}(\varphi^{q},w^{q})}\|\chi_{B}\|_{L\mathcal{M}^{p}(\varphi^{p},w^{p})^{\prime}}}{|B|^{1-\alpha/n}}<\infty.$

For the characterization of the boundedness of $I_{\alpha}$ we need a stronger
condition as happened for singular integrals. In this case the condition reads

(1.5)
$\sup_{B}\frac{\|\chi_{B}\|_{L\mathcal{M}^{q}(\varphi^{q},w^{q})}\|(M\chi_{B})^{1-\alpha/n}\|_{L\mathcal{M}^{p}(\varphi^{p},w^{p})^{\prime}}}{|B|^{1-\alpha/n}}<\infty.$

A term of the form $(M\chi_{B})^{1-\alpha/n}$ already appeared in the
characterization of two-weighted Lebesgue inequalities for $M_{\alpha}$ and
$I_{\alpha}$ (see [19] and [21]).

It is known that in the case of unweighted global Morrey spaces the range of
$p$ and $q$ can be extended to $1/p-1/q=\alpha/(n-\lambda)$ (where
$\varphi(B)=r_{B}^{\lambda}$), due to a result of D. R. Adams. Weighted
estimates on global Morrey spaces in the Adams range were obtained by T. Iida,
Y. Komori-Furuya, and E. Sato in [8], and more generally by S. Nakamura, Y.
Sawano, and H. Tanaka in [15]. It is known that for unweighted local Morrey
spaces the result in the Adams range does not hold (see [10]). We extend the
impossibility of Adams-type estimates also to the weighted setting (Theorem
5.2).

In our approach in [4] a crucial role was played by the local Hardy-Littlewood
maximal operator, where local refers to the use of balls separated from the
origin. These operators and their weighted inequalities were studied in [13]
and [6]. In this paper we also need local versions of the operators we study.
The weighted results we use were given in [7] and are presented in Subsection
2.4.

In Section 7 we consider power weights and obtain sharp results for the
special cases of weighted local Morrey spaces of Samko and Komori-Shirai type.

### Notation

  * •

Given a set $A$, $\chi_{A}$ is its characteristic function, $|A|$ is its
Lebesgue measure, and for a weight $v$, $v(A)$ is the integral of $v$ over $A$
(but notice that $\varphi$ is not a weight in the definition of the norm of
the Morrey space);

  * •

for a ball $B$, $r_{B}$ will denote its radius and $c_{B}$ its center; for
$\lambda>0$, $\lambda B$ denotes the ball with center $c_{B}$ and radius
$\lambda r_{B}$; $\widetilde{B}$ is the smallest ball centered at the origin
containing $B$, that is, $\widetilde{B}=B(0,|c_{B}|+r_{B})$;

  * •

For $\varphi$ defined on balls we denote as $\varphi^{p}$ the function such
that $\varphi^{p}(B)=\varphi(B)^{p}$;

  * •

$F\lesssim G$ means that the inequality $F\leq c\,G$ holds with a constant $c$
depending only on the ambient parameters; and $F\sim G$ means that both
$F\lesssim G$ and $G\lesssim F$ hold.

##  2\. Preliminaries

The most used versions of weighted Morrey spaces correspond to the choices
$\varphi(B)=r_{B}^{\lambda}$ ($0<\lambda<n$), introduced by N. Samko in [17],
and $\varphi(B)=w(B)^{\lambda/n}$ ($0<\lambda<n$), introduced by Komori and
Shirai in [9]. Generalized weighted (global and local) Morrey spaces have been
considered by several authors. For general weighted local Morrey spaces we
refer for instance to [16].

###  2.1. Conditions on $\varphi$

We assume that $\varphi$ is doubling, that is,

$\varphi(2B)\lesssim\varphi(B)$

for $B$ centered at the origin. We also assume that it satisfies a reverse
doubling property of the form

(2.1)
$\frac{\varphi(B(0,r))}{\varphi(B(0,R))}\lesssim\left(\frac{r}{R}\right)^{\delta},$

for some $\delta>0$. From this property it follows that if $0<b<1$ it holds
that

(2.2) $\sum_{j=0}^{\infty}\varphi(B(0,b^{j}R))\lesssim\varphi(B(0,R)).$

For a measure $\mu$ the doubling condition on all the Euclidean balls of
$\mathbb{R}^{n}$ implies the reverse doubling property (see [20, Lemma 20]).
When $\varphi$ is of the form $\varphi(B)=\mu(B)^{\lambda}$ for a doubling
measure $\mu$, then (2.1) is satisfied.

######  Proposition 2.1.

Assume that $\varphi$ satisfies (2.2). Then the norm (1.1) is equivalent to

(2.3)
$\sup_{B:r_{B}=\kappa|c_{B}|}\left(\frac{1}{\varphi(\widetilde{B})}\int_{B}|f|^{p}w\right)^{1/p}<\infty,$

where $\kappa\in(0,1)$ is fixed.

We recall that $\widetilde{B}$ denotes the smallest ball centered at the
origin containing $B$. The main idea for this reduction comes from [16]. We
adapt here to local Morrey spaces the proof given in [3, Proposition 2.1] for
global Morrey spaces.

###### Proof.

It is clear that (2.3) is not greater than (1.1) because
$B\subset\widetilde{B}$.

Let $b>1/3$. Set $A_{j}:=B(0,b^{j}R)\setminus B(0,b^{j+1}R)$. There exists
$C(n)$ depending only on the dimension $n$ such that the annulus $A_{j}$ can
be covered by $C(n)$ balls $B_{k,j}:=B(y_{k,j},r_{j})$, $k=1,\dots,C(n)$, for
which $|y_{k,j}|=b^{j}(1+b)R/2$ (that is, $y_{k,j}$ is equidistant from the
spheres of the boundary of the annulus) and $r_{j}=b^{j}(1-b)R$ (the width of
the annulus). By construction $\widetilde{B_{k,j}}$ is independent of $k$ and
satisfies $B(0,b^{j}R)\subset\widetilde{B_{k,j}}\subset B(0,b^{j-1}R)$. Then
$\varphi(\widetilde{B_{k,j}})\lesssim\varphi(B(0,b^{j-1}R))$. Consequently,

$\displaystyle\frac{1}{\varphi(B(0,R))}\int_{B(0,R)}|f|^{p}w=\frac{1}{\varphi(B(0,R))}\sum_{j=0}^{\infty}\int_{A_{j}}|f|^{p}w$
$\displaystyle\qquad\leq\frac{1}{\varphi(B(0,R))}\,C(n)\sum_{j=0}^{\infty}\varphi(B(0,b^{j-1}R))\sup_{k,j}\frac{1}{\varphi(\widetilde{B_{k,j}})}\int_{B_{k,j}}|f|^{p}w.$

The balls $B_{k,j}$ satisfy $r_{B}=\kappa|c_{B}|$ for $\kappa=2(1-b)/(1+b)$.
Using (2.2) we obtain the control of (1.1) by (2.3). ∎

######  Remark 2.2.

For a function $f$ supported in a ball $B$ with $r_{B}\leq\kappa|c_{B}|$
($0<\kappa<1$) it holds that

(2.4)
$\|f\|_{\mathcal{LM}^{p}(\varphi,w)}\sim\left(\frac{1}{\varphi(\widetilde{B})}\int_{B}|f|^{p}w\right)^{1/p}.$

This is because the balls $B(0,R)$ with $R<|c_{B}|-r_{B}$ do not intersect $B$
and $\varphi(B(0,R))$ is essentially constant for
$|c_{B}|-r_{B}<R<|c_{B}|+r_{B}$, due to the properties of $\varphi$. In
particular, if $f$ is the characteristic function of $B$,

(2.5)
$\|\chi_{B}\|_{\mathcal{LM}^{p}(\varphi,w)}\sim\left(\frac{w(B)}{\varphi(\widetilde{B})}\right)^{1/p}.$

######  Proposition 2.3.

Let $\gamma\in\mathbb{R}$. Define $\psi(B)=\varphi(B)r_{B}^{\gamma}$ and
$v(x)=w(x)|x|^{\gamma}$. If both $\psi$ and $\varphi$ satisfy (2.2), then
$\mathcal{LM}^{p}(\varphi,w)=\mathcal{LM}^{p}(\psi,v)$.

###### Proof.

It is enough to realize that for the balls $B$ involved in (2.3) if $x\in B$
we have $|x|\sim r_{B}$, and also $r_{B}\sim r_{\widetilde{B}}$. ∎

The spaces need not be the same without the reduction of the norm to (2.3).
For instance, if $\varphi(B)=r_{B}^{\lambda}$ with $\lambda>0$ and $w(x)\equiv
1$, the space $\mathcal{LM}^{p}(\varphi,w)$ is not trivial, but
$\mathcal{LM}^{p}(\psi,v)=\\{0\\}$ if $\lambda+\gamma$ is negative.

###  2.2. The Köthe dual (associate space) of a Banach lattice

Let $X$ be a Banach lattice of measurable functions in $\mathbb{R}^{n}$ (a
Banach space in which $|g|\leq|f|$ a.e. and $f\in X$ implies $g\in X$ and
$\|g\|_{X}\leq\|f\|_{X}$). We define its Köthe dual as the space $X^{\prime}$
collecting the measurable functions $g$ such that

(2.6) $\|g\|_{X^{\prime}}:=\sup\left\\{\int_{\mathbb{R}^{n}}|fg|:f\in X\text{
and }\|f\|_{X}\leq 1\right\\}<\infty.$

It is also a Banach lattice. An immediate consequence of (2.6) is the Hölder-
type inequality

(2.7) $\int_{\mathbb{R}^{n}}|fg|\leq\|f\|_{X}\|g\|_{X^{\prime}}.$

###  2.3. Weak-type local Morrey spaces

We define for $1\leq p<\infty$ the weighted weak-type local Morrey space
$W\mathcal{LM}^{p}(\varphi,w)$ as the collection of measurable functions in
$\mathbb{R}^{n}$ for which

(2.8)
$\|f\|_{W\mathcal{LM}^{p}(\varphi,w)}=\sup_{B}\sup_{t>0}\frac{t(w(\\{x\in
B:|f(x)|>t\\})^{1/p}}{\varphi(B)^{1/p}}<\infty.$

It is clear that $\mathcal{LM}^{p}(\varphi,w)$ is continuously embedded in
$W\mathcal{LM}^{p}(\varphi,w)$. Also that for a measurable set $E$, it holds

(2.9)
$\|\chi_{E}\|_{W\mathcal{LM}^{p}(\varphi,w)}=\|\chi_{E}\|_{\mathcal{LM}^{p}(\varphi,w)}.$

In particular, we have (2.5) with $W\mathcal{LM}^{p}(\varphi,w)$ instead of
$\mathcal{LM}^{p}(\varphi,w)$.

###  2.4. The local Hardy-Littlewood maximal operator, local fractional
operators and local singular integrals

For fixed $\kappa\in(0,1)$ we define the basis
$\mathcal{B}_{\kappa,\text{loc}}$ as the family of balls $B$ such that
$r_{B}<\kappa|c_{B}|$. For $\alpha\in[0,n)$ we define the operator
$M_{\kappa,\alpha,\text{loc}}$ associated to $\mathcal{B}_{\kappa,\text{loc}}$
acting on $f$ at $x\neq 0$ as

(2.10)
$M_{\kappa,\alpha,\text{loc}}f(x)=\sup_{x\in\mathcal{B}_{\kappa,\text{loc}}}\frac{1}{|B|^{1-\alpha/n}}\int_{B}|f|.$

For $\alpha=0$ this is the local Hardy-Littlewood maximal operator, which we
denote simply as $M_{\kappa,\text{loc}}$. For $\alpha\in(0,n)$ it is the local
fractional maximal operator. Notice that the term local here is associated to
a family of balls different from the one used in the definition of the local
Morrey spaces.

C.-C. Lin and K. Stempak studied in [13] a variant of $M_{\kappa,\text{loc}}$
and characterized its weighted inequalities by a local $A_{p}$ condition. E.
Harboure, O. Salinas, and B. Viviani obtained in [6] a similar result in a
more general setting, and in [7] they studied weighted inequalities for the
fractional case and for singular integrals. Their result applied to the
operator in (2.10) says the following: let $1<p<n/\alpha$ and $q$ given by
$1/p-1/q=\alpha/n$. Then $M_{\kappa,\alpha,\text{loc}}$ is bounded from
$L^{p}(w^{p})$ to $L^{q}(w^{q})$ if and only if $w$ satisfies the
$A_{p,q,\text{loc}}$ condition

(2.11)
$\sup_{B\in\mathcal{B}_{\kappa,\text{loc}}}\frac{w^{q}(B)^{1/q}\,w^{-p^{\prime}}(B)^{1/p^{\prime}}}{|B|^{1-\frac{\alpha}{n}}}<\infty.$

For $\alpha=0$ this is the usual $A_{p}$ condition adapted to the local
setting (written for $w^{p}$ instead of $w$), and for $\alpha>0$ this is the
local version of the condition for fractional operators in [14]. For $p=1$ and
$q=n/(n-\alpha)$ the condition is modified as usual,

(2.12) $\frac{w^{n/(n-\alpha)}(B)}{|B|}\lesssim\inf_{x\in
B}w(x)^{\frac{n}{n-\alpha}}\quad\text{for}\quad
B\in\mathcal{B}_{\kappa,\text{loc}},$

and it characterizes the weak-type boundedness of
$M_{\kappa,\alpha,\text{loc}}$ from $L^{1}(w)$ to
$L^{\frac{n}{n-\alpha},\infty}(w)$.

In the mentioned papers the authors also proved that the conditions are
independent of $\kappa$, so that we do not use this parameter in the notation
of the weight class. It is clear that the usual $A_{p,q}$ weights are in
$A_{p,q,\text{loc}}$ and it is immediate to check that $|x|^{\alpha}w(x)\in
A_{p,q,\text{loc}}$ whenever $w\in A_{p,q}$ and $\alpha\in\mathbb{R}$.

