Calderón–Zygmund Type Singular Operators in Weighted Generalized Morrey Spaces

Abstract

We find conditions for the weighted boundedness of a general class of multidimensional singular integral operators in generalized Morrey spaces $\mathcal {L}^{p,\varphi }(\mathbb {R}^n,w),$ defined by a function $\varphi (x,r)$ and radial type weight $w(|x-x_0|), x_0\in {\mathbb {R}}^{n}.$ These conditions are given in terms of inclusion into $\mathcal {L}^{p,\varphi }(\mathbb {R}^n,w),$ of a certain integral constructions defined by $\varphi$ and w. In the case of $\varphi =\varphi (r)$ we also provide easy to check sufficient conditions for that in terms of indices of $\varphi$ and w.

1 Introduction

The generalized Morrey space $\mathcal {L}^{p,\varphi }(\mathbb {R}^n )$ with which we work in this paper is defined in Subsect. 2.1.

Weighted estimates for multi-dimensional singular integral operators, obtained up to now, concern specific situations where the weight w and the function $\varphi$ defining the generalized Morry space are dependent on each other, or the weight is subject to the unnatural condition that it should belong to the Muckenhoupt class for Lebesgue spaces. Note in this relation that the Muckenhoupt class expected for Morrey spaces is different from the Muckenhoupt class $A_p$ corresponding to Lebesgue spaces.

In this paper we present a new approach, which allows us to prove boundedness of weighted Caderón–Zygmund operators in the general case when the weight w and the function $\varphi$ are independent on each other and the weight is not supposed to be in $A_p$.

It is also worthwhile to note that, for the class of admitted weights, we manage to give the boundedness conditions in terms of numerical characteristics of the weight w and the function $\varphi$, which is of importance for applications.

The classical Morrey spaces $\mathcal {L}^{p,\lambda },$ introduced in relation to the study of regularity properties of solutions to partial differential equations (PDE), are well known, see for instance the books [3, 8, 21] and references therein. The generalized Morrey spaces $\mathcal {L}^{p,\varphi }$ are obtained by replacing $r^\lambda$ by a function $\varphi = \varphi (r)$ in the definition of the Morrey space, see Definition 2.1; Morrey spaces, classical or generalized, have a long history which is well presented in various sources, we refer for instance to the survey paper [15].

The weighted Morrey spaces are treated in the usual sense: $\mathcal {L}^{p,\varphi }(\Omega , w): = \{ f: \ \ w f \in \mathcal {L}^{p,\varphi }(\Omega )\}, \ \Omega \subseteq {\mathbb {R}}^{n}.$

Non-weighted estimates for singular integral operators (SIO) in Morrey spaces are well studied, see for instance, [1, 4, 12, 16, 20] and the references therein.

The weighted Morrey spaces are much less studied. In the paper [17] there were proved weighted estimates in Morrey spaces $\mathcal {L}^{p,\lambda }$ for Hardy operators on $\mathbb {R}_+$ with applications to weighted estimates of the Hilbert transform on $\mathbb {R}^1$ (even on Carleson curves in the complex plane), see also an application of the latter result in [10] for the study of singular integral equations in weighted generalized Morrey spaces. There were used radial type weights $w(|x-x_0|)$, generated by quasi-monotone functions w(r); these weights have possible decay or growth at $r=0$ and $r=\infty$, characterized by the condition that the weight becomes almost increasing or almost decreasing after the multiplication by a power function. Such weights oscillate between two powers at the origin and infinity (with different exponents for the origin and infinity, in general), see an overview on such weights in [14].

In the paper [7] by Komori and Shirai, there where admitted general weights but generalized Morrey spaces were considered in a particular setting: the function defining the space was a power of the weight. Later in [5] there was studied the boundedness of singular type operators in weighted generalized Morrey spaces where, as well as in [7], the authors restricted their studies by the assumption that weights are in the Muckenhoupt class $A_p.$ This assumption is unnatural for Morrey spaces. For instance, as shown in [17] in the one-dimensional case, and as follows from the result of this paper in the multi-dimensional case, the singular operator is bounded in the classical Morrey space $L^{p, \lambda }(w)$ with the power weight $w= |x|^\gamma$ if

$$\begin{aligned} \lambda - n < \gamma < \lambda + n(p-1) \end{aligned}$$

(if and only if in the one-dimensional case; “only if” may be shown to hold also in the multi-dimensional case under some assumption on the non-degeneracy of the singular operator). However the restriction to $w \in A_p$ gives a different interval

$$\begin{aligned} - n < \gamma < n(p-1). \end{aligned}$$

Thus, the class of weights governing the boundedness of the singular operator in weighted generalized Morrey spaces is a certain class $A_{p, \varphi }$ depending both on p and $\varphi .$ An intersection of this class with $A_p$ essentially restricts the study. The class $A_{p, \varphi }$ is expected to be more narrow than $A_p$ with respect to the growth of weights but larger with respect to their vanishing. A characterization of the class $A_{p, \varphi }$ is an open problem. Note that in the case of the maximal operator in [18] there were given some separate necessary and sufficient conditions on weights w with a “small” gap between them, for local Morrey spaces.

