Regularity of solutions of elliptic equations in divergence form in modified local generalized Morrey spaces

Abstract

Aim of this paper is to prove regularity results, in some Modified Local Generalized Morrey Spaces, for the first derivatives of the solutions of a divergence elliptic second order equation of the form

$$\begin{aligned} \mathscr {L}u{:}{=}\sum _{i,j=1}^{n}\left( a_{ij}(x)u_{x_{i}}\right) _{x_{j}}=\nabla \cdot f,\qquad \hbox {for almost all }x\in \Omega \end{aligned}$$

where the coefficients $a_{ij}$ belong to the Central (that is, Local) Sarason class CVMO and f is assumed to be in some Modified Local Generalized Morrey Spaces $\widetilde{LM}_{\{x_{0}\}}^{p,\varphi }$. Heart of the paper is to use an explicit representation formula for the first derivatives of the solutions of the elliptic equation in divergence form, in terms of singular integral operators and commutators with Calderón–Zygmund kernels. Combining the representation formula with some Morrey-type estimates for each operator that appears in it, we derive several regularity results.

1 Introduction and mathematical background

In this note we consider the following divergence form elliptic equation

$$\begin{aligned} \mathscr {L}u{:}{=}\sum _{i,j=1}^{n}\left( a_{ij}(x)u_{x_{i}}\right) _{x_{j}}=\nabla \cdot f,\qquad \hbox {for almost all }x\in \Omega \end{aligned}$$

(1.1)

in a bounded set $\Omega \subset {\mathbb {R}}^{n}$, $n\ge 3$.

We assume that $\mathscr {L}$ is a linear elliptic operator and its coefficients belong to the space VMO and the vectorial field $f=(f_{1},f_{2},\ldots ,f_{n})$ is such that $f_{i}\in LM^{p,\varphi }$ for $i=1,\ldots ,n$, with $1<p<\infty$ and $\varphi$ positive and measurable function. The space VMO was introduced by Sarason and it is the proper subspace of the John-Nirenberg space BMO whose BMO norm over a ball vanishes as the radius of the ball tends to zero.

In the last few years have been studied several differential problems on nonstandard function spaces (see for instance [21,22,23]) and, in particular, several results have been obtained on Generalized Morrey Spaces (see, for instance, [12]).

Recently, in [5, 27, 28] the authors studied some regularity results for solutions of linear partial differential equations with discontinuous coefficients in nondivergence form.

Our main result in this paper is the study of local regularity in the Generalized Morrey Spaces $LM^{p,\varphi }$ of the first derivatives of the solutions of the equation under consideration as in the past has been done in $L^{p}-$spaces and in $L^{p,\lambda }-$spaces.

See, for instance, [2] where the author obtains local regularity in the classical Lebesgue spaces $L^p$ for the first derivatives of the solutions of the equation with discontinuous coefficients. See, also, [24] in which has been done the same in the Morrey spaces $L^{p,\lambda }$. Hearth of the technique is the use of an integral representation formula for the first derivatives of the solutions of Equation (1.1) and the boundedness in $L^{p,\varphi }$ of some integral operators and commutators appearing in this formula.

Precisely, in this work we apply the boundedness on Generalized local Morrey Spaces of singular integral operators and its commutators obtained in [13]. We would like to point out that in the last decades a lot of authors studied the boundedness of such operators in several functional spaces (see e.g. [1, 4, 14]).

Throughout the paper, we set $d=\displaystyle \sup \nolimits _{x,y\in \Omega } |x-y|$, $B(x,r)=\{y\in {\mathbb {R}}^n: |x-y|<r\}$ and $\Omega (x,r)=\Omega \cap B(x,r)$. Furthermore, by $A \lesssim B$ we mean that $A \le C B$ with some positive constant C independent of appropriate quantities. If $A \lesssim B$ and $B \lesssim A$, we write $A\approx B$ and say that A and B are equivalent.

Let $\Omega$ be an open bounded subset of ${\mathbb {R}}^{n}$, with $n\ge 3$, and f be a locally integrable function on $\Omega$. We say that f belongs to the John-Nirenberg space BMO of the functions with bounded mean oscillation if

$$\begin{aligned} \Vert f\Vert _{*} {:}{=}\sup _{B}\frac{1}{|B|}\int _{B}|f(x)-f_{B}|\,\mathrm {d}x<\infty \end{aligned}$$

where B ranges in the set of the balls contained in $\Omega$ and $f_{B}$ is the integral average of f over B, namely

$$\begin{aligned} f_{B}{:}{=}\frac{1}{B}\int _{B}f(x)\,\mathrm {d}x. \end{aligned}$$

We say that the number $\Vert f\Vert _{*}$ is the BMO-norm of f.

If $f\in BMO$ and r is a positive number, we set

$$\begin{aligned} \eta (r){:}{=}\sup _{{\mathop {\rho \le r}\limits ^{x\in \mathbb {R}^{n}}}}\frac{1}{|B_{\rho }|}\int _{B_{\rho }}|f(x)-f_{B_{\rho }}|\,\mathrm {d}x, \end{aligned}$$

where $B_{\rho }$ stands for a ball with radius $\rho$ less than or equal to r. The function $\eta (r)$ is called VMO-modulus of f. We say that $f\in BMO$ is in the space VMO of functions with vanishing mean oscillation if

$$\begin{aligned} \lim _{r\rightarrow 0^{+}}\eta (r)=0. \end{aligned}$$

In the sequel we denote $\eta _{ij}$ the VMO-modulus of the coefficient $a_{ij}$ and

$$\begin{aligned} \eta (r)=\left( \sum _{i,j=1}^{n}\eta _{ij}^{2}(r)\right) ^{\frac{1}{2}}. \end{aligned}$$

For further details on the VMO space, we refer the reader to [25] and to the references therein.

The definition of local BMO space is as follows.

Definition 1.1

Let $1\le q<\infty$. A function $f\in L^q_\mathrm{loc}(\mathbb {R}^{n})$ is said to belong to the $CBMO^q_{\{x_0\}}(\mathbb {R}^{n})$ (central BMO space), if

$$\begin{aligned} \Vert f\Vert _{CBMO^q_{\{x_0\}}}=\sup _{r>0} \Big (\frac{1}{|B(x_0,r)|} \int _{B(x_0,r)}|f(y)-f_{B(x_0,r)}|^q dy \Big )^{1/q}<\infty . \end{aligned}$$

We set

$$\begin{aligned} CBMO^q_{\{x_0\}}(\mathbb {R}^{n})=\{f \in L^q_\mathrm{loc}(\mathbb {R}^{n})~ :~ \Vert f \Vert _{CBMO^q_{\{x_0\}}} < \infty \}. \end{aligned}$$

In [16], Lu and Yang introduced the central BMO space $CBMO^q(\mathbb {R}^{n})=CBMO^q_{\{0\}}(\mathbb {R}^{n})$. Note that, $BMO(\mathbb {R}^{n}) \subset CBMO^q_{\{x_0\}}(\mathbb {R}^{n})$, $1\le q<\infty$. The space $CBMO^q_{\{x_0\}}(\mathbb {R}^{n})$ can be regarded as a local version of $BMO(\mathbb {R}^{n})$, the space of bounded mean oscillation, at the origin. But, they have quite different properties. The classical John-Nirenberg inequality shows that functions in $BMO(\mathbb {R}^{n})$ are locally exponentially integrable. This implies that, for any $1\le q<\infty$, the functions in $BMO(\mathbb {R}^{n})$ can be described by means of the condition:

$$\begin{aligned} \sup _{r>0} \Big ( \frac{1}{|B|} \int _{B}|f(y)-f_{B}|^qdy \Big )^{1/q}<\infty , \end{aligned}$$

where B denotes an arbitrary ball in $\mathbb {R}^{n}$. However, the space $CBMO^q_{\{x_0\}}(\mathbb {R}^{n})$ depends on q. If $q_1 < q_2$, then $CBMO^{q_2}_{\{x_0\}}(\mathbb {R}^{n}) \subsetneqq CBMO^{q_1}_{\{x_0\}}(\mathbb {R}^{n})$. Therefore, there is no analogy of the famous John-Nirenberg inequality of $BMO(\mathbb {R}^{n})$ for the space $CBMO^q_{\{x_0\}}(\mathbb {R}^{n})$. One can imagine that the behavior of $CBMO^q_{\{x_0\}}(\mathbb {R}^{n})$ may be quite different from that of $BMO(\mathbb {R}^{n})$.