In [7] the local fractional integral $I_{\kappa,\alpha,\text{loc}}$ is defined
as

(2.13)
$I_{\kappa,\alpha,\text{loc}}f(x)=\int_{\\{y:|y-x|<\kappa|x|\\}}\frac{f(y)}{|x-y|^{n-\alpha}}\,dy.$

It is proved that the boundedness from $L^{p}(w^{p})$ to $L^{q}(w^{q})$ of
$I_{\kappa,\alpha,\text{loc}}$ is characterized also by (2.11), and the weak-
type for $p=1$ by (2.12).

A local version of Calderón-Zygmund operators was also introduced in [7]. We
do not need this general concept for our work, but only the following
particular result.

######  Theorem 2.4.

[7, Corollary 5.2] Let $T$ be a Calderón-Zygmund operator in $\mathbb{R}^{n}$
and $\kappa\in(0,1)$. Define

(2.14) $T_{\kappa}f(x)=T(f\chi_{B(x,\kappa|x|)})(x).$

Then $T_{\kappa}$ is bounded on $L^{p}(w)$ for $1<p<\infty$ and $w\in
A_{p,\text{loc}}$, and of weak type (1,1) with respect to weights in
$A_{1,\text{loc}}$.

##  3\. Necessary condition for singular integrals

A Calderón-Zygmund operator $T$ is an operator bounded on $L^{2}$ and
associated with a standard kernel $K$ in the sense that if $f$ is a compactly
supported $\mathcal{C}^{\infty}$ function, and $x\notin\text{supp}\,f$,
$Tf(x)$ is given by

(3.1) $Tf(x)=\int_{\mathbb{R}^{n}}K(x,y)f(y)\,dy.$

A standard kernel $K$ must satisfiy a size condition

(3.2) $|K(x,y)|\lesssim\frac{1}{|x-y|^{n}},$

and a regularity condition which we will not need to detail here (see [5,
Subsection 8.1.1], for instance). Using the $L^{2}$ boundedness of $T$ it is
shown in [5, Proposition 8.1.9] that if $f$ is bounded and compactly
supported, and $x\notin\text{supp}\,f$, the representation (3.1) holds as an
absolutely convergent integral. We will assume that when we extend the
operator to local Morrey spaces we keep such representation for bounded
compactly supported functions.

We obtain the necessary condition for Riesz transforms in the general setting
of Banach lattices. We recall that the Riesz transform $R_{j}$ ($1\leq j\leq
n$) is defined (usually up to a constant factor) as the following principal
value integral:

(3.3)
$R_{j}f(x)=\text{p.v.}\int_{\mathbb{R}^{n}}\frac{x_{j}-y_{j}}{|x-y|^{n+1}}\,f(y)dy.$

The definition makes sense for smooth functions. It is a Calderón-Zygmund
operator whose kernel is given by

$K(x,y)=\frac{x_{j}-y_{j}}{|x-y|^{n+1}}.$

######  Theorem 3.1.

Let $X$ be a Banach lattice of measurable functions in $\mathbb{R}^{n}$ that
contains the characteristic functions of balls. If the Riesz transforms are
bounded on $X$, then

(3.4) $\sup_{B}\frac{\|\chi_{B}\|_{X}\|M\chi_{B}\|_{X^{\prime}}}{|B|}<\infty,$

where the supremum is taken over all balls in $\mathbb{R}^{n}$.

We give first a lemma, which will be used in the proof of the theorem.

######  Lemma 3.2.

Let $B$ and $B^{\prime}$ balls with the same radius such that
$c_{B^{\prime}}=c_{B}+4r_{B}\mathbf{e}_{j}$, where $\mathbf{e}_{j}$ is the
unit vector in the direction of the $j$-th component. If $R_{j}$ is bounded on
$X$, then $\|\chi_{B}\|_{X}\sim\|\chi_{B^{\prime}}\|_{X}$.

###### Proof.

Let $x\in B$ and $y\in B^{\prime}$. We have $y_{j}>x_{j}$ and
$y_{j}-x_{j}\sim|x-y|\sim r_{B}$. Then

$|R_{j}\chi_{B^{\prime}}(x)|=\int_{B^{\prime}}\frac{y_{j}-x_{j}}{|x-y|^{n+1}}dy\sim
1.$

Hence $\chi_{B}(x)\lesssim|R_{j}\chi_{B^{\prime}}(x)|$ and taking norms in $X$
and using the boundedness of $R_{j}$ we deduce that
$\|\chi_{B}\|_{X}\lesssim\|\chi_{B^{\prime}}\|_{X}$. Changing the roles of $B$
and $B^{\prime}$ we obtain the reverse inequality and the lemma is proved. ∎

######  Remark 3.3.

In the case of the Morrey spaces, the norm of $\chi_{B}$ in
$L\mathcal{M}^{p}(\varphi,w)$ and in $WL\mathcal{M}^{p}(\varphi,w)$ is the
same. It is enough to assume that $R_{j}$ is bounded from
$L\mathcal{M}^{p}(\varphi,w)$ to $WL\mathcal{M}^{p}(\varphi,w)$ to get the
conclusion of the lemma.

###### Proof of Theorem 3.1.

Let $B$ be a ball and $Q$ the cube with sides parallel to the coordinate axes
and the same center and radius as $B$. Let $2Q$ be the cube with double
radius. We can decompose $\mathbb{R}^{n}\setminus 2Q$ into $2n$ regions,
$\\{P_{j}^{+},P_{j}^{-}:j=1,\dots,n\\}$ in such a way that for $x\in B$ and
$y\in P_{j}^{+}$ (resp. $P_{j}^{-}$), we have

(3.5) $0<y_{j}-x_{j}\sim|x-y|\quad(\text{resp. }0<x_{j}-y_{j}\sim|x-y|).$

(See Figure 1 for $n=2$.)

$B$$Q$$2Q$$P_{1}^{+}$$P_{1}^{-}$$P_{2}^{+}$$P_{2}^{-}$ Figure 1. The
decomposition used in the proof of Theorem 3.1 for $n=2$.

Let $f$ be a bounded compactly supported function. Then for $x\in B$ we have

$|R_{j}(|f|\chi_{P_{j}^{+}})(x)|=\int_{P_{j}^{+}}\frac{y_{j}-x_{j}}{|x-y|^{n+1}}|f(y)|\,dy\sim\int_{P_{j}^{+}}\frac{|f(y)|}{|c_{B}-y|^{n}}dy,$

and similarly for $P_{j}^{-}$. Therefore,

$\int_{P_{j}^{+}\cup
P_{j}^{-}}|f(y)|\frac{M\chi_{B}(y)}{|B|}dy\chi_{B}(x)\lesssim|R_{j}(|f|\chi_{P_{j}^{+}\cup
P_{j}^{-}})(x)|.$

Taking norms in $X$ and using the assumed boundedness of $R_{j}$ we deduce
that

(3.6) $\frac{1}{|B|}\int_{P_{j}^{+}\cup P_{j}^{-}}|f(y)|{M\chi_{B}(y)}dy\
\|\chi_{B}\|_{X}\lesssim\|f\|_{X}.$

Let $B^{\prime}$ be the ball with the same radius as $B$ and center
$c_{B^{\prime}}=c_{B}+4r_{B}\mathbf{e}_{1}$. For $x\in B^{\prime}$ and $y\in
2Q$ we have $0<x_{1}-y_{1}\sim r_{B}\sim|x-y|$. Then

$|R_{1}(|f|\chi_{2Q})(x)|=\int_{2Q}\frac{x_{1}-y_{1}}{|x-y|^{n+1}}|f(y)|dy\sim\int_{2Q}\frac{|f(y)|}{r_{B}^{n}}dy.$

Moreover, $M\chi_{B}(y)\sim 1$ for $y\in 2Q$. Hence

$\int_{2Q}|f(y)|\frac{M\chi_{B}(y)}{|B|}dy\
\chi_{B^{\prime}}(x)\lesssim|R_{1}(|f|\chi_{2Q})(x)|.$

Taking norms in $X$, using the boundedness of $R_{1}$, and Lemma 3.2 we obtain

$\frac{1}{|B|}\int_{2Q}|f(y)|{M\chi_{B}(y)}dy\
\|\chi_{B}\|_{X}\lesssim\|f\|_{X}.$

Adding this estimate to the sum for $j=1,\dots,n$ of the estimates (3.6) we
get

(3.7) $\frac{1}{|B|}\int_{\mathbb{R}^{n}}|f(y)|{M\chi_{B}(y)}dy\
\|\chi_{B}\|_{X}\lesssim\|f\|_{X}.$

We obtained (3.7) for bounded functions with compact support. Given a general
$f$ in $X$, for each $N>1$ we define

$f_{N}(x)=\min\\{|f(x)|,N\\}\chi_{|x|\leq N}(x).$

Since $X$ is a Banach lattice, $f_{N}\in X$ and $\|f_{N}\|_{X}\leq\|f\|_{X}$.
We can apply (3.7) to $f_{N}$ to obtain

$\frac{1}{|B|}\int_{\mathbb{R}^{n}}|f_{N}(y)|{M\chi_{B}(y)}dy\
\|\chi_{B}\|_{X}\lesssim\|f_{N}\|_{X}\leq\|f\|_{X}.$

Using the monotone convergence theorem we get (3.7) for every $f\in X$. Taking
the supremum over the functions $f$ such that $\|f\|_{X}\leq 1$ we obtain
(3.4). ∎

##  4\. Calderón-Zygmund operators on weighted local Morrey spaces

Calderón-Zygmund operators are first defined on smooth functions and can be
extended by density to other spaces like weighted $L^{p}$ spaces, for
instance. Nevertheless, due to the lack of density of smooth functions on the
weighted local Morrey spaces used in the results of this section, we need to
give an appropriate definition. To this end, we proceed as follows. Let $T$ be
a Calderón-Zygmund operator associated to the kernel $K$ and $f\in
L\mathcal{M}^{p}(\varphi,w)$ with $w$ satisfying (1.3). Since (1.3) is
stronger than (1.2), we know that $w\in A_{p,\textrm{loc}}$ (see [4, Lemma
6.2]). Let $B$ be a ball such that $|c_{B}|=6r_{B}$. Decompose $f$ as
$f=f_{1}+f_{2}$, where $f_{1}=f\chi_{2B}$. We have that $f_{1}$ is in $L^{1}$,
hence $Tf_{1}$ is well defined. The integrability of $f_{1}$ is a consequence
of the inequality

$\int_{2B}|f|\leq\left(\int_{2B}|f|^{p}w\right)^{1/p}w^{1-p^{\prime}}(2B)^{1/p^{\prime}},$

where the last term is finite because $w\in A_{p,\text{loc}}$. (For $p=1$ the
last term is replaced by $\sup_{x\in 2B}w(x)^{-1}$.)

On the other hand, for $x\in B$, we define

(4.1) $Tf_{2}(x)=\int_{(2B)^{c}}K(x,y)f(y)dy.$

This integral is well defined because for $x\in B$ we have

$|K(x,y)\chi_{(2B)^{c}}(y)|\lesssim\frac{1}{|B|}M\chi_{B}(y),$

using the size condition (3.2) of the kernel $K$, and $M\chi_{B}\in
L\mathcal{M}^{p}(\varphi,w)^{\prime}$ by (1.3). Moreover, we have the
following pointwise estimate:

(4.2)
$\left|\int_{(2B)^{c}}K(x,y)f(y)dy\right|\lesssim\frac{1}{|B|}\|M\chi_{B}\|_{L\mathcal{M}^{p}(\varphi,w)^{\prime}}\|f\|_{L\mathcal{M}^{p}(\varphi,w)}.$

Thus we define $Tf$ on $B$ as $Tf_{1}+Tf_{2}$. We need to check that if
$B^{\prime}$ is another ball such that $|c_{B^{\prime}}|=6r_{B^{\prime}}$ and
$B\cap B^{\prime}\neq\emptyset$, both definitions of $Tf$ coincide almost
everywhere on $B\cap B^{\prime}$, that is,

$T(f\chi_{2B})(x)+\int_{(2B)^{c}}K(x,y)f(y)dy=T(f\chi_{2B^{\prime}})(x)+\int_{(2B^{\prime})^{c}}K(x,y)f(y)dy,$

for almost every $x\in B\cap B^{\prime}$. This can be written as

$\displaystyle T(f(\chi_{2B}-\chi_{2B^{\prime}}))(x)$
$\displaystyle=\int_{2B\setminus
2B^{\prime}}K(x,y)f(y)dy-\int_{2B^{\prime}\setminus 2B}K(x,y)f(y)dy$
$\displaystyle=\int_{\mathbb{R}^{n}}K(x,y)\,[f(\chi_{2B}-\chi_{2B^{\prime}})](y)dy.$

The equality holds because $f(\chi_{2B}-\chi_{2B^{\prime}})$ is in $L^{1}$ and
$x\in B\cap B^{\prime}$ is outside the support of $f$. Indeed, the
representation formula (3.1) can be used for $L^{1}$ functions as proved in
[5, Proposition 8.2.2].

A priori estimates with Morrey norms are not enough to extend singular
integral operators (and others) to the whole Morrey space. This question,
disregarded in a number of papers, was addressed already in [1], where duality
was used to define the Calderón-Zygmund operator on the corresponding Morrey
space. Since then different ways had been considered as can be seen, for
instance, in [18, Section 10.4, vol. I]. Our definition can be viewed as the
natural extension to the weighted local Morrey space of the definition of the
operator in $L^{1}$ in the sense that for functions in $L^{1}\cap
L\mathcal{M}^{p}(\varphi,w)$ they coincide. In [12], for instance, a similar
definition appears in a quite general context.

We notice that for $x\in B$ we have $2B\subset B(x,3/5|x|)$. Indeed, if $y\in
2B$,

$|y-x|\leq|y-c_{B}|+|c_{B}-x|\leq 3r_{B}\leq\frac{3}{5}|x|,$

because $|c_{B}|=6r_{B}$. Then $f\chi_{2B}=(f\chi_{2B})\chi_{B(x,3/5|x|)}$,
hence

(4.3) $T(f\chi_{2B})=T_{3/5}(f\chi_{2B})$

in the sense of (2.14). By using [7, Corollary 5.2] we know that $T_{3/5}$ is
bounded on $L^{p}(w)$ for $w\in A_{p,\text{loc}}$ for $1<p<\infty$, and is of
weak-type $(1,1)$ with respect to $w$.