A study of weighted problem in Morrey spaces, free of the restriction $w \in A_p$ which was started in [17] in the case of radial weights, was continued in [10, 13, 19] for potential, singular and Hardy type operators in generalized Morrey spaces.

We deal with weights of radial type which allows us to use the technique of numerical characteristics of quasi-monotone weights, which is effective for applications. Details of such numerical characteristics may be found in [14].

The paper is organized as follows. In Sect. 2 we give definitions and necessary preliminaries. The main results (Theorems 3.2–3.5) are presented, proved and discussed in Sect. 3. Condition of boundedness in terms of indices of $\varphi$ and w are proved in Subsect. 3.3.

2 Preliminaries

2.1 Morrey Spaces

Let $\varphi (x,r)$ be a non-negative function on ${\mathbb {R}}^{n}\times \mathbb {R}_+$ such that

$$\begin{aligned} \inf \limits _{x\in {\mathbb {R}}^{n}}\inf \limits _{r>\delta }\varphi (x,r)>0 \end{aligned}$$

(2.1)

for every $\delta >0.$ We will use the notation

$$\begin{aligned} \mathfrak {M}_{p,\varphi }(f;x,r):\,=\frac{1}{\varphi (x,r)}\int \limits _{ B (x,r)}|f(y)|^p\, dy. \end{aligned}$$

(2.2)

where $B(x,r)=\{y\in \mathbb {R}^n: |x-y|<r\}$ and $1\le p<\infty$.

Definition 2.1

The generalized Morrey space $\mathcal {L}^{p,\varphi }(\mathbb {R}^n )$ is defined as the space of functions $f\in L^p_\mathrm{loc}(\mathbb {R}^n )$ such that

$$\begin{aligned} \sup \limits _{x\in \mathbb {R}^n , r>0}\mathfrak {M}_{p,\varphi }(f;x,r)<\infty . \end{aligned}$$

(2.3)

Equipped with the norm

$$\begin{aligned} \Vert f\Vert _{p,\varphi }:\,=\sup \limits _{x\in \mathbb {R}^n , r>0} \mathfrak {M}^\frac{1}{p}_{p,\varphi }(f;x,r) \end{aligned}$$

(2.4)

this is a Banach space. The corresponding weighted space $\mathcal {L}^{p,\varphi }(\mathbb {R}^n ,w)$ is defined as

$$\begin{aligned} \mathcal {L}^{p,\varphi }(\mathbb {R}^n ,w):\,=\left\{ f : wf\in \mathcal {L}^{p,\varphi }(\mathbb {R}^n), \right\} , \end{aligned}$$

(2.5)

where w is a weight. with $\Vert f\Vert _{\mathcal {L}^{p,\varphi }(\mathbb {R}^n ,w)}:\,=\Vert wf\Vert _{\mathcal {L}^{p,\varphi }(\mathbb {R}^n )}.$

We assume that

$$\begin{aligned} \inf \limits _{x\in {\mathbb {R}}^{n}}\varphi (x,r) \ge c r^n, \quad 0<r<1, \end{aligned}$$

(2.6)

which makes the space $\mathcal {L}^{p,\varphi }(\mathbb {R}^n)$ nontrivial, since bounded functions with compact support belong then to $\mathcal {L}^{p,\varphi }(\mathbb {R}^n)$.

Note that the conditions (2.1) and (2.6) are assumed to be satisfied everywhere in the sequel without further reference.

The classical Morrey space corresponds to the case $\varphi (x,r)\equiv r^\lambda , \ 0<\lambda <n$. It will be denoted by

$$\begin{aligned} \mathcal {L}^{p,\lambda }(\mathbb {R}^n) \end{aligned}$$

without danger of confusion with notation.

We will need the following statement on inclusion of radial functions into Morrey spaces on bounded sets. Bounded functions are always in Morrey spaces under the assumption (2.6). For possibly unbounded functions we will use the following result from [13, Proposition 3.3] in the case of $\varphi (x,r)=\varphi (r)$ and radial weights. This result is given in terms of Mtuszewska-Orlicz-type indices of functions $\varphi$ and w,  (the definition and some properties of the indices we give in the next Subsect. 2.2, we also refer to our paper [14] for more detailed information about their properties).

Lemma 2.2

Let $\ell =diam \, \Omega < \infty$, $\varphi (r)$ be an almost increasing function on $[0,\ell ]$ satisfying the condition $\varphi (r)\ge c r^n$, $V\in \overline{W}([0,\ell ])$ and let V(r) be doubling. The function $V(|x-x_0|)$, $x_0\in \Omega$, belongs to the space $\mathcal {L}^{p,\varphi }$, if

$$\begin{aligned} M(\varphi ) - pm(V) < n. \end{aligned}$$

(2.7)

The following simple fact of independent interest is also useful for our goals.