Lemma 1.2

([17]) Let b be a function in $CBMO^p_{\{x_0\}}(\mathbb {R}^{n})$, $1 \le p < \infty$ and $r_1, r_2 > 0$. Then

$$\begin{aligned} \left( \frac{1}{|B(x_0,r_1)|} \int _{B(x_0,r_1)} |b(y)-b_{B(x_0,r_2)}|^p dy\right) ^{\frac{1}{p}} \le C \left( 1+ \Big |\ln \frac{r_1}{r_2} \Big | \right) \Vert b \Vert _{CBMO^p_{\{x_0\}}}, \end{aligned}$$

where $C>0$ is independent of b, $r_1$ and $r_2$.

We say that $f\in CBMO^p_{\{x_0\}}$ is in the space $CVMO^p_{\{x_0\}}$ of functions with vanishing mean oscillation if

$$\begin{aligned} \lim _{r\rightarrow 0^{+}}\eta (r)=0. \end{aligned}$$

The following condition is essential to the proof of the main result of the paper: A function b is said to satisfy the well known mean value inequality if there exists a constant $C>0$ such that for any ball $B \subset \mathbb {R}^n$

$$\begin{aligned} \Vert b(\cdot )-b_{B}\Vert _{L^{\infty }(\mathbb {R}^n)} \lesssim \frac{1}{|B|} \int _{B} |b(x)-b_{B}| dx. \end{aligned}$$

(1.2)

Also, we recall the definition of the classical Morrey Spaces, formulated by Morrey in 1938 in [19].

For $1<p<\infty$, $0<\lambda <n$, we say that a measurable function f belong to the Morrey space $L^{p,\lambda }(\Omega )$ if its norm, defined by

$$\begin{aligned} \Vert f\Vert _{L^{p,\lambda }(\Omega )}^{p}=\sup _{{\mathop {\rho >0}\limits ^{x\in \Omega }}} \frac{1}{\rho ^{\lambda }}\int _{B(x,\rho )\cap \Omega }|f(y)|^{p}\mathrm {d}y \end{aligned}$$

is finite.

The first author, Mizuhara and Nakai [6, 18, 20] extended the previous definition of Morrey Space, introducing the Generalized Morrey Spaces (see, also [7, 8, 26]).

Definition 1.3

Let $\varphi (x,r)$ be a positive measurable function on $\Omega \times (0,\infty )$ and $1\le p<\infty$. We denote by $M^{p,\varphi }(\Omega )$ $(WM^{p,\varphi }(\Omega ))$ the Generalized Morrey space (the weak Generalized Morrey space), the space of all functions $f\in L^{p}_{\mathrm{loc}}(\Omega )$ with finite quasinorm

$$\begin{aligned}&\Vert f\Vert _{M^{p,\varphi }(\Omega )}=\sup _{\underset{0<r<d}{x\in \Omega }} \frac{1}{\varphi (x,r)}\frac{1}{|B(x,r)|^{\frac{1}{p}}}\Vert f\Vert _{L^{p}(\Omega (x,r))}\\&\Big (\Vert f\Vert _{WM^{p,\varphi }(\Omega )}=\sup _{\underset{0<r<d}{x\in \Omega }} \frac{1}{\varphi (x,r)}\frac{1}{|B(x,r)|^{\frac{1}{p}}}\Vert f\Vert _{WL^{p}(\Omega (x,r))}\Big ). \end{aligned}$$

According to this definition we obtain, for $0\le \lambda <n$, the Morrey space $L^{p,\lambda }$ under the choice $\varphi (x,r)=r^{\frac{\lambda -n}{p}}$:

$$\begin{aligned} L^{p,\lambda }=M^{p,\varphi }\Big |_{\varphi (x,r)=r^{\frac{\lambda -n}{p}}}. \end{aligned}$$

In this note we are interested in the study of regularity properties of solutions to elliptic equations in the local version of Generalized Morrey Spaces. In order to achieve this purpose we need the following definitions.

Definition 1.4

Let $\varphi (x,r)$ be a positive measurable function on $\Omega \times (0,d)$ and $1\le p<\infty$. Fixed $x_{0}\in \Omega$, we denote by by $LM_{\{x_{0}\}}^{p,\varphi }(\Omega )$ $(WLM_{\{x_{0}\}}^{p,\varphi }(\Omega ))$ the local Generalized Morrey space (the weak local Generalized Morrey space), the space of all functions $f\in L^{p}_{\mathrm{loc}}(\Omega )$ with finite quasinorm

$$\begin{aligned}&\Vert f\Vert _{LM_{\{x_{0}\}}^{p,\varphi }(\Omega )}=\sup _{0<r<d}\frac{1}{\varphi (x_0,r)} \frac{1}{|B(x_0,r)|^{\frac{1}{p}}}\Vert f\Vert _{L^p(\Omega (x_0,r))}\\&\Big (\Vert f\Vert _{WLM_{\{x_{0}\}}^{p,\varphi }(\Omega )}=\sup _{0<r<d}\frac{1}{\varphi (x_0,r)} \frac{1}{|B(x_0,r)|^{\frac{1}{p}}}\Vert f\Vert _{WL^p(\Omega (x_0,r))}\Big ). \end{aligned}$$

Definition 1.5

Let $\varphi (x,r)$ be a positive measurable function on $\Omega \times (0,d)$ and $1\le p<\infty$. We denote by $\widetilde{M}^{p,\varphi }(\Omega )$ $\big (W\widetilde{M}^{p,\varphi }(\Omega )\big )$ the modified Generalized Morrey space (the modified weak Generalized Morrey space), the space of all functions $f\in L^p(\Omega )$ with finite norm

$$\begin{aligned}&\Vert f\Vert _{\widetilde{M}^{p,\varphi }(\Omega )}=\Vert f\Vert _{M^{p,\varphi }(\Omega )} + \Vert f\Vert _{L^p(\Omega )}\\&\Big (\Vert f\Vert _{W\widetilde{M}^{p,\varphi }(\Omega )}=\Vert f\Vert _{WM^{p,\varphi }(\Omega )} + \Vert f\Vert _{WL^p(\Omega )}\Big ). \end{aligned}$$

According to this definition we obtain, for $\lambda \ge 0$, the local Morrey Space $LM_{\{x_{0}\}}^{p,\lambda }$ under the choice $\varphi (x_{0},r)=r^{\frac{\lambda -n}{p}}$:

$$\begin{aligned} LM_{\{x_{0}\}}^{p,\lambda }(\Omega )=LM_{\{x_{0}\}}^{p,\varphi }(\Omega ) \Big |_{\varphi (x_{0},r)=r^{\frac{\lambda -n}{p}}}. \end{aligned}$$

Definition 1.6

Let $\varphi (x,r)$ be a positive measurable function on $\Omega \times (0,\infty )$ and $1\le p<\infty$. Fixed $x_{0}\in \Omega$, we denote by $\widetilde{LM}_{\{x_{0}\}}^{p,\varphi }(\Omega )$ $\big (\widetilde{LM}_{\{x_{0}\}}^{p,\varphi }(\Omega )\big )$ the modified local Generalized Morrey space (the modified weak local Generalized Morrey space), the space of all functions $f\in L^p(\Omega )$ with finite norm

$$\begin{aligned}&\Vert f\Vert _{\widetilde{LM}_{\{x_{0}\}}^{p,\varphi }(\Omega )}=\Vert f\Vert _{LM_{\{x_{0}\}}^{p,\varphi } (\Omega )} + \Vert f\Vert _{L^p(\Omega )}\\&\Big (\Vert f\Vert _{W\widetilde{LM}_{\{x_{0}\}}^{p,\varphi }(\Omega )}=\Vert f\Vert _{WLM_{\{x_{0}\}}^ {p,\varphi }(\Omega )} + \Vert f\Vert _{WL^p(\Omega )}\Big ). \end{aligned}$$

Remark 1.7

For further details on Local Generalized Morrey Spaces, see for instance [10, 11, 15].

