We consider the integral functional of Hörmander's vector fields

$ \begin{equation}\label{eq1.1} I(u) = \int_\Omega {f(x, Xu(x))} dx, x \in \Omega , \end{equation} $

(1.1)

where $\Omega \subset {\mathbb{R}^n}(n\geq 3)$ is a bounded open set, $X = \left\{ {{X_1}, \cdots, {X_m}} \right\}$ $\left({m \geq n} \right)$ are ${C^\infty }$ vector fields in $\Omega $ satisfying the Hörmander's finite rank condition[11], rank Lie$\left[{{X_1}, \cdots, {X_m}} \right] = n, $ where ${X_j} = \sum\limits_{i = 1}^n {{b_{ij}}(x)\frac{\partial } {{\partial {x_i}}}}, {b_{ij}}(x) \in {C^\infty }(\Omega), j = {\text{1}}, {\text{ }} \cdots, m$. Note that, when $f\left({x, z} \right)$ in (1.1) is a Carathéodory function and satisfies the standard growth condition ${\left| z \right|^p} \leq f\left({x, z} \right) \leq c\left({1 + {{\left| z \right|}^p}} \right), 1 < p < Q, $ where $c$ is a positive constant, $Q$ is homogeneous dimension of $\Omega $ relative to $\left\{ {{X_1}, \cdots, {X_m}} \right\}$ and $Q \geq n$, Xu[16] proved the existence of minimizers of (1.1) by direct method and obtained Hölder continuity by Moser's method. Furthermore, Xu[17] obtained ${C^\infty }$ continuity by similar method. Afterwards, Giannetti[7] obtained higher integrability of the minimizers of (1.1) under the growth condition

$ \left| {f(\xi ) - f(\eta )} \right| \leq c\left| {\xi - \eta } \right|\left( {{{\left| \xi \right|}^{p - 1}} + {{\left| {\xi - \eta } \right|}^{p - 1}}} \right), 1 < p < Q. $

Namely, he proved $u \in W_{loc}^{1, p}(\Omega)$ if $u \in W_{loc}^{1, r}(\Omega), {r_{\text{1}}} \leq r < p, {\text{max}}\{ {\text{1}}, p - {\text{1}}\} < {r_{\text{1}}} < p.$ When $m = n$, $Xu = \left({\frac{{\partial u}} {{\partial {x_1}}}, \cdots, \frac{{\partial u}} {{\partial {x_n}}}} \right)$, (1.1) is an integral function in Euclidean spaces, the study on the regularity and boundedness of minimizers of such integral function can also be seen in [4, 8, 10].

In this paper we assume that $f\left({x, z} \right)$ in (1.1) is a Carathéodory function and satisfies the standard growth condition

$ \begin{equation}\label{eq1.2} {c_1}{\left| z \right|^p} - {g_1}(x) \leq f\left( {x, z} \right) \leq {c_2}{\left| z \right|^p} + {g_2}(x), 1 < p < Q, \end{equation} $

(1.2)

where ${c_1}$ and ${c_2}$ are positive constants, functions ${g_1}(x), {g_2}(x) \in {L^{\frac{1} {{1 - b}}}}(\Omega), b = \frac{{t - p}} {t}, 1 < p < t \leq q\leq Q$. By virtue of the Sobolev inequality (see Lemma 2.5) related to Hörmander's vector fields and the iteration formula of Stampacchia (see Lemma 2.6), it is proved that the minimizers of integral functional have higher integrability with the boundary data allowing the higher integrability. Moreover, the ${L^1 }(\Omega)$ and ${L^\infty }(\Omega)$ boundedness of minimizers are also given, which extends the results of Leonetti and Siepe[12] and Leonetti and Petricca[13] from Euclidean spaces to Hörmander's vector fields. Throughout this paper $c$ denotes the different positive constants in different places.

