\documentclass[reqno]{amsart} \usepackage{hyperref} \usepackage{amssymb} \AtBeginDocument{{\noindent\small \emph{Electronic Journal of Differential Equations}, Vol. 2018 (2018), No. 110, pp. 1--24.\newline ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu} \thanks{\copyright 2018 Texas State University.} \vspace{8mm}} \begin{document} \title[\hfilneg EJDE-2018/110\hfil Global regularity in Orlicz-Morrey spaces] {Global regularity in Orlicz-Morrey spaces of solutions to nondivergence elliptic equations with VMO coefficients} \author[V. S. Guliyev, A. A. Ahmadli, M. N. Omarova, L. Softova \hfil EJDE-2018/110\hfilneg] {Vagif S. Guliyev, Aysel A. Ahmadli,\\ Mehriban N. Omarova, Lubomira Softova} \address{Vagif S. Guliyev \newline Ahi Evran University, Department of Mathematics, 40100 Kirsehir, Turkey. \newline S.M. Nikol'skii Institute of Mathematics at RUDN University, Moscow, 117198, Russia.\newline Institute of Mathematics and Mechanics, Az 1141 Baku, Azerbaijan} \email{vagif@guliyev.com} \address{Aysel A. Ahmadli \newline Dumlupinar University, Department of Mathematics, 40100 Kytahya, Turkey} \email{aysel.ahmadli@gmail.com} \address{Mehriban N. Omarova \newline Baku State University, AZ1141 Baku, Azerbaijan. \newline Institute of Mathematics and Mechanics, Az 1141 Baku, Azerbaijan} \email{mehribanomarova@yahoo.com} \address{Lubomira G. Softova \newline Department of Mathematics, University of Salerno, Fisciano, Italy} \email{lsoftova@unisa.it} \dedicatory{Communicated by Vicentiu D. Radulescu} \thanks{Submitted September 11, 2017. Published May 10, 2018.} \subjclass[2010]{35J25, 35B40, 42B20, 42B35, 46E30} \keywords{Generalized Orlicz-Morrey spaces; Calder\'on-Zygmund integrals; \hfill\break\indent commutators; VMO; elliptic equations; Dirichlet problem} \begin{abstract} We show continuity in generalized Orlicz-Morrey spaces $M_{\Phi,\varphi}(\mathbb{R}^n)$ of sublinear integral operators generated by Calder\'on-Zygmund operator and their commutators with BMO functions. The obtained estimates are used to study global regularity of the solution of the Dirichlet problem for linear uniformly elliptic operator $\mathcal{L}=\sum_{i,j=1}^n a^{ij}(x)D_{ij}$ with discontinuous coefficients. We show that $\mathcal{L} u\in M_{\Phi,\varphi}$ implies the second-order derivatives belong to $M_{\Phi,\varphi}$. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{corollary}[theorem]{Corollary} \newtheorem{definition}[theorem]{Definition} \newtheorem{remark}[theorem]{Remark} \newtheorem{example}[theorem]{Example} \allowdisplaybreaks \section{Introduction} The classical Morrey spaces $L_{p,\lambda}$ are originally introduced in \cite{Morrey} to study the local behavior of solutions to elliptic partial differential equations. In fact, the better inclusion between the Morrey and the H\"older spaces permits to obtain higher regularity of the solutions to different elliptic and parabolic boundary problems. Recall that for a bounded domain $\Omega\subset\mathbb{R}^n$ satisfying the cone property, the space $L_{p,\lambda}$ with $1\leq p<\infty$ consists of all functions $f\in L_p(\Omega)$ such that $$ \|f\|_{L_{p,\lambda}(\Omega)}=\Big( \sup_{{\mathcal{B}}_r}\frac1{r^\lambda} \int_{{\mathcal{B}}_r\cap\Omega}|f(y)|^p\,dy \Big)^{1/p}<\infty, $$ where ${\mathcal{B}}_r$ ranges over all balls in $\mathbb{R}^n$ centered in some point $x\in \Omega$ and of radius $r>0$. For the properties and applications of the classical Morrey spaces, we refer the readers to \cite{Camp, Morrey, Peetre, Pi} and the references there. Chiarenza and Frasca \cite{ChFra} showed the boundedness of the Hardy-Littlewood maximal operator in $L_{p,\lambda}(\mathbb{R}^n)$ that allows them to prove continuity of fractional and classical Calder\'on-Zygmund operators in these spaces. Recall that integral operators of that kind appear in the representation formulae of the solutions of elliptic/parabolic equations and systems. Thus the continuity of the Calder\'on-Zygmund integrals implies regularity of the solutions in the corresponding spaces. Mizuhara\cite{Mi} gave a generalization of these spaces considering a weight function $\omega(x,r):{\mathbb{R}^n}\times\mathbb{R}_+\to\mathbb{R}_+$ instead of $r^\lambda$. He studied also a continuity in $L_{p,\omega}$ of some classical integral operators. Later Nakai extended the results of Chiarenza and Frasca in $L_{p,\omega}$ imposing certain integral and doubling conditions on $\omega$ (see \cite{Na}). Taking a weight $\omega=\varphi^pr^n$ the conditions of Mizuhara-Nakai become $$ \int_r^\infty \varphi(x,t)^p \frac{dt}{t} \le C \, \varphi(x,r)^p,\quad C^{-1} \le \frac{\varphi(x,t)}{\varphi(x,r)} \le C, \quad \forall\, r \le t \le 2r, $$ where the constants do not depend on $t$, $r$ and $x\in \mathbb{R}^n$. In series of works, the first author studies the continuity in generalized Morrey spaces of sublinear operators generated by various integral operators as Calder\'on-Zygmund, Riesz potental and others (see \cite{GulDoc, GulJIA, GAKSh}). The following theorem obtained in \cite{GulDoc} extends the results of Nakai in Morrey-type spaces with weight $\omega=\varphi r^n$ (for the definition of the spaces see \S~\ref{sec2}) \begin{theorem}[\cite{GulDoc, GulJIA}] \label{th1} Let $1\le p<\infty$ and $(\varphi_1,\varphi_2)$ satisfy the condition \begin{equation}\label{GulZ} \int_t^{\infty} \varphi_1(x,r)\frac{dr}{r} \le C \varphi_2(x,t), \end{equation} where $C$ does not depend on $x$ and $t$. Then the maximal operator $M$ and the Calder\'on-Zygmund integral operators $K$ are bounded from $M_{p,\varphi_1}$ to $M_{p,\varphi_2}$ for $p > 1$ and from $M_{1,\varphi_1}$ to the weak space $WM_{1,\varphi_2}$. \end{theorem} Later this result was extended on spaces with weaker condition on the weight pair $(\varphi_1, \varphi_2)$ (see \cite{GAKSh}, see also \cite{Eroglu,ErogAz,ErogGulAz_MN2017}). For more recent results on boundedness and continuity of singular integral operators in generalized Morrey and new functional spaces and their application in the differential equations theory see \cite{AkbGulMus1, BonMolRad1, BonMolRad, FazioPR, FanLuYa, GuHaSam, GulSoft1, GulSoft2, MihRadRep, Pal, PeSam, RadRep, Sf1, Sf2, Xiao} and the references there. Throughout this paper the following notation will be used:\\ $D_iu=\partial u/ \partial x_i$, $Du=(D_1u,\ldots,D_nu)$ means the gradient of $u$,\\ $D_{ij}u= \partial^2 u/\partial x_i\partial x_j$, $D^2u=\{D_{ij}u\}_{ij=1}^n$ is the Hessian matrix of $u$,\\ ${\mathcal{B}}_r={\mathcal{B}}(x_0,r)=\{x\in{\mathbb{R}}^n:\ |x-x_0|0\}$,\\ ${{\mathcal{B}}}_r^+\equiv {{\mathcal{B}}}^+(x^0,r)= {{\mathcal{B}}}(x^0,r)\cap \mathbb{R}^n_+$, ${2{\mathcal{B}}}_r^+= {\mathcal{B}}^+(x^0,2r)$ where $x^0=(x',0)$. The standard summation convention on repeated upper and lower indices is adopted. The letter $C$ is used for various positive constants and may change from one occurrence to another. In this paper, we shall use the symbol $A \lesssim B$ to indicate that there exists a universal positive constant $C$, independent of all important parameters, such that $A\le CB$. $A \approx B$ means that $A \lesssim B$ and $B \lesssim A$. \section{Preliminaries on Orlicz and Orlicz-Morrey spaces} \begin{definition}\label{def2} \rm A function $\Phi : [0,+\infty] \to [0,\infty]$ is called a Young function if $\Phi$ is convex, left-continuous, $\lim_{r\to +0} \Phi(r) = \Phi(0) = 0$ and $\lim_{r\to +\infty} \Phi(r) = \Phi(\infty) = \infty$. \end{definition} From the convexity and $\Phi(0) = 0$ it follows that any Young function is increasing. If there exists $s \in (0,+\infty)$ such that $\Phi(s) = +\infty$, then $\Phi(r) = +\infty$ for $r \geq s$. We say that $\Phi \in \Delta_2$, if for any $a > 1$, there exists a constant $C_a>0$ such that $\Phi(at) \leq C_a\Phi(t)$ for all $t > 0$. A Young function $\Phi$ is said to satisfy the $\nabla_2$-condition, denoted also by $\Phi \in \nabla_2$, if $$ \Phi(r)\leq \frac{1}{2k}\Phi(kr),\quad r\geq 0, $$ for some $k>1$. The function $\Phi(r) = r$ satisfies the $\Delta_2$-condition but does not satisfy the $\nabla_2$-condition. If $1 < p < \infty$, then $\Phi(r) = r^p$ satisfies both the conditions. The function $\Phi(r) = e^r - r - 1$ satisfies the $\nabla_2$-condition but does not satisfy the $\Delta_2$-condition. The following two indices $$ q_{\Phi}=\inf_{t>0}\frac{t\varphi(t)}{\Phi(t)},\quad p_{\Phi}=\sup_{t>0}\frac{t\varphi(t)}{\Phi(t)} $$ of $\Phi$, where $\varphi(t)$ is the right-continuous derivative of $\Phi$, are well known in the theory of Orlicz spaces. As is well known, $$ p_{\Phi} < \infty \; \Longleftrightarrow \; \Phi \in \Delta_2, $$ and the function $\Phi$ is strictly convex if and only if $q_{\Phi} > 1$. If $0 < q_{\Phi} \leq p_{\Phi} < \infty$, then $\frac{\Phi(t)}{t^{q_{\Phi}}}$ is increasing and $\frac{\Phi(t)}{t^{p_{\Phi}}}$ is decreasing on $(0,\infty)$. \begin{lemma}[{\cite[Lemma 1.3.2]{KokKrbec}}] \label{propsing} Let $\Phi\in\Delta_2$. Then there exist $p>1$ and $b>1$ such that $$ \frac{\Phi(t_2)}{t_2^p}\leq b \frac{\Phi(t_1)}{t_1^p} $$ for $01+\log_{2}A$, then $$ \int^{\infty}_t\frac{\varphi(s)}{s^{p}}ds\lesssim\frac{\Phi(t)}{t^{p}}, \quad t>0. $$ \item[(2)] Assume that $\Phi\in\nabla_{2}$. Then $$ \int_{0}^{t}\frac{\varphi(s)}{s}ds\lesssim\frac{\Phi(t)}{t}, \quad t>0. $$ \end{itemize} \end{lemma} Recall that a function $\Phi$ is said to be quasiconvex if there exist a convex function $\omega$ and a constant $c > 0$ such that $$ \omega(t) \le \Phi(t) \le c \omega(c t), ~ t \in [0,\infty). $$ Let $\mathcal{Y}$ be the set of all Young functions $\Phi$ such that \begin{equation}\label{2.1} 0<\Phi(r)<+\infty\quad \text{for } 00$ }\big\} $$ is called Orlicz space. The space $L_{\Phi}^{\rm loc}({\mathbb{R}^n})$ endowed with the natural topology is defined as the set of all functions $f$ such that $f\chi_{_B}\in L_{\Phi}({\mathbb{R}^n})$ for all balls $B \subset {\mathbb{R}^n}$. \end{definition} Note that $L_{\Phi}({\mathbb{R}^n})$ is a Banach space with respect to the norm $$ \|f\|_{L_{\Phi}}=\inf\big\{\lambda>0:\int_{{\mathbb{R}^n}}\Phi \Big(\frac{|f(x)|}{\lambda}\Big)dx\leq 1\big\}, $$ see, for example \cite[Section 3, Theorem 10]{RaoRen}, so that $$ \int_{{\mathbb{R}^n}}\Phi\Big(\frac{|f(x)|}{\|f\|_{L_{\Phi}}}\Big)dx\leq 1. $$ For a measurable set $\Omega\subset \mathbb{R}^{n}$, a measurable function $f$ and $t>0$, let $$ m(\Omega, f, t)=|\{x\in\Omega:|f(x)|>t\}|. $$ In the case $\Omega=\mathbb{R}^{n}$, we shortly denote it by $m(f,\ t)$. \begin{definition} \rm The weak Orlicz space $$ WL_{\Phi}(\mathbb{R}^{n})=\{f\in L_{\rm loc}^{1} (\mathbb{R}^{n}):\| f\|_{WL_{\Phi}}<+\infty\} $$ is defined by the norm $$ \| f\|_{WL_{\Phi}}=\inf\Big\{\lambda>0\ : \sup_{t>0}\Phi(t)m\big(\frac{f}{\lambda}, t\big) \leq 1\Big\}. $$ \end{definition} For Young functions $\Phi$ and $\Psi$, we write $\Phi\approx\Psi$ if there exists a constant $C \geq 1$ such that $$ \Phi(C^{-1}r)\leq \Psi(r) \leq \Phi(Cr)\quad \text{for all }r\geq 0. $$ If $\Phi\approx\Psi$, then $L_{\Phi}({\mathbb{R}^n})=L_{\Psi}({\mathbb{R}^n})$ with equivalent norms. We note that, for Young functions $\Phi$ and $\Psi$, if there exist $C,R \geq 1$ such that $$ \Phi(C^{-1}r)\leq \Psi(r) \leq \Phi(Cr)\quad \text{for } r \in (0, R^{-1})\cup (R,\infty) , $$ then $\Phi\approx\Psi$. For a Young function $\Phi$ and $0 \leq s \leq +\infty$, let $$ \Phi^{-1}(s)=\inf\{r\geq 0: \Phi(r)>s\}\quad (\inf\emptyset=+\infty). $$ If $\Phi \in \mathcal{Y}$, then $\Phi^{-1}$ is the usual inverse function of $\Phi$. We note that $$ \Phi(\Phi^{-1}(r))\leq r \leq \Phi^{-1}(\Phi(r)) \quad \text{for } 0\leq r<+\infty. $$ For a Young function $\Phi$, the complementary function $\widetilde{\Phi}(r)$ is defined by \begin{equation}\label{2.2} \widetilde{\Phi}(r)=\begin{cases} \sup\{rs-\Phi(s): s\in [0,\infty)\}, & r\in [0,\infty) \\ +\infty, & r=+\infty. \end{cases} \end{equation} The complementary function $\widetilde{\Phi}$ is also a Young function and $\widetilde{\widetilde{\Phi}}=\Phi$. If $\Phi(r)=r$, then $\widetilde{\Phi}(r)=0$ for $0\leq r \leq 1$ and $\widetilde{\Phi}(r)=+\infty$ for $r>1$. If $1 < p < \infty$, $1/p+1/p^\prime= 1$ and $\Phi(r) = r^p/p$, then $\widetilde{\Phi}(r) = r^{p^\prime}/p^\prime$. If $\Phi(r) = e^r-r-1$, then $\widetilde{\Phi}(r) = (1+r) \log(1+r)-r$. \begin{remark}\label{retwc} \rm Note that $\Phi \in \nabla_2$ if and only if $\widetilde{\Phi} \in \Delta_2$. Also, if $\Phi$ is a Young function, then $\Phi \in \nabla_2$ if and only if $\Phi^{\gamma}$ be quasiconvex for some $\gamma \in (0,1)$ (see, for example \cite[ p. 15]{KokKrbec}). \end{remark} It is known that \begin{equation}\label{2.3} r\leq \Phi^{-1}(r)\widetilde{\Phi}^{-1}(r)\leq 2r \quad \text{for } r\geq 0. \end{equation} The following analogue of the H\"older inequality is known. \begin{theorem}[\cite{Weiss}] \label{HolderOr} For a Young function $\Phi$ and its complementary function $\widetilde{\Phi}$, the following inequality is valid $$ \|fg\|_{L_{1}({\mathbb{R}^n})} \leq 2 \|f\|_{L_{\Phi}} \|g\|_{L_{\widetilde{\Phi}}}. $$ \end{theorem} Note that Young functions satisfy the property \begin{equation} \label{sam1} \Phi(\alpha t)\leq \alpha \Phi(t) \end{equation} for all $0<\alpha<1$ and $0 \le t < \infty$, which is a consequence of the convexity: $\Phi(\alpha t) = \Phi(\alpha t + (1-\alpha)0) \leq \alpha \Phi(t) + (1-\alpha)\Phi(0)=\alpha \Phi(t)$. \begin{lemma}[\cite{BenSharp, LiuHouW}]\label{lem4.0} Let $\Phi$ be a Young function and $B$ a ball in $\mathbb{R}^n$. Then $$ \|\chi_{_B}\|_{WL_{\Phi}({\mathbb{R}^n})} = \|\chi_{_B}\|_{L_{\Phi}({\mathbb{R}^n})} = \frac{1}{\Phi^{-1}(|B|^{-1})}. $$ \end{lemma} In the next sections where we prove our main estimates, we use the following lemma, which follows from Theorem \ref{HolderOr} and Lemma \ref{lem4.0}. \begin{lemma}\label{lemHold} For a Young function $\Phi$ and $B=B(x,r)$, we have $$ \|f\|_{L_{1}(B)} \leq 2 |B| \Phi^{-1}\left(|B|^{-1}\right) \|f\|_{L_{\Phi}(B)} . $$ \end{lemma} \begin{definition} \label{def2.4}\rm Let $\varphi(x,r)$ be a positive measurable function on ${\mathbb{R}^n} \times (0,\infty)$ and $\Phi$ any Young function. We denote by $M_{\Phi,\varphi}({\mathbb{R}^n})$ the generalized Orlicz-Morrey space, the space of all functions $f\in L_{\Phi}^{\rm loc}({\mathbb{R}^n})$ with finite quasinorm $$ \|f\|_{M_{\Phi,\varphi}} = \sup_{x\in{\mathbb{R}^n}, r>0} \varphi(x,r)^{-1} \Phi^{-1}(|B(x,r)|^{-1}) \, \|f\|_{L_{\Phi}(B(x,r))}. $$ Also by $WM_{\Phi,\varphi}({\mathbb{R}^n})$ we denote the weak generalized Orlicz-Morrey space of all functions $f\in WL_{\Phi}^{\rm loc}({\mathbb{R}^n})$ for which $$ \|f\|_{WM_{\Phi,\varphi}} = \sup_{x\in{\mathbb{R}^n}, r>0} \varphi(x,r)^{-1} \Phi^{-1}(|B(x,r)|^{-1}) \, \|f\|_{WL_{\Phi}(B(x,r))} < \infty, $$ where $WL_{\Phi}(B(x,r))$ denotes the weak $L_\Phi$-space of measurable functions $f$ for which \[ \|f\|_{WL_{\Phi}(B(x,r))} \equiv \|f \chi_{_{B(x,r)}}\|_{WL_{\Phi}({\mathbb{R}^n})}. \] \end{definition} According to this definition, we recover the spaces $M_{p,\varphi}$ and $WM_{p,\varphi}$ under the choice $\Phi(r)=r^{p}$: $$ M_{p,\varphi}=M_{\Phi,\varphi}\big|_{\Phi(r)=r^{p}}, \quad WM_{\Phi,\lambda}= WM_{\Phi,\varphi}\big|_{\Phi(r)=r^{p}}. $$ \section{Definitions and statement of the problem}\label{sec2} In the present section we give the definitions of the functional spaces to which the coefficients and the data of the problem belong. The domain $\Omega\subset \mathbb{R}^n$ supposed to be bounded with $\partial\Omega\in C^{1,1}$. \begin{definition} \label{def1} \rm Let $\varphi:\Omega\times\mathbb{R}_+\to \mathbb{R}_+$ be a measurable function and $1 \le p < \infty$. The generalized Orlicz-Morrey space $M_{\Phi,\varphi}(\Omega)$ consists of all $f\in L_{\Phi}^{\rm loc}(\Omega)$ $$ \|f\|_{M_{\Phi,\varphi}(\Omega)} = \sup_{x\in\Omega, r>0} \varphi(x,r)^{-1} \Phi^{-1}(|{\mathcal{B}}(x,r)|^{-1}) \|f\|_{L_{\Phi}(\Omega \cap {\mathcal{B}}(x,r))} $$ For any bounded domain $\Omega$ we define $M_{\Phi,\varphi}(\Omega)$ taking $f\in L_{\Phi}(\Omega)$ and $\Omega_r$ instead of ${\mathcal{B}}(x,r)$ in the norm above. The generalized Sobolev-Orlicz-Morrey space $W_{2,\Phi,\varphi}(\Omega)$ consists of all Sobolev functions $u\in W_{2,\Phi}(\Omega)$ with distributional derivatives $D^su\in M_{\Phi,\varphi}(\Omega)$, endowed with the norm $$ \|u\|_{W_{2,\Phi,\varphi}(\Omega)} =\sum_{0\leq |s|\leq 2}\|D^s f\|_{M_{\Phi,\varphi}(\Omega)}. $$ The space $ W_{2,\Phi,\varphi}(\Omega)\cap W_{1,\Phi}^0(\Omega)$ consists of all functions $u\in W_{2,\Phi}(\Omega)\cap W^0_{1,\Phi}(\Omega)$ with $D^su\in M_{\Phi,\varphi}(\Omega)$, and is endowed by the same norm. Recall that $W^0_{1,\Phi}(\Omega)$ is the closure of $C_0^\infty(\Omega)$ with respect to the norm in $W_{1,\Phi}$. \end{definition} \begin{definition} \label{def2b} \rm Let $\varphi:\Omega \times\mathbb{R}_+\to \mathbb{R}_+$ be a measurable function, the generalized weak Morrey space ${WM}_{\Phi,\varphi}(\Omega)$ consists of all measurable functions such that $$ \|f\|_{{WM}_{\Phi,\varphi}(\Omega)} = \sup_{x\in \Omega, r>0} \varphi(x,r)^{-1} \Phi^{-1}(|{\mathcal{B}}(x,r)|^{-1}) \|f\|_{WL_{\Phi}(\Omega \cap {\mathcal{B}}(x,r))}, $$ where $WL_{\Phi}(\Omega \cap {\mathcal{B}}(x,r))$ denotes the weak $L_\Phi$-space of measurable functions $f$ for which \[ \|f\|_{WL_{\Phi}({\mathcal{B}}(x,r))} \equiv \|f \chi_{_{\Omega \cap {\mathcal{B}}(x,r)}}\|_{WL_{\Phi}({\mathbb{R}^n})}. \] For a bounded domain $\Omega$ we define the space ${WM}_{\Phi,\varphi}(\Omega)$ taking $f\in WL_{\Phi}(\Omega)$. \end{definition} \begin{definition} \rm Let $a\in L_1^{\rm loc}({\mathbb{R}}^n)$ and $a_{{{\mathcal{B}}}_r}= \frac{1}{|{{\mathcal{B}}}_r|}\int_{{{\mathcal{B}}}_r} a(y) dy $ is the mean integral of $a$. We say that \begin{itemize} \item $a\in BMO$ (bounded mean oscillation, \cite{JN}) if $$ \|a\|_{\ast}=\sup_{R>0}\sup_{{{\mathcal{B}}}_r, r\leq R} \frac1{|{{\mathcal{B}}}_r|}\int_{{{\mathcal{B}}}_r}|a(y)-a_{{{\mathcal{B}}}_r}|dy<+\infty. $$ The quantity $\|a\|_\ast$ is a norm in $BMO$ modulo constant function under which $BMO$ is a Banach space; \item $a\in VMO$ ({\it vanishing mean oscillation}, \cite{Sar}) if $a\in BMO$ and $$ \lim_{R\to 0}\gamma_a(R)=\lim_{R\to 0} \sup_{{\mathcal{B}}_r, r\leq R}\frac1{|{\mathcal{B}}_r|}\int_{{\mathcal{B}}_r}|a(y)-a_{{{\mathcal{B}}}_r}|dy=0. $$ The quantity $\gamma_a(R)$ is called $VMO$-modulus of $a$. \end{itemize} For any bounded domain $\Omega\subset {\mathbb{R}}^n$ we define $BMO(\Omega)$ and $VMO(\Omega)$ taking $a\in L_1(\Omega)$ and $\Omega_r $ instead of ${\mathcal{B}}_r$ in the definition above. \end{definition} According to \cite{A,Jones}, having a function $a\in BMO(\Omega)$ or $VMO(\Omega)$ it is possible to extend it in the whole ${\mathbb{R}}^n$ preserving its $BMO$-norm or $VMO$-modulus, respectively. In the following we use this property without explicit references. Any bounded uniformly continuous function $f\in BUC$ with modulus of continuity $\omega_f(r)$ is also $VMO$ and $\gamma_f(r)\equiv \omega_f(r)$. Besides that, $BMO$ and $VMO$ contain also discontinuous functions and the following example shows the inclusion $W_{1,n}({\mathbb{R}}^n)\subset VMO\subset BMO$. \begin{example} \label{examp3.1} \rm $f_\alpha(x)= |\log|x||^\alpha \in VMO$ for any $\alpha\in(0,1)$; $f_\alpha\in W_{1,n}({\mathbb{R}}^n)$ for $\alpha\in(0,1-1/n)$, $f_\alpha\notin W_{1,n}({\mathbb{R}}^n)$ for $\alpha\in[1-1/n, 1)$;\ $f(x)=|\log|x||\in BMO\setminus VMO$; $\sin f_\alpha(x)\in VMO\cap L_\infty({\mathbb{R}^n})$. \end{example} In the Sections~\ref{sec2a}, \ref{sec3} and \ref{sec4} we study continuity in the spaces $M_{\Phi,\varphi}$ of certain sublinear integrals and their commutators with $BMO$ functions. These results unified withe known estimates in $L_p(\mathbb{R}^n)$ permit to obtain continuity of the Calder\'on-Zygmund operators in $M_{p,\varphi}(\mathbb{R}^n)$ that is shown in \S~\ref{sec5}. The last section is dedicated to the Dirichlet problem for a linear uniformly elliptic operator with $VMO$ coefficients. This problem is firstly studied by Chiarenza, Frasca and Longo. In their pioneer works \cite{ChFraL1}, \cite{ChFraL2} they