Lemma 2.3

Let $\ell =diam \, \Omega < \infty$, $\varphi (r)$ be an almost increasing function on $[0,\ell ]$. and V(r) be almost decreasing. The function $V(|x-x_0|)$, $x_0\in \Omega$, belongs to the space $\mathcal {L}^{p,\varphi }$, if $V\in \overline{V}([0,\ell ])$ and is almost decreasing, and the following conditions hold:

$$\begin{aligned} m(V)>-\frac{n}{p} \quad \mathrm{and} \quad \sup \limits _{0<r<\ell }\frac{r^n V^p(r)}{\varphi (r)}<\infty . \end{aligned}$$

(2.8)

Proof

From the fact that V(r) is almost decreasing, it is easily derived that

$$\begin{aligned} \int \limits _{B(x,r)}V^p(|y|)\,dy \le C \int \limits _{B(0,r)}V^p(|y|)\,dy. \end{aligned}$$

(2.9)

Indeed,

$$\begin{aligned} \int \limits _{B(x,r)}V^p(|y|)\,dy= \int \limits _{E_1\cap B(x,r)}V^p(|y|)\,dy+\int \limits _{E_2\cap B(x,r)}V^p(|y|)\,dy, \end{aligned}$$

where $E_1=\{y: |y-x|\ge |y|\}$ and $E_1=\{y: |y-x|\le |y|\}.$ It is obvious that $E_1\cap B(x,r) \subseteq B(0,1)$ and for the second integral we have $V(|y|)\le V(|x-y|)$, which completes the proof of (2.9). By (2.9) and the assumption on m(w) we have

$$\begin{aligned} \frac{1}{\varphi (r)} \int \limits _{B(x,r)}V^p(|y|)\,dy \le \frac{C}{\varphi (r)} \int \limits _0^r t^{n-1}V^p(t)\,dt \le C\frac{r^{n}V^p(r)}{\varphi (r)}, \end{aligned}$$

which completes the proof. $\square$

2.2 On Some Classes of Weight Functions

In the sequel, a non-negative function f on $[0,\ell ], 0<\ell \le \infty ,$ is called almost increasing (almost decreasing), if there exists a constant $C (\ge 1)$ such that $f(x)\le Cf(y)$ for all $x\le y$ ($x\ge y$, respectively). Equivalently, a function f is almost increasing (almost decreasing), if it is equivalent to an increasing (decreasing, resp.) function g, i.e. $c_1 f(x)\le g(x)\le c_2 f(x), c_1>0,c_2>0$.

Definition 2.4

Let $0<\ell <\infty .$

• (1) By $W=W([0,\ell ])$ we denote the class of continuous and positive functions $\varphi$ on $(0,\ell ]$ such that there exists finite or infinite limit $\lim \limits _{x\rightarrow 0}\varphi (x)$;

• (2) by $W_0=W_0([0,\ell ])$ we denote the class of almost increasing functions $\varphi \in W$ on $(0,\ell )$;

• (3) by $\overline{W}=\overline{W}([0,\ell ])$ we denote the class of functions $\varphi \in W$ such that $x^a\varphi (x)\in W_0$ for some $a =a(\varphi )\in \mathbb {R}^1$;

• (4) by $\underline{W}=\underline{W}([0,\ell ])$ we denote the class of functions $\varphi \in W$ such that $\frac{\varphi (t)}{t^b}$ is almost decreasing for some $b\in \mathbb {R}^1$.

Definition 2.5

Let $0<\ell <\infty .$

• (1) By $W_\infty =W_\infty ([\ell ,\infty ])$ we denote the class of functions $\varphi$ which are continuous and positive and almost increasing on $[\ell ,\infty )$ and which have the finite or infinite limit $\lim _{x\rightarrow \infty }\varphi (x),$

• (2) by $\overline{W}_\infty =\overline{W}_\infty ([\ell ,\infty ))$ we denote the class of functions $\varphi \in W_\infty$ such $x^a\varphi (x)\in W_\infty$ for some $a =a(\varphi )\in \mathbb {R}^1$.

By $\overline{W}(\mathbb {R}_+)$ we denote the set of functions on $\mathbb {R}_+$ whose restrictions onto (0, 1) are in $\overline{W}([0,1])$ and restrictions onto $[1,\infty )$ are in $\overline{W}_\infty ([1,\infty )).$ Similarly, the set $\underline{W}(\mathbb {R}_+)$ is defined.

For a function $\varphi \in \overline{W}$, the numbers

$$\begin{aligned} m(\varphi )=\sup _{0<x<1}\frac{\ln \left( \limsup \limits _{h\rightarrow 0} \frac{\varphi (hx)}{\varphi (h)}\right) }{\ln x}= \lim _{x\rightarrow 0}\frac{\ln \left( \limsup \limits _{h\rightarrow 0} \frac{\varphi (hx)}{\varphi (h)}\right) }{\ln x} \end{aligned}$$

(2.10)

and

$$\begin{aligned} M(\varphi )=\sup _{x>1}\frac{\ln \left( \limsup \limits _{h\rightarrow 0} \frac{\varphi (hx)}{\varphi (h)}\right) }{\ln x}=\lim _{x\rightarrow \infty }\frac{\ln \left( \limsup \limits _{h\rightarrow 0} \frac{\varphi (hx)}{\varphi (h)}\right) }{\ln x} \end{aligned}$$