Definition 2.1[3, 6] For any $1 < p < Q$, Sobolev space ${W^{{\text{1}}, p}}\left(\Omega \right)$ is defined by

$ {W^{{\text{1}}, p}}\left( \Omega \right) = \left\{ {u \in {L^p}\left( \Omega \right), {X_j}u \in {L^p}\left( \Omega \right), j = {\text{1}}, {\text{ }} \cdots , m} \right\}, $

whose norm is ${\left\| u \right\|_{{W^{{\text{1}}, p}}\left(\Omega \right)}} = {\left({\int_\Omega {\left({{{\left| u \right|}^p} + {{\left| {Xu} \right|}^p}} \right)dx} } \right)^{\frac{1} {p}}}, $ where $\left| {Xu} \right| = {\left({\sum\limits_{j = 1}^m {{{\left({{X_j}u} \right)}^2}} } \right)^{\frac{1} {2}}}.$ We also denote the closure of $C_0^\infty (\Omega)$ in ${W^{{\text{1}}, p}}\left(\Omega \right)$ by $W_0^{1, p}(\Omega)$, whose norm is ${\left\| u \right\|_{W_0^{1, p}(\Omega)}} = {\left({\int_\Omega {{{\left| {Xu} \right|}^p}dx} } \right)^{\frac{1} {p}}}.$

Definition 2.2[1, 2, 9] The weak ${L^p}(\Omega)$ space which is also known as the Marcinkiewicz space, denoted by $L_{weak}^p(\Omega), $ is the set of all measurable functions $f(x)$ satisfying

$\begin{equation}\label{eq1.3} meas\left\{ {x \in \Omega :\left| {f(x)} \right| > {t_0}} \right\} \leq \frac{c} {{{t_0}^p}} \end{equation} $

(2.1)

for any ${t_0} > 0$ and some positive constants $c = c\left(f \right)$, where $meas{\kern 1pt} {\kern 1pt} E{\kern 1pt} $ denotes the $n$-dimensional Lebesgue measure of $E \subset {\mathbb{R}^n}$. If $f \in L_{weak}^p(\Omega)$, then $f \in {L^{{q_0}}}(\Omega)$ for any $1 \leq {q_0} < p$. Moreover, when $p = \infty $, it follows that $L_{weak}^\infty (\Omega) = {L^\infty }(\Omega)$.

Definition 2.3 A function $u \in {u_ * } + W_0^{1, p}(\Omega)$ is called a minimizer of (1.1) with the condition (1.2) if

$ \begin{equation}\label{eq1.4} \int_\Omega {f(x, Xu)dx} \leq \int_\Omega {f(x, Xw)dx} \end{equation} $

(2.2)

for any $w \in {u_ * } + W_0^{1, p}(\Omega)$, where ${u_ * }$ denotes the boundary data.

In this paper, our mian results are sated as follows.

Theorem 2.4 Let $u \in {u_ * } + W_0^{1, p}(\Omega)$ be a minimizer of (1.1) with the condition (1.2), where ${u_ * } \in {W^{1, q}}(\Omega), $ $1 < p < q \leq Q$. Suppose that ${g_1}(x), {g_2}(x) \in {L^{\frac{1} {{1 - b}}}}(\Omega), b = \frac{{t - p}} {t}, p < t \leq q$. Hence,

(ⅰ) (integrability) if $b < \frac{{Q - p}}{Q}$, then $u - {u_ * } \in L_{weak}^{\frac{{Qp}} {{Q - p - bQ}}}(\Omega), $ where $\frac{{Qp}} {{Q - p - bQ}} > {p^ * } = \frac{{Qp}} {{Q - p}} > p$;

(ⅱ) (${L^1 }(\Omega)$ boundedness) if $b = \frac{{Q - p}} {Q}$, then there exists a positive constant $\theta $ such that ${{e^{\theta \left| {u - {u_*}} \right|}} \in {L^1}(\Omega)}; $

(ⅲ) (${L^\infty }(\Omega)$ boundedness) if $b > \frac{{Q - p}} {Q}$, then $u - {u_ * } \in {L^\infty }(\Omega).$

Inspired by Leonetti and Siepe[12], for a minimizer $u$ of (1.1) with the condition (1.2), we can rewrite $u$ as $u = {u_ * } + (u - {u_ * }), $ our aim is to prove when the boundary datum ${u_ * }$ has the higher integrability, $u - {u_ * }$ also has the higher integrability. The following two lemmas are needed for the proof of Theorem 2.4.