prove unique strong solvability of \begin{equation} \label{DP} \begin{gathered} \mathcal{L} u \equiv a^{ij}(x)D_{ij} u=f(x) \quad \text{a.a. } x\in \Omega,\\ u\in\ W_{2,p}(\Omega)\cap W^0_{1,p}(\Omega), \quad p\in(1,\infty) \end{gathered} \end{equation} providing such way the classical theory on operators with continuous coefficients to those with discontinuous ones. Later their results are extended in the Sobolev-Morrey spaces $W_{2,p,\lambda}(\Omega)\cap W^0_{1,p}(\Omega)$, $\lambda\in(1,n)$ (see \cite{FazioPR}, \cite{FanLuYa}). In the present work we show that $\mathcal{L} u\in M_{\Phi,\varphi}(\Omega)$ implies the same regularity of the second order derivatives $D_{ij}u$. The weight $\varphi(x,r)$ satisfies an integral condition weaker than \eqref{GulZ}. \section{Sublinear operators and commutators generated by singular integrals in the space $M_{\Phi,\varphi}(\mathbb{R}^n)$} \label{sec2a} In this section we present results obtained in \cite{HasJFSA} concerning continuity of sublinear operators generated by singular integrals as Calder\'on-Zygmund. Let $T$ be a sublinear operator such that for any $f\in L_1(\mathbb{R}^n)$ with compact support and $x\notin {\operatorname{supp}} f$ holds \begin{equation}\label{subl} |Tf(x)|\le C \int_{\mathbb{R}^n} \frac{|f(y)|}{|x-y|^{n}} \,dy, \end{equation} where $C$ is independent of $f$. \begin{theorem}\label{3.4.a} Let $\Phi$ any Young function, $\varphi_1,\varphi_2:\mathbb{R}^n\times\mathbb{R}_+\to \mathbb{R}_+$ be measurable functions such that for any $x\in{\mathbb{R}^n}$ and for any $t>0$, \begin{equation}\label{sal21} \int_r^{\infty} \Big(\operatorname{ess\,inf}_{t0$, \begin{equation}\label{ComCond} \int_{r}^{\infty} \Big(1+\ln \frac{t}{r}\Big) \Big(\operatorname{ess\,inf}_{t0$ and define $f_1=\chi_{\{|f|> \lambda\}}\cdot f$ and $f_2=\chi_{\{|f| \le \lambda\}}\cdot f$. Then $f=f_1+f_2$. We have \begin{gather*} |\{|\widetilde K f|>\lambda\}| \leq|\{|\widetilde K f_1|>{\lambda}/{2}\}|+|\{|\widetilde K f_2|>{\lambda}/{2}\}|, \\ \Phi(\lambda)|\{|\widetilde K f|>\lambda\}| \leq|\Phi(\lambda)\{|\widetilde K f_1|>{\lambda}/{2}\}|+\Phi(\lambda)|\{|\widetilde K f_2| >{\lambda}/{2}\}|. \end{gather*} We know that from the weak (1,1) boundedness and $L_p$, $p>1$ boundedness of $\widetilde K$, \begin{gather*} \{|{\widetilde K}(\chi_{\{|f|>\lambda\}}\cdot f)|>\lambda\}| \lesssim\frac{1}{\lambda}\int_{\{|f|>\lambda\}}|f|, \\ \{|{\widetilde K}(\chi_{\{|f|\leq\lambda\}}\cdot f)|>\lambda\}| \lesssim\frac{1}{\lambda^p}\int_{\{|f|\leq\lambda\}}|f|^p. \end{gather*} Since $f_1\in WL_1(\mathbb{R}^n_+)$ and $\frac{\Phi(\lambda)}{\lambda}$ increasing we have \begin{align*} \Phi(\lambda)\big|\big\{x\in\mathbb{R}^n_+:|\widetilde K f_1(x)|>\frac{\lambda}{2}\big\}\big| & \lesssim \frac{\Phi(\lambda)}{\lambda}\int_{\mathbb{R}^n_+} |f_1(x)|dx \\ &=\frac{\Phi(\lambda)}{\lambda}\int_{\{x \in \mathbb{R}^n_+ : |f(x)|>\lambda\}}|f(x)|dx \\ &\lesssim \int_{\mathbb{R}^n_+}|f(x)|\frac{\Phi(|f(x)|)}{|f(x)|}dx\\ &=\int_{\mathbb{R}^n_+} \Phi(|f(x)|)dx. \end{align*} By Lemma \ref{propsing} and $f_2\in L_p(\mathbb{R}^n_+)$ we have \begin{align*} \Phi(\lambda) \, \big|\big\{x\in\mathbb{R}^n_+ : |\widetilde K f_2(x)|>\frac{\lambda}{2}\big\}\big| & \lesssim \frac{\Phi(\lambda)}{\lambda^p}\int_{\mathbb{R}^n_+}|f_2(x)|^p dx \\ &=\frac{\Phi(\lambda)}{\lambda^p}\int_{\{x \in \mathbb{R}^n_+ : |f(x)| \leq \lambda\}} |f(x)|^p dx \\ & \lesssim \int_{\mathbb{R}^n_+}|f(x)|^p\frac{\Phi(|f(x)|)}{|f(x)|^p}dx\\ &=\int_{\mathbb{R}^n_+}\Phi(|f(x)|)dx. \end{align*} Thus we obtain $$ |\{x\in\mathbb{R}^n_+ : |\widetilde K f(x)|>\lambda\}| \leq \frac{C}{\Phi(\lambda)} \int_{\mathbb{R}^n_+} \Phi(|f(x)|)dx \leq \frac{1}{\Phi\left(\frac{\lambda}{C\|f\|_{L_\Phi}}\right)}. $$ Since $\|\cdot\|_{L_\Phi}$ norm is homogeneous this inequality is true for every $f\in L_{\Phi}(\mathbb{R}^n_+)$. Now proved that for $\Phi \in \Delta_2 \cap \nabla_2$ the nonsingular integral operator $\widetilde K$ is bounded in $L_{\Phi}(\mathbb{R}^n_+)$. As before we use distribution functions. \begin{align*} \int_{\mathbb{R}^n_+}\Phi\Big(\frac{\widetilde K f(x)}{\Lambda}\Big) dx & = \frac{1}{\Lambda} \int_0^{\infty} \varphi\big(\frac{\lambda}{\Lambda}\big)|\{x \in \mathbb{R}^n_+ : |\widetilde K f(x)| > \lambda\}| d\lambda \\ & = \frac{2}{\Lambda} \int_0^{\infty} \varphi\big(\frac{2\lambda}{\Lambda}\big) |\{x \in \mathbb{R}^n_+ : |\widetilde K f(x)| > 2\lambda\}|d\lambda. \end{align*} What is different from the estimate for the maximal operator is the point that $\widetilde K$ is not $L_{\infty}(\mathbb{R}^n_+)$ bounded. Let $p > 1$ be sufficiently large. Then \begin{align*} |\{x\in \mathbb{R}^n_+ : \widetilde K f(x)>2\lambda\}| & \leq |\{x \in \mathbb{R}^n_+ : |{\widetilde K}(\chi_{\{|f|>\lambda\}}\cdot f)(x)|>\lambda\}| \\ &\quad + |\{x \in \mathbb{R}^n_+ : |{\widetilde K}(\chi_{\{|f|\leq\lambda\}}\cdot f)(x)|>\lambda\}|. \end{align*} By the weak $(1, 1)$ boundedness and $L_p$-boundedness of $\widetilde K$ (see Theorem \ref{ChFrLTr93}) we have \begin{gather*} |\{x \in \mathbb{R}^n_+ : |{\widetilde K}(\chi_{\{|f|>\lambda\}}\cdot f)(x)|>\lambda\}| \lesssim\frac{1}{\lambda}\int_{\{x \in \mathbb{R}^n_+ : |f(x)|>\lambda\}}|f(x)| dx, \\ |\{x \in \mathbb{R}^n_+ : |{\widetilde K}(\chi_{\{|f|\leq\lambda\}}\cdot f)(x)|>\lambda\}| \lesssim\frac{1}{\lambda^p}\int_{\{x \in \mathbb{R}^n_+ : |f(x)|\leq\lambda\}}|f(x)|^p dx. \end{gather*} Using the same calculation used for the maximal operator works for the first term, \begin{equation}\label{63.44} \frac{1}{\Lambda}\int_0^\infty\varphi\Big(\frac{2\lambda}{\Lambda}\Big) \{|{\widetilde K}(\chi_{\{|f|>\lambda\}}\cdot f)|>\lambda\}|d\lambda \leq \int_{\mathbb{R}^n_+} \Phi\Big(\frac{c|f|}{\Lambda}\Big). \end{equation} For the second term a similar computation still works, but we use that $\Phi \in \Delta_2$, \begin{align*} & \frac{1}{\Lambda}\int_0^\infty\varphi\Big(\frac{2\lambda}{\Lambda}\Big) \{|{\widetilde K}(\chi_{\{|f|\leq\lambda\}}\cdot f)(x)|>\lambda\}|d\lambda \\ & \lesssim \frac{1}{\Lambda}\int_0^\infty\varphi\Big(\frac{2\lambda}{\Lambda}\Big) \Big(\int_{\{x \in \mathbb{R}^n_+ : |f(x)|\leq\lambda\}}|f(x)|^p dx\Big) \frac{d\lambda}{\lambda^p} \\ & \lesssim \frac{1}{\Lambda}\int_{\mathbb{R}^n_+}|f(x)|^p \Big(\int_{|f(x)|}^\infty\varphi\Big(\frac{2\lambda}{\Lambda}\Big) \frac{d\lambda}{\lambda^p}\Big)dx. \end{align*} Using Lemma \ref{prop62.20} (1), we have \begin{equation}\label{63.45} \begin{aligned} &\frac{1}{\Lambda}\int_0^\infty\varphi\Big(\frac{2\lambda}{\Lambda}\Big) \{|{\widetilde K}(\chi_{\{|f|\leq\lambda\}}\cdot f)(x)|>\lambda\}|d\lambda \\ &\lesssim\int_{\mathbb{R}^n_+} \Phi\Big(\frac{2|f(x)|}{\Lambda}\Big) dx \leq \int_{\mathbb{R}^n_+} \Phi\Big(\frac{c|f(x)|}{\Lambda}\Big) dx. \end{aligned} \end{equation} Thus, putting together \eqref{63.44} and \eqref{63.45}, we obtain $$ \int_{\mathbb{R}^n_+} \Phi\Big(\frac{\widetilde K f(x)}{\Lambda}\Big)dx \leq \int_{\mathbb{R}^n_+} \Phi\Big(\frac{c_0|f(x)|}{\Lambda}\Big) dx. $$ Again we shall label the constant we want to distinguish from other less important constants. As before, if we set $\Lambda = c_2\|f\|_{L_\Phi}(\mathbb{R}^n_+)$, then we obtain $$ \int_{\mathbb{R}^n_+} \Phi\Big(\frac{\widetilde K f(x)}{\Lambda}\Big) dx \leq 1. $$ Hence the operator norm of $\widetilde T$ is less than $c_2$. \end{proof} \section{Sublinear operators generated by nonsingular integral operators in the space $M_{\Phi,\varphi}(\mathbb{R}^n_+)$} \label{sec3} We use the following statement on the boundedness of the weighted Hardy operator $$ H^{\ast}_{w} g(t):=\int_t^{\infty} g(s) w(s) ds,~ \ \ 00} v_2(t) H^{\ast}_{w} g(t) \leq C \sup _{t>0} v_1(t) g(t) \end{equation} holds for some $C>0$ for all non-negative and non-decreasing $g$ on $(0,\infty)$ if and only if \begin{equation} \label{vav02} B:= \sup _{t>0} v_2(t)\int_t^{\infty} \frac{w(s) ds}{\sup _{s<\tau<\infty} v_1(\tau)}<\infty. \end{equation} Moreover, the value $C=B$ is the best constant for \eqref{vav01}. \end{theorem} \begin{remark}\label{rem2.3} \rm In \eqref{vav01} and \eqref{vav02} it is assumed that $\frac{1}{\infty}=0$ and $0 \cdot \infty=0$. \end{remark} For any $x\in \mathbb{R}^n_+$ define $\tilde x=(x',-x_n)$ and recall that $x^0=(x',0)$. Let $\widetilde{T}$ be a sublinear operator such that for any $f\in L_1(\mathbb{R}^n_+)$ with a compact support holds \begin{equation}\label{tlT} |\widetilde Tf(x)|\leq C \int_{\mathbb{R}^n_+}\frac{|f(y)|}{|\tilde x-y|^n}\,dy. \end{equation} \begin{lemma} \label{lem3.3} Let $\Phi$ any Young function, $f\in L_{\Phi}^{\rm loc}(\mathbb{R}^n_+)$, be such that \begin{equation}\label{eq3.5.nm} \int_1^{\infty} \|f\|_{L_{\Phi}({\mathcal{B}}^+(x^0,t))} \Phi^{-1}\big(t^{-n}\big) \frac{dt}{t}<\infty \end{equation} and $\widetilde T$ be a sublinear operator satisfying \eqref{tlT}. \begin{itemize} \item[(i)] If $\widetilde T$ bounded on $L_{\Phi}(\mathbb{R}^n_+)$, then \begin{equation}\label{eq3.5} \|\widetilde T f\|_{L_{\Phi}({\mathcal{B}}^+(x^0,r))} \le \frac{C}{\Phi^{-1}\big(r^{-n}\big)} \int_{2r}^{\infty} \|f\|_{L_{\Phi}({\mathcal{B}}^+(x^0,t))} \Phi^{-1}\big(t^{-n}\big) \frac{dt}{t}. \end{equation} \item[(ii)] If $\widetilde T$ bounded from $L_{\Phi}(\mathbb{R}^n_+)$ on $WL_{\Phi}(\mathbb{R}^n_+)$, then \begin{equation}\label{weak} \|\widetilde T f\|_{WL_{\Phi}({\mathcal{B}}^+(x^0,r))} \le \frac{C}{\Phi^{-1}\big(r^{-n}\big)} \int_{2r}^{\infty} \|f\|_{L_{\Phi}({\mathcal{B}}^+(x^0,t))} \Phi^{-1}\big(t^{-n}\big) \frac{dt}{t}, \end{equation} where the constants are independent of $x^0$, $r$ and $f$. \end{itemize} \end{lemma} \begin{proof} (i) Denote ${\mathcal{B}}_r^+={\mathcal{B}}^+(x^0,r)$, ${\mathcal{B}}_t^+={\mathcal{B}}^+(x^0,t)$ and for any $f\in L_{\Phi}^{\rm loc}(\mathbb{R}^n_+)$ write $f=f_1+f_2$ with $f_1=f\chi_{2{\mathcal{B}}_r^+}$ and $f_2=f\chi_{(2{\mathcal{B}}_r^+)^c}$. Because of the $(\Phi,\Phi)$-boundedness of the operator $\widetilde T$ (see Theorem \ref{fffN}) and $f_1\in L_{\Phi}(\mathbb{R}^n_+)$ we have \[ \|\widetilde T f_1\|_{L_{\Phi}({\mathcal{B}}_r^+)}\leq \|\widetilde T f_1\|_{L_{\Phi}(\mathbb{R}^n_+)}\leq C\|f_1\|_{L_{\Phi}(\mathbb{R}^n_+)}=C\|f\|_{L_{\Phi}(2{\mathcal{B}}_r^+)}. \] It is easy to see that for arbitrary points $x \in {\mathcal{B}}_r^+$ and $y\in (2{\mathcal{B}}_r^+)^c$ it holds \begin{equation}\label{xy} \frac{1}{2}|x^0-y|\le |\tilde x-y| \le \frac{3}{2}|x^0-y|. \end{equation} Applying \eqref{tlT} and the Fubuni theorem to $\widetilde T f_2$ we obtain \begin{align*} |\widetilde T f_2(x)| &\leq C \int_{\mathbb{R}^n_+}\frac{|f_2(y)|}{|\tilde x-y|^n}\, dy\\ &\leq C \int_{(2{\mathcal{B}}_r^+)^c}\frac{|f(y)|}{|x^0-y|^{n}} dy \leq C\int_{(2{\mathcal{B}}_r^+)^c}|f(y)|\int_{|x^0-y|}^\infty \frac{dt}{t^{n+1}}\\ &\leq C \int_{2r}^\infty \Big(\int_{2r\leq |x^0-y|0}\varphi_2(x^0,r)^{-1} \int_r^{\infty} \|f\|_{L_{\Phi}({\mathcal{B}}^+(x^0,t))} \Phi^{-1}\big(t^{-n}\big) \frac{dt}{t}. \] Applying the Theorem \ref{thm3.2} to the above integral with \begin{gather*} w(r)=\Phi^{-1}\big(r^{-n}\big), \quad v_2(x^0,r)=\varphi_2(x^0,r)^{-1} ,\\ v_1(x^0,r)=\varphi_1(x^0,r)^{-1} \Phi^{-1}\big(r^{-n}\big), \quad g(x^0,r)=\|f\|_{L_{\Phi}({\mathcal{B}}^+(x^0,r))},\\ H^{\ast}_{w}g(x^0,r)=\int_r^{\infty} \|f\|_{L_{\Phi}({\mathcal{B}}^+(x^0,t))} w(t) dt, \end{gather*} where condition \eqref{vav02} is equivalent to \eqref{sal21}, we obtain \[ \|\widetilde T f\|_{M_{\Phi,\varphi_2}(\mathbb{R}^n_+)} \lesssim \sup_{x\in{\mathbb{R}^n},\,r>0} \varphi_1(x^0,r)^{-1} \Phi^{-1}\big(r^{-n}\big) \|f\|_{L_{\Phi}({\mathcal{B}}^+(x^0,r))} = \|f\|_{M_{\Phi,\varphi_1}(\mathbb{R}^n_+)}. \] The case $p=1$ is treated in the same manner using \eqref{weak} and \eqref{vav02}, \begin{align*} \|\widetilde T f\|_{WM_{1,\varphi_2}(\mathbb{R}^n_+)} & \lesssim \sup_{x^0,\,r>0}\varphi_2(x^0,r)^{-1} \int_r^{\infty} \|f\|_{L_{\Phi}({\mathcal{B}}^+(x^0,t))} \Phi^{-1}\big(t^{-n}\big) \frac{dt}{t} \\ &= \sup_{x^0,\,r>0} \varphi_1(x^0,r)^{-1} \Phi^{-1}\big(r^{-n}\big) \|f\|_{L_{\Phi}({\mathcal{B}}^+(x^0,r))} \\ &= \|f\|_{M_{\Phi,\varphi_1}(\mathbb{R}^n_+)}. \end{align*} \end{proof} \section{Commutators of sublinear operators generated by nonsingular integrals in the space $M_{\Phi,\varphi}(\mathbb{R}^n_+)$} \label{sec4} For a function $a\in BMO$ and sublinear operator $\widetilde T$ satisfying \eqref{tlT} define the commutator ${\widetilde T}_{a}=[a,\widetilde T] f=a\widetilde T f-\widetilde T(af)$. Suppose that for any $f\in L_1(\mathbb{R}^n_+)$ with compact support and $x\notin \operatorname{supp} f$, it holds \begin{equation}\label{sublcomm} |{\widetilde T}_{a}f(x)|\le C\int_{\mathbb{R}^n_+} |a(x)-a(y)| \frac{|f(y)|}{ |\tilde x-y|^{n}}\,dy, \end{equation} with a constant independent of $f$ and $x$. Suppose in addition that ${\widetilde T}_a$ is bounded in $L_{\Phi}(\mathbb{R}^n_+)$ satisfying $\|{\widetilde T}_af\|_{L_{\Phi}(\mathbb{R}^n_+)}\leq C\|a\|_\ast\|f\|_{L_{\Phi}(\mathbb{R}^n_+)}$. Our aim is to show boundedness of ${\widetilde T}_a$ in $M_{\Phi,\varphi}(\mathbb{R}^n_+)$. For this goal we recall some well known properties of the $BMO$ functions. \begin{lemma}[John-Nirenberg lemma \cite{JN}] \label{lem2.4} Let $a\in BMO$ and $p\in (1,\infty)$. Then for any ball ${\mathcal{B}}$ it holds \begin{equation} \label{sov21p} \Big( \frac{1}{|{\mathcal{B}}|}\int_{{\mathcal{B}}}|a(y)-a_{{\mathcal{B}}}|^p dy\Big)^{1/p} \leq C(p) \|a\|_{*} . \end{equation} \end{lemma} \begin{definition} \label{def7.1} \rm A Young function $\Phi$ is said to be