######  Theorem 4.1.

Let $1<p<\infty$. Let $T$ be a Calderón-Zygmund operator and let $w$ be a
weight satisfying (1.3). Then the operator $T$ defined as indicated is bounded
on $L\mathcal{M}^{p}(\varphi,w)$.

Let $p=1$. Then $T$ is of weak type $(1,1)$, that is, it is bounded from
$L\mathcal{M}^{1}(\varphi,w)$ to $WL\mathcal{M}^{1}(\varphi,w)$.

###### Proof.

Let $B$ be a ball such that $|c_{B}|=6r_{B}$. Decompose $f$ as $f_{1}+f_{2}$
as before. On the one hand, using (4.2) we have

$\displaystyle\left(\frac{1}{\varphi(B)}\int_{B}|Tf_{2}|^{p}w\right)^{\frac{1}{p}}$
$\displaystyle\lesssim\left(\frac{w(B)}{\varphi(B)}\right)^{\frac{1}{p}}\frac{1}{|B|}\|M\chi_{B}\|_{L\mathcal{M}^{p}(\varphi,w)^{\prime}}\|f\|_{L\mathcal{M}^{p}(\varphi,w)}$
$\displaystyle\lesssim\|\chi_{B}\|_{L\mathcal{M}^{p}(\varphi,w)}\frac{1}{|B|}\|M\chi_{B}\|_{L\mathcal{M}^{p}(\varphi,w)^{\prime}}\|f\|_{L\mathcal{M}^{p}(\varphi,w)}.$

On the other hand, taking into account (4.3) and that $T_{3/5}$ is bounded on
$L^{p}(w)$ for $w\in A_{p,\text{loc}}$, we have

$\displaystyle\frac{1}{\varphi(B)}\int_{B}|Tf_{1}|^{p}w$
$\displaystyle=\frac{1}{\varphi(B)}\int_{B}|T_{3/5}f_{1}|^{p}w$
$\displaystyle\lesssim\frac{1}{\varphi(B)}\int_{2B}|f|^{p}w\lesssim\|f\|_{L\mathcal{M}^{p}(\varphi,w)}^{p}.$

The theorem is proved for $1<p<\infty$.

In the case $p=1$ the part corresponding to $Tf_{2}$ is still valid, and for
$Tf_{1}$ we replace the $L^{p}(w)$-boundedness by the weak-type (1,1) estimate
for $T_{3/5}$. ∎

Combining this result with the necessary condition of the previous section we
obtain the following corollary.

######  Corollary 4.2.

The Riesz transforms are bounded on $L\mathcal{M}^{p}(\varphi,w)$ for
$1<p<\infty$ if and only if $w$ satisfies (1.3). The Riesz transforms are
bounded from $L\mathcal{M}^{1}(\varphi,w)$ to $WL\mathcal{M}^{1}(\varphi,w)$
if and only if (1.3) holds for $p=1$.

##  5\. Necessary conditions for fractional operators

We recall that the fractional maximal operator $M_{\alpha}$ is defined for
locally integrable functions in $\mathbb{R}^{n}$ as in (2.10), but without
restrictions on the balls, that is, the supremum is taken over all balls
containing $x$. The fractional integral $I_{\alpha}$ is defined as in (2.13)
with the integral extended to all of $\mathbb{R}^{n}$. We also recall the
pointwise bound $M_{\alpha}f(x)\leq I_{\alpha}f(x)$.

######  Theorem 5.1.

Let $X$ be a Banach lattice and $Y$ a normed space.

(a) If the fractional maximal operator $M_{\alpha}$ is bounded from $X$ to
$Y$, then

(5.1)
$\sup_{B}\frac{\|\chi_{B}\|_{Y}\|\chi_{B}\|_{X^{\prime}}}{|B|^{1-\alpha/n}}<\infty.$

(b) If the fractional integral operator $I_{\alpha}$ is bounded from $X$ to
$Y$, then

(5.2)
$\sup_{B}\frac{\|\chi_{B}\|_{Y}\|(M\chi_{B})^{1-\alpha/n}\|_{X^{\prime}}}{|B|^{1-\alpha/n}}<\infty.$

###### Proof.

(a) Let $B$ be a ball. By definition of $M_{\alpha}$ we have

$\frac{1}{|B|^{1-\alpha/n}}\int_{B}|f|\leq M_{\alpha}f(x)\quad\text{for all
}x\in B,$

that is,

$\left(\frac{1}{|B|^{1-\alpha/n}}\int_{B}|f|\right)\chi_{B}(x)\leq
M_{\alpha}f(x).$

Taking the norm in $Y$ and assuming that $M_{\alpha}$ is bounded from $X$ to
$Y$,

$\left(\frac{1}{|B|^{1-\alpha/n}}\int\chi_{B}|f|\right)\|\chi_{B}\|_{Y}\leq\|M_{\alpha}f\|_{Y}\lesssim\|f\|_{X}.$

Taking the supremum over $f$ such that $\|f\|_{X}\leq 1$ we obtain (5.1).

(b) Let $B$ be a ball and $x\in B$. Then $|x-y|\lesssim r_{B}+|c_{B}-y|$.
Hence,

$M\chi_{B}(y)\sim\frac{|B|}{(r_{B}+|c_{B}-y|)^{n}}\lesssim\frac{|B|}{|x-y|^{n}}.$

Therefore, for $x\in B$,

$I_{\alpha}(|f|)(x)=\int\frac{|f(y)|}{|x-y|^{n-\alpha}}\,dy\gtrsim\int|f(y)|\left(\frac{M\chi_{B}(y)}{|B|}\right)^{1-\frac{\alpha}{n}}\,dy.$

We deduce that

$\left[\frac{1}{|B|^{1-\frac{\alpha}{n}}}\int|f|(M\chi_{B})^{1-\frac{\alpha}{n}}\right]\chi_{B}(x)\lesssim
I_{\alpha}(|f|)(x),$

and proceeding as in part (a) we obtain (5.2). ∎

We obtain in the following theorem a restriction on the exponents of the local
Morrey spaces for the boundedness of the fractional operators. It is enough to
state it for $M_{\alpha}$ because of the pointwise bound between $M_{\alpha}$
and $I_{\alpha}$.

######  Theorem 5.2.

If the fractional maximal operator $M_{\alpha}$ is bounded from
$\mathcal{LM}^{p}(\varphi,u)$ to $W\mathcal{LM}^{q}(\psi,v)$, then

(5.3) $\frac{1}{p}-\frac{1}{q}\leq\frac{\alpha}{n}.$

###### Proof.

Let $x\neq 0$ be a Lebesgue point of both $u$ and $v$, such that $v(x)\neq 0$.
Let $B_{r}=B(x,r)$ with $4r\leq|x|$. Let $\widehat{B}=B(0,2|x|)$. Then

$\|\chi_{B_{r}}\|_{\mathcal{LM}^{p}(\varphi,u)}\sim\left(\frac{u(B_{r})}{\varphi(\widetilde{B}_{r})}\right)^{1/p}\sim\left(\frac{u(B_{r})}{\varphi(\widehat{B})}\right)^{1/p},$

and

$\|\chi_{B_{r}}\|_{W\mathcal{LM}^{q}(\psi,v)}=\|\chi_{B_{r}}\|_{\mathcal{LM}^{q}(\psi,v)}\sim\left(\frac{v(B_{r})}{\psi(\widehat{B})}\right)^{1/q}.$

Using that $r^{\alpha}\chi_{B_{r}}(y)\leq M_{\alpha}\chi_{B_{r}}(y)$, from the
boundedness assumption we deduce

$r^{\alpha}\left(\frac{v(B_{r})}{\psi(\widehat{B})}\right)^{1/q}\lesssim\left(\frac{u(B_{r})}{\varphi(\widehat{B})}\right)^{1/p},$

which can be written as

(5.4)
$r^{\alpha+n/q-n/p}\left(\frac{v(B_{r})}{r^{n}}\right)^{1/q}\lesssim\left(\frac{u(B_{r})}{r^{n}}\right)^{1/p}\frac{\psi(\widehat{B})^{1/q}}{\varphi(\widehat{B})^{1/p}}.$

Since $x$ is a Lebesgue point of $u$ and $v$, we have

$\lim_{r\to 0}\frac{v(B_{r})}{r^{n}}=v(x)\quad\text{and}\quad\lim_{r\to
0}\frac{u(B_{r})}{r^{n}}=u(x).$

Hence, letting $r$ tend to zero in (5.4), we see that the inequality can only
hold if $\alpha+n/q-n/p\geq 0$. ∎

This theorem extends to a general weighted setting a result which is known in
the unweighted setting with $\varphi(B)=\psi(B)=r_{B}^{\lambda}$. A
counterexample is in [10]; see also [18, Example 158]. This is in contrast
with the case of global Morrey spaces, where Adams extended the unweighted
result to the range $1/p-1/q=\alpha/(n-\lambda)$. For weighted inequalities in
the Adams range on global Morrey spaces see [8] and [15]. Let us mention that
it is possible to obtain an Adams-type result on (unweighted) local Morrey
spaces if we restrict the action of the operator to radial functions (see [10]
and [11]).

##  6\. Sufficient conditions for fractional operators

Before characterizing the weights for $M_{\alpha}$ we prove a lemma, similar
to Lemma 6.2 in [4]. We use the notation $\varphi^{p}$ to denote the function
such that $\varphi^{p}(B)=\varphi(B)^{p}$ as mentioned in the introduction.

######  Lemma 6.1.

Let $1\leq p<n/\alpha$ and $1/p-1/q=\alpha/n$. Then $w\in A_{p,q,\text{loc}}$
as defined in (2.11) if and only if $w$ satisfies (1.4) restricted to the
balls $B$ such that $r_{B}\leq\kappa|c_{B}|$ for some fixed $\kappa\in(0,1)$.

###### Proof.

Let $B$ be a ball such that $r_{B}\leq\kappa|c_{B}|$. We have from (2.5) that

(6.1)
$\|\chi_{B}\|_{\mathcal{LM}^{q}(\varphi^{q},w^{q})}\sim\left(\frac{w^{q}(B)}{\varphi^{q}(\widetilde{B})}\right)^{1/q}=\frac{w^{q}(B)^{1/q}}{\varphi(\widetilde{B})}.$

Let $p>1$. We claim that

(6.2) $\|\chi_{B}\|_{\mathcal{LM}^{p}(\varphi^{p},w^{p})^{\prime}}\sim
w^{-p^{\prime}}(B)^{1/p^{\prime}}\varphi(\widetilde{B}).$

On the one hand we have

$\|\chi_{B}\|_{\mathcal{LM}^{p}(\varphi^{p},w^{p})^{\prime}}\geq\frac{1}{\|w^{-p^{\prime}}\chi_{B}\|_{\mathcal{LM}^{p}(\varphi^{p},w^{p})}}\int_{B}w^{-p^{\prime}}\sim
w^{-p^{\prime}}(B)^{1/p^{\prime}}\varphi(\widetilde{B}),$

where we used (2.4) for the norm of $w^{-p^{\prime}}\chi_{B}$.

On the other hand,

$\int_{B}|f|\leq\left(\int_{B}|f|^{p}w^{p}\right)^{1/p}w^{-p^{\prime}}(B)^{1/p^{\prime}}\leq\varphi(\widetilde{B})\|f\|_{\mathcal{LM}^{p}(\varphi^{p},w^{p})}w^{-p^{\prime}}(B)^{1/p^{\prime}}.$

Taking the supremum over the functions $f$ with
$\|f\|_{\mathcal{LM}^{p}(\varphi^{p},w^{p})}\leq 1$ we prove the claim (6.2).

Therefore,

$\frac{\|\chi_{B}\|_{\mathcal{LM}^{q}(\varphi^{q},w^{q})}\|\chi_{B}\|_{\mathcal{LM}^{p}(\varphi^{p},w^{p})^{\prime}}}{|B|^{1-\alpha/n}}\sim\frac{w^{q}(B)^{1/q}w^{-p^{\prime}}(B)^{1/p^{\prime}}}{|B|^{1-\alpha/n}},$

and this proves the lemma for $p>1$.

Let $p=1$. We have (6.1) with $q=n/(n-\alpha)$. Let $w\in
A_{1,n/(n-\alpha),\text{loc}}$. Then

$\displaystyle\int_{B}|f|$ $\displaystyle\lesssim\int_{B}|f|w\
\left(\frac{|B|}{w^{n/(n-\alpha)}(B)}\right)^{1-\alpha/n}$
$\displaystyle\leq\varphi(\widetilde{B})\left(\frac{|B|}{w^{n/(n-\alpha)}(B)}\right)^{1-\alpha/n}\|f\|_{\mathcal{LM}^{1}(\varphi,w)}.$

From here we obtain

$\|\chi_{B}\|_{\mathcal{LM}^{1}(\varphi,w)^{\prime}}\lesssim\varphi(\widetilde{B})\left(\frac{|B|}{w^{n/(n-\alpha)}(B)}\right)^{1-\alpha/n},$

which together with (6.1) implies that $w$ satisfies (1.4) for the selected
balls.

Let $E$ be a subset of $B$ of positive measure. Then

$\|\chi_{B}\|_{\mathcal{LM}^{1}(\varphi,w)^{\prime}}\geq\frac{1}{\|\chi_{E}\|_{\mathcal{LM}^{1}(\varphi,w)}}\int_{B}\chi_{E}\sim\frac{\varphi(\widetilde{B})|E|}{w(E)}.$

Using (6.1) and (1.4) we obtain

(6.3)
$\left(\frac{w^{n/(n-\alpha)}(B)}{|B|}\right)^{1-\alpha/n}\lesssim\frac{w(E)}{|E|}.$

With $E=\\{x\in B:w(x)\leq\inf_{B}w+\epsilon\\}$ we obtain

$\left(\frac{w^{n/(n-\alpha)}(B)}{|B|}\right)^{1-\alpha/n}\lesssim\inf_{B}w+\epsilon.$

Since this holds for any $\epsilon>0$ we get the
$A_{1,n/(n-\alpha),\text{loc}}$ condition. ∎

In the following theorem we obtain the characterization of weighted
inequalities for $M_{\alpha}$ and $I_{\alpha}$.

