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
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.
Similar content being viewed by others
Avoid common mistakes on your manuscript.
1 Introduction and mathematical background
In this note we consider the following divergence form elliptic equation
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
where B ranges in the set of the balls contained in \(\Omega \) and \(f_{B}\) is the integral average of f over B, namely
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
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
In the sequel we denote \(\eta _{ij}\) the VMO-modulus of the coefficient \(a_{ij}\) and
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
We set
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:
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
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
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 \(B \subset \mathbb {R}^n\)
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
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
According to this definition we obtain, for \(0\le \lambda <n\), the Morrey space \(L^{p,\lambda }\) under the choice \(\varphi (x,r)=r^{\frac{\lambda -n}{p}}\):
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
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
According to this definition we obtain, for \(\lambda \ge 0\), the local Morrey Space \(LM_{\{x_{0}\}}^{p,\lambda }\) under the choice \(\varphi (x_{0},r)=r^{\frac{\lambda -n}{p}}\):
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
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
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:
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
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
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 \(1< q < \infty \)
where c does not depend on \(x_0\) and f and for \(q = 1\)
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
where C does not depend on r and \(x_0\).
If \(a\in CBMO^{\max \{p,p'\}}_{\{x_0\}}(\mathbb {R}^{n})\), the commutator
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
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
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
holds for some \(C>0\) for all non-negative and non-decreasing g on (0, d) if and only if
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
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
We get
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:
From (1.2) and Lemma 1.2 (see [29] [inequality (1.3)]) for \(J_2\) we have
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
Hence, from Lemma 1.2 we get
For \(x\in B\) by Fubini’s theorem applying Hölder inequality and from Lemma 1.2 we have
\(\square \)
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
(i) Let \(1< q <\infty \). If \(g \in L^q(\Omega )\) such that
then for any \(r \in (0,d)\) the inequality
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
holds with constant \(c>0\) independent of g, \(x_0\) and r.
Proof
Let \(1 \le q< p < \infty \). Since
we get that
(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
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
where the constant is independent of g, \(x_0\) and r.
We have
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
By Fubini’s theorem, we get that
Applying Hölder’s inequality, we obtain
Thus the inequality
holds for all \(r \in (0,d/2)\) for \(q \ge 1\).
Finally, combining (2.10) and (2.11), we obtain that
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
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
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
where the constant is independent of f, \(x_0\) and r.
On the other hand, since
using (2.11), we get that
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
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:
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
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
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
and for \(q=1\)
\(\square \)
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:
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
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
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
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 \(B_{R}\subset \Omega \),
Then if u is a solution of (1.3) and \(v=\varphi u\) we have
where
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
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:
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
Now we apply the hypothesis \(\varphi _2 > rsim \varphi _1\), obtaining the following estimate for the norm \(\Vert f\Vert _{\widetilde{LM}_{\{x_{0}\}}^{q,\varphi _2}}\):
Using the previous estimate we finally obtain that
\(\square \)
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
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:
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
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:
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
where \(\frac{1}{p}=\frac{1}{q}+\frac{1}{n}\).
References
Deringoz, F., Guliyev, V.S., Hasanov, S.G.: Characterizations for the Riesz potential and its commutators on generalized Orlicz-Morrey spaces. J. Inequal. Appl. 2016, 248 (2016). https://doi.org/10.1186/s13660-016-1192-z
Di Fazio, G.: \(L^p\) estimates for divergence form elliptic equations with discontinuous coefficients. Boll. Un. Mat. Ital. 7(10–A), 409–420 (1996)
Di Fazio, G., Ragusa, M.A.: Interior estimates in Morrey spaces for strong solutions to nondivergence form elliptic equations with discontinuous coefficients. J. Funct. Anal. 112, 241–256 (1993)