of upper type p (resp. lower type p) for some $p\in[0,\infty)$, if there exists a positive constant $C$ such that, for all $t\in[1,\infty)$(resp. $t\in[0,1]$) and $s\in[0,\infty)$, $$ \Phi(st)\le Ct^p\Phi(s). $$ \end{definition} \begin{remark}\label{remlowup} \rm We know that if $\Phi$ is lower type $p_0$ and upper type $p_1$ with $10$ such that \[ \left|b_{{\mathcal{B}}_r}-b_{{\mathcal{B}}_t}\right| \le C \|b\|_\ast \ln \frac{t}{r} \quad \text{for } 0<2r0}\Phi^{-1}\big(r^{-n}\big) \|b(\cdot)-b_{B(x,r)}\|_{L_{\Phi}(B(x,r))}. $$ \end{lemma} For the variable exponent Lebesgue space $L_{p(\cdot)}$ Lemma \ref{Bmo-orlicz} was proved in \cite{IzukiSaw}. For a Young function $\Phi$, let $$ a_{\Phi}:=\inf_{t\in(0,\infty)}\frac{t\Phi^{\prime}(t)}{\Phi(t)}, \quad b_{\Phi}:=\sup_{t\in(0,\infty)}\frac{t\Phi^{\prime}(t)}{\Phi(t)}. $$ \begin{remark}\label{indorl} \rm It is known that $\Phi\in \Delta_2\cap\nabla_2$ if and only if $10$ holds \begin{equation} \label{sov21} \int_1^{\infty} \Big(1+\ln \frac{t}{r}\Big) \|f\|_{L_{\Phi}({\mathcal{B}}^+_t(x^0,t))} \Phi^{-1}\big(t^{-n}\big) \frac{dt}{t}<\infty. \end{equation} Then $$ \|{\widetilde T}_{a} f\|_{L_{\Phi}({\mathcal{B}}^+_r)} \lesssim \frac{\|a\|_{*}}{\Phi^{-1}\big(r^{-n}\big)} \int_{2r}^{\infty} \Big(1+\ln \frac{t}{r}\Big) \|f\|_{L_{\Phi}({\mathcal{B}}^+(x^0,t))}\, \Phi^{-1}\big(t^{-n}\big) \frac{dt}{t}. $$ \end{lemma} \begin{proof} Decompose $f$ as $f= f\chi_{2{\mathcal{B}}_r^+}+ f\chi_{(2{\mathcal{B}}_r^+)^c}= f_1+f_2$. From the boundedness of ${\widetilde T}_{a}$ in $L_{\Phi}(\mathbb{R}^n_+)$ it follows that $$ \|{\widetilde T}_{a} f_1\|_{L_{\Phi}({\mathcal{B}}_r^+)} \leq \|{\widetilde T}_{a} f_1\|_{L_{\Phi}(\mathbb{R}^n_+)} \lesssim \|a\|_\ast \|f_1\|_{L_{\Phi}(\mathbb{R}^n_+)} = \|a\|_{*} \|f\|_{L_{\Phi}(2{\mathcal{B}}_r^+)}. $$ On the other hand, because of \eqref{xy}, we can write \begin{align*} \|{\widetilde T}_{a} f_2\|_{L_{\Phi}({\mathcal{B}}_r^+)} & \lesssim \Big(\int_{{\mathcal{B}}_r^+}\Big(\int_{(2{\mathcal{B}}_r^+)^c} \frac{|a(x)-a(y)||f(y)|}{|x^0-y|^{n}}dy\Big)^pdx\Big)^{1/p} \\ & \lesssim \Big(\int_{{\mathcal{B}}_r^+}\Big(\int_{(2{\mathcal{B}}_r^+)^c}\frac{|a(y) -a_{{\mathcal{B}}_r^+}||f(y)|}{|x^0-y|^{n}}dy\Big)^pdx\Big)^{1/p} \\ &\quad + \Big(\int_{{\mathcal{B}}_r^+}\Big(\int_{(2{\mathcal{B}}_r^+)^c}\frac{|a(x) -a_{{\mathcal{B}}_r^+}||f(y)|}{|x^0-y|^{n}}dy\Big)^pdx\Big)^{1/p}=I_1+I_2. \end{align*} We estimate $I_1$ as follows \begin{align*} I_1 & \lesssim \frac{1}{\Phi^{-1}\big(r^{-n}\big)} \int_{(2{\mathcal{B}}_r^+)^c}\frac{|a(y)-a_{{\mathcal{B}}_r^+}||f(y)|}{|x^0-y|^{n}}\,dy\\ &= \frac{1}{\Phi^{-1}\big(r^{-n}\big)} \int_{(2{\mathcal{B}}_r^+)^c}|a(y)-a_{{\mathcal{B}}_r^+}||f(y)| \int_{|x^0-y|}^{\infty}\frac{dt}{t^{n+1}}\,dy\\ &= \frac{1}{\Phi^{-1}\big(r^{-n}\big)} \int_{2r}^{\infty}\int_{2r\leq |x^0-y|\leq t} |a(y)-a_{{\mathcal{B}}_r^+}|\, |f(y)|dy\,\frac{dt}{t^{n+1}}\\ & \lesssim \frac{1}{\Phi^{-1}\big(r^{-n}\big)} \int_{2r}^{\infty} \int_{{\mathcal{B}}_t^+}|a(y)-a_{{\mathcal{B}}_r^+}||f(y)|dy\,\frac{dt}{t^{n+1}}. \end{align*} Applying H\"older's inequality, Lemma~\ref{lem2.4} and \eqref{propBMO}, we obtain \begin{align*} I_1 &\lesssim \Big( \frac{1}{\Phi^{-1}\big(r^{-n}\big)} \int_{2r}^{\infty}\int_{{\mathcal{B}}_t^+} |a(y)-a_{{\mathcal{B}}_t^+}||f(y)|dy\,\frac{dt}{t^{n+1}} \\ &\quad + \frac{1}{\Phi^{-1}\big(r^{-n}\big)} \int_{2r}^{\infty}|a_{{\mathcal{B}}_r^+}-a_{{\mathcal{B}}_t^+}| \int_{{\mathcal{B}}_t^+} |f(y)|dy\,\frac{dt}{t^{n+1}}\Big) \\ &\lesssim \Big( \frac{1}{\Phi^{-1}\big(r^{-n}\big)} \int_{2r}^{\infty} \left\|a(\cdot)-a_{{\mathcal{B}}_t^+}\right\|_{L_{\widetilde{\Phi}}({\mathcal{B}}_t^+)} \|f\|_{L_{\Phi}({\mathcal{B}}_t^+)}\,\frac{dt}{t^{n+1}} \\ &\quad + \frac{1}{\Phi^{-1}\big(r^{-n}\big)} \int_{2r}^{\infty}|a_{{\mathcal{B}}_r^+}-a_{{\mathcal{B}}_t^+}| \|f\|_{L_{\Phi}({\mathcal{B}}_t^+)}\, \Phi^{-1}\big(t^{-n}\big) \frac{dt}{t} \Big) \\ & \lesssim \|a\|_{*}\,\frac{1}{\Phi^{-1}\big(r^{-n}\big)} \int_{2r}^{\infty}\Big(1+\ln \frac{t}{r}\Big) \|f\|_{L_{\Phi}({\mathcal{B}}_t^+)}\,\Phi^{-1}\big(t^{-n}\big) \frac{dt}{t}. \end{align*} To estimate $I_2$ note that \[ I_2 = \|a(\cdot)-a_{{\mathcal{B}}_r^+}\|_{L_{\Phi}({\mathcal{B}}_r^+)} \int_{(2{\mathcal{B}}_r^+)^c} \frac{|f(y)|}{|x^0-y|^{n}}dy. \] By Lemma \ref{lem2.4} and \eqref{sal00} we obtain \[ I_2 \lesssim \frac{\|a\|_{*}}{\Phi^{-1}\big(r^{-n}\big)} \int_{(2{\mathcal{B}}_r^+)^c}\frac{|f(y)|}{|x^0-y|^{n}}dy \lesssim \frac{\|a\|_{*}}{\Phi^{-1}\big(r^{-n}\big)} \int_{2r}^{\infty}\|f\|_{L_{\Phi}({\mathcal{B}}_t^+)} \Phi^{-1}\big(t^{-n}\big) \frac{dt}{t}. \] Summing $I_1$ and $I_2$ we obtain that for all $p \in (1,\infty)$, $$ \|{\widetilde T}_{a} f_2\|_{L_{\Phi}({\mathcal{B}}_r^+)} \lesssim \frac{\|a\|_{*}}{\Phi^{-1}\big(r^{-n}\big)} \int_{2r}^{\infty} \Big(1+\ln \frac{t}{r}\Big) \|f\|_{L_{\Phi}({\mathcal{B}}_t^+)}\,\Phi^{-1}\big(t^{-n}\big) \frac{dt}{t}. $$ Finally, $$ \|{\widetilde T}_{a} f\|_{L_{\Phi}({\mathcal{B}}_r^+)} \lesssim \|a\|_{*}\,\|f\|_{L_{\Phi}(2{\mathcal{B}}_r^+)}+ \frac{\|a\|_{*}}{\Phi^{-1}\big(r^{-n}\big)} \int_{2r}^{\infty} \Big(1+\ln \frac{t}{r}\Big) \|f\|_{L_{\Phi}({\mathcal{B}}_t^+)}\,\Phi^{-1}\big(t^{-n}\big) \frac{dt}{t} $$ and the statement follows by \eqref{sal01}. \end{proof} \begin{theorem} \label{theor3.3F} Let $\Phi$ be a Young function with $\Phi\in \Delta_2\cap\nabla_2$, $a \in BMO$ and $\varphi_1,\varphi_2:\mathbb{R}^n\times\mathbb{R}_+\to \mathbb{R}_+$ be measurable functions satisfying \eqref{ComCond}. Suppose ${\widetilde T}_{a}$ be a sublinear operator bounded on $L_{\Phi}(\mathbb{R}^n_+)$ and satisfying \eqref{sublcomm}. Then ${\widetilde T}_{a}$ is bounded from $M_{\Phi,\varphi_1}(\mathbb{R}^n_+)$ to $M_{\Phi,\varphi_2}(\mathbb{R}^n_+)$ and \begin{equation}\label{normTaf} \|{\widetilde T}_{a} f\|_{M_{\Phi,\varphi_2}(\mathbb{R}^n_+)} \leq C \|a\|_{*} \|f\|_{M_{\Phi,\varphi_1}(\mathbb{R}^n_+)} \end{equation} with a constant independent of $f$. \end{theorem} The statement of the theorem follows by Lemma~\ref{lem5.1} and Theorem~\ref{thm3.2} in the same manner as the proof of Theorem~\ref{3.4}. \section{Singular and nonsingular integral operators in the spaces $M_{\Phi,\varphi}$} \label{sec5} In this section we deal with Calder\'on-Zygmund type integrals and their commutators with $BMO$ functions. We start with the definition of the corresponding kernel. \begin{definition}\label{CZK} \rm A measurable function $\mathcal{K}(x,\xi):{\mathbb{R}^n}\times{\mathbb{R}^n}\setminus\{0\}\to \mathbb{R}$ is called a variable Calder\'on-Zygmund kernel if: \begin{itemize} \item[(i)] $\mathcal{K}(x,\cdot)$ is a Calder\'on-Zygmund kernel for almost all $x\in{\mathbb{R}^n}$: \begin{itemize} \item[(a)] $\mathcal{K}(x,\cdot)\in C^\infty({\mathbb{R}^n}\setminus\{0\})$, \item[(b)] $\mathcal{K}(x,\mu\xi)=\mu^{-n}\mathcal{K}(x,\xi)$ for all $\mu>0$, \item[(c)] $\int_{{\mathbb{S}}^{n-1}}\mathcal{K}(x,\xi)d\sigma_\xi=0$, $\int_{{\mathbb{S}}^{n-1}}|\mathcal{K}(x,\xi)|d\sigma_\xi<+\infty$, \end{itemize} \item[(ii)] $\max_{|\beta|\leq 2n}\|D^\beta_\xi \mathcal{K}(x,\xi) \|_{L_\infty({\mathbb{R}^n}\times{\mathbb{S}}^{n-1})}=M<\infty$ independently of $x$. \end{itemize} \end{definition} The singular integrals \begin{gather*} \mathfrak{K} f(x)= \operatorname{P.V.