Let $\Omega$ be a bounded open set in $\mathbb {R}^{n}$, $n\ge 3$, let us consider

$$\begin{aligned} \mathscr {L}u\equiv -\sum _{i,j=1}^{n} \left( a_{ij}(x)u_{x_{i}}\right) _{x_{j}} =\nabla \cdot f,\qquad \hbox {a.e. }x\in \Omega , \end{aligned}$$

(1.3)

and, fixed $x_{0}\in \mathbb {R}^{n}$, we suppose that there exists $p\in ]1,+\infty [$ and a positive measurable function $\varphi$ defined on $\mathbb {R}^{n}\times (0,\infty )$ such that:

$$\begin{aligned} f=(f_{1},\ldots ,f_{n})\in \big [LM_{\{x_{0}\}}^{p,\varphi }(\Omega )\big ]^{n}; \end{aligned}$$

(1.4)

$$\begin{aligned} a_{ij}(x)\in L^{\infty }\cap {CVMO^{\max \{p,p'\}}_{\{x_0\}}},\forall i,j=1,\ldots ,n; \end{aligned}$$

(1.5)

$$\begin{aligned} a_{ij}(x)=a_{ji}(x),\quad \forall i,j=1,\ldots ,n,\;\hbox {a.a. }x\in \Omega ; \end{aligned}$$

(1.6)

$$\begin{aligned} \exists \kappa >0:\kappa ^{-1}|\xi |^{2}\le a_{ij}\xi _{i}\xi _{j}\le \kappa |\xi |^{2},\quad \forall \xi \in \mathbb {R}^{n},\;\hbox {a.a. }x\in \Omega . \end{aligned}$$

(1.7)

We say that a function u is a solution of (1.3) if $u,\partial _{x_{i}}u\in L^{p}(\Omega )$, $\forall i=1,\ldots ,n$ and for some $1<p<\infty$ and

$$\begin{aligned} \int _{\Omega }a_{ij}u_{x_{i}}\varphi _{x_{j}}\,\mathrm {d}x=-\int _{\Omega }f_{i} \varphi _{x_{i}}\,\mathrm {d}x,\qquad \forall \varphi \in C_{0}^{\infty }(\Omega ). \end{aligned}$$

2 Calderón–Zygmund kernel and preliminary results

In order to present the representation formula for the first derivatives of a solution of 1.3, we find it convenient to present the definition of Calderón–Zygmund kernel:

Definition 2.1

Let $k:\mathbb {R}^{n}{\setminus }\{0\}\rightarrow \mathbb {R}$. We say that k(x) is a Calderón–Zygmund kernel (C-Z kernel) if: .

• (1) $k\in C^{\infty }(\mathbb {R}^{n}{\setminus }\{0\})$;

• (2) k(x) is homogeneous of degree $-n$;

• (3) $\int _{\Sigma }k(x)\,\mathrm {d}x=0$, where $\Sigma =\{x\in \mathbb {R}^{n}:|x|=1\}$.

Many authors obtained several boundedness results for integral operators involving Calderón–Zygmund kernels. For instance, in [3] the authors studied the boundedness of Calderón–Zygmund singular integral operators and commutators on Morrey Spaces. Recently, in [13] the authors extended the previous results in Generalized Local Morrey Spaces.

The previous theorem was proved using the following important result contained in [10].

Theorem 2.2

Let $x_0 \in \mathbb {R}^n$, $1 \le q < \infty$, K be a Calderón–Zygmund singular integral operator and the functions $\varphi _1,\varphi _2$ satisfy the condition

$$\begin{aligned} \int _{r}^{\infty } \frac{\mathop \mathrm{ess \; inf}\limits _{t<\tau <\infty }\varphi _1(x_0,\tau ) \, \tau ^{\frac{n}{q}}}{t^{\frac{n}{q}+1}} \, dt \le C \, \varphi _2(x_0,r), \end{aligned}$$

(2.1)

where C does not depend on r. Then for $1< q < \infty$ the operator K is bounded from $LM^{q,\varphi _1}_{\{x_0\}}(\mathbb {R}^n)$ to $LM^{q,\varphi _2}_{\{x_0\}}(\mathbb {R}^n)$ and for $1 \le q < \infty$ the operator K is bounded from $LM^{q,\varphi _1}_{\{x_0\}}(\mathbb {R}^n)$ to $WLM^{q,\varphi _2}_{\{x_0\}}(\mathbb {R}^n)$. Moreover, for $1< q < \infty$

$$\begin{aligned} \Vert K f\Vert _{LM^{q,\varphi _2}_{\{x_0\}}} \le c \, \Vert f\Vert _{LM^{q,\varphi _1}_{\{x_0\}}}, \end{aligned}$$

where c does not depend on $x_0$ and f and for $q = 1$

$$\begin{aligned} \Vert K f\Vert _{WLM^{1,\varphi _2}_{\{x_0\}}} \le c \, \Vert f\Vert _{LM^{1,\varphi _1}_{\{x_0\}}}, \end{aligned}$$

where c does not depend on $x_0$ and f.

Precisely, using the boundedness of the Calderón–Zygmund singular integral operators from $LM_{\{x_{0}\}}^{p,\varphi }(\mathbb {R}^{n})$ in itself (see [10]), the following theorem is valid that will be crucial in the sequel.

Theorem 2.3

Let $x_{0}\in \mathbb {R}^{n}$, $1<p<+\infty$, K be a Calderón–Zygmund singular integral operator and the measurable function $\varphi :\mathbb {R}^{n}\times (0,\infty )\rightarrow \mathbb {R}^{+}$ satisfy the conditions

$$\begin{aligned} \int _{r}^{\infty } \Big (1+\ln \frac{t}{r}\Big ) \frac{\mathop \mathrm{ess \; inf}\limits _{t<s<\infty } \varphi _1(x_0,s) s^{\frac{n}{p}}}{t^{\frac{n}{p}+1}} dt \le C \,\varphi _2(x_0,r), \end{aligned}$$

(2.2)

where C does not depend on r and $x_0$.

If $a\in CBMO^{\max \{p,p'\}}_{\{x_0\}}(\mathbb {R}^{n})$, the commutator

$$\begin{aligned}{}[a,K](f)=aKf-K(af) \end{aligned}$$

is a bounded operator from $LM_{\{x_{0}\}}^{p,\varphi }(\mathbb {R}^{n})$ in itself.

Precisely, for every $f\in LM_{\{x_{0}\}}^{p,\varphi }(\mathbb {R}^{n})$, we have

$$\begin{aligned} \Vert [a,K](f)\Vert _{LM_{\{x_{0}\}}^{p,\varphi }}\le c \Vert a\Vert _{{CBMO^{{\max \{p,p'\}}}_{\{x_0\}}}}\Vert f\Vert _{LM_{\{x_{0}\}}^{p,\varphi }}. \end{aligned}$$

To prove Theorem 2.3, we first give some auxiliary lemmas.

In this section we are going to use the following statement on the boundedness of the weighted Hardy operator

$$\begin{aligned} H^{*}_{w} g(t){:}{=}\int _t^{d} g(s) w(s) ds,~ 0<t<d <\infty , \end{aligned}$$

where w is a fixed function non-negative and measurable on (0, d).

The following lemma was proved in [10], see also [9].

Lemma 2.4

Let $v_1$, $v_2$ and w be positive almost everywhere and measurable functions on (0, d). The inequality

$$\begin{aligned} \mathop \mathrm{ess \; sup}\limits _{0<t<d} v_2(t) H^{*}_{w} g(t) \le C \mathop \mathrm{ess \; sup}\limits _{0<t<d} v_1(t) g(t) \end{aligned}$$

(2.3)

holds for some $C>0$ for all non-negative and non-decreasing g on (0, d) if and only if

$$\begin{aligned} B:= \mathop \mathrm{ess \; sup}\limits _{0<t<d} v_2(t)\int _t^{d} \frac{w(s) ds}{\mathop \mathrm{ess \; sup}\limits _{s< \tau<d} v_1(\tau )}< \infty . \end{aligned}$$

(2.4)

Moreover, if $C^{*}$ is the minimal value of C in (2.3), then $C^{*}=B$.

Remark 2.5

In (2.3) and (2.4) it is assumed that $\frac{1}{\infty }=0$ and $0 \cdot \infty =0$.

Lemma 2.6

Let $x_0 \in \mathbb {R}^{n}$, $1< p < \infty$, $b \in CBMO^{\max \{p,p'\}}_{\{x_0\}}(\mathbb {R}^{n})$ and K be a Calderón–Zygmund singular integral operator. Then the inequality

$$\begin{aligned} \Vert [b,K](f)\Vert _{L^p(B)} \lesssim \Vert b\Vert _{CBMO^{\max \{p,p'\}}_{\{x_0\}}} \, r^{\frac{n}{p}} \int _{2r}^{\infty } \Big (1+\ln \frac{t}{r}\Big ) t^{-\frac{n}{p}-1} \Vert f\Vert _{L^p(B(x_0,t))} dt \end{aligned}$$

holds for any ball $B=B(x_0,r)$ and for all $f\in L^p_\mathrm{loc}(\mathbb {R}^{n})$.