(2.11)

are known as the Matuszewska-Orlicz-type lower and upper indices of the function $\varphi (r)$. We refer to [11, 14] for the properties of the indices of such a type. Note that in this definition $\varphi (x)$ need not to be an N-function: only its behavior at the origin is of importance. Observe that $0\le m(\varphi )\le M(\varphi ) \le \infty \ \ \text {for}\ \ \varphi \in W_0,$ and $-\infty <m(\varphi )\le M(\varphi )\le \infty \quad \text {for} \quad \varphi \in \overline{W},$ and the following formulas are valid:

$$\begin{aligned} m[x^a \varphi (x)]=a + m(\varphi ), \quad M[x^a \varphi (x)]=a + M(\varphi ), \quad \quad a\in \mathbb {R}^1, \end{aligned}$$

(2.12)

$$\begin{aligned} m([\varphi (x)]^a)=a m(\varphi ), \quad M([\varphi (x)]^a)=a M(\varphi ), \quad a\ge 0 \end{aligned}$$

(2.13)

$$\begin{aligned} m\left( \frac{1}{\varphi }\right) = -M(\varphi ), \quad \quad M\left( \frac{1}{\varphi }\right) = - m(\varphi ). \end{aligned}$$

(2.14)

$$\begin{aligned} m(uv)\ge m(u)+m(v), \quad \quad M(uv)\le M(u)+M(v) \end{aligned}$$

(2.15)

for $\varphi , u,v\in \overline{W}.$

The following classes of weight functions were introduced in [17], see also [13].

Definition 2.6

By $\mathbf {V}_{\pm }$, we denote the classes of functions w non-negative on $[0,\infty )$ and positive on $(0,\infty ),$ defined by the conditions:

$$\begin{aligned} \mathbf {V_{+}} : \qquad \frac{|w(t)- w(\tau )|}{|t-\tau |} \le C \frac{ w(t_+)}{t_+} , \qquad \end{aligned}$$

(2.16)

$$\begin{aligned} \mathbf {V_{-}} : \qquad \frac{| w(t)- w(\tau )|}{|t-\tau |} \le C \frac{ w(t_-)}{t_+} , \qquad \end{aligned}$$

(2.17)

where $t,\tau \in (0,\infty ), t\ne \tau ,$ and $t_+ = \max (t,\tau ), \ t_- = \min (t,\tau ).$

Lemma 2.7

Functions $\varphi \in \mathbf {V}_+$ are almost increasing and functions $\varphi \in \mathbf {V}_-$ are almost decreasing.

Proof

Let $w \in \mathbf {V}_+$ and $\tau \le t$. By (2.16) we have $|w(t)-w(\tau )|\le C w(t)\left( 1-\frac{\tau }{t}\right) \le Cw(t).$ Then $w(\tau )\le |w(t)-w(\tau )|+w(t)\le (C+1)w(t).$ The case $w \in \mathbf {V}_-$ is treated similarly. $\square$

Note that for power weights we have

$$\begin{aligned} t^\gamma \in \mathbf {V_+} \ \Longleftrightarrow \ \gamma \ge 0 , \ \quad \ \ t^\gamma \in \mathbf {V_-} \ \Longleftrightarrow \ \gamma \le 0. \end{aligned}$$

2.3 On Integral Operators with a Kernel Homogeneous of Degree $-n$

We will use the following known result on the boundedness in $L^p({\mathbb {R}}^{n})$ of integral operators

$$\begin{aligned} K f(x):\,=\int \limits _{{\mathbb {R}}^{n}} k(x,y) f(y)\ dy , \qquad x \in {\mathbb {R}}^{n}, \end{aligned}$$

(2.18)

with the kernel k(x, y), which is homogeneous of degree $-n$, i.e. $k(\lambda x, \lambda y) = \lambda ^{-n} k(x,y), \ \lambda >0,$ and invariant with respect to notaions: $k(\omega (x),\omega (y)) = k(x,y), \ \omega \in SO(n), \ x,y\in {\mathbb {R}}^{n}$. For the proof we refer to [6, Theorem 6.4].

Theorem 2.8

Let the kernel k(x, y) be invariant with respect to rotations and homogeneous of degree $-n$. The operator K is bounded in $L^p({\mathbb {R}}^{n}), 1\le p < \infty ,$ if

$$\begin{aligned} \kappa : = \int _{R^n} \left| k(e_1,y)\right| |y|^{-\frac{n}{p}} dy <+ \infty , \quad e_1 = (1,0,\dots , 0) \end{aligned}$$

(2.19)

and then $\Vert K\varphi \Vert ^p \le \kappa \Vert \varphi \Vert ^p.$ If $k(x,y)\ge 0$, then the condition (2.19) is necessary for the boundedness and $\Vert K\Vert _{L^p\rightarrow L^p}=\kappa .$

2.4 On Non-weighted Boundedness of SIO in the Generalized Morrey Space

To formulate a result from [20], we need the following notion, where T is a sub-linear operator, i.e. $|T(f+g)|\le T|f|+T|g|.$

Definition 2.9

Let $1<p<\infty .$ A sub-linear operator T will be called p-admissible singular type operator, if:

• (1) T satisfies a “size condition” of the form

  $$\begin{aligned} \chi _{B(x,r)}(z)|T(f\chi _{{\mathbb {R}}^{n}\backslash B(x,2r)})(z)|\le C \chi _{B(x,r)}(z)\int \limits _{{\mathbb {R}}^{n}\backslash B(x,2r)}\frac{|f(y)|\,dy}{|y-z|^n} \end{aligned}$$

  for $x\in {\mathbb {R}}^{n}$ and $r>0$;

• (2) T is bounded in $L^{p}({\mathbb {R}}^{n}).$

The maximal operator

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

(2.20)

is an example of p-admissible singular type operators, as can be easily obtained from the definition of the maximal operator.