Eroglu, A., Guliyev, V.S., Omarova, M.N.: Characterizations for the Nonsingular Integral Operator and its Commutators on Generalized Orlicz-Morrey Spaces. Azerb. J. Math. 7(2), 67–89 (2017)
Gadjiev, T.S., Galandarova, Sh, Guliyev, V.S.: Regularity in generalized Morrey spaces of solutions to higher order nondivergence elliptic equations with VMO coefficients. Electron. J. Qual. Theory Differ. Equ. 55, 17 (2019)
Guliyev, V.S. Integral operators on function spaces on the homogeneous groups and on domains in \(\mathbb{R}^n\), [Russian Doctor’s degree dissertation]. Moscow: Mat. Inst. Steklov. p. 329 (1994)
Guliyev, V.S.: Function spaces, integral operators and two weighted inequalities on homogeneous groups, some applications, p. 332. Casioglu, (Russian) Baku (1999)
Guliyev, V.S.: Boundedness of the maximal, potential and singular operators in the generalized Morrey spaces. J Inequal Appl. Art. ID 503948, 1–20 (2009)
Guliyev, V.S.: Generalized weighted Morrey spaces and higher order commutators of sublinear operators. Eurasian Math. J. 3(3), 33–61 (2012)
Guliyev, V.S.: Local generalized Morrey spaces and singular integrals with rough kernel. Azerb. J. Math. 3(2), 79–94 (2013)
Guliyev, V.S.: Generalized local Morrey spaces and fractional integral operators with rough kernel, Problems in mathematical analysis. No. 71. J. Math. Sci. (N.Y.) 193(2), 211–227 (2013)
Guliyev, V.S., Omarova, M.N.: Parabolic oblique derivative problem with discontinuous coefficients in generalized weighted Morrey spaces. Open Mathematics 14(1), 49–61 (2016)
Guliyev, V.S., Omarova, M.N., Ragusa, M.A., Scapellato, A.: Commutators and Generalized local Morrey spaces. J. Math. Anal. Appl. 457(2), 1388–1402 (2018)
Guliyev, V.S., Guliyev, R.V., Omarova, M.N.: Riesz transforms associated with Schrödinger operator on vanishing generalized Morrey spaces. Appl Comput Math. 17(1), 56–71 (2018)
Kokilashvili, V., Meskhi, A., Rafeiro, H., Samko, S.: Integral operators in non-standard function spaces. Vol. II: Variable Exponent Hölder, Morrey-Campanato and Grand Spaces, Operator theory: advances and applications. Birkhäuser, Basel (2016)
Lu, S.Z., Yang, D.C.: The central BMO spaces and Littlewood-Paley operators. Approx. Theory Appl. (N.S.) 11, 72–94 (1995)
Lu, S.Z., Wu, Q.: CBMO estimates for commutators and multilinear singular integrals. Math. Nachr. 276, 75–88 (2004)
Mizuhara, T.: Boundedness of some classical operators on generalized Morrey spaces. In: Igari, S. (ed.) Harmonic Analysis, pp. 183–189. Springer, ICM 90 Satellite Proceedings, (1991)
Morrey, C.B.: On the solutions of quasi-linear elliptic partial differential equations. Trans. Amer. Math. Soc. 43, 126–166 (1938)
Nakai, E.: Hardy-Littlewood maximal operator, singular integral operators and Riesz potentials on generalized Morrey spaces. Math. Nachr. 166, 95–103 (1994)
Polidoro, S., Ragusa, M.A.: Harnack inequality for hypoelliptic ultraparabolic equations with a singular lower order term. Rev. Mat. Iberoam. 24(3), 1011–1046 (2008)
Rădulescu, V.D., Repovš, D.D.: Partial Differential Equations with Variable Exponents - Variational Methods and Qualitative Analysis. CRC Press, Boca Raton, FL (2015)
Rădulescu, V.: Nonlinear elliptic equations with variable exponent: old and new. Nonlinear Anal. 121, 336–369 (2015)
Ragusa, M.A.: Regularity of solutions of divergence form elliptic equations. Proc. Amer. Math. Soc. 128, 533–540 (1999)
Ragusa, M.A.: Necessary and sufficient condition for a \(VMO\) function. Appl. Math. Comput. 218(24), 11952–11958 (2012)
Sawano, Y.: A thought on generalized Morrey spaces. J. Indonesian Math. Soc. 25(3), 210–281 (2019)
Scapellato, A.: On some qualitative results for the solution to a Dirichlet problem in local generalized Morrey spaces. AIP Conference Proceedings 1798, art.n.020138 (2017). https://doi.org/10.1063/1.4972730
Scapellato, A.: Some properties of integral operators on generalized Morrey spaces. AIP Conference Proceedings 1863, 510004 (2017). https://doi.org/10.1063/1.4992662
Shi, S., Lu, Sh: Characterization of the central Campanato space via the commutator operator of Hardy type. J. Math. Anal. Appl. 429(2), 713–732 (2015)
Acknowledgements
The first and the third authors were partially supported by the Ministry of Education and Science of the Russian Federation (5-100 program of the Russian Ministry of Education). The first author was also partially supported by the Grant of Cooperation Program 2532 TUBITAK - RFBR (RUSSIAN foundation for basic research) (Agreement number no. 119N455). The first and the second authors were partially supported by the Grant of 1st Azerbaijan-Russia Joint Grant Competition (Agreement number no. EIF-BGM-4-RFTF-1/2017-21/01/1-M-08). The fourth author was supported by “Piano di incentivi per la ricerca di Ateneo 2020/2022 (Pia.ce.ri)” - Università degli Studi di Catania.
Funding
Open access funding provided by Università degli Studi di Catania within the CRUI-CARE Agreement.
Author information
Authors and Affiliations
Contributions
All authors contributed to the study conception and design. All authors read and approved the final manuscript.
Corresponding author
Ethics declarations
Conflict of interest
The authors declare that they have no conflict of interest.
Availability of data and material
Not applicable.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
About this article
Cite this article
Guliyev, V.S., Omarova, M.N., Ragusa, M.A. et al. Regularity of solutions of elliptic equations in divergence form in modified local generalized Morrey spaces. Anal.Math.Phys. 11, 13 (2021). https://doi.org/10.1007/s13324-020-00433-9
Received:
Revised:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13324-020-00433-9