Proof

Let $1< p < \infty$, $b \in BMO(\mathbb {R}^n)$, and K be a Calderón–Zygmund singular integral operator. For arbitrary $x_0 \in \mathbb {R}^n$, set $B=B(x_0,r)$ for the ball centered at $x_0$ and of radius r. Write $f=f_1+f_2$ with $f_1=f\chi _{2B}$ and $f_2=f\chi _{\,^{^{\complement }}\!{(2B)}}$. Hence

$$\begin{aligned}&[b, K](f)(x) \equiv J_1 + J_2 + J_3 + J_4 = \big (b(x)-b_{B}\big ) K(f_{1})(x)\\&- K \Big (\big (b(\cdot )-b_{B}\big )f_{1}\Big )(x) + \big (b(x)-b_{B}\big ) K(f_{2})(x) - K \Big (\big (b(\cdot )-b_{B}\big )f_{2}\Big )(x). \end{aligned}$$

We get

$$\begin{aligned} \Vert [b, K](f)\Vert _{L^p(B)}\le \Vert J_1\Vert _{L^p(B)}+\Vert J_2\Vert _{L^p(B)}+\Vert J_3\Vert _{L^p(B)}+\Vert J_4\Vert _{L^p(B)}. \end{aligned}$$

From the boundedness of K on $L^p(\mathbb {R}^n)$, (1.2) and Lemma 1.2 (see [29] [inequality (1.3)]) it follows that:

$$\begin{aligned} \Vert J_1\Vert _{L^p(B)}&\le \Vert \big (b(\cdot )-b_{B}\big ) K(f_{1})(\cdot ) \Vert _{L^{p}(B)}\\&\le \Vert b(\cdot )-b_{B}\Vert _{L^{\infty }(B)} \Vert K(f_{1})\Vert _{L^{p}(B)}\\&\lesssim |B|^{-1} \Vert b(\cdot )-b_{B}\Vert _{L^{1}(B)} \, \Vert f_1\Vert _{L^{p}(\mathbb {R}^n)}\\&\approx |B|^{-1+\frac{1}{p'}} \, \Vert b(\cdot )-b_{B}\Vert _{L^{p}(B)} \, \Vert f\Vert _{L^{p}(2B)} \, r^{\frac{n}{p}} \, \int _{2r}^{\infty } t^{-1-\frac{n}{p}} dt\\&\lesssim \Vert b\Vert _{CBMO^{p}_{\{x_0\}}} \, r^{\frac{n}{p}} \, \int _{2r}^{\infty } t^{-\frac{n}{p}-1} \Vert f \Vert _{L^{p}(B(x_0,t))} dt. \end{aligned}$$

From (1.2) and Lemma 1.2 (see [29] [inequality (1.3)]) for $J_2$ we have

$$\begin{aligned} \Vert J_2\Vert _{L^{p}(B)}&\le \Vert K\big (b(\cdot )-b_{B}\big ) f_{1} \Vert _{L^{p}(B)}\\&\lesssim \Vert b(\cdot )-b_{B}\Vert _{L^{\infty }(B)} \Vert K(f_{1})\Vert _{L^{p}(B)}\\&\lesssim |B|^{-1} \Vert b(\cdot )-b_{B}\Vert _{L^{1}(B)} \, \Vert f\Vert _{L^{p}(2B)}\\&\approx |B|^{-1+\frac{1}{p'}} \, \Vert b(\cdot )-b_{B}\Vert _{L^{p}(B)} \, \Vert f\Vert _{L^{p}(2B)} \, r^{\frac{n}{p}} \, \int _{2r}^{\infty } t^{-1-\frac{n}{p}} dt\\&\lesssim \Vert b\Vert _{CBMO^{p}_{\{x_0\}}} \, r^{\frac{n}{p}} \, \int _{2r}^{\infty } t^{-\frac{n}{p}-1} \Vert f \Vert _{L^{p}(B(x_0,t))} dt. \end{aligned}$$

For $J_3$, it is known that $x\in B$, $y\in {\,^{^{\complement }}\!(2B)}$, which implies $\frac{1}{2}|x_0-y| \le |x-y| \le \frac{3}{2}|x_0-y|$.

By Fubini’s theorem and applying Hölder inequality we have

$$\begin{aligned} |K(f_2)(x)|&\lesssim \int _{\,^{^{\complement }}\!(2B)} \frac{|f(y)|}{|x_0-y|^{n}}dy\\&\approx \int _{2r}^{\infty }\int _{2r<|x_0-y|<t} |f(y)|dy \, t^{-1-n} dt\\&\lesssim \int _{2r}^{\infty }\int _{B(x_0,t)} |f(y)|dy \, t^{-1-n} dt\\&\lesssim \int _{2r}^{\infty } \Vert f\Vert _{L^{p}(B(x_0,t))} \, |B(x_0,t)|^{1-\frac{1}{p}} \, \frac{dt}{t^{n+1}}\\&\lesssim \int _{2r}^{\infty } t^{-\frac{n}{p}-1} \, \Vert f \Vert _{L^{p}(B(x_0,t))} \, dt. \end{aligned}$$

Hence, from Lemma 1.2 we get

$$\begin{aligned} \Vert J_3\Vert _{L^{p}(B)}&= \Vert \big (b(\cdot )-b_{B}\big ) K(f_{2})(\cdot ) \Vert _{L^{p}(B)}\\&\lesssim \Vert b(\cdot )-b_{B}\Vert _{L^{p}(B)} \int _{2r}^{\infty } t^{-\frac{n}{p}-1} \, \Vert f \Vert _{L^{p}(B(x_0,t))} \, dt\\&\lesssim \Vert b\Vert _{CBMO^{p}_{\{x_0\}}} \, r^{\frac{n}{p}} \, \int _{2r}^{\infty } t^{-\frac{n}{p}-1} \, \Vert f \Vert _{L^{p}(B(x_0,t))} \, dt. \end{aligned}$$

For $x\in B$ by Fubini’s theorem applying Hölder inequality and from Lemma 1.2 we have

$$\begin{aligned}&|K \Big (\big (b(\cdot )-b_{B}\big )f_{2}\Big )(x)| \lesssim \int _{\,^{^{\complement }}\!(2B)} |b(y)-b_{B}| \, \frac{|f(y)|}{|x-y|^{n}}dy\\&\lesssim \int _{\,^{^{\complement }}\!(2B)} |b(y)-b_{B}| \, \frac{|f(y)|}{|x_0-y|^{n}}dy\\&\approx \int _{2r}^{\infty }\int _{2r<|x_0-y|<t}|b(y)-b_{B}| \, |f(y)|dy \, \frac{dt}{t^{n+1}}\\&\lesssim \int _{2r}^{\infty }\int _{B(x_0,t)} |b(y)-b_{B(x_0,t)}|\, |f(y)|dy \frac{dt}{t^{n+1}}\\&\quad + \int _{2r}^{\infty }|b_{B(x_0,r)}-b_{B(x_0,t)}| \int _{B(x_0,t)} |f(y)|dy \frac{dt}{t^{n+1}}\\&\lesssim \int _{2r}^{\infty } \, \Vert (b(\cdot )-b_{B(x_0,t)})\Vert _{L^{p'}(B(x_0,t))} \, \Vert f\Vert _{L^{p}(B(x_0,t))} \, \frac{dt}{t^{n+1}}\\&\quad + \int _{2r}^{\infty }|b_{B(x_0,r)}-b_{B(x_0,t)}| \Vert f\Vert _{L^{p}(B(x_0,t))} \, |B(x_0,t)|^{1-\frac{1}{p}} \, t^{-n-1} dt\\&\lesssim \Vert b\Vert _{CBMO^{p'}_{\{x_0\}}} \, \int _{2r}^{\infty } \, |B(x_0,t)|^{\frac{1}{p'}} \, \Vert f\Vert _{L^{p}(B(x_0,t))} t^{-n-1} \, dt\\&\quad + \Vert b\Vert _{CBMO^{p'}_{\{x_0\}}} \, \int _{2r}^{\infty }\Big (1+\ln \frac{t}{r}\Big ) \, t^{-\frac{n}{p}-1}~ \Vert f\Vert _{L^{p}(B(x_0,t))} dt\\&\lesssim \Vert b\Vert _{CBMO^{p'}_{\{x_0\}}} \, \int _{2r}^{\infty }\Big (1+\ln \frac{t}{r}\Big ) ~ t^{-\frac{n}{p}-1}~ \Vert f\Vert _{L^{p}(B(x_0,t))} dt. \end{aligned}$$

$\square$

Remark 2.7

The statement of Theorem 2.3 follows by Lemmas 2.4 and 2.6.

In order to achieve the regularity results, we must prove the following theorem.