}\int_{{\mathbb{R}^n}}\mathcal{K}(x,x-y)f(y)dy,\\ \begin{aligned} \mathfrak{C}[a, f](x)&= \operatorname{P.V.}\int_{{\mathbb{R}^n}}\mathcal{K}(x,x-y)f(y)[a(x)-a(y)]dy \\ &=a(x)\mathfrak{K} f(x)-\mathfrak{K}(af)(x) \end{aligned} \end{gather*} are bounded in $L_{\Phi}({\mathbb{R}^n})$ (see \cite{Nakai2}), moreover $$ |\mathcal{K}(x,\xi)|\leq |\xi|^{-n}\big|\mathcal{K}\big(x,\frac{\xi}{|\xi|}\big)\big| \leq M|\xi|^{-n} $$ which implies $$ |\mathfrak{K} f(x)|\leq C \int_{{\mathbb{R}^n}}\frac{|f(y)|}{|x-y|^n}\,dy,\quad |\mathfrak{C}[a, f](x)|\leq C \int_{{\mathbb{R}^n}} \frac{|a(x)-a(y)||f(y)|}{|x-y|^n}\,dy $$ and hence the validity of all results from \S~\ref{sec2a}. Let us note that any measurable function $\varphi:{\mathbb{R}^n}\times\mathbb{R}_+\to\mathbb{R}_+$ satisfying the condition \eqref{ComCond} satisfies also \eqref{sal21} with $\varphi_1\equiv\varphi_2\equiv\varphi$. Hence the following results hold as a simple application of the estimates from $\S~\ref{sec2a}$. \begin{theorem}\label{CZcont} Let $\Phi$ be a Young function with $\Phi\in \Delta_2\cap\nabla_2$ and $\varphi:{\mathbb{R}^n}\times\mathbb{R}_+\to\mathbb{R}_+$ be measurable function such that for all $x\in{\mathbb{R}^n}$ and $r>0$ \begin{equation}\label{weight} \int_{r}^{\infty} \Big(1+\ln \frac{t}{r}\Big) \Big(\operatorname{ess\,inf}_{t0$ there exists a positive number $\rho_0=\rho_0(\varepsilon,\gamma_a)$ such that for any ball ${\mathcal{B}}_r$ with a radius $r\in(0,\rho_0)$ and all $f\in M_{\Phi,\varphi}({\mathcal{B}}_r)$ holds \begin{equation}\label{normB} \|\mathfrak{C}[a,f]\|_{M_{\Phi,\varphi}({\mathcal{B}}_r^+)} \leq C\varepsilon\|f\|_{M_{\Phi,\varphi}({\mathcal{B}}_r^+)}, \end{equation} with $C=C(n,p,\varphi,\Omega, \mathcal{K})$. \end{corollary} To obtain the above estimates it suffices to extend $\mathcal{K}(x,\cdot)$ and $f(\cdot)$ as zero outside $\Omega$ (see \cite[Theorem 2.11]{ChFraL1} for details). Recall that the extension of $a$ keeps its $BMO $ norm or $VMO$-modulus according to \cite{A,Jones}. For any $x,y\in {\mathbb R}^n_+$, $\tilde x=(x',-x_n)$ define the {\it generalized reflection} ${\mathcal{T}}(x;y)$ as $$ {\mathcal{T}}(x;y) =x-2x_n\frac{{\bf a}^n(y)}{a^{nn}(y)}\quad {\mathcal{T}}(x)={\mathcal{T}}(x;x):{\mathbb{R}}^n_+\to {\mathbb{R}}^n_-, $$ where ${\bf a}^n$ is the last row of the coefficients matrix ${\bf a}$. Then there exist positive constants $C_1, C_2$ depending on $n$ and $\Lambda$, such that $$ C_1|\widetilde x - y| \leq |{\mathcal{T}}(x)-y| \leq C_2 |\widetilde x -y|,\quad \forall x,y\in{\mathbb{R}}_+^{n}. $$ For any $f\in M_{\Phi,\varphi}({\mathbb{R}}^{n}_+)$ and $a\in BMO$ consider the nonsingular integral operators $$ \widetilde {\mathfrak{K}} f(x)=\int_{{\mathbb{R}}^{n}_+} \mathcal{K} (x,{\mathcal{T}}(x)-y)f(y) dy,\quad \widetilde {\mathfrak{C}} [a,f](x)= a(x)\mathfrak{K} f(x) -\mathfrak{K}(af)(x). $$ The kernel $\mathcal{K}(x,{\mathcal{T}}(x)-y):\mathbb{R}^n\times\mathbb{R}^n_+\to\mathbb{R}$ is not singular and verifies the conditions (i)(b) and (ii) from Definition~\ref{CZK}. Moreover $$ |\mathcal{K}(x, \mathcal{T}(x)-y)|\leq M|\mathcal{T}(x)-y|^{-n}\leq C|\tilde x-y|^{-n}, $$ which implies $$ |\widetilde{\mathfrak{K}} f(x)| \leq C\int_{\mathbb{R}^n_+} \frac{|f(y)|}{|\tilde x-y|}\,dy,\quad |\widetilde{\mathfrak{C}}[a, f](x)| \leq C\int_{\mathbb{R}^n_+} |a(x)-a(y)| \frac{|f(y)|}{|\tilde x-y|}\,dy. $$ The following estimates are simple consequence of the results in \S~\ref{sec3} and \S~\ref{sec4}. \begin{theorem}\label{nonsing} Let $\Phi$ be a Young function with $\Phi\in \Delta_2\cap\nabla_2$, $a \in BMO(\mathbb{R}^n_+)$ and $\varphi$ be measurable function satisfying \eqref{weight}. Then the operators $\widetilde{\mathfrak{K}} f$ and $\widetilde{\mathfrak{C}}[a, f]$ are continuous in $M_{\Phi,\varphi}$ and for all $f\in M_{\Phi,\varphi}(\mathbb{R}^n_+) $ holds \begin{equation} \label{KC} \begin{gathered} \|\widetilde{\mathfrak{K}} f\|_{M_{\Phi,\varphi}(\mathbb{R}^n_+)} \leq C \|f\|_{M_{\Phi,\varphi}(\mathbb{R}^n_+)}, \\ \|\widetilde{\mathfrak{C}}[a, f]\|_{M_{\Phi,\varphi}(\mathbb{R}^n_+)} \leq C \|a\|_\ast \|f\|_{M_{\Phi,\varphi}(\mathbb{R}^n_+)} \end{gathered} \end{equation} with a constant dependent on known quantities only. \end{theorem} \begin{corollary} \label{localnonsing} Let $\Phi$ and $\varphi$ be as in Theorem~\ref{nonsing} and $a\in VMO$ with a $VMO$-modulus $\gamma_a$. Then for any $\varepsilon>0$ there exists a positive number $\rho_0=\rho_0(\varepsilon,\gamma_a)$ such that for any ball ${\mathcal{B}}_r^+$ with a radius $r\in(0,\rho_0)$ and all $f\in M_{\Phi,\varphi}({\mathcal{B}}_r^+)$ holds \begin{equation}\label{tlKl} \|\widetilde{\mathfrak{C}}[a,f]\|_{M_{\Phi,\varphi}({\mathcal{B}}^+_r)} \leq C\varepsilon\|f\|_{M_{\Phi,\varphi}({\mathcal{B}}^+_r)}, \end{equation} where $C$ is independent of $\varepsilon$, $f$ and $r$. \end{corollary} The proof of the above corollary is as that of \cite[Theorem 2.13]{ChFraL1}. \section{Dirichlet problem}\label{s5} We consider the Dirichlet problem for second order linear equations \begin{equation} \label{sal11} \begin{gathered} \mathcal{L} u := a^{ij}(x)D_{ij} u=f(x) \quad \text{a.a. } x\in \Omega,\\ u\in\ W_{2,\Phi,\varphi}(\Omega) \cap W^0_{1,\Phi}(\Omega) \end{gathered} \end{equation} subject to the following conditions: \begin{itemize} \item[(H1)] Uniform ellipticity of $\mathcal{L}$: there exists a constant $\Lambda>0$, such that \begin{gather*} \Lambda^{-1}|\xi|^2\leq a^{ij}(x)\xi_i\xi_j\leq\Lambda|\xi|^2 \quad \text{a.a. } x\in\Omega,\; \forall \xi\in \mathbb{R}^n\\ a^{ij}(x)=a^{ji}(x) \quad 1\leq i,j\leq n. \end{gather*} This assumption implies immediately essential boundedness of the coefficients $a^{ij}\in L_{\infty}(\Omega)$. \item[(H2)] Regularity of the data: $a^{ij}\in VMO(\Omega)$ and $f\in M_{\Phi,\varphi}(\Omega)$ with $10$ there exists $r_0(\varepsilon)$ such that for any $r\theta$ and $s=0,1,2$ we have $$ \eta(x)=\begin{cases} 1 & x\in {{\mathcal{B}}}_{\theta r}\\ 0 & x\not\in {{\mathcal{B}}}_{\theta'r} \end{cases} \quad \eta(x)\in C_0^\infty({{\mathcal{B}}}_r),\quad |D^s\eta|\leq C [\theta(1-\theta)r]^{-s}. $$ Applying \eqref{22} to $v(x)=\eta(x) u(x)\in W_{2,\Phi,\varphi}({\mathcal{B}}_r)\cap