######  Theorem 6.2.

Let $0<\alpha<n$, $1<p<n/\alpha$, and $1/p-1/q=\alpha/n$.

(a) The fractional maximal operator $M_{\alpha}$ is bounded from
$L\mathcal{M}^{p}(\varphi^{p},w^{p})$ to $L\mathcal{M}^{q}(\varphi^{q},w^{q})$
if and only if (1.4) holds.

(b) The fractional integral operator $I_{\alpha}$ is bounded from
$L\mathcal{M}^{p}(\varphi^{p},w^{p})$ to $L\mathcal{M}^{q}(\varphi^{q},w^{q})$
if and only if (1.5) holds.

(c) For $p=1$ and $q=n/(n-\alpha)$ both results hold with $M_{\alpha}$ and
$I_{\alpha}$ bounded from $L\mathcal{M}^{1}(\varphi,w)$ to
$WL\mathcal{M}^{\frac{n}{n-\alpha}}(\varphi^{\frac{n}{n-\alpha}},w^{\frac{n}{n-\alpha}})$.

###### Proof.

The necessity of all cases has been proved in the previous section. Let us
prove the sufficiency.

(a) Let $B$ with $|c_{B}|=6r_{B}$. Given a nonnegative function $f$ we
decompose it as $f=f_{1}+f_{2}$, where $f_{1}=f\chi_{2B}$. Using the
subadditivity of $M_{\alpha}$ we have

$M_{\alpha}f(y)\leq M_{\alpha}f_{1}(y)+M_{\alpha}f_{2}(y).$

Since $f_{1}$ is supported in $2B$, to compute $M_{\alpha}f_{1}(y)$ for $y\in
B$ we only need to consider balls $B(y,\rho)$ with $\rho<3r_{B}$. Moreover,
for $r_{B}<\rho<3r_{B}$ we can replace $B(y,\rho)$ with $2B$. Therefore,

$\displaystyle M_{\alpha}f_{1}(y)$ $\displaystyle\sim\sup_{\rho\leq
r_{B}}\frac{1}{|B(y,\rho)|^{1-\frac{\alpha}{n}}}\int_{B(y,\rho)}|f|+\frac{1}{|2B|^{1-\frac{\alpha}{n}}}\int_{2B}|f|$
$\displaystyle\sim
M_{1/5,\alpha,\textrm{loc}}f_{1}(y)+\frac{1}{|2B|^{1-\frac{\alpha}{n}}}\int_{2B}|f|.$

On the other hand, since $f_{2}$ is supported on $(2B)^{c}$, to compute
$M_{\alpha}f_{2}(y)$ for $y\in B$ one only needs to consider balls of radius
greater than $r_{B}$, and as a consequence, $M_{\alpha}f_{2}$ is almost
constant on $B$. Then

$M_{\alpha}f_{2}(y)\sim\sup_{B^{\prime}\supset
B}\frac{1}{|B^{\prime}|^{1-\frac{\alpha}{n}}}\int_{B^{\prime}}|f_{2}|\quad\text{
for }y\in B.$

Altogether, we have for all $y\in B$ that

(6.4) $M_{\alpha}f(y)\sim
M_{1/5,\alpha,\textrm{loc}}(f\chi_{2B})(y)+\sup_{B^{\prime}\supset
B}\frac{1}{|B^{\prime}|^{1-\frac{\alpha}{n}}}\int_{B^{\prime}}|f_{2}|.$

For the last term we have

$\displaystyle\frac{1}{|B^{\prime}|^{1-\frac{\alpha}{n}}}\int_{B^{\prime}}|f|$
$\displaystyle\leq\frac{1}{|B^{\prime}|^{1-\frac{\alpha}{n}}}\|f\|_{L\mathcal{M}^{p}(\varphi^{p},w^{p})}\|\chi_{B^{\prime}}\|_{L\mathcal{M}^{p}(\varphi^{p},w^{p})^{\prime}}$
$\displaystyle\lesssim\frac{\|f\|_{L\mathcal{M}^{p}(\varphi^{p},w^{p})}}{\|\chi_{B^{\prime}}\|_{L\mathcal{M}^{q}(\varphi^{q},w^{q})}}\leq\frac{\|f\|_{L\mathcal{M}^{p}(\varphi^{p},w^{p})}}{\|\chi_{B}\|_{L\mathcal{M}^{q}(\varphi^{q},w^{q})}},$

where we used successively (2.7), (1.4) for $B^{\prime}$, and $B\subset
B^{\prime}$.

Now we have

$\displaystyle\left(\frac{1}{\varphi(\widetilde{B})^{q}}\int_{B}(M_{\alpha}f)^{q}w\right)^{1/q}$
$\displaystyle\qquad\lesssim\left(\frac{1}{\varphi(\widetilde{B})^{q}}\int_{B}\left(M_{1/5,\alpha,\textrm{loc}}(f\chi_{2B})+\frac{\|f\|_{L\mathcal{M}^{p}(\varphi^{p},w^{p})}}{\|\chi_{B}\|_{L\mathcal{M}^{q}(\varphi^{q},w^{q})}}\right)^{q}w^{q}\right)^{1/q}$
$\displaystyle\qquad\lesssim\left(\frac{1}{\varphi(\widetilde{B})^{q}}\int_{2B}|f|^{p}w^{p}\right)^{1/p}+\|f\|_{L\mathcal{M}^{p}(\varphi^{p},w^{p})},$

where we used that $w\in A_{p,q,\text{loc}}$, which is implied by (1.4)
according to Lemma 6.1. Taking into account that $\widetilde{2B}\subset
2\widetilde{B}$, we obtain the required bound using the doubling property of
$\varphi$.

(b) Let $f$ be nonnegative. We take a ball $B$ as in part (a) and decompose
$f$ as $f_{1}+f_{2}$. From the linearity of $I_{\alpha}$ we have

$I_{\alpha}f=I_{\alpha}f_{1}+I_{\alpha}f_{2}.$

For $x\in B$ we have $I_{\alpha}f_{1}(x)=I_{3/5,\alpha,\text{loc}}f_{1}(x)$,
where the local fractional operator has been defined in (2.13). Condition
(1.5) is stronger than (1.4), hence $w\in A_{p,q,\text{loc}}$. Using the
weighted boundedness of $I_{3/5,\alpha,\text{loc}}$ from $L^{p}(w^{p})$ to
$L^{q}(w^{q})$ for $w\in A_{p,q,\text{loc}}$, this term works as the first
term in the right-hand side of (6.4).

On the other hand, for $x\in B$ and $y\in\mathbb{R}^{n}\setminus 2B$ we have
$M\chi_{B}(y)\sim|B||x-y|^{-n}$. Hence,

$I_{\alpha}f_{2}(x)=\int_{\mathbb{R}^{n}\setminus
2B}\frac{f(y)}{|x-y|^{n-\alpha}}\lesssim\int_{\mathbb{R}^{n}}f(y)\left(\frac{M\chi_{B}(y)}{|B|}\right)^{1-\frac{\alpha}{n}}\,dy.$

Using (2.7) we obtain

$\displaystyle\left(\frac{1}{\varphi(\widetilde{B})^{q}}\int_{B}(I_{\alpha}f_{2})^{q}w^{q}\right)^{1/q}$
$\displaystyle\qquad\leq\frac{\|f\|_{L\mathcal{M}^{p}(\varphi^{p},w^{p})}\|(M\chi_{B})^{1-\alpha/n}\|_{L\mathcal{M}^{p}(\varphi^{p},w^{p})^{\prime}}}{|B|^{1-\alpha/n}}\left(\frac{1}{\varphi(\widetilde{B})^{q}}\int_{B}w^{q}\right)^{1/q}$

and (1.5) suffices to conclude.

(c) For $p=1$ we proceed as in (a) and (b), but when dealing with the local
operators acting on $f_{1}$ the weighted inequalities are of weak-type. The
rest of the proof is the same. ∎

######  Remark 6.3.

For weights $w$ in $A_{p,q,\text{loc}}$ one only needs to verify (1.4) and
(1.5) for balls centered at the origin. Balls $B$ such that $|c_{B}|\leq
6r_{B}$, can be replaced by $\widetilde{B}$, because $B\subset\widetilde{B}$
and $|B|\sim|\widetilde{B}|$. For balls with $|c_{B}|>6r_{B}$,
$A_{p,q,\text{loc}}$ is enough for $M_{\alpha}$ due to Lemma 6.1, but also for
$I_{\alpha}$, as a consequence of the proof of Theorem 6.2, where to obtain
the bound for $I_{\alpha}f_{1}$ we only use that $w\in A_{p,q,\text{loc}}$.

The proof of the theorem can be adapted to the case $1/p-1/q<\alpha/n$ with
appropriate weights.

######  Theorem 6.4.

Let $0<\alpha<n$, $0\leq\gamma<\alpha$, $1\leq p<n/\gamma$ and
$1/p-1/q=\gamma/n$. Assume that $w\in A_{p,q,\text{loc}}$. Set
$v_{\gamma}(x)=|x|^{\gamma-\alpha}w(x)$.

Let $p>1$. The fractional maximal operator $M_{\alpha}$ is bounded from
$L\mathcal{M}^{p}(\varphi^{p},w^{p})$ to
$L\mathcal{M}^{q}(\varphi^{q},v_{\gamma}^{q})$ if

(6.5)
$\sup_{B}\frac{\|\chi_{B}\|_{L\mathcal{M}^{q}(\varphi^{q},v_{\gamma}^{q})}\|\chi_{B}\|_{L\mathcal{M}^{p}(\varphi^{p},w^{p})^{\prime}}}{|B|^{1-\alpha/n}}<\infty$

for balls centered at the origin.

The fractional integral operator $I_{\alpha}$ is bounded from
$L\mathcal{M}^{p}(\varphi^{p},w^{p})$ to
$L\mathcal{M}^{q}(\varphi^{q},v_{\gamma}^{q})$ if

(6.6)
$\sup_{B}\frac{\|\chi_{B}\|_{L\mathcal{M}^{q}(\varphi^{q},v_{\gamma}^{q})}\|(M\chi_{B})^{1-\alpha/n}\|_{L\mathcal{M}^{p}(\varphi^{p},w^{p})^{\prime}}}{|B|^{1-\alpha/n}}<\infty.$

for balls centered at the origin.

For $p=1$ and $q=n/(n-\gamma)$ the results hold for the boundedness from
$L\mathcal{M}^{1}(\varphi,w)$ to
$WL\mathcal{M}^{\frac{n}{n-\gamma}}(\varphi^{\frac{n}{n-\gamma}},v_{\gamma}^{\frac{n}{n-\gamma}})$.

###### Proof.

First we proof the following pointwise inequalities:

(6.7)
$M_{\kappa,\alpha,\text{loc}}f(x),I_{\kappa,\alpha,\text{loc}}(|f|)(x)\lesssim|x|^{\alpha-\gamma}M_{\kappa,\gamma,\text{loc}}f(x).$

Let $B$ a ball containing $x$ and such that $r_{B}<\kappa|c_{B}|$. Then

$\frac{1}{|B|^{1-\alpha/n}}\int_{B}|f(y)|\,dy\leq\frac{r_{B}^{\alpha-\gamma}}{|B|^{1-\gamma/n}}\int_{B}|f(y)|\,dy\lesssim\frac{|x|^{\alpha-\gamma}}{|B|^{1-\gamma/n}}\int_{B}|f(y)\,dy,$

from which (6.7) follows for $M_{\kappa,\alpha,\text{loc}}$.

On the other hand,

$\displaystyle\int_{\\{y:|y-x|<\kappa|x|\\}}$
$\displaystyle\frac{|f(y)|}{|x-y|^{n-\alpha}}\,dy=\sum_{j=0}^{\infty}\int_{\\{y:2^{-j-1}\kappa|x|\leq|y-x|<2^{-j}\kappa|x|\\}}\frac{|f(y)|}{|x-y|^{n-\alpha}}\,dy$
$\displaystyle\lesssim\sum_{j=0}^{\infty}(2^{-j}\kappa|x|)^{\alpha-\gamma}\frac{1}{(2^{-j}\kappa|x|)^{n-\gamma}}\int_{\\{y:|y-x|<2^{-j}\kappa|x|\\}}|f(y)|\,dy$
$\displaystyle\lesssim|x|^{\alpha-\gamma}M_{\kappa,\gamma,\text{loc}}f(x).$

We proceed as in the proof of the previous theorem. The action of the
operators on $f_{2}$ is treated in the same way as before. When working with
$f_{1}$, using (6.7) the needed estimate is shifted to
$M_{\kappa,\gamma,\text{loc}}$. ∎

From Theorem 5.1 we know that (6.5) and (6.6) for all balls in
$\mathbb{R}^{n}$ are necessary for the boundedness of the operators. In the
case $\gamma<\alpha$ of Theorem 6.4 we add the condition $w\in
A_{p,q,\text{loc}}$ to get the sufficiency. Working as in Lemma 6.1 we can
prove that $w\in A_{p,q,\text{loc}}$ implies (6.5), but we do not know whether
the opposite holds for $\gamma<\alpha$. This is why we do not get a full
characterization as was the case for $\gamma=\alpha$.

##  7\. Power weights

In this section we obtain sharp power weighted inequalities in the case of the
Samko and Komori-Shirai versions of the weighted local Morrey spaces. We use
the notation $w_{\beta}(x)=|x|^{\beta}$.

###  7.1. Weighted spaces of Samko type

The Samko type spaces are defined with the condition
$\varphi_{\lambda}(B)=r_{B}^{\lambda}$ with $\lambda>0$. First we give a
lemma.