Theorem 2.8

Let $\Omega$ be an open bounded subset of $\mathbb {R}^n$, $d=\sup _{x,y\in \Omega }|x-y|<\infty$, $\Omega (x_0,r)=\Omega \cap B(x_0,r)$, $x_0 \in \Omega$, $0<r\le d$, $1 \le q< p< \infty$, $\frac{1}{q}=\frac{1}{p}+\frac{1}{n}$ and

$$\begin{aligned} Tg(x)=\int _{\Omega }\frac{g(y)}{|x-y|^{n-1}}dy. \end{aligned}$$

(i) Let $1< q <\infty$. If $g \in L^q(\Omega )$ such that

$$\begin{aligned} \int _{r}^d t^{-\frac{n}{p}-1}\Vert g\Vert _{L^q(\Omega (x_0,t))}\,dt < \infty \quad \text{ for } \text{ all }\quad r\in (0,d), \end{aligned}$$

(2.5)

then for any $r \in (0,d)$ the inequality

$$\begin{aligned} \Vert Tg\Vert _{L^p(\Omega (x_0,r))}\le c r^{\frac{n}{p}}\int _{r}^d t^{-\frac{n}{p}-1}\Vert g\Vert _{L^q(\Omega (x_0,t))}\,dt + c r^{\frac{n}{p}}\Vert g\Vert _{L^q(\Omega )} \end{aligned}$$

(2.6)

holds with constant $c>0$ independent of g, $x_0$ and r.

(ii) Let $q = 1$. If $g \in L^1(\Omega )$ satisfies condition (2.5), then for any $r \in (0,d)$ the inequality

$$\begin{aligned} \Vert Tg\Vert _{WL^p(\Omega (x_0,r))}\le c r^{\frac{n}{p}}\int _{r}^d t^{-\frac{n}{p}-1}\Vert g\Vert _{L^1(\Omega (x_0,t))}\,dt + c r^{\frac{n}{p}}\Vert g\Vert _{L^1(\Omega )} \end{aligned}$$

(2.7)

holds with constant $c>0$ independent of g, $x_0$ and r.

Proof

Let $1 \le q< p < \infty$. Since

$$\begin{aligned} r^{\frac{n}{p}} \int _{r}^d t^{-\frac{n}{p}-1}\Vert g\Vert _{L^q(\Omega (x_0,t))}\,dt&\ge r^{\frac{n}{p}}\Vert g\Vert _{L^q(\Omega (x_0,r))}\int _{r}^d t^{-\frac{n}{p}-1}\,dt \\&\thickapprox \Vert g\Vert _{L^q(\Omega (x_0,r))} (d^{\frac{n}{p}} - r^{\frac{n}{p}}), \quad r \in (0,d), \end{aligned}$$

we get that

$$\begin{aligned} \Vert g\Vert _{L^q(\Omega (x_0,r))} \lesssim r^{\frac{n}{p}} \int _{r}^d t^{-\frac{n}{p}-1}\Vert g\Vert _{L^q(\Omega (x_0,t))}\,dt + r^{\frac{n}{p}} \Vert g\Vert _{L^q(\Omega )}, \quad r \in (0,d). \end{aligned}$$

(2.8)

(i). Assume that $1< q < \infty$. Let $r \in (0,d/2)$. We write $g = g_1 + g_2$ with $g_1 = g \chi _{\Omega (x_0,2r)}$ and $g_2 = g \chi _{\Omega \backslash \Omega (x_0,2r)}$. Taking into account the linearity of T, we have

$$\begin{aligned} \Vert Tg\Vert _{L^p(\Omega (x_0,r))} \le \Vert Tg_1\Vert _{L^p(\Omega (x_0,r))} + \Vert Tg_2\Vert _{L^p(\Omega (x_0,r))}. \end{aligned}$$

(2.9)

Since $g_1 \in L^q(\Omega )$, in view of (2.8), the boundedness of T from $L^q(\Omega )$ to $L^p(\Omega )$ implies that

$$\begin{aligned} \Vert Tg_1\Vert _{L^p(\Omega (x_0,r))}&\le \Vert Tg_1\Vert _{L^p(\Omega )} \lesssim \Vert g_1\Vert _{L^q(\Omega )} \thickapprox \Vert g\Vert _{L^q(\Omega (x_0,2r))} \nonumber \\&\lesssim r^{\frac{n}{p}} \int _{r}^d t^{-\frac{n}{p}-1}\Vert g\Vert _{L^q(\Omega (x_0,t))} \,dt + r^{\frac{n}{p}} \Vert g\Vert _{L^q(\Omega )}, \end{aligned}$$

(2.10)

where the constant is independent of g, $x_0$ and r.

We have

$$\begin{aligned} |Tg_2(x)| \lesssim \int _{\Omega \backslash \Omega (x_0,2r)} \frac{|g(y)|}{|x-y|^{n-1}}\,dy,\qquad x\in \Omega (x_0,r). \end{aligned}$$

It is clear that $x\in \Omega (x_0,r)$, $y\in \Omega \backslash (\Omega (x_0,2r))$ implies $\frac{1}{2}|x_0-y| \le |x-y| < \frac{3}{2} |x_0-y|$. Therefore we obtain that

$$\begin{aligned} \Vert Tg_2\Vert _{L^p(\Omega (x_0,r))} \lesssim r^{\frac{n}{p}}\int _{\Omega \backslash (\Omega (x_0,2r))} \frac{|g(y)|}{|x_0-y|^{n-1}}\,dy. \end{aligned}$$

By Fubini’s theorem, we get that

$$\begin{aligned}&\int _{\Omega \backslash \Omega (x_0,2r)} \frac{|g(y)|}{|x_0-y|^{n-1}}\,dy \\&\quad \thickapprox \int _{\Omega \backslash \Omega (x_0,2r)} |g(y)| \left( 1+\int _{|x_0-y|}^d \frac{ds}{s^{n}}\right) \,dy \\&\quad = \int _{\Omega \backslash \Omega (x_0,2r)} |g(y)|\,dy + \int _{\Omega \backslash \Omega (x_0,2r)}|g(y)|\left( \int _{|x_0-y|}^d \frac{ds}{s^n}\right) \,dy \\&\quad = \int _{\Omega \backslash \Omega (x_0,2r)} |g(y)|\,dy + \int _{2r}^d \left( \int _{2r \le |x_0-y|\le s}|g(y)|\,dy\right) \frac{ds}{s^n} \\&\quad \le \int _{\Omega } |g(y)|\,dy + \int _{2r}^d \left( \int _{\Omega (x_0,s)}|g(y)|\,dy\right) \frac{ds}{s^n}. \end{aligned}$$

Applying Hölder’s inequality, we obtain

$$\begin{aligned} \int _{\Omega \backslash \Omega (x_0,2r)} \frac{|g(y)|}{|x_0-y|^n}\,dy \lesssim \Vert g\Vert _{L^q(\Omega )} + \int _{2r}^d s^{-\frac{n}{p}-1}\Vert g\Vert _{L^q(\Omega (x_0,s))}\,ds. \end{aligned}$$

Thus the inequality

$$\begin{aligned} \Vert Tg_2\Vert _{L^p(\Omega (x_0,r))} \lesssim r^{\frac{n}{p}} \int _{r}^d s^{-\frac{n}{p}-1}\Vert g\Vert _{L^q(\Omega (x_0,s))}\,ds + r^{\frac{n}{p}} \Vert g\Vert _{L^q(\Omega )} \end{aligned}$$

(2.11)

holds for all $r \in (0,d/2)$ for $q \ge 1$.

Finally, combining (2.10) and (2.11), we obtain that

$$\begin{aligned} \Vert Tg\Vert _{L^p(\Omega (x_0,r))} \lesssim r^{\frac{n}{p}} \int _{r}^d s^{-\frac{n}{p}-1}\Vert g\Vert _{L^q(\Omega (x_0,s))}\,ds + r^{\frac{n}{p}} \Vert g\Vert _{L^q(\Omega )} \end{aligned}$$

holds for all $r \in (0,d/2)$ with a constant independent of f, $x_0$ and r.

Let now $r \in [d/2,d)$. Then, using $(L^q(\Omega ),L^p(\Omega ))$-boundedness of T, we obtain

$$\begin{aligned} \Vert Tg\Vert _{L^p(\Omega (x_0,r))} \le \Vert Tg\Vert _{L^p(\Omega )} \lesssim \Vert g\Vert _{L^q(\Omega )} \thickapprox r^{\frac{n}{p}}\Vert g\Vert _{L^q(\Omega )}, \end{aligned}$$

and inequality (2.6) holds.