W_{1,\Phi}^0({{\mathcal{B}}}_r)$ we obtain \begin{align*} \|D^2u\|_{\Phi,\varphi;{\theta r}} & \leq C\|\mathcal{L} v \|_{\Phi,\varphi;{\theta' r}}\\ & \leq C\Big(\|\mathcal{L} u\|_{\Phi,\varphi;{\theta' r}} +\frac{\|Du\|_{\Phi,\varphi;{\theta' r}}}{\theta(1-\theta)r} +\frac{\|u\|_{\Phi,\varphi;{\theta' r}}}{[\theta(1-\theta)r]^2} \Big). \end{align*} Define the weighted semi-norm $$ \Theta_s = \sup_{0<\theta<1} \big[\theta(1-\theta)r \big]^s \|D^s u \|_{\Phi,\varphi;\theta r}, \quad s=0,1,2. $$ Because of the choice of $\theta'$ we have $\theta(1-\theta)\leq 2\theta'(1-\theta')$. Thus, after standard transformations and taking the supremum with respect to $\theta\in(0,1)$ the last inequality rewrites as \begin{equation}\label{theta} \Theta_2 \leq C \left(r^2\|\mathcal{L} u\|_{\Phi,\varphi;r} +\Theta_1+\Theta_0 \right)\,. \end{equation} \begin{lemma}[Interpolation inequality]\label{interpolation} There exists a constant $C$ independent of $r$ such that $$ \Theta_1\leq \varepsilon \Theta_2+\frac{C}{\varepsilon}\Theta_0\quad \text{for any } \varepsilon\in(0,2). $$ \end{lemma} \begin{proof} By simple scaling arguments we obtain in $M_{\Phi,\varphi}({\mathbb{R}}^n)$ an interpolation inequality analogous to \cite[Theorem 7.28]{GT} $$ \|D u\|_{\Phi,\varphi;r}\leq \delta\|D^2 u\|_{\Phi,\varphi;r} +\frac{C}{\delta}\|u\|_{\Phi,\varphi;r}, \quad \delta\in(0,r)\,. $$ We can always find some $\theta_0\in(0,1)$ such that \begin{align*} \Theta_1 &\leq 2[\theta_0(1-\theta_0)r] \|Du\|_{\Phi,\varphi;\theta_0r}\\ &\leq 2[\theta_0(1-\theta_0)r]\Big( \delta \|D^2 u\|_{\Phi,\varphi;\theta_0r} +\frac{C}{\delta}\|u\|_{\Phi,\varphi;\theta_0r} \Big)\,. \end{align*} The assertion follows choosing $\delta =\frac\varepsilon2[\theta_0(1-\theta_0)r]<\theta_0r$ for any $\varepsilon\in(0,2)$. \end{proof} Interpolating $\Theta_1$ in \eqref{theta}, we obtain $$ \frac{r^2}{4}\|D^2u\|_{\Phi,\varphi;r/2}\leq \Theta_2 \leq C\left( r^2\|\mathcal{L} u\|_{\Phi,\varphi;r}+ \|u\|_{\Phi,\varphi;r} \right) $$ and hence the Caccioppoli-type estimate \begin{equation}\label{locest} \|D^2u\|_{\Phi,\varphi; r/2} \leq C\Big( \|\mathcal{L} u\|_{\Phi,\varphi;r}+ \frac1{r^2}\|u\|_{\Phi,\varphi;r} \Big). \end{equation} Let ${\mathbf{v}}=\{v_{ij}\}_{ij=1}^n\in[ L_{\Phi,\omega}({{\mathcal{B}}}_r)]^{n^2}$ be arbitrary function matrix. Define the operators $$ {\mathcal{S}}_{ijhk}(v_{hk})(x) = {\mathfrak{C}}_ {ij}[a^{hk},v_{hk}](x) \quad i,j,h,k=1,\ldots, n. $$ Because of the $VMO$ properties of $a^{ij}$'s we can choose $r$ so small that \begin{equation}\label{S} \sum_{i,j,h,k=1}^n \|{\mathcal{S}}_{ijhk}\|<1. \end{equation} Now for a given $u\in W_{2,\Phi}({{\mathcal{B}}}_r)\cap W_{1,\Phi}^0({{\mathcal{B}}}_r)$ with $\mathcal{L}u\in M_{\Phi,\varphi}({{\mathcal{B}}}_r)$ define $$ {{\mathcal{H}}}_{ij}(x)={\mathfrak{K}}_{ij}\mathcal{L}u(x) +\mathcal{L}u(x)\int_{{{\mathbb{S}}}^{n-1}} \Gamma_j(x;y)y_id\sigma_y $$ and \eqref{normO} implies ${{\mathcal{H}}}_{ij}\in M_{\Phi,\varphi}({{\mathcal{B}}}_r)$. Define the operator ${\mathcal{W}}$ by the setting $$ {\mathcal{W}}{\mathbf{v}}= \Big\{ \sum_{h,k=1}^n\big( {\mathcal{S}}_{ijhk}v_{hk}+ {{\mathcal{H}}}_{ij}(x) \big) \Big\}_{ij=1}^n : \big [M_{\Phi,\varphi}({{\mathcal{B}}}_r) \big]^{n^2}\to \big [M_{\Phi,\varphi}({{\mathcal{B}}}_r) \big]^{n^2}. $$ By \eqref{S} the operator ${\mathcal{W}}$ is a contraction mapping and there exists a unique fixed point $\widetilde{\mathbf{v}}=\{\widetilde v_{ij}\}_{ij=1}^n\in [M_{\Phi,\varphi}({{\mathcal{B}}}_r)]^{n^2}$ of ${\mathcal{W}}$ such that $\mathcal{W} \widetilde{\mathbf{v}}=\widetilde{\mathbf{v}}$. On the other hand it follows from the representation formula \eqref{IF} that also $D^2u=\{D_{ij}u\}_{ij=1}^n$ is a fixed point of ${\mathcal{W}}$. Hence $D^2u\equiv \widetilde{\mathbf{v}}$, that is $D_{ij}u\in M_{\Phi,\varphi}({{\mathcal{B}}}_r)$ and in addition \eqref{locest} holds. The interior estimate \eqref{upl} follows from \eqref{locest} by a finite covering of $\Omega'$ with balls ${{\mathcal{B}}}_{r/2}$, $r<\operatorname{dist} (\Omega',\partial\Omega'')$. \end{proof} To prove a local boundary estimate for the norm of $D_{ij}u$ we define the space $W_{2,\Phi}^{\gamma_0}({{\mathcal{B}}}^+_r)$ as a closure of $C_{\gamma_0}=\{u\in C_0^\infty({{\mathcal{B}}}(x^0,r)): u(x)=0 \text{ for } x_n\leq 0 \}$ with respect to the norm of $W_{2,p}$. \begin{theorem}[Boundary estimate]\label{th7} Let $u\in W_{2,\Phi}^{\gamma_0}({{\mathcal{B}}}_r^+)$ and suppose that $\mathcal{L} u\in M_{\Phi,\varphi}({{\mathcal{B}}}_r^+)$ with $\Phi\in \Delta_2\cap\nabla_2$ and $\varphi$ satisfying \eqref{weight}. Then $D_{ij}u\in M_{\Phi,\varphi}({{\mathcal{B}}}_r^+)$ and for each $\varepsilon>0$ there exists $r_0(\varepsilon)$ such that \begin{equation}\label{bdrest} \|D_{ij}u\|_{\Phi,\varphi;{{\mathcal{B}}}_r^+} \leq C\|\mathcal{L} u\|_{\Phi,\varphi;{{\mathcal{B}}}^+_r}, \quad \forall r\in(0,r_0). \end{equation} \end{theorem} \begin{proof} For $u\in W_{2,\Phi}^{\gamma_0}({{\mathcal{B}}}_r^+)$ the boundary representation formula holds (see \cite{ChFraL2}) \begin{equation} \label{bdrep} \begin{aligned} D_{ij}u(x)&= \operatorname{P.V.} \int_{{{\mathcal{B}}}_r^+}\Gamma_{ij}(x,x-y)\mathcal{L} u(y)dy\\ &\quad + \operatorname{P.V.} \int_{{{\mathcal{B}}}_r^+}\Gamma_{ij}(x,x-y) \big[a^{hk}(x)-a^{hk}(y)\big]D_{hk} u(y)dy\\ &\quad + \mathcal{L} u(x)\int_{{{\mathbb{S}}}^{n-1}}\Gamma_j(x,y)y_i d\sigma_y +I_{ij}(x), \quad \forall i,j=1,\ldots,n, \end{aligned} \end{equation} where we have set \begin{gather*} \begin{aligned} I_{ij}(x) &= \int_{{{\mathcal{B}}}_r^+}\Gamma_{ij}(x,{\mathcal{T}}(x)-y)\mathcal{L} u(y)dy\\ &\quad + \int_{{{\mathcal{B}}}_r^+}\Gamma_{ij}(x,{\mathcal{T}}(x)-y)\big[a^{hk}(x)-a^{hk}(y)\big] D_{hk} u(y)dy\\ \quad\forall i,j=1,\ldots,n-1, \end{aligned}\\ \begin{aligned} I_{in}(x) &= I_{ni}(x)\\ &= \int_{{{\mathcal{B}}}_r^+} \Gamma_{il}(x,{\mathcal{T}}(x)-y)(D_n{\mathcal{T}}(x))^l\\ &\quad \times\left\{\big[a^{hk}(x)-a^{hk}(y) \big]D_{hk}u(y)+\mathcal{L} u(y) \right\}dy, \quad \forall i=1,\ldots, n-1, \end{aligned}\\ \begin{aligned} I_{nn}(x) &= \int_{{{\mathcal{B}}}_r^+} \Gamma_{ls}(x,{\mathcal{T}}(x)-y)(D_n{\mathcal{T}}(x))^l (D_n{\mathcal{T}}(x))^s \\ &\quad \times \left\{\big[a^{hk}(x)-a^{hk}(y) \big]D_{hk}u(y)+\mathcal{L} u(y)\right\}dy, \end{aligned} \end{gather*} where $D_n{\mathcal{T}}(x)=\left( (D_n{\mathcal{T}}(x))^1,\ldots,(D_n{\mathcal{T}}(x))^n \right) ={\mathcal{T}}(e_n,x)$. Applying estimates \eqref{KC} and \eqref{tlKl}, taking into account the $VMO$ properties of the coefficients $a^{ij}$'s, it is possible to choose $r_0$ so small that $$ \|D_{ij}u\|_{p,\varphi;{{\mathcal{B}}}_r^+}\leq C \|\mathcal{L} u\|_{p,\varphi; {{\mathcal{B}}}_r^+}\quad \text{for each } r