######  Lemma 7.1.

Let $1\leq p<p_{0}<\infty$ and $\lambda,\mu>0$. Let $\beta$ and $\delta$ be
related by

(7.1) $\beta+\frac{n-\lambda}{p}=\delta+\frac{n-\mu}{p_{0}}.$

Then
$\|f\|_{\mathcal{LM}^{p}(\varphi_{\lambda},w_{\beta}^{p})}\lesssim\|f\|_{\mathcal{LM}^{p_{0}}(\varphi_{\mu},w_{\delta}^{p_{0}})}$.

###### Proof.

We can use (2.3) for the norm in the Morrey spaces. Let $B$ be a ball such
that $|c_{B}|=4r_{B}$. Then we have $r_{\widetilde{B}}\sim|c_{B}|\sim|x|$ for
$x\in B$. Using these equivalences and Hölder’s inequality we get

$\displaystyle\left(\frac{1}{r_{\widetilde{B}}^{\lambda}}\int_{B}|f(x)|^{p}|x|^{\beta
p}dx\right)^{1/p}\lesssim|c_{B}|^{-\frac{\lambda}{p}+\beta+\frac{n}{p}-\frac{n}{p_{0}}}\left(\int_{B}|f|^{p_{0}}\right)^{1/p_{0}}$
$\displaystyle\qquad\qquad\lesssim|c_{B}|^{-\frac{\lambda}{p}+\beta+\frac{n}{p}-\frac{n}{p_{0}}+\frac{\mu}{p_{0}}-\delta}\left(\frac{1}{r_{\widetilde{B}}^{\mu}}\int_{B}|f(x)|^{p_{0}}|x|^{\delta
p_{0}}dx\right)^{1/p_{0}}$
$\displaystyle\qquad\qquad\leq\|f\|_{\mathcal{LM}^{p_{0}}(\varphi_{\mu},w_{\delta}^{p_{0}})},$

where we used (7.1). The result follows. ∎

######  Proposition 7.2.

Let $\varphi_{\lambda}(B)=r_{B}^{\lambda}$ with $\lambda>0$. If $M_{\alpha}$
is bounded from $\mathcal{LM}^{p}(\varphi_{\lambda_{1}},w_{\beta_{1}}^{p})$ to
$W\mathcal{LM}^{q}(\varphi_{\lambda_{2}},w_{\beta_{2}}^{q})$, then

(7.2)
$\beta_{2}-\beta_{1}-\frac{n-\lambda_{1}}{p}+\frac{n-\lambda_{2}}{q}+\alpha=0.$

Moreover,

(7.3) $\alpha\leq\beta_{1}+\frac{n-\lambda_{1}}{p}<n.$

If $I_{\alpha}$ is bounded from
$\mathcal{LM}^{p}(\varphi_{\lambda_{1}},w_{\beta_{1}}^{p})$ to
$W\mathcal{LM}^{q}(\varphi_{\lambda_{2}},w_{\beta_{2}}^{q})$ the same
conditions are necessary, and moreover the left-hand side inequality in (7.3)
is strict.

Using (7.2) we can write (7.3) in the equivalent form

$0\leq\beta_{2}+\frac{n-\lambda_{2}}{q}<n-\alpha.$

This implies that $\beta_{2}q\geq\lambda_{2}-n$, and from (7.3) we also have
$\beta_{1}p>\lambda_{1}-n$.

###### Proof.

Let $B$ be such that $4r_{B}\leq|c_{B}|$. Then

$\|\chi_{B}\|_{\mathcal{LM}^{p}(\varphi_{\lambda_{1}},w_{\beta_{1}}^{p})}\sim\left(\frac{w_{\beta_{1}}^{p}(B)}{\varphi_{\lambda_{1}}(\widetilde{B})}\right)^{1/p}\sim\left(\frac{|c_{B}|^{\beta_{1}p}r_{B}^{n}}{|c_{B}|^{\lambda_{1}}}\right)^{1/p}=|c_{B}|^{\beta_{1}-\lambda_{1}/p}r_{B}^{n/p}.$

Similarly,

$\|r_{B}^{\alpha}\chi_{B}\|_{W\mathcal{LM}^{q}(\varphi_{\lambda_{2}},w_{\beta_{2}}^{q})}=r_{B}^{\alpha}\|\chi_{B}\|_{\mathcal{LM}^{q}(\varphi_{\lambda_{2}},w_{\beta_{2}}^{q})}\sim|c_{B}|^{\beta_{2}-\lambda_{2}/q}r_{B}^{\alpha+n/q}.$

Using $r_{B}^{\alpha}\chi_{B}\leq M_{\alpha}\chi_{B}$ we have

$|c_{B}|^{\beta_{2}-\lambda_{2}/q}r_{B}^{\alpha+n/q}\lesssim|c_{B}|^{\beta_{1}-\lambda_{1}/p}r_{B}^{n/p},$

that is,

$|c_{B}|^{\beta_{2}-\lambda_{2}/q-\beta_{1}+\lambda_{1}/p}r_{B}^{\alpha+n/q-n/p}\lesssim
1.$

For fixed $c_{B}$ letting $r_{B}$ tend to zero we obtain the condition (5.3).
Take now $r_{B}=|c_{B}|/4$ and we obtain

$|c_{B}|^{\beta_{2}-\lambda_{2}/q-\beta_{1}+\lambda_{1}/p+\alpha+n/q-n/p}\lesssim
1.$

Since $|c_{B}|$ can be any number in $(0,\infty)$ we deduce the necessity of
(7.2).

An alternative proof of (7.2) can be obtained using a scale argument.

The necessity of the first inequality in (7.3) is deduced from the fact that
the characteristic function of a ball centered at the origin is not in
$W\mathcal{LM}^{q}(\varphi_{\lambda_{2}},w_{\beta_{2}}^{q})$ for
$\beta_{2}q+n-\lambda_{2}<0$ (see [2, Proposition 5.3]). On the other hand,
the function $|x|^{-n}\chi_{B(0,1)}$ is in
$\mathcal{LM}^{p}(\varphi_{\lambda_{1}},w_{\beta_{1}}^{p})$ for
$\beta_{1}+(n-\lambda_{1})/p\geq n$, and it is not locally integrable, hence
the necessity of the second inequality in (7.3) follows.

From the pointwise inequality $M_{\alpha}f(x)\leq I_{\alpha}(|f|)(x)$ it is
clear that the conditions are necessary for $I_{\alpha}$. We next show that
the equality in (7.3) is not possible for $I_{\alpha}$.

Assume that $\beta_{1}p+n-\lambda_{1}=\alpha p$ (equivalently,
$\beta_{2}q+n-\lambda_{2}=0$ due to (7.2)). Take
$f(y)=|y|^{-\alpha}\chi_{B(0,1)}$. Then
$f\in{\mathcal{LM}^{p}(\varphi_{\lambda_{1}},w_{\beta_{1}}^{p})}$. For
$|x|<1/2$,

$I_{\alpha}f(x)\geq\int_{2|x|<|y|<1}\frac{|y|^{-\alpha}}{|x-y|^{n-\alpha}}\,dy\gtrsim\int_{2|x|<|y|<1}|y|^{-n}\,dy\sim-\log(2|x|).$

The function $-\log(2|x|)\chi_{|x|<1/2}$ does not belong to
$W\mathcal{LM}^{q}(\varphi_{\lambda_{2}},w_{\beta_{2}}^{q})$ for
$\beta_{2}q+n-\lambda_{2}=0$. This rules out the equality case in (7.3) for
$I_{\alpha}$. ∎

######  Theorem 7.3.

Let $0<\alpha<n$, $0\leq\gamma\leq\alpha$, $1\leq p<n/\gamma$, and
$1/p-1/q=\gamma/n$. Set $\varphi_{\lambda}(B)=r_{B}^{\lambda}$ for
$\lambda>0$.

  1. (1)

Let $1<p<n/\gamma$. Then $M_{\alpha}$ is bounded from
$\mathcal{LM}^{p}(\varphi_{\lambda_{1}},w_{\beta_{1}})$ to
$\mathcal{LM}^{q}(\varphi_{\lambda_{2}},w_{\beta_{2}})$ if and only if (7.2)
and (7.3) hold. If $p=1$ and $q=n/(n-\gamma)$ in (7.2) and (7.3), then
$M_{\alpha}$ is bounded from
$\mathcal{LM}^{1}(\varphi_{\lambda_{1}},w_{\beta_{1}})$ to
$W\mathcal{LM}^{n/(n-\gamma)}(\varphi_{\lambda_{2}},w_{\beta_{2}})$.

  2. (2)

The same result holds for $I_{\alpha}$ instead of $M_{\alpha}$ except that in
the left-hand side condition of (7.3) the inequality is strict.

###### Proof.

The necessity of the conditions have been proved in Proposition 7.2. Let us
check the sufficiency.

Case $\gamma=\alpha$ and $\beta_{1}=\beta_{2}:=\beta$. Then (7.2) reduces to
$\lambda_{1}/p=\lambda_{2}/q$. Using Theorem 6.2 we need to check (1.4) for
$\varphi(B)=r_{B}^{\lambda_{1}/p}=r_{B}^{\lambda_{2}/q}$. We only need to
verify it for balls centered at the origin, because $w_{\beta}\in
A_{p,q,\text{loc}}$.

Let $B$ be a ball centered at the origin. Then

$\|\chi_{B}\|_{\mathcal{LM}^{q}(\varphi_{\lambda_{2}},w_{\beta}^{q})}\sim
r_{B}^{\frac{n-\lambda_{2}}{q}+\beta}$

and

(7.4)
$\|\chi_{B}\|_{\mathcal{LM}^{p}(\varphi_{\lambda_{1}},w_{\beta}^{p})^{\prime}}\lesssim
r_{B}^{n-\frac{n-\lambda_{1}}{p}-\beta}.$

The last one is similar to [3, Lemma 2.12 (b)]. Consequently, (1.4) holds for
balls centered at the origin.

For $I_{\alpha}$ we need to check (1.5). By Remark 6.3 we only need to
consider $B$ centered at the origin. We use that

$M\chi_{B}(x)\sim\chi_{B}(x)+\sum_{k=1}^{\infty}2^{-kn}\chi_{2^{k}B\setminus
2^{k-1}B}(x).$

With (7.4) for $2^{k}B$ we have

$\|(M\chi_{B})^{1-\alpha/n}\|_{\mathcal{LM}^{p}(\varphi_{\lambda_{1}},w_{\beta}^{p})^{\prime}}\lesssim\sum_{k=0}^{\infty}2^{-k(n-\alpha)}(2^{k}r_{B})^{n-\frac{n-\lambda_{1}}{p}-\beta}\lesssim
r_{B}^{n-\frac{n-\lambda_{1}}{p}-\beta},$

where in the last step we use that the exponent of $2^{k}$ is negative due to
the strict inequality assumed in (7.3). Consequently, (1.5) is satisfied.

Case $\gamma=\alpha$ and $\beta_{2}\neq\beta_{1}$. If $\beta_{2}>\beta_{1}$,
let $\lambda_{1}^{\prime}=\lambda_{1}+(\beta_{2}-\beta_{1})p$. Then using
Proposition 2.3 we have

$\mathcal{LM}^{p}(\varphi_{\lambda_{1}},w_{\beta_{1}}^{p})=\mathcal{LM}^{p}(\varphi_{\lambda_{1}^{\prime}},w_{\beta_{2}}^{p}).$

Condition (7.2) becomes $\lambda_{1}^{\prime}/p=\lambda_{2}/q$ and (7.3) is
satisfied with $\beta_{2}$ and $\lambda_{1}^{\prime}$ instead of $\beta_{1}$
and $\lambda_{1}$. Thus we can apply the result in the first part of the proof
to deduce the boundedness from
$\mathcal{LM}^{p}(\varphi_{\lambda_{1}^{\prime}},w_{\beta_{2}}^{p})$ to
$\mathcal{LM}^{q}(\varphi_{\lambda_{2}},w_{\beta_{2}}^{q})$, which is the
desired result.

The proof for $\beta_{1}>\beta_{2}$ is similar, setting
$\lambda_{2}^{\prime}=\lambda_{2}+(\beta_{1}-\beta_{2})q$ and taking into
account that

$\mathcal{LM}^{q}(\varphi_{\lambda_{2}},w_{\beta_{2}}^{q})=\mathcal{LM}^{q}(\varphi_{\lambda_{2}^{\prime}},w_{\beta_{1}}^{q}).$

Case $\gamma<\alpha$. Although it would be possible to use Theorem 6.4 we
prefer to proceed using Lemma 7.1 to provide a different insight.

Given $p,q,\lambda_{1},\lambda_{2},\beta_{1}$ and $\beta_{2}$ such that (7.2)
and (7.3) hold, we choose $p_{0},q_{0},\mu_{1},\mu_{2},\delta_{1}$ and
$\delta_{2}$ such that

$\displaystyle p\geq p_{0},\ q\leq q_{0},\
\frac{1}{p_{0}}-\frac{1}{q_{0}}=\frac{\alpha}{n},\ \mu_{1},\mu_{2}>0,$
$\displaystyle\beta_{1}+\frac{n-\lambda_{1}}{p}=\delta_{1}+\frac{n-\mu_{1}}{p_{0}},\text{
and }\beta_{2}+\frac{n-\lambda_{2}}{q}=\delta_{2}+\frac{n-\mu_{2}}{q_{0}}.$

Then the corresponding conditions (7.2) and (7.3) hold for
$p_{0},q_{0},\mu_{1},\mu_{2},\delta_{1}$ and $\delta_{2}$, and $p_{0}$ and
$q_{0}$ satisfy the conditions in the previous part of the proof. Using Lemma
7.1 we obtain

$\displaystyle\|M_{\alpha}f\|_{\mathcal{LM}^{q}(\varphi_{\lambda_{2}},w_{\beta_{2}}^{q})}$
$\displaystyle\lesssim\|M_{\alpha}f\|_{\mathcal{LM}^{q_{0}}(\varphi_{\mu_{2}},w_{\delta_{2}}^{q_{0}})}$
$\displaystyle\lesssim\|f\|_{\mathcal{LM}^{p_{0}}(\varphi_{\mu_{1}},w_{\delta_{1}}^{p_{0}})}\lesssim\|f\|_{\mathcal{LM}^{p}(\varphi_{\lambda_{1}},w_{\beta_{1}}^{p})}.$

In the case $p=1$ one has to work with weak-type inequalities, but the proof
has only small changes and we do not write the details here. ∎

The result for $n/p-n/q=\alpha$ and $\beta_{1}=\beta_{2}$ is in [16]. When
$\beta_{1}=\beta_{2}=0$ this is known as Spanne’s result (usually stated for
global Morrey spaces).