(ii). Assume that $q = 1$. Let again $r \in (0,d/2)$. We write $g = g_1 + g_2$ with $g_1 = g \chi _{\Omega (x_0,2r)}$ and $g_2 = g \chi _{\Omega \backslash \Omega (x_0,2r)}$. Taking into account the linearity of T, we have

$$\begin{aligned} \Vert Tg\Vert _{L^p(\Omega (x_0,r))} \le \Vert Tg_1\Vert _{L^p(\Omega (x_0,r))} + \Vert Tf_2\Vert _{L^p(\Omega (x_0,r))}. \end{aligned}$$

(2.12)

Since $g_1 \in L^q(\Omega )$, in view of (2.8), the boundedness of T from $L^1(\Omega )$ to $WL^p(\Omega )$ implies that

$$\begin{aligned} \Vert Tg_1\Vert _{WL^p(\Omega (x_0,r))}&\le \Vert Tg_1\Vert _{WL^p(\Omega )} \lesssim \Vert g_1\Vert _{L^1(\Omega )} \thickapprox \Vert g\Vert _{L^1(\Omega (x_0,2r))} \nonumber \\&\lesssim r^{\frac{n}{p}} \int _{r}^d t^{-\frac{n}{p}-1}\Vert g\Vert _{L^1(\Omega (x_0,t))} \,dt + r^{\frac{n}{p}} \Vert g\Vert _{L^1(\Omega )}, \end{aligned}$$

(2.13)

where the constant is independent of f, $x_0$ and r.

On the other hand, since

$$\begin{aligned} \Vert Tg_2\Vert _{WL^p(\Omega (x_0,r))} \le \Vert Tg_2\Vert _{L^p(\Omega (x_0,r))} \end{aligned}$$

using (2.11), we get that

$$\begin{aligned} \Vert Tg_2\Vert _{WL^p(\Omega (x_0,r))} \lesssim r^{\frac{n}{p}} \int _{r}^d s^{-\frac{n}{p}-1}\Vert g\Vert _{L^1(\Omega (x_0,s))}\,ds + r^{\frac{n}{p}} \Vert g\Vert _{L^1(\Omega )} \end{aligned}$$

(2.14)

holds true for all $r \in (0,d/2)$.

Combining (2.12), (2.13) and (2.14), we see that inequality (2.7) holds true for all $r \in (0,d/2)$ with a constant independent of g, $x_0$ and r.

If $r \in [d/2,d)$, then, using the boundedness of T from $L^1(\Omega )$ to $WL^p(\Omega )$, we obtain that

$$\begin{aligned} \Vert Tg\Vert _{WL^p(\Omega (x_0,r))} \le \Vert Tg\Vert _{WL^p(\Omega )} \lesssim \Vert g\Vert _{L^1(\Omega )} \thickapprox r^{\frac{n}{p}}\Vert g\Vert _{L^1(\Omega )}, \end{aligned}$$

and, inequality (2.7) holds. $\square$

In order to achieve the regularity results, we must prove the following theorem.

Theorem 2.9

Let $\Omega$ be an open bounded subset of $\mathbb {R}^n$, $x_0 \in \Omega$, $1 \le q< p< \infty$, $\frac{1}{q}=\frac{1}{p}+\frac{1}{n}$. Let also $\varphi _{1}(x,r)$ and $\varphi _{2}(x,r)$ two positive measurable functions defined on $\Omega \times (0,d)$ such that the following condition is fulfilled:

$$\begin{aligned} \int _r^{d} \frac{\mathop \mathrm{ess \; inf}\limits _{t<\tau <\infty }\varphi _2(x_0,\tau ) \, \tau ^{\frac{n}{q}}}{t^{\frac{n}{p}+1}} \, dt \le C \, \varphi _1(x_0,r), \end{aligned}$$

(2.15)

where C does not depend on r. Then, in the case $q>1$ for every $g\in \widetilde{LM}_{\{x_{0}\}}^{q,\varphi _{2}}(\Omega )$, the function Tg(x) is a.e. defined, Tg belongs to the space $\widetilde{LM}_{\{x_{0}\}}^{p,\varphi _{1}}(\Omega )$ and there exists $c=c(q,\varphi _1,\varphi _2,n)>0$ such that

$$\begin{aligned} \Vert Tg\Vert _{\widetilde{LM}_{\{x_{0}\}}^{p,\varphi _{1}}(\Omega )}\le c\Vert g\Vert _{\widetilde{LM}_{\{x_{0}\}}^{q,\varphi _{2}}(\Omega )}. \end{aligned}$$

In the case $q=1$ the function Tg belongs to the space $\widetilde{LM}_{\{x_{0}\}}^{p,\varphi _{1}}(\Omega )$ and there exists $c=c(\varphi _1,\varphi _2,n)>0$ such that

$$\begin{aligned} \Vert Tg\Vert _{\widetilde{LM}_{\{x_{0}\}}^{p,\varphi _{1}}(\Omega )}\le c\Vert g\Vert _{\widetilde{LM}_{\{x_{0}\}}^{1,\varphi _{2}}(\Omega )}. \end{aligned}$$

Proof

By Theorem 2.8 and Theorem 2.4 with $v_2(r)=\varphi _1(x_0,r)^{-1}$, $v_1(r)=\varphi _2(x_0,r)^{-1} r^{-\frac{n}{q}}$ and $w(r)=r^{-\frac{n}{p}}$ for $q>1$ we have

$$\begin{aligned} \Vert T g\Vert _{\widetilde{LM}_{\{x_{0}\}}^{p,\varphi _{1}}(\Omega )}&\lesssim \sup _{0<r<d}\varphi _1(x_0,r)^{-1} \int _r^{d}\Vert f\Vert _{L^q(\Omega (x_0,t))}\frac{dt}{t^{\frac{n}{p}+1}} + \Vert T g\Vert _{L^p(\Omega )}\\&\lesssim \sup _{0<r<d}\varphi _2(x_0,r)^{-1} \, r^{-\frac{n}{q}} \,\Vert g\Vert _{L^q(\Omega (x_0,r))} + \Vert g\Vert _{L^{q}(\Omega )}\\&= \Vert g\Vert _{LM^{q,\varphi _2}_{\{x_0\}}(\Omega )} + \Vert g\Vert _{L^{q}(\Omega )}\\&= \Vert g\Vert _{\widetilde{LM}^{q,\varphi _2}_{\{x_0\}}(\Omega )} \end{aligned}$$

and for $q=1$

$$\begin{aligned} \Vert T g\Vert _{\widetilde{LM}_{\{x_{0}\}}^{p,\varphi _{1}}(\Omega )}&\lesssim \sup _{0<r<d}\varphi _1(x_0,r)^{-1} \int _r^{d}\Vert f\Vert _{L^1(\Omega (x_0,t))}\frac{dt}{t^{\frac{n}{p}+1}} + \Vert T g\Vert _{L^p(\Omega )}\\&\lesssim \sup _{0<r<d}\varphi _2(x_0,r)^{-1} \, r^{-n} \,\Vert g\Vert _{L^1(\Omega (x_0,r))} + \Vert g\Vert _{L^{1}(\Omega )}\\&= \Vert g\Vert _{LM^{1,\varphi _2}_{\{x_0\}}(\Omega )} + \Vert g\Vert _{L^{1}(\Omega )}\\&= \Vert g\Vert _{\widetilde{LM}^{1,\varphi _2}_{\{x_0\}}(\Omega )}. \end{aligned}$$

$\square$

From Theorem 2.9 we get the following corollary.