We also need the following subclass of p-admissible integral operators, where we require that the size condition holds for all $x \ne y.$

Definition 2.10

We say that a singular integral operator

$$\begin{aligned} T f(x)=\int _{{\mathbb {R}}^{n}}K(x,y) f(y) dy=\lim _{\varepsilon \rightarrow 0}\int _{|x-y|>\varepsilon }K(x,y) f(y) dy, \end{aligned}$$

(2.21)

belongs to the class $\mathfrak {S}_p$, if

$$\begin{aligned} |K(x,y)|\le \frac{C}{|x-y|^n} \end{aligned}$$

(2.22)

for all $x\ne y$ and T is bounded in $L^p({\mathbb {R}}^{n}), 1<p<\infty .$

Calderon–Zygmund operators with a standard kernel K(x, y) belong to the class $\mathfrak {S}_p, 1 <p<\infty$. The latter means (see for instance [2], p. 99) that

• (1) K(x, y) satisfies the size condition $|K(x,y)|\le C|x-y|^{-n}$ for $x\ne y$;

• (2) K(x, y) is a continuous function on $\{(x,y)\in \mathbb {R}^n \times \mathbb {R}^n : x\ne y\},$ satisfying the estimates $\ \ |K(x,y)-K(x,z)|\le C \frac{|y-z|^\sigma }{|x-y|^{n+\sigma }},\;\; \text{ if }\;\;|x-y|>2|y-z|,$ and $|K(x,y)-K(\xi ,y)|\le C \frac{|x-\xi |^\sigma }{|x-y|^{n+\sigma }},\;\; \sigma >0,\;\; \text{ if }\;\;|x-y|>2|x-\xi |;$

• (3) T is bounded in $L^2({\mathbb {R}}^{n})$.

As is well known, under the conditions 1–3 the operator T is bounded in $L^p({\mathbb {R}}^{n})$ for all $p\in (1,\infty ).$

The boundedness of non-weighted multi-dimensional singular integral operators within the frameworks of generalized Morrey spaces was first considered by Guliev in [4], where weak Morrey spaces were also included. With the aid of the results of [4], similar results for vanishing generalized Morrey spaces were obtained in [20, Theorem 5.1].

For the proof of the following theorem, under the assumptions given in that theorem, we refer to [20, Theorem 5.1].

Theorem 2.11

Let $1<p<\infty$. Every p-admissible singular type operator T is bounded in the Morrey space $\mathcal {L}^{p,\varphi }(\mathbb {R}^n )$, if $\varphi$ satisfies the condition

$$\begin{aligned} \int \limits _r^\infty \frac{\varphi ^\frac{1}{p}(x,t)\, dt}{t^{1+\frac{n}{p}}} \le C_0 \,\frac{\varphi ^\frac{1}{p}(x,r)}{r^\frac{n}{p}}, \end{aligned}$$

(2.23)

where $C_0$ does not depend on $x\in {\mathbb {R}}^{n}$ and $r>0.$

From Theorem 2.11 the following corollary is derived.

Corollary 2.12

Let $1<p<\infty$ and $\varphi$ satisfy the condition (2.23). Then every operator $T\in \mathfrak {S}_p$ is bounded in the Morrey space $\mathcal {L}^{p,\varphi }(\mathbb {R}^n )$.

3 The Main Results

As is well known, the boundedness of any operator A in a weighted Banach function space $X(w) = \{ f: wf \in X \}$ is equivalent to the boundedness of the weighted operator $w A\frac{1}{w}$ in the non-weighted space X,  which is evident and used without any additional explanation.

We will work with the following weighted multi-dimensional Hardy operators:

$$\begin{aligned} H_w f(x)= \frac{w (|x|)}{|x|^n}\int \limits _{|y|<|x|} \frac{f(y)dy}{w (|y|)}, \quad \mathcal {H}_w f(x) = w (|x|)\int \limits _{|y|>|x|} \frac{f(y)dy}{|y|^nw (|y|)}. \end{aligned}$$

(3.1)

For information of Hardy type operators we refer to the book [9].

Note that the non-weighted Hardy operators

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

(3.2)

are also examples of operators of the class $\mathfrak {S}_p, 1<p<\infty$; indeed their boundedness in $L^p({\mathbb {R}}^{n})$ is well known, while the size condition is seen from the fact that $|x-y|\le 2|x|$ for the operator H and $|x-y|\le 2|y|$ for the operator $\mathcal {H}$.