###  7.2. Weighted spaces of Komori-Shirai type

Given a weight $w$, let us denote $\psi_{w,\lambda}(B)=w(B)^{\lambda/n}$. The
weighted local Morrey spaces of Komori-Shirai type are defined by
$L\mathcal{M}^{p}(\psi_{w,\lambda},w)$ with $\lambda>0$. The typical estimates
for fractional operators are not exactly between two spaces of Komori-Shirai
type. For instance, when we say that $M_{\alpha}$ and $I_{\alpha}$ are bounded
from $L\mathcal{M}^{p}(\psi_{w^{q},\lambda},w^{p})$ to
$L\mathcal{M}^{q}(\psi_{w^{q},\lambda},w^{q})$ (see [9] for global Morrey
spaces) the first space is not of Komori-Shirai type as defined above. To
agree with the definition it should be
$L\mathcal{M}^{p}(\psi_{w^{p},\lambda},w^{p})$.

Nevertheless, in the case of power weights, the results can be stated in such
a way that both spaces are of Komori-Shirai type. This holds because those
spaces can be represented as Samko-type spaces as shown in the following
proposition.

######  Proposition 7.4.

Let $\lambda>0$ and $\beta>-n$. Then

(7.5)
$\mathcal{LM}^{p}(\psi_{w_{\beta},\lambda},w_{\beta})=\mathcal{LM}^{p}(\varphi_{\lambda(1+\beta/n)},w_{\beta})=\mathcal{LM}^{p}(\varphi_{\lambda},w_{\beta(1-\lambda/n)}).$

A similar result holds for weak-type spaces.

###### Proof.

In all cases we can use the restriction of the Morrey norm to (2.3). Notice
that for the balls in (2.3) we have the equivalence $|c_{B}|\sim r_{B}\sim|x|$
for $x\in B$. Then $w_{\beta}(B)\sim r_{B}^{\beta+n}$ and the result follows.
∎

Using this equality of spaces we can rewrite Theorem 7.3 in terms of Komori-
Shirai type spaces. We state it as a corollary.

######  Corollary 7.5.

Let $0<\alpha<n$, $0\leq\gamma\leq\alpha$, and $1/p-1/q=\gamma/n$. Let
$1<p<n/\gamma$. Then $I_{\alpha}$ is bounded from
$\mathcal{LM}^{p}(\psi_{w_{\beta_{1}}^{p},\lambda_{1}},w_{\beta_{1}}^{p})$ to
$\mathcal{LM}^{q}(\psi_{w_{\beta_{2}}^{q},\lambda_{2}},w_{\beta_{2}}^{q})$ if
and only if

(7.6)
$(n-\lambda_{2})\left(\frac{1}{q}+\frac{\beta_{2}}{n}\right)-(n-\lambda_{1})\left(\frac{1}{p}+\frac{\beta_{1}}{n}\right)+\alpha=0.$

and

(7.7) $\alpha<(n-\lambda_{1})\left(\frac{1}{p}+\frac{\beta_{1}}{n}\right)<n.$

With $p=1$ and $q=n/(n-\gamma)$ the above conditions are necessary and
sufficient for $I_{\alpha}$ to be bounded from
$\mathcal{LM}^{1}(\psi_{w_{\beta_{1}},\lambda_{1}},w_{\beta_{1}})$ to
$W\mathcal{LM}^{n/(n-\gamma)}(\psi_{w_{\beta_{2}}^{n/(n-\gamma)},\lambda_{2}},w_{\beta_{2}}^{n/(n-\gamma)})$.

Similar results hold for $M_{\alpha}$ adding the equality at the left-hand
side of (7.7).

Notice that for the local integrability of $w_{\beta_{1}}^{p}$ and
$w_{\beta_{2}}^{q}$ we need $\beta_{1}p+n>0$ and $\beta_{2}q+n>0$. From (7.7)
we deduce $\lambda_{1}<n$ and together with (7.6) we obtain the necessity of
$\lambda_{2}<n$. For $M_{\alpha}$ in the equality case of the left-hand side
of (7.7) $\lambda_{2}=n$ is admissible too.

## References

  * [1] Alvarez, J.: Continuity of Calderón-Zygmund type operators on the predual of a Morrey space. Clifford algebras in analysis and related topics (Fayetteville, AR, 1993), 309–319, Stud. Adv. Math., CRC, Boca Raton, FL (1996)
  * [2] Duoandikoetxea, J.; Rosenthal, M.: Boundedness of operators on certain weighted Morrey spaces beyond the Muckenhoupt range. Potential Anal. 53, 1255–1268 (2020)
  * [3] Duoandikoetxea, J.; Rosenthal, M.: Boundedness properties in a family of weighted Morrey spaces with emphasis on power weights. J. Funct. Anal. 279, no. 8, 108687, 26 pp. (2020)
  * [4] Duoandikoetxea, J.; Rosenthal, M.: Muckenhoupt-type conditions on weighted Morrey spaces. J. Fourier Anal. Appl. 27, no. 2, Paper No. 32, 33 pp. (2021)
  * [5] Grafakos, L.: Modern Fourier Analysis. Springer Verlag (2014)
  * [6] Harboure, E.; Salinas, O.; Viviani, B.: Local maximal function and weights in a general setting. Math. Ann. 358, 609–628 (2014)
  * [7] Harboure, E.; Salinas, O.; Viviani, B.: Local fractional and singular integrals on open subsets. J. Analyse Math. 138, 301–324 (2019)
  * [8] Iida, T.; Komori-Furuya, Y.; Sato, E.: The Adams inequality on weighted Morrey spaces. Tokyo J. Math. 34, 535–545 (2011)
  * [9] Komori, Y.; Shirai, S.: Weighted Morrey spaces and a singular integral operator. Math. Nachr. 282, 219–231 (2009)
  * [10] Komori-Furuya, Y.; Sato, E.: Fractional integral operators on central Morrey spaces. Math. Inequal. Appl. 20, 801–813 (2017)
  * [11] Komori-Furuya, Y.; Sato, E.: Weighted estimates for fractional integral operators on central Morrey spaces. Math. Nachr. 290, 2901–2908 (2017)
  * [12] Ho, K.-P.: Definability of singular integral operators on Morrey-Banach spaces. J. Math. Soc. Japan 72, 155–170 (2020)
  * [13] Lin, C.-C.; Stempak, K.: Local Hardy-Littlewood maximal operator. Math. Ann. 348, 797–813 (2010)
  * [14] Muckenhoupt, B.; Wheeden, R.: Weighted norm inequalities for fractional integrals. Trans. Amer. Math. Soc. 192, 261–274 (1974)
  * [15] Nakamura, S.; Sawano, Y.; Tanaka H.: The fractional operators on weighted Morrey spaces. J. Geom. Anal. 28, 1502–1524 (2018)
  * [16] Nakamura, S.; Sawano, Y.; Tanaka H.: Weighted local Morrey spaces. Ann. Acad. Sci. Fenn. Math., 45, 67–93 (2020)
  * [17] Samko, N.: Weighted Hardy and singular operators in Morrey spaces. J. Math. Anal. Appl. 350, 56–72 (2009)
  * [18] Sawano, Y.; Di Fazio, G.; Hakim, D. I.: Morrey Spaces. Introduction and Applications to Integral Operators and PDE’s, vol. I and II. Chapman & Hall/CRC Monographs and Research Notes in Mathematics, Boca Raton (2020)
  * [19] Sawyer E,; Wheeden, R. L.: Weighted inequalities for fractional integrals on Euclidean and homogeneous spaces, Amer. J. Math. 114, 813–874 (1992)
  * [20] Strömberg, J.-O., Torchinsky, A.: Weighted Hardy spaces. Lecture Notes in Mathematics, 1381. Springer-Verlag, Berlin (1989)
  * [21] Wheeden, R.: A characterization of some weighted norm inequalities for the fractional maximal function. Studia Math. 107, 257–272 (1993)


# Tripotential MOND theories

Mordehai Milgrom  Department of Particle Physics and Astrophysics, Weizmann
Institute

###### Abstract

I present a new class of nonrelativistic, modified-gravity MOND theories. The
three gravitational degrees of freedom of these “TRIMOND” theories are the
MOND potential and two auxiliary potentials, one of which emerges as the
Newtonian potential. Their Lagrangians involve a function of three
acceleration variables – the gradients of the potentials. So, the transition
from the Newtonian to the MOND regime is rather richer than in the aquadratic-
Lagrangian theory (AQUAL) and the quasilinear MOND theory (QUMOND), which are
special cases of TRIMOND, each defined by a Lagrangian function of a single
variable. In particular, unlike AQUAL and QUMOND whose deep-MOND limit (DML)
is fully dictated by the required scale invariance, here, the scale-invariant
DML still requires specifying a function of two variables. For one-dimensional
(e.g., spherical) mass distributions, in all TRIMOND theories the MOND
acceleration is a (theory specific, but system independent) function of the
Newtonian acceleration; their variety appears in nonsymmetric situations.
Also, they all make the salient, primary MOND predictions. For example, they
predict the same DML virial relation as AQUAL and QUMOND, and thus the same
DML $M-\sigma$ relation, and the same DML two-body force. Yet they can differ
materially on secondary predictions. Such TRIMOND theories may be the
nonrelativistic limits of scalar-bimetric relativistic formulations of MOND,
such as BIMOND with an added scalar.

##  I Introduction

The “aquadratic-Lagrangian” (AQUAL) theory [1] was the first full-fledged,
nonrelativistic theory that embodies the basic tenets of MOND [2]. (For
reviews of MOND see Refs. [3, 4, 5, 6, 7, 8].) AQUAL modifies the Poisson
equation for the gravitational potential of a given distribution of masses,
without modifying the equation of motion of the masses; it is thus a
“modified-gravity” (MG) theory. Some twenty five years later another full-
fledged nonrelativistic, MG Lagrangian theory for MOND has been proposed – the
quasilinear MOND theory (QUMOND) [9]. These two theories – disparate in their
amenability to solution, but yielding very similar prediction – have become
the workhorses for studying various aspects of MOND, including many detailed
studies that involve numerical solutions of the field equations of these
theories (see reviews [3, 4, 5, 6, 7, 8]).

AQUAL and QUMOND satisfy the basic tenets of MOND – the involvement of one
new, acceleration constant, $a_{0}$; a Newtonian limit at high accelerations
($a\gg a_{0}$, or $a_{0}\rightarrow 0$); and scale invariance in the low-
acceleration, deep-MOND limit (DML), $a_{0}\rightarrow\infty$. They thus share
all the primary predictions of MOND – those that follow from only the basic
tenets [10]. Examples of such predictions are, the asymptotic flatness of
rotation curves $V(r)\rightarrow V_{\scriptscriptstyle\infty}$; the relation
between the total mass, $M$, and the asymptotic speed:
$MGa_{0}=V_{\scriptscriptstyle\infty}^{4}$, (underlying the “baryonic Tully-
Fisher relation”); a good approximation to the full rotation curves from the
mass distribution; enhanced stability of disc galaxies; a relation between the
total mass of a system in the DML and its mass-average, root-mean-squared
velocity dispersion, $\sigma$: $MGa_{0}=\xi\sigma^{\scriptscriptstyle 4}$,
with $\xi\sim 1$; and the qualitative prediction of an “external-field effect”
(EFE).

Perhaps as a result of these two theories remaining the main tools for
deriving quantitative results in MOND, the notion may have taken root, that
they represent MOND in its entirety (in the nonrelativistic regime).

It has, however, been stressed repeatedly that there may well be other
theories that satisfy the basic MOND tenets, and thus share its primary
predictions, but that may differ from AQUAL and QUMOND, and among themselves,
in making other, secondary predictions. Among such second-tier predictions one
may count some fine details of the rotation curves (e.g., Refs. [11, 12, 13,
14, 15]); the exact value of $\xi$ in the $M~{}-~{}\sigma$ relation, and its
dependence on dimensionless attributes of the system; the exact dependence of
the effective two-body force on the masses; the exact nature and strength of
the EFE – such as the exact effects of the galactic field on the dynamics of
the solar system or of wide binaries in the galaxy; etc.

One possible avenue for looking for such other MOND formulations is that of
so-called “modified-inertia” theories, which as described, e.g., in the recent
Ref. [16], can make very different secondary predictions than AQUAL and
QUMOND.

But secondary predictions can differ also among MG theories, as was
demonstrated recently in Ref. [17].

AQUAL and QUMOND are quite restricted in their scope in the following sense:
They each involves one “interpolating function” of a single acceleration
variable that is introduced, by hand, in their Lagrangian. As a result, all
the (secondary) predictions of these theories involve, and are dictated, by
this one function. For example, this function dictates the prediction of the
“mass-discrepancy-acceleration relation” (MDAR also known as the RAR) and can
be extracted from the observed relation. It then largely determines all the
other predictions of these theories. This is much too restrictive, and is not
a situation that characterizes other, well known modifications of Newtonian
dynamics.

For example, in quantum theory we do not introduce an interpolation function
at the basic level. We do encounter different “interpolating functions”
between the classical and the quantum regimes for different phenomena: The
black-body function, the expression for the specific heat of solids, and the
barrier penetration probability are only a few examples. The same is true in
relativity vs Newtonian dynamics.