Corollary 2.10

Let $\Omega$ be an open bounded subset of $\mathbb {R}^n$, $1 \le q< p< \infty$, $\frac{1}{q}=\frac{1}{p}+\frac{1}{n}$. Let also $\varphi _{1}(x,r)$ and $\varphi _{2}(x,r)$ two positive measurable functions defined on $\Omega \times (0,d)$ such that the following condition is fulfilled:

$$\begin{aligned} \int _r^{d} \frac{\mathop \mathrm{ess \; inf}\limits _{t<\tau <d}\varphi _2(x,\tau ) \, \tau ^{\frac{n}{q}}}{t^{\frac{n}{p}+1}} \, dt \le C \, \varphi _1(x,r), \end{aligned}$$

(2.16)

where C does not depend on x and r. Then, in the case $q>1$ for every $g\in \widetilde{M}^{q,\varphi _{2}}(\Omega )$, the function Tg(x) is a.e. defined, Tg belongs to the space $\widetilde{M}^{p,\varphi _{1}}(\Omega )$ and there exists $c=c(q,\varphi _1,\varphi _2,n)>0$ such that

$$\begin{aligned} \Vert Tg\Vert _{\widetilde{M}^{p,\varphi _{1}}(\Omega )}\le c\Vert g\Vert _{\widetilde{M}^{q,\varphi _{2}}(\Omega )}. \end{aligned}$$

In the case $q=1$ the function Tg belongs to the space $W\widetilde{M}^{p,\varphi _{1}}(\Omega )$ and there exists $c=c(\varphi _1,\varphi _2,n)>0$ such that

$$\begin{aligned} \Vert Tg\Vert _{W\widetilde{M}^{p,\varphi _{1}}(\Omega )}\le c\Vert g\Vert _{\widetilde{M}^{1,\varphi _{2}}(\Omega )}. \end{aligned}$$

3 Application to partial differential equations

Let us consider the divergence form elliptic equation (1.3), in a bounded set $\Omega \subset \mathbb {R}^{n}$, $n\ge 3$. We set

$$\begin{aligned}&\Gamma (x,t)=\frac{1}{n(2-n)\omega _{n}\sqrt{\det \{a_{ij}(x)\}}}\left( \sum _{i,j=1}^{n} A_{ij}(x)t_{i}t_{j}\right) ^{\frac{2-n}{2}},\\&\Gamma _{i}(x,t)=\frac{\partial }{\partial t_{i}}\Gamma (x,t),\qquad \Gamma _{ij}(x,t) =\frac{\partial }{\partial t_{i}\partial t_{j}}\Gamma (x,t),\\&M=\max _{i,j=1,\ldots ,n}\max _{|\alpha |\le 2n}\left\| \frac{\partial ^{\alpha } \Gamma _{ij}(x,t)}{\partial t^{\alpha }}\right\| _{L^{\infty }(\Omega \times \Sigma )}, \end{aligned}$$

for a.a. $x\in B$ and $\forall t\in \mathbb {R}^{n}{\setminus }\{0\}$, where $A_{ij}$ denote the entries of the inverse matrix of the matrix $\{a_{ij}(x)\}_{i,j=1,\ldots ,n}$, and $\omega _{n}$ is the measure of the unit ball in $\mathbb {R}^{n}$.

It is well known that $\Gamma _{ij}(x,t)$ are Calderón–Zygmund kernels in the t variable.

Let $r,R\in \mathbb {R}^{+}$, $r<R$ and $\varphi \in C^{\infty }(\Omega )$ be a standard cut-off function such that for every ball $B_{R}\subset \Omega$,

$$\begin{aligned} \varphi (x)=1\quad \hbox {in }B_{r},\qquad \varphi (x)=0,\quad \hbox {in }\Omega {\setminus } B_{R}. \end{aligned}$$

Then if u is a solution of (1.3) and $v=\varphi u$ we have

$$\begin{aligned} L(v)=\nabla \cdot G+g , \end{aligned}$$

where

$$\begin{aligned} G&=\varphi f+uA\nabla \varphi ,\\ g&=\langle A\nabla u,\nabla \varphi \rangle -\langle f,\nabla \varphi \rangle . \end{aligned}$$

Using the notations above, we are able to recall an integral representation formula for the first derivatives of a solution u of (1.3).

Lemma 3.1

For every $i=1,\ldots ,n$, let $a_{ij}\in L^{\infty }(\mathbb {R}^{n})\cap {CBMO^{\max \{p,p'\}}_{\{x_0\}}}$ satisfy (1.6) and (1.7), let u be a solution of (1.3) and let $\varphi$, g and G defined as above. Then, for every $i=1,\ldots ,n$ we have

setting $c_{ih}=\int _{|t|=1}\Gamma _{i}(x,t)t_{h}\,\mathrm {d}\sigma _{t}$.

Using the representation formula stated in Lemma 3.1, we can obtain a regularity result for the solutions to (1.3).

Theorem 3.2

Let $a_{ij}$ be such that (1.5), (1.6), (1.7) are true, we assume that the condition (2.15) is fulfilled and that $\varphi _2 > rsim \varphi _1$. Let also suppose that u is a solution of (1.3) such that $\partial _{x_{i}}u\in \widetilde{LM}_{\{x_{0}\}}^{q,\varphi _{2}}(\Omega )$, for all $i=1,\ldots ,n$, $f\in [ \widetilde{LM}_{\{x_{0}\}}^{q,\varphi _{1}}(\Omega )]^{n}$, $x_0 \in \Omega$. Let $\varphi \in C^\infty (\Omega )$ a standard cut-off function. Then, for any $K\subset \Omega$ compact there exists a constant $c(n,p,\varphi _1,\varphi _2,dist(K,\partial \Omega ))$ such that

$$\begin{aligned}&(i)\qquad \partial _{x_{i}}u\in \widetilde{LM}_{\{x_{0}\}}^{p,\varphi _{1}}(K), \qquad \forall i=1,\ldots ,n,\\&(ii)\qquad \Vert \partial _{x_{i}}u\Vert _{\widetilde{LM}_{\{x_{0}\}}^{p,\varphi _{1}}(K)} \lesssim \Vert u\Vert _{\widetilde{LM}_{\{x_{0}\}}^{p,\varphi _{1}}(\Omega )} +\Vert \partial _{x_{i}}u\Vert _{\widetilde{LM}_{\{x_{0}\}}^{q,\varphi _{2}}(\Omega )} + \Vert f\Vert _{\widetilde{LM}_{\{x_{0}\}}^{q,\varphi _1}(\Omega ) },\\&\qquad \forall i=1,\ldots ,n, \end{aligned}$$

where $\frac{1}{p}=\frac{1}{q}+\frac{1}{n}$.

Proof

Let $K\subset \Omega$ be a compact set. Using Lemma and the boundedness of the commutator proved in [13], we obtain the following estimate:

$$\begin{aligned} \Vert \partial _{x_i}( \varphi u)\Vert _{\widetilde{LM}_{\{x_{0}\}}^{p,\varphi _1}(K)}\le & {} \Vert C[a_{ij}, \varphi ]\partial _{x_h}(u\varphi )\Vert _{\widetilde{LM}_{\{x_{0}\}}^{p,\varphi _1}(K)} +\Vert KG\Vert _{\widetilde{LM}_{\{x_{0}\}}^{p,\varphi _1}(K)}\\&+\Vert Tg\Vert _{\widetilde{LM}_{\{x_{0}\}}^ {p,\varphi _1}(K)}+\Vert G\Vert _{\widetilde{LM}_{\{x_{0}\}}^{p,\varphi _1}(K)}\\\le & {} c\Vert a\Vert _{{CVMO^{\max \{p,p'\}}_{\{x_0\}}}}\Vert \partial _{x_h}(u\varphi )\Vert _{\widetilde{LM} _{\{x_{0}\}}^{p,\varphi _1}(K)}+\Vert G\Vert _{\widetilde{LM}_{\{x_{0}\}}^{p,\varphi _1}(K)}\\&+\Vert g\Vert _{\widetilde{LM}_{\{x_{0}\}}^{q,\varphi _2}(K)}\\&+\Vert G\Vert _{\widetilde{LM}_{\{x_{0}\}}^{p,\varphi _1} (K)}, \end{aligned}$$

where the norm $\Vert a\Vert _{CVMO^{\max \{p,p'\}}_{\{x_0\}}}$ is taken in the set $B_R$.

Taking into account that $a\in CVMO^{\max \{p,p'\}}_{\{x_0\}}$, we can choose the radius R of the ball $B_R$ such that $c\Vert a\Vert _{CVMO^{\max \{p,p'\}}_{\{x_0\}}}<\frac{1}{2}$. This remark allow us to write

$$\begin{aligned}&\Vert \partial _{x_i}( \varphi u)\Vert _{\widetilde{LM}_{\{x_{0}\}}^{p,\varphi _1}(K)} \\&\quad \le \Vert G\Vert _{\widetilde{LM}_ {\{x_{0}\}}^{p,\varphi _1}(K)}+\Vert g\Vert _{\widetilde{LM}_{\{x_{0}\}}^{q,\varphi _2}(K)} +\Vert G\Vert _{\widetilde{LM}_{\{x_{0}\}}^{p,\varphi _1}(K)}\\&\quad \approx \Vert G\Vert _{\widetilde{LM}_{\{x_{0}\}}^{p,\varphi _1}(K)}+\Vert g\Vert _{\widetilde{LM}_ {\{x_{0}\}}^{q,\varphi _2}(K)}\\&\quad = \Vert \varphi f+uA\nabla \varphi \Vert _{\widetilde{LM}_{\{x_{0}\}}^{p,\varphi _1}(K)} +\Vert \langle A\nabla u,\nabla \varphi \rangle -\langle f, \nabla \varphi \rangle \Vert _ {\widetilde{LM}_{\{x_{0}\}}^{q,\varphi _2}(K)}\\&\quad \le \Vert f\Vert _{\widetilde{LM}_{\{x_{0}\}}^{p,\varphi _1}(K)}+\Vert u\Vert _{\widetilde{LM}_ {\{x_{0}\}}^{p,\varphi _1}(K)}+\Vert \partial _{x_i}u\Vert _{\widetilde{LM}_{\{x_{0}\}}^ {q,\varphi _2}(K)}+\Vert f\Vert _{\widetilde{LM}_{\{x_{0}\}}^{q,\varphi _2}(K)}. \end{aligned}$$