3.1 On Operators $K_m$ and $\mathcal {K}_m$

We will use the following hybrids of Hardy and potential operators:

$$\begin{aligned} K_m f(x)= \frac{1}{|x|^{m}}\int \limits _{|y|<|x|} \frac{f(y)}{|x-y|^{n-m}}dy, \quad \mathcal {K}_m f(x) =\int \limits _{|y|>|x|} \frac{f(y)dy}{|y|^m|x-y|^{n-m}}, \end{aligned}$$

(3.3)

where $0<m\le n.$

We need the following Lemma of independent interest:

Lemma 3.1

The operators $K_m$ and $\mathcal {K}_m, \ 0<m\le n,$ belong to the class $\mathfrak {S}_p, 1<p<\infty .$

Proof

The validity of the size condition is obvious: $|x|^m|x-y|^{n-m}> 2^{-m}|x-y|^n$ for $K_m$ and $|y|^m|x-y|^{n-m}> 2^{-m}|x-y|^n$ for $\mathcal {K}_m.$ As regards the boundedness in $L^p,$ these operators are examples of integral operators (2.18) with kernels homogeneous of degree $-n$. The operators $K_m$ and $\mathcal {K}_m$ have the kernels

$$\begin{aligned} k_+(x,y):\,= \frac{\theta _+(x,y)}{|x|^m|x-y|^{n-m}} \quad \mathrm{and} \quad k_-(x,y):\,= \frac{\theta _-(x,y)}{|y|^m|x-y|^{n-m}}, \end{aligned}$$

respectively, where we denoted

$$\begin{aligned} \theta _+(x,y):\,= \chi _{E_+}(x,y), \quad \theta _-(x,y):\,= \chi _{E_-}(x,y) \end{aligned}$$

with $E_+: =\{(x,y)\in \mathbb {R}^{2n}: |y|<|x|\}, \ \ E_-: =\{(x,y)\in \mathbb {R}^{2n}: |y|>|x|\},$ for brevity. The condition (2.19) for these kernels holds for all $m\in (0,n]$ and $1<p<\infty ,$ which is verified directly. Then the proof is complete by Theorem 2.8. $\square$

Our first main result is the following pointwise estimate, which reduces the boundedness of singular operators with weight $w \in \mathbf {V}_-\cup \mathbf {V}_+$ to the boundedness of Hardy operators with the same weight.

3.2 Main Theorems

Theorem 3.2

Let T be an operator (2.21) with the size condition (2.22) and $w \in \mathbf {V}_-\cup \mathbf {V}_+$. Then

$$\begin{aligned}&\left| w T\frac{1}{w }f(x)- Tf(x)\right| \nonumber \\&\quad \le C\left( H_w(|f|)(x) + \sum \limits _{m=1}^{n-1} K_m(|f|)(x)+ \mathcal {K}_1 (|f|)(x)\right) , \qquad \mathrm{if} \quad w \in \mathbf {V}_+,\qquad \quad \end{aligned}$$

(3.4)

and

$$\begin{aligned}&\left| w T\frac{1}{w }f(x)- Tf(x)\right| \nonumber \\&\quad \le C\left( \mathcal {H}_w(|f|)(x)+ \sum \limits _{m=1}^{n-1} \mathcal {K}_m(|f|)(x)+ K_1(|f|)(x)\right) , \qquad \mathrm{if} \quad w \in \mathbf {V}_-.\qquad \quad \end{aligned}$$

(3.5)

Proof

We prove the estimate (3.4). In the relation

$$\begin{aligned} w T\frac{1}{w }f(x)- Tf(x)=\int _{{\mathbb {R}}^{n}}\frac{w(|x|)-w(|y|)}{w(|y|)}K(x,y)f(y)\,dy \end{aligned}$$

we use the size condition for K(x, y) and split the integration into the cases $|y|<|x|$ and $|y|>|x|$ and, by (2.16), we obtain that

$$\begin{aligned}&\left| w T\frac{1}{w }f(x)- Tf(x)\right| \\&\quad \le C\left( \frac{w(|x|)}{|x|}\int \limits _{|y|<|x|}\frac{|f(y)|\,dy}{w(|y|)|x-y|^{n-1}} + \int \limits _{|y|>|x|}\frac{|f(y)|\,dy}{|y|\cdot |x-y|^{n-1}} \right) \\&\quad = C\left( \frac{1}{|x|}\int \limits _{|y|<|x|}\frac{w(|x|)-w(|y|)}{w(|y|)} \frac{|f(y)|\,dy}{|x-y|^{n-1}} + K_1 (|f|)(x)+\mathcal {K}_1(|f|)(x) \right) . \end{aligned}$$

Applying again (2.16), now in the fraction $\frac{w(|x|)-w(|y|)}{w(|y|)},$ we arrive at

$$\begin{aligned}&\!\!\!\!\left| w T\frac{1}{w }f(x)- Tf(x)\right| \\&\le C\left( \frac{w(|x|)}{|x|^2}\int \limits _{|y|<|x|}\frac{|f(y)|\,dy}{w(|y|) |x-y|^{n-2}} + K_1 (|f|)(x)+\mathcal {K}_1(|f|)(x) \right) . \end{aligned}$$

By iterating this procedure, we obtain (3.4). The proof of (3.5) is similar so we leave out the details. $\square$