Likewise, one expects that in a more fundamental MOND theory, interpolating
functions will not be put in by hand in the fundamental equations. They will
be derivable from the theory, and will also differ among the different
applications of the theory.

In distancing ourselves at least one step from the restricted AQUAL/QUMOND, we
should realize that there are MOND theories whose Newtonian-DML interpolation
scheme is richer, even if it is still implanted at the basic level of the
theory.

Here I present a new class of nonrelativistic, MG formulations of MOND that
involve three gravitational potentials, one that dictates the acceleration of
masses – and is thus the “observed” MOND potential – with two auxiliary ones.
The interpolating function depends on the gradients of the last two and is a
function of three scalar, acceleration variables, offering a rather richer
scope than AQUAL/QUMOND, which are special cases in this new class.

In Sec. II, I describe the TRIMOND Lagrangian and the resulting field
equations for the three potentials, and consider their high- and low-
acceleration limits. In Sec. III, I describe some of the properties of the
solutions and various predictions of the TRIMOND theories. Section IV is a
brief discussion of potential relativistic extensions.

##  II Tripotential formulations

The tripotential theories involve three potentials as the gravitational
degrees of freedom, with one of them, the MOND potential $\phi$ coupling to
matter directly. The other two are called $\varphi$ and $\psi$. The Lagrangian
density of these theories is of the form

$\mathcal{L}=\mathcal{L}_{G}+\rho(\frac{1}{2}{\bf v}^{2}-\phi),$ (1)

where its gravitational part is

$\mathcal{L}_{G}=-\frac{1}{8\pi
G}[2\vec{\nabla}(\phi-\epsilon\varphi)\cdot\vec{\nabla}\psi-
a_{0}^{2}\mathcal{F}(x,y,z)],$ (2)

with the scalar, acceleration variables

$x\equiv(\vec{\nabla}\psi)^{2}/a_{0}^{2},~{}~{}~{}y\equiv(\vec{\nabla}\varphi)^{2}/a_{0}^{2},~{}~{}~{}z=2\vec{\nabla}\psi\cdot\vec{\nabla}\varphi/a_{0}^{2},$
(3)

and $\mathcal{F}$ is a dimensionless function satisfying the basic tenets of
MOND (see Secs. II.2 and II.3).

The constant $\epsilon$ is chosen, after $\mathcal{F}$ is defined, such that
the normalization of the MOND acceleration far away from a bounded mass, $M$,
is the conventional one, i.e., $g\rightarrow(MGa_{0})^{1/2}/r$. Its choice
will be explained in Sec. III.2, where we discuss the asymptotic behavior.

Variation over the particle degrees of freedom gives

$\ddot{\bf r}_{i}=-\vec{\nabla}\phi({\bf r}_{i});$ (4)

so $\phi$ is the (MOND) gravitational potential, which dictates particle
accelerations.

Variation over $\phi$ gives

$\Delta\psi=4\pi G\rho;$ (5)

so, $\psi$ equals the Newtonian potential sourced by $\rho$. (As usual, we
seek solutions for which the three gradients vanish at infinity for isolated
systems.)

Varying over $\varphi$ gives

$\vec{\nabla}\cdot(\mathcal{F}_{y}\vec{\nabla}\varphi)+\vec{\nabla}\cdot(\mathcal{F}_{z}\vec{\nabla}\psi)=-\epsilon\Delta\psi=-4\pi
G\epsilon\rho.$ (6)

Varying over $\psi$ gives

$\Delta\phi=\vec{\nabla}\cdot(\mathcal{F}_{x}\vec{\nabla}\psi)+\vec{\nabla}\cdot(\mathcal{F}_{z}\vec{\nabla}\varphi)+\epsilon\Delta\varphi\equiv\hat{\rho}.$
(7)

After $\psi$ is solved for from Eq. (5), it is substituted in Eq. (6), which
becomes a nonlinear, second order equation in $\varphi$, with coefficients
that depend on position through $\vec{\nabla}\psi({\bf r})$. The choice of
$\mathcal{F}$ should be such that Eq. (6) is elliptic, and then it has a
unique solution for $\varphi$ with our boundary conditions. Then, $\varphi$
and $\psi$ are substituted in the right-hand side of Eq. (7) – a Poisson
equation for $\phi$ – to get the effective density, $\hat{\rho}$, that sources
it.

As in AQUAL, the nonlinearity of Eq. (6) involves functions of only up to the
first derivatives, the second derivatives appearing in a linear fashion with
coefficients that are functions of position and the first derivatives111Such
equations are called “quasilinear” in the mathematical literature, to be
distinguished from what we call “quasilinear” in describing QUMOND.. Thus,
after the substitution of $\vec{\nabla}\psi$, equation (6) is of the form

$a^{\scriptscriptstyle ij}({\bf r},\vec{\nabla}\varphi)\varphi_{,i,j}+b({\bf
r},\vec{\nabla}\varphi)=0.$ (8)

Then, the ellipticity condition is that the matrix $a^{\scriptscriptstyle ij}$
has a definite sign [19].

The solution scheme is rather simple in the case where $\mathcal{F}$ does not
depend on $z$. Eqs. (6)(7) can then be written as

$\vec{\nabla}\cdot(\mathcal{F}_{y}\vec{\nabla}\varphi)=-4\pi G\epsilon\rho.$
(9)
$\Delta\phi=\vec{\nabla}\cdot(\mathcal{F}_{x}\vec{\nabla}\psi)+\epsilon\Delta\varphi.$
(10)

Equation (5) is solved for $\psi$, which is then substituted in Eq. (9), to
get an AQUAL-type equation of the form
$\vec{\nabla}\cdot[\tilde{\mu}(|\vec{\nabla}\varphi|,{\bf
r})\vec{\nabla}\varphi]=-4\pi G\epsilon\rho$ (with the added dependence of
$\tilde{\mu}$ on position, which does not complicate the solution).
Substituting the solutions for $\psi$ and $\varphi$ in the right-hand side of
Eq. (10) gives the Poisson equation for the MOND potential.

In this case, the ellipticity condition for Eq. (9) can be shown to be that
$y\mathcal{F}_{y}$ is a monotonic function of $y$

###  II.1 AQUAL and QUMOND as special cases

AQUAL is a special case of the tripotential class, gotten by putting
$\mathcal{F}_{x}=\mathcal{F}_{z}=0$, and the DML of $\mathcal{F}(y)$ is such
that for $y\rightarrow 0$, $\mathcal{F}\rightarrow-qy^{\scriptscriptstyle
3/2}$. The value of $\epsilon$, as determined from the procedure described in
Sec. III.2, is $\epsilon=(3q/2)^{1/3}$. With these choices we have
$\phi=\epsilon\varphi$, while $\varphi$ satisfies an AQUAL equation

$\vec{\nabla}\cdot(\mathcal{F}_{y}\vec{\nabla}\varphi)=-4\pi G\epsilon\rho.$
(11)

In the high-acceleration limit
$\mathcal{F}_{y}\rightarrow-\epsilon^{\scriptscriptstyle 2}$.222Without loss
of generality we can put $\epsilon=1$, $q=2/3$, and $\eta=-1$.

QUMOND is a special case gotten for the choice
$\mathcal{F}_{y}=\mathcal{F}_{z}=0$, $\epsilon=0$, and $\mathcal{F}(x)$ having
the appropriate low- and high-acceleration limits. For these, Eq. (6) is
satisfied trivially, and Eq. (7) becomes

$\Delta\phi=\vec{\nabla}\cdot(\mathcal{F}_{x}\vec{\nabla}\psi)\equiv\hat{\rho},$
(12)

which is the QUMOND field equation.333Remember that the ellipticity condition
for AQUAL is that $X\mu(X)$ be monotonic, and the QUMOND equations, which
require solving only the Poisson equation are elliptic.

###  II.2 Newtonian limit

To recover the Newtonian limit when $a_{0}\rightarrow 0$ – one of the basic
MOND axioms – we require that in this limit $\mathcal{F}_{z}\rightarrow 0$,
$\mathcal{F}_{y}\rightarrow\eta$ and $\mathcal{F}_{x}\rightarrow\beta$, with
$\beta-\epsilon^{2}/\eta=1$. This can be seen from the field equations, or by
noting that with these choices the Lagrangian can be written as

$\mathcal{L}=-\frac{1}{8\pi
G}[(\vec{\nabla}\phi)^{2}-(\vec{\nabla}\theta)^{2}-(\vec{\nabla}\zeta)^{2}]+\rho(\frac{1}{2}{\bf
v}^{2}-\phi),$ (13)

where, $\theta=\phi-\psi$, and
$\zeta=\epsilon\eta^{\scriptscriptstyle-1/2}\psi+\eta^{\scriptscriptstyle
1/2}\varphi$, which is the Poisson Lagrangian for $\phi$ (and implying the
irrelevant $\psi=\phi$, and $\zeta=0$)444A newtonian limit is also obtained
with $\mathcal{F}_{z}\rightarrow 0$, $\mathcal{F}_{y}\rightarrow\infty$ (so
that $\varphi\rightarrow 0$), and $\mathcal{F}_{x}\rightarrow 1$..

###  II.3 The deep-MOND limit

Another basic MOND tenet is that MOND gravity becomes space-time scale
invariance in the limit $a_{0}\rightarrow\infty$ (while $G\rightarrow 0$, with
$\mathcal{A}_{0}=Ga_{0}$ fixed) [18]. By scaling we mean transforming $(t,{\bf
r})\rightarrow(\lambda t,\lambda{\bf r})$. All terms in the limiting
Lagrangian density should then transform in the same way under scaling. Since
the left-hand side of Eq. (4) transforms as $\lambda^{-1}$, $\phi$ must be
invariant. The last term of Eq. (1) then dictates that all terms in
$\mathcal{L}$ are multiplied by $\lambda^{-3}$ under scaling (because $\rho$
does; ${\bf v}$ is invariant). Thus, $\varphi$ is invariant, and
$\psi\rightarrow\lambda^{-1}\psi$. Thus,
$x,y,z\rightarrow\lambda^{-4}x,\lambda^{-2}y,\lambda^{-3}z$, and thus
$\mathcal{F}$ must satisfy

$\mathcal{F}(\lambda^{-4}x,\lambda^{-2}y,\lambda^{-3}z)=\lambda^{-3}\mathcal{F}(x,y,z).$
(14)

By taking, for example, $\lambda=y^{1/2}$, we see that $\mathcal{F}$ can be
written as

$\mathcal{F}(x,y,z)=y^{3/2}\mathcal{F}(x/y^{2},1,z/y^{3/2})\equiv
y^{3/2}\bar{\mathcal{F}}(x/y^{2},z/y^{3/2}).$ (15)

If more convenient, we can use the alternative, equivalent forms

$\mathcal{F}(x,y,z)=x^{3/4}\mathcal{F}(1,y/x^{1/2},z/x^{3/4})\equiv
x^{3/4}\bar{\mathcal{F}}^{(1)}(y/x^{1/2},z/x^{3/4}),$ (16)
$\mathcal{F}(x,y,z)=z\mathcal{F}(x/z^{4/3},y/z^{2/3},1)\equiv
z\bar{\mathcal{F}}^{(2)}(x/z^{4/3},y/z^{2/3}).$ (17)

Such functions are the most general that ensure scale invariance. So, unlike
AQUAL or QUMOND where there is no freedom do dictate an interpolating function
in the deep-MOND limit, here we are left with the freedom to dictate a
function of two variables.

##  III Some consequences and predictions

###  III.1 One-dimensional configurations

As in AQUAL and QUMOND, for systems of one-dimensional symmetry (planar,
cylindrical, or spherical) the MOND acceleration $g$ is a system-independent
(but theory specific) function of the Newtonian acceleration,
$g_{\scriptscriptstyle N}$:

$g=\nu(g_{\scriptscriptstyle N}/a_{0})g_{\scriptscriptstyle N}.$ (18)

To see this, note that in this one-dimensional case we can strip the
divergences in the field Eqs. (6)(7) (e.g., by applying Gauss’s theorem) and
we can replace the vectorial relations by scalar ones with the absolute values
of the respective accelerations replacing the potential gradient:
$\vec{\nabla}\psi=g_{\scriptscriptstyle N}{\bf
e},~{}\vec{\nabla}\varphi=g_{\scriptscriptstyle\varphi}{\bf
e},~{}\vec{\nabla}\phi=g{\bf e}$ (where ${\bf e}$ is the radial unit vector).
Equation (6) than reads (putting $a_{0}=1$ for the nonce)

$\mathcal{F}_{y}(g_{\scriptscriptstyle N}^{\scriptscriptstyle
2},g_{\scriptscriptstyle\varphi}^{\scriptscriptstyle 2},2g_{\scriptscriptstyle
N}g_{\scriptscriptstyle\varphi})g_{\scriptscriptstyle\varphi}+[\mathcal{F}_{z}(g_{\scriptscriptstyle
N}^{\scriptscriptstyle 2},g_{\scriptscriptstyle\varphi}^{\scriptscriptstyle
2},2g_{\scriptscriptstyle
N}g_{\scriptscriptstyle\varphi})+\epsilon]g_{\scriptscriptstyle N}=0,$ (19)

from which $g_{\scriptscriptstyle\varphi}$ can be solved for as a function of
$g_{\scriptscriptstyle N}$. $\mathcal{F}$ has to be such that the solution for
$g_{\scriptscriptstyle\varphi}$ exists and is unique (the ellipticity of Eq.
(6) shoud ensure this). Equation (7) gives, in the same vein,

$g=\mathcal{F}_{x}(g_{\scriptscriptstyle N}^{\scriptscriptstyle
2},g_{\scriptscriptstyle\varphi}^{\scriptscriptstyle 2},2g_{\scriptscriptstyle
N}g_{\scriptscriptstyle\varphi})g_{\scriptscriptstyle
N}+[\mathcal{F}_{z}(g_{\scriptscriptstyle N}^{\scriptscriptstyle
2},g_{\scriptscriptstyle\varphi}^{\scriptscriptstyle 2},2g_{\scriptscriptstyle
N}g_{\scriptscriptstyle\varphi})+\epsilon]g_{\scriptscriptstyle\varphi}.$ (20)

Substituting the $g_{\scriptscriptstyle\varphi}$ values gotten from Eq. (19),
we then get an equation of the form (18).