Now we apply the hypothesis $\varphi _2 > rsim \varphi _1$, obtaining the following estimate for the norm $\Vert f\Vert _{\widetilde{LM}_{\{x_{0}\}}^{q,\varphi _2}}$:

$$\begin{aligned} \Vert f\Vert _{\widetilde{LM}_{\{x_{0}\}}^{q,\varphi _2}(K)}\le & {} \sup _{0<r<d} \frac{1}{\varphi _2(x_0,r)}\frac{1}{|B(x_0,r)| ^{\frac{1}{q}}}\Vert f\Vert _{L^{q} (|B(x_0,r)\cap K)}+\Vert f\Vert _{L^q(K)}\\\lesssim & {} \sup _{0<r<d} \frac{1}{\varphi _1(x_0,r)}\frac{1}{|B(x_0,r)|^{\frac{1}{q}}} \Vert f\Vert _{L^{q}(|B(x_0,r)\cap K)} + \Vert f\Vert _{L^q(K)}\\= & {} \Vert f\Vert _{LM_{\{x_0\}}^{q,\varphi _1}(K)}+\Vert f\Vert _{L^q(K)} = \Vert f\Vert _{\widetilde{LM}_ {\{x_{0}\}}^{q,\varphi _1}(K) }. \end{aligned}$$

Using the previous estimate we finally obtain that

$$\begin{aligned} \Vert \partial _{x_{i}}u\Vert _{\widetilde{LM}_{\{x_{0}\}}^{p,\varphi _{1}}(K)}&\le C \left( \Vert u\Vert _{\widetilde{LM}_{\{x_{0}\}}^{p,\varphi _{1}}(\Omega )} +\Vert \partial _{x_{i}}u\Vert _{\widetilde{LM}_{\{x_{0}\}}^{q,\varphi _{2}}(\Omega )} + \Vert f\Vert _{\widetilde{LM}_{\{x_{0}\}}^{q,\varphi _1}(\Omega ) }\right) , \\&\qquad \forall i=1,\ldots ,n, \end{aligned}$$

$\square$

From Theorem 3.2 we get the following corollary.

Corollary 3.3

Let $a_{ij}\in L^{\infty } (\mathbb {R}^n)\cap {VMO}$ such that (1.6), (1.7) are true, we assume that the condition (2.16) is fulfilled and that $\varphi _2 > rsim \varphi _1$. Let also suppose that u is a solution of (1.3) such that $\partial _{x_{i}}u\in \widetilde{LM}_{\{x_{0}\}}^{q,\varphi _{2}}(\Omega )$, for all $i=1,\ldots ,n$, $f\in [ \widetilde{M}^{p,\varphi _{1}}(\Omega )]^{n}$. Let $\varphi \in C^\infty (\Omega )$ a standard cut-off function. Then, for any $K\subset \Omega$ compact there exists a constant $c(n,p,\varphi _1,\varphi _2,dist(K,\partial \Omega ))$ such that

$$\begin{aligned}&(i)\qquad \partial _{x_{i}}u\in \widetilde{M}^{p,\varphi _{1}}(K),\qquad \forall i=1,\ldots ,n,\\&(ii)\qquad \Vert \partial _{x_{i}}u\Vert _{\widetilde{M}^{p,\varphi _{1}}(K)} \lesssim \Vert u \Vert _{\widetilde{M}^{p,\varphi _{1}}(\Omega )} +\Vert \partial _{x_{i}}u\Vert _{\widetilde{M}^{q,\varphi _{2}}(\Omega )} + \Vert f\Vert _{\widetilde{M}^ {q,\varphi _1}(\Omega ) },\\&\qquad \forall i=1,\ldots ,n, \end{aligned}$$

where $\frac{1}{p}=\frac{1}{q}+\frac{1}{n}$.

In the case $\varphi _1(x,r)=\varphi _2(x,r)$ we get the following corollaries.

Corollary 3.4

Let $a_{ij}$ be such that (1.5), (1.6), (1.7) are true, we assume that $\varphi (x,r)$ positive measurable function defined on $\Omega \times (0,d)$ and the following condition is fulfilled:

$$\begin{aligned} \int _r^{d} \frac{\mathop \mathrm{ess \; inf}\limits _{t<\tau <\infty }\varphi (x_0,\tau ) \, \tau ^{\frac{n}{q}}}{t^{\frac{n}{p}+1}} \, dt \le C \, \varphi (x_0,r), \end{aligned}$$

where C does not depend on r.

Let also suppose that u is a solution of (1.3) such that $\partial _{x_{i}}u\in \widetilde{LM}_{\{x_{0}\}}^{q,\varphi }(\Omega )$, for all $i=1,\ldots ,n$, $f\in [ \widetilde{LM}_{\{x_{0}\}}^{q,\varphi }(\Omega )]^{n}$, $x_0 \in \Omega$. Let $\varphi \in C^\infty (\Omega )$ a standard cut-off function. Then, for any $K\subset \Omega$ compact there exists a constant $c(n,p,\varphi ,dist(K,\partial \Omega ))$ such that

$$\begin{aligned}&(i)\qquad \partial _{x_{i}}u\in \widetilde{LM}_{\{x_{0}\}}^{p,\varphi }(K),\qquad \forall i=1,\ldots ,n,\\&(ii)\qquad \Vert \partial _{x_{i}}u\Vert _{\widetilde{LM}_{\{x_{0}\}}^{p,\varphi }(K)} \lesssim \Vert u\Vert _{\widetilde{LM}_{\{x_{0}\}}^{p,\varphi }(\Omega )} +\Vert \partial _{x_{i}}u\Vert _{\widetilde{LM}_{\{x_{0}\}}^{q,\varphi }(\Omega )}+ \Vert f \Vert _{\widetilde{LM}_{\{x_{0}\}}^{q,\varphi }(\Omega ) },\\&\qquad \forall i=1,\ldots ,n, \end{aligned}$$

where $\frac{1}{p}=\frac{1}{q}+\frac{1}{n}$.

Corollary 3.5

Let $a_{ij}\in L^{\infty }(\mathbb {R}^n)\cap {VMO}$ satisfy (1.6), (1.7) are true, we assume that $\varphi (x,r)$ positive measurable function defined on $\Omega \times (0,d)$ and the following condition is fulfilled:

$$\begin{aligned} \int _r^{d} \frac{\mathop \mathrm{ess \; inf}\limits _{t<\tau <\infty }\varphi (x,\tau ) \, \tau ^{\frac{n}{q}}}{t^{\frac{n}{p}+1}} \, dt \le C \, \varphi (x,r), \end{aligned}$$

where C does not depend on x, r.

Let also suppose that u is a solution of (1.3) such that $\partial _{x_{i}}u\in \widetilde{M}^{q,\varphi }(\Omega )$, for all $i=1,\ldots ,n$, $f\in [ \widetilde{M}^{q,\varphi }(\Omega )]^{n}$. Let $\varphi \in C^\infty (\Omega )$ a standard cut-off function. Then, for any $K\subset \Omega$ compact there exists a constant $c(n,p,\varphi ,dist(K,\partial \Omega ))$ such that

$$\begin{aligned}&(i)\qquad \partial _{x_{i}}u\in \widetilde{M}^{p,\varphi }(K),\qquad \forall i=1,\ldots ,n,\\&(ii)\qquad \Vert \partial _{x_{i}}u\Vert _{\widetilde{M}^{p,\varphi }(K)} \lesssim \Vert u \Vert _{\widetilde{M}^{p,\varphi }(\Omega )} +\Vert \partial _{x_{i}}u\Vert _{\widetilde{M}^{q,\varphi }(\Omega )}+ \Vert f\Vert _{\widetilde{M}^ {q,\varphi }(\Omega ) },\\&\qquad \forall i=1,\ldots ,n, \end{aligned}$$

where $\frac{1}{p}=\frac{1}{q}+\frac{1}{n}$.