Theorem 3.3

Let $1<p<\infty ,$ the function $\varphi (x,t)$ satisfy the condition (2.23), $w \in \mathbf {V}_-\cup \mathbf {V}_+$ and $T\in \mathfrak {S}_p$. Then the conditions

$$\begin{aligned} H_w \quad \mathrm{is\,bounded\,in} \quad L^p({\mathbb {R}}^{n}) \quad \mathrm{when } \quad w \in \mathbb {V}_+, \end{aligned}$$

(3.6)

$$\begin{aligned} \mathcal {H}_w \quad \mathrm{is\,bounded\,in} \quad L^p({\mathbb {R}}^{n}) \quad \mathrm{when } \quad w \in \mathbb {V}_-, \end{aligned}$$

(3.7)

are sufficient for the weighted operator $wT\frac{1}{w}$ to be bounded in the Morrey space $\mathcal {L}^{p,\varphi }(\mathbb {R}^n )$.

Proof

In view of the estimates (3.4), and (3.5) in the Theorem 3.2, we only need to prove that the operators $T, K_m,\mathcal {K}_m, H_w$ and $\mathcal {H}_w$ are bounded in the space $\mathcal {L}^{p,\varphi }(\mathbb {R}^n )$. The boundedness of $T\in \mathfrak {S}_p$ is provided by Corollary 2.12, the boundedness of $K_m$ and $\mathcal {K}_m$ follows from Lemma 3.1 and the same Corollary 2.12, while the boundedness of $H_w$ and $\mathcal {H}_w$ are just assumed in (3.6) and (3.7). The proof is complete. $\square$

We continue by recall into some known results concerning the boundedness in (3.6) and (3.7). They were obtained in [13, Theorem 4.4] and use the following à priori assumptions:

$$\begin{aligned} w \in \overline{W}(\mathbb {R}_+), \ w (2t) \le cw (t), \ \frac{\varphi ^\frac{1}{p}(x,\cdot )}{w(\cdot ) }\in \underline{W}(\mathbb {R}_+) \ \ \ \mathrm{uniformly \ in} \ \ x \end{aligned}$$

(3.8)

for the operator $H_w$, and

$$\begin{aligned} w\in \underline{W}(\mathbb {R}_+), \quad \text {or} \quad w \in \overline{W}(\mathbb {R}_+) \quad \text {and} \quad w (2t)\le Cw (t) \end{aligned}$$

(3.9)

for the operator $\mathcal {H}_w.$ Under these assumptions the sufficient boundedness conditions from [13, Theorem 4.4] are the following:

$$\begin{aligned} \sup \limits _{x\in \Omega ,r>0}\frac{1}{\varphi (x,r)} \int _{B(x,r)} w^p (|y|)|y|^{-np} \left( \int _0^{|y|} \frac{t^{\frac{n}{p^\prime } -1} \varphi ^{\frac{1}{p}}(y,t)}{w (t)}dt \right) ^p dy < \infty \end{aligned}$$

(3.10)

for the operator $H_w$ and

$$\begin{aligned} \sup \limits _{x\in \Omega ,r>0}\frac{1}{\varphi (x,r)} \int _{B(x,r)}w ^p(|y|) \left( \int _{|y|}^\infty \frac{t^{- \frac{n}{p} -1} \varphi ^{\frac{1}{p}}(y,t)}{w (t)}dt \right) ^p dy < \infty , \end{aligned}$$

(3.11)

for the operator $\mathcal {H}_w$.

Remark 3.4

The above conditions were obtained in [13] for $\varphi (x,r)\equiv \varphi (r)$ not depending on x, but the analysis of the proof in [13] shows that, with our assumptions on $\varphi (x,r)$, Theorem 4.4 from [13] remains valid also in this situation.

Taking the above conditions into account, we arr ive at the following final statement for singular operators, where by $\Delta _2$ we denote the class of all non-negative functions w satisfying the doubling condition $w(2t)\le Cw(t.)$

Theorem 3.5

Let $1<p<\infty ,$ $T\in \mathfrak {S}_p$ and let the following à priori assumptions be satisfied:

• (i) $w\in \mathbf {V}_+\cup \mathbf {V}_-$,

• (ii) $\frac{\varphi (x,\cdot )}{w^p(\cdot ) }\in \underline{W}(\mathbb {R}_+)$ uniformly in x and $\varphi (x,t)$ satisfies the condition (2.23).

Then the weighted operator

$$\begin{aligned} wT\frac{1}{w} f = w(|x-x_0|)\int _{{\mathbb {R}}^{n}}\frac{K(x,y)}{w(|y-x_0|)}f(y)\, dy, \ \ \ \ x_0\in {\mathbb {R}}^{n}, \end{aligned}$$

(3.12)

is bounded in the Morrey space $\mathcal {L}^{p,\varphi }(\mathbb {R}^n),$ if

$w\in \Delta _2$ and (3.10) holds when $w\in \mathbf {V}_+$,

(3.11) holds, when $w\in \mathbf {V}_-$.