The function $\nu$ is an example of an emerging interpolating function. For
the general TRIMOND theories it is relevant only for one-dimensional
configurations. It can be used as some tool for comparison with AQUAL and
QUMOND, where it is the only such function that appears and that is relevant
to all phenomena and configurations.

We can define for this one-dimensional case

$\mathcal{F}^{*}(g_{\scriptscriptstyle
N},g_{\scriptscriptstyle\varphi})\equiv\mathcal{F}(g_{\scriptscriptstyle
N}^{\scriptscriptstyle 2},g_{\scriptscriptstyle\varphi}^{\scriptscriptstyle
2},2g_{\scriptscriptstyle N}g_{\scriptscriptstyle\varphi}).$ (21)

Then Eqs. (19) (20) can be written as

$\frac{\partial\mathcal{F}^{*}}{\partial
g_{\scriptscriptstyle\varphi}}=-2\epsilon g_{\scriptscriptstyle N},$ (22)
$g=\epsilon
g_{\scriptscriptstyle\varphi}+\frac{1}{2}\frac{\partial\mathcal{F}^{*}}{\partial
g_{\scriptscriptstyle N}}.$ (23)

For nonsymmetric systems – e.g., in predicting the rotation curves of disc
galaxies – TRIMOND theories do differ from each other. This is true even when
$\mathcal{F}$ does not depend on $z$. In fact, AQUAL and QUMOND themselves
differ in such predictions (e.g., Ref. [15]). But the fact that general
TRIMOND theories revolve on Lagrangian functions of three variables can
greatly increase the variety in predictions. To better bring out the possible
variety in behaviors for nonsymmetric systems we can work with the
acceleration variable $w\equiv(xy)^{\scriptscriptstyle 1/2}-z/2$ instead of
with $z$ itself. Since $w=0$ for one-dimensional configurations, the one-
dimensional “interpolating function”, $\nu$, is oblivious to the dependence of
$\mathcal{F}$ on $w$.

###  III.2 Asymptotic behavior

Consider now the asymptotic behavior of the MOND potential at radii, $r$, far
outside a bounded mass distribution of total mass $M$. While the Newtonian
acceleration is $\vec{\nabla}\psi=g_{\scriptscriptstyle N}{\bf e}$, with
$g_{\scriptscriptstyle N}=MG/r^{2}$, it follows from the above that
$\vec{\nabla}\varphi=\gamma(g_{\scriptscriptstyle N}a_{0})^{1/2}{\bf e}$ and
$\vec{\nabla}\phi=\kappa(g_{\scriptscriptstyle N}a_{0})^{1/2}{\bf e}$, with
$\gamma$ and $\kappa$ numbers that depend on the choice of $\mathcal{F}$ and
on $\epsilon$. This is the only dependence on $M$ and $r$ that is compatible
with the dimensions and scaling dimensions of the potentials.555In fact, from
the general discussion of spherical systems it follows that all accelerations
are functions of $MG/r^{2}$.

Use the DML form of $\mathcal{F}$ form Eq. (15) (which applies in the
asymptotic regime)

$\mathcal{F}(x,y,z)=y^{3/2}\bar{\mathcal{F}}(u,v),$ (24)

where, $u\equiv x/y^{2},~{}~{}v\equiv z/y^{3/2}$.

$u$ and $v$, which are dimensionless and scale invariant, must be numbers;
indeed, $u=\gamma^{\scriptscriptstyle-4}$, and
$v=2\gamma^{\scriptscriptstyle-2}$. We thus have for the asymptotic
expressions of derivatives of $\mathcal{F}$ appearing in the field equations
(6) (7),

$\mathcal{F}_{x}\rightarrow y^{\scriptscriptstyle-1/2}\bar{\mathcal{F}}_{u}.$
(25) $\mathcal{F}_{y}\rightarrow\frac{3}{2}y^{\scriptscriptstyle
1/2}[\bar{\mathcal{F}}-\frac{4}{3}\gamma^{\scriptscriptstyle-4}\bar{\mathcal{F}}_{u}-\gamma^{\scriptscriptstyle-2}\bar{\mathcal{F}}_{v}].$
(26) $\mathcal{F}_{z}\rightarrow\bar{\mathcal{F}}_{v}.$ (27)

(Note that
$\bar{\mathcal{F}},~{}~{}\bar{\mathcal{F}}_{u},~{}~{}\bar{\mathcal{F}}_{v}$
become constants that depend on $\gamma$ in the asymptotic regime.) As in the
spherically symmetric case, Eqs. (6) (7) can be stripped of all the
divergences. Then, substituting from Eqs. (25)-(27) we get (after a little
additional algebra)

$\epsilon=-\frac{3}{2}\gamma^{\scriptscriptstyle
2}\bar{\mathcal{F}}\left[1-\frac{4}{3}\frac{\partial\ln{\bar{\mathcal{F}}}}{\partial\ln{u}}-\frac{2}{3}\frac{\partial\ln{\bar{\mathcal{F}}}}{\partial\ln{v}}\right]$
(28) $\kappa=-\frac{3}{2}\gamma^{\scriptscriptstyle
3}\bar{\mathcal{F}}\left[1-2\frac{\partial\ln{\bar{\mathcal{F}}}}{\partial\ln{u}}-\frac{\partial\ln{\bar{\mathcal{F}}}}{\partial\ln{v}}\right].$
(29)

It is standard to normalize $a_{0}$ so that asymptotically we have the
equality

$g=(g_{\scriptscriptstyle N}a_{0})^{1/2}{\bf e},$ (30)

namely, $\kappa=1$. Here, this is ensured as follows: Given our choice of
$\mathcal{F}$, with its DML behavior encapsulated in $\bar{\mathcal{F}}$,
equation (29) with $k=1$ is used to determine $\gamma$. Then, Eq. (28) fixes
the value of $\epsilon$ that we have to take. This yields

$\epsilon=\frac{2}{3\gamma}-\frac{1}{2}\gamma^{\scriptscriptstyle
2}\bar{\mathcal{F}}(\gamma^{\scriptscriptstyle-4},2\gamma^{\scriptscriptstyle-2}).$
(31)

###  III.3 Deep-MOND virial relation and consequences

The tripotential theories, with the MOND limit defined by Eq. (14) fall under
the general class of MG, MOND theories dealt with in Ref. [20]. It was shown
there that such theories generally satisfy a very useful virial relation: For
a deep-MOND, self-gravitating, isolated system of pointlike masses, $m_{p}$,
at positions ${\bf r}_{p}$, subject to (gravitational) forces ${\bf F}_{p}$,
the following holds:

$\sum_{p}{\bf r}_{p}\cdot{\bf
F}_{p}=-(2/3)(Ga_{0})^{1/2}[(\sum_{p}m_{p})^{3/2}-\sum_{p}m_{p}^{3/2}].$ (32)

This then leads to the deep-MOND two-body force for arbitrary masses,

$F(m_{1},m_{2},\ell)=\frac{2}{3}\frac{(a_{0}G)^{1/2}}{\ell}[(m_{1}+m_{2})^{3/2}-m_{1}^{3/2}-m_{2}^{3/2}]$
(33)

($\ell$ is the distance between the masses), and to a general mass-velocity-
dispersion relation

$\sigma^{2}=\frac{2}{3}(MGa_{0})^{1/2}[1-\sum_{p}(m_{p}/M)^{3/2}],$ (34)

where $\sigma^{2}=M^{-1}\sum_{p}m_{p}{\bf v}_{p}^{2}$ is the mass-weighted,
three-dimensional, mean-squared velocity dispersion, and $M=\sum_{p}m_{p}$ is
the total mass.

This also implies that all these theories predict the same value of the $Q$
parameter, defined in Ref. [12], which was proposed as a possible
discriminator between MOND theories. This parameter thus cannot discriminate
between the different TRIMOND theories.

The reason that the full detail of the DML of the tripotential theories does
not enter the virial relation and its corollaries is that what effectively
enters the derivation of this relation is only the asymptotic behavior of the
MOND field from a single mass. And, from Eq. (30), this is oblivious to the
details of the theory,

##  IV Discussion

The large variety afforded by the freedom to choose the three-variable
$\mathcal{F}$ can potentially lead to a considerable variety in secondary
predictions of TRIMOND theories. However, to demonstrate this in some
quantitative detail would require numerical solutions, which I cannot provide
here.

AQUAL and QUMOND were shown to be the nonrelativistic limits of certain
versions of the bimetric, relativistic formulation of MOND (BIMOND) [21, 22].
It seems that generalizations of BIMOND with an additional, scalar degree of
freedom could give TRIMOND theories as their Newtonian limits. BIMOND theories
use in their Lagrangians scalars that are quadratic in the tensors

$C^{\scriptscriptstyle\lambda}_{\scriptscriptstyle\mu\nu}=\Gamma^{\scriptscriptstyle\lambda}_{\scriptscriptstyle\mu\nu}-\hat{\Gamma}^{\scriptscriptstyle\lambda}_{\scriptscriptstyle\mu\nu},$
(35)

where $\Gamma^{\scriptscriptstyle\lambda}_{\scriptscriptstyle\mu\nu}$ and
$\hat{\Gamma}^{\scriptscriptstyle\lambda}_{\scriptscriptstyle\mu\nu}$ are the
connections of the two metrics that are the gravitational degrees of freedom
in BIMOND. Such quadratic scalars reduce in the nonrelativistic limit to
$(\vec{\nabla}\chi)^{2}$, for some potential $\chi$.

With an additional, scalar degree of freedom, $\varphi$, we can construct
scalar variables such as
$\varphi_{\scriptscriptstyle,\mu}\varphi_{\scriptscriptstyle,}^{\scriptscriptstyle\mu}$
and the mixed $\varphi_{\scriptscriptstyle,\mu}\bar{C}^{\mu}$,
$\varphi_{\scriptscriptstyle,\mu}C^{\mu}$, where
$C^{\lambda}=g^{\scriptscriptstyle\mu\nu}C^{\scriptscriptstyle\lambda}_{\scriptscriptstyle\mu\nu}$,
$\bar{C}^{\mu}=g^{\lambda\nu}C^{\scriptscriptstyle\mu}_{\scriptscriptstyle\mu\nu}$
are the two traces of
$C^{\scriptscriptstyle\lambda}_{\scriptscriptstyle\mu\nu}$. Such variable are
expected to reduce in the nonrelativistic limit to some
$(\vec{\nabla}\varphi)^{2}$ and $\vec{\nabla}\varphi\cdot\vec{\nabla}\chi$. So
it may be possible to construct such scalar-bimetric theories that reduce to
some TRIMOND theories.

## References

  * [1] J. Bekenstein and M. Milgrom, 1984, Does the missing mass problem signal the breakdown of Newtonian gravity? Astrophys. J. 286, 7
  * [2] M. Milgrom, 1983, A modification of the Newtonian dynamics as a possible alternative to the hidden mass hypothesis. Astrophys. J. 270, 365
  * [3] B. Famaey and S.S. McGaugh, 2012, Modified Newtonian Dynamics (MOND): Observational Phenomenology and Relativistic Extensions. Living Rev. Relativ. 15, 10
  * [4] M. Milgrom, 2014, continually updated, The MOND paradigm of modified dynamics. Scholarpedia, 9, 31410
  * [5] M. Milgrom, 2020, MOND vs. dark matter in light of historical parallels. Stud. Hist. Philos. Mod. Phys. 71, 170
  * [6] S. McGaugh, 2020, Predictions and Outcomes for the Dynamics of Rotating Galaxies. Galaxies 8, 35
  * [7] D. Merritt, 2020, A Philosophical Approach to MOND: Assessing the Milgromian Research Program in Cosmology, Cambridge University Press
  * [8] I. Banik and HS. Zhao, 2022, From galactic bars to the Hubble tension – weighing up the astrophysical evidence for Milgromian gravity. Symmetry 14, 1331
  * [9] M. Milgrom, 2010, Quasi-linear formulation of MOND. Mon. Not. R. Astron. Soc. 403, 886
  * [10] M. Milgrom, 2014, MOND laws of galactic dynamics. Mon. Not. R. Astron. Soc. 437, 2531
  * [11] R. Brada and M. Milgrom, 1995, Exact solutions and approximations of MOND fields of disk galaxies. Mon. Not. R. Astron. Soc. 276, 453
  * [12] M. Milgrom, 2012, Global Deep-MOND Parameter as a Theory Discriminant. Phys. Rev. Lett. 109, 251103
  * [13] K. Brown, R. Abraham, L. Kell, and H. Mathur, 2018, The radial acceleration relation and a magnetostatic analogy in quasilinear MOND. New J. Phys. 20, 063042
  * [14] J. Petersen and F. Lelli, 2020 A first attempt to differentiate between modified gravity and modified inertia with galaxy rotation curves. Astron. Astrophys. 636, A56
  * [15] KH. Chae, 2022, Distinguishing Dark Matter, Modified Gravity, and Modified Inertia with the Inner and Outer Parts of Galactic Rotation Curves. Astrophys. J. 941, 55
  * [16] M. Milgrom, 2022, Models of modified-inertia formulation of MOND. Phys. Rev. D 106, 064060
  * [17] M. Milgrom, 2023, Generalizations of Quasilinear MOND (QUMOND). arXiv:2305.01589
  * [18] M. Milgrom, 2009, The Mond Limit from Spacetime Scale Invariance . Astrophys. J. 698, 1630
  * [19] D. Gilbarg and N.S. Trudinger, 1977, Elliptic Partial Differential Equations of Second Order. Springer-Verlag
  * [20] M. Milgrom, 2014, General virial theorem for modified-gravity MOND. Phys. Rev. D 89, 024016
  * [21] M. Milgrom, 2009, Bimetric MOND gravity. Phys. Rev. D 80, 123536
  * [22] M. Milgrom, 2022, Broader view of bimetric MOND. Phys. Rev. D 106, 084010