Remark 3.6

Theorem 3.5 remains valid for $\mathbb {R}^n$ replaced by an open set $\Omega \subset {\mathbb {R}}^{n}$ and $\mathbb {R}_+$ by $[0,\ell ], \ell =$ diam $\Omega$ and under the corresponding interpretation of the class $\mathfrak {S}_p.$

3.3 Boundedness Conditions in Terms of the Indices of Weights

In the next theorem, for the case $\varphi (x,r)\equiv \varphi (r)$ and bounded sets $\Omega$, we provide conditions, easy to check in applications, for the weighted boundedness of singular operators, in terms of the Matuszewska-Orlicz indices of the functions $\varphi (r)$ and w(r). Note that the assumption on the boundedness of $\Omega$ is made only for simplicity, to avoid the use of Matuszewska-Orlicz indices related to infinity.

Theorem 3.7

Let $\ell =$ diam  $\Omega <\infty ,$ $1<p<\infty ,$ $T\in \mathfrak {S}_p,$ and the following a priori assumptions hold:

• (i) $\varphi \in \underline{W}(0,\ell )$, $\varphi$ is almost increasing and $0<m(\varphi )\le M(\varphi )<n$,

• (ii) $w\in \mathbf {V}_+\cup \mathbf {V}_-$.

Then the weighted operator $wT\frac{1}{w}$ is bounded in the Morrey space $\mathcal {L}^{p,\varphi }(\Omega ),$ if

$$\begin{aligned} -\frac{n}{p} < m\left( \frac{w}{\varphi ^\frac{1}{p}}\right) \le M\left( \frac{w}{\varphi ^\frac{1}{p}}\right) <\frac{n}{p^\prime }. \end{aligned}$$

(3.13)

Proof

By Theorem 3.5 and Remark 3.6 it only suffices to check that the written inequalities for the Matuszewska-Orlicz indices guarantee the validity of the conditions (3.10) and (3.11) under the assumptions of this theorem,. Note that the inequality $M(\varphi )<n$ is equivalent to (2.23), so (2.23) holds.

We start with the case $w\in \mathbf {V}_+$. The condition (3.10) is nothing else but the statement that

$$\begin{aligned} V\in \mathcal {L}^{p,\varphi }(\Omega ) \end{aligned}$$

(3.14)

for the radial function V(|x|), where

$$\begin{aligned} V(r)= w (r)r^{-n} \int _0^r \frac{t^{\frac{n}{p^\prime } -1} \varphi ^{\frac{1}{p}}(t)}{w (t)}dt. \end{aligned}$$

Since $\frac{n}{p^\prime }+m\left( \frac{\varphi ^\frac{1}{p}}{w}\right) >0,$ we have $V(r)\le C \frac{\varphi ^\frac{1}{p} (r)}{r^\frac{n}{p} }.$ To verify (3.14), it remains to apply Lemma 2.3 to the function $V(r)=\frac{\varphi ^\frac{1}{p} (r)}{r^\frac{n}{p} }$. If the conditions of that Lemma are fulfilled, then the function V is almost decreasing, since $M(\varphi )< n$, the condition $m(V)>-\frac{n}{p}$ holds by the inequality $m(\varphi )>0$ and the second of the conditions in (2.8) turns out to be trivial.

In the case $w\in \mathbf {V}_-$, to check the condition (3.11) (with $\int _{|y|}^\infty$ replaced by $\int _{|y|}^\ell$ in our case), we similarly observe that (3.11) is nothing else but $\mathbb {V} \in \mathcal {L}^{p,\varphi }(\Omega ),$ for $\mathbb {V}(|x|)$, where

$$\begin{aligned} \mathbb {V}(r)= w (r) \int _r^\ell \frac{t^{-\frac{n}{p} -1} \varphi ^{\frac{1}{p}}(t)}{w (t)}dt\le C r^{-\frac{n}{p} } \varphi ^{\frac{1}{p}}(r), \end{aligned}$$

after which we proceed as in the case $w\in \mathbf {V}_+$. $\square$

Remark 3.8

From the properties $m(u^{-1})=M(u), \ m(uv)\ge m(u)+m(v), M(uv)\le M(u)+M(v)$ for the Matuszewska-Orlicz-type indices it easily follows that the conditions

$$\begin{aligned} \frac{M(\varphi )-n}{p}<m(w)\le M(w) <\frac{m(\varphi )}{p}+\frac{n}{p^\prime } \end{aligned}$$

(3.15)

are sufficient for the validity of (3.13).

For the classical Morrey space $\mathcal {L}^{p,\lambda }({\mathbb {R}}^{n})$ and power weight $w(r)=r^\gamma ,$ from Theorem 3.7 and (3.15), we obtain the following corollary.

Corollary 3.9

Let $1<p<\infty , T\in \mathfrak {S}_p$ and $w(r)=r^\gamma$. Then the weighted singular integral operator $wT\frac{1}{w}$ is bounded in the space $\mathcal {L}^{p,\lambda }({\mathbb {R}}^{n})$, if

$$\begin{aligned} 0\le \lambda <n \quad \mathrm{and } \qquad \frac{\lambda -n}{p}<\gamma <\frac{\lambda }{p}+\frac{n}{p^\prime }. \end{aligned}$$

(3.16)