Lemma 2.5[3, 6] Let $\Omega \subset {\mathbb{R}^n}$ be a bounded open set. Then for any $u \in W_0^{1, p}(\Omega), 1 < p < Q, $ there holds $W_0^{1, p}(\Omega) \subset {L^{{p^ * }}}(\Omega), $ where ${p^ * } = \frac{{Qp}} {{Q - p}}$, namely there exists a positive constant $c$ such that

$ {\left( {\int_\Omega {{{\left| u \right|}^{{p^ * }}}dx} } \right)^{\frac{1} {{{p^ * }}}}} \leq c{\left( {\int_\Omega {{{\left| {Xu} \right|}^p}dx} } \right)^{\frac{1} {p}}} $

for any $u \in W_0^{1, p}(\Omega), 1 < p < Q$.

Lemma 2.6[5, 14, 15] Let $\varphi (t)$ be a nonnegative and nonincreasing function on $\left[{{k_0}, + \infty } \right)$ satisfying $ \varphi (h) \le \frac{c}{{{{\left({h - k} \right)}^\alpha }}}{\left[{\varphi (k)} \right]^\beta }, h > k \ge {k_0}, $ where $c, \alpha$ and $\beta $ are positive constants. If $\beta < 1, {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {k_0} > 0, $ then

$ \begin{equation}\label{eq1.5} \varphi (h) \le \left[ {{c^{\frac{1}{{1 - \beta }}}} + {{(2{k_0})}^{\frac{\alpha }{{1 - \beta }}}}\varphi ({k_0})} \right]{2^{\frac{\alpha }{{{{(1 - \beta )}^2}}}}}{\left( {\frac{1}{h}} \right)^{\frac{\alpha }{{1 - \beta }}}}; \end{equation} $

(2.3)

if $\beta {\rm{ = }}1, $ then

$ \begin{equation}\label{eq1.6} \varphi (h) \le {e^{1 - \tau (h - {k_0})}}\varphi ({k_0}), \end{equation} $

(2.4)

where $\tau = {\left({ec} \right)^{ - {\kern 1pt} \frac{1}{\alpha }}} > 0$; if $\beta > 1, $ then

$ \begin{equation}\label{eq1.7} \varphi ({k_0} + d) = 0, \end{equation} $

(2.5)

where $d = c{\left({\varphi ({k_0})} \right)^{\frac{{\beta - 1}}{\alpha }}}{2^{\frac{\beta }{{\beta - 1}}}}.$

Proof of Theorem 2.4 For any $k \in \left({0, + \infty } \right)$, suppose that ${T_k}: \mathbb{R} \to \mathbb{R}$ is a function such that

$ \begin{equation}\label{eq1.8} {T_k}(u - {u_ * }) = \left\{ {\begin{array}{*{20}{c}} {u - {u_ * }, {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left| {u - {u_ * }} \right| \leq k, } \\ {k\frac{{u - {u_ * }}} {{\left| {u - {u_ * }} \right|}}, {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \left| {u - {u_ * }} \right| > k, } \\ \end{array} } \right. \end{equation} $

(3.1)

setting $\psi = u - {u_ * } - {T_k}(u - {u_ * })$, it follows from (3.1) that

$ \begin{equation}\label{eq1.9} \psi = \left\{ {\begin{array}{*{20}{c}} {u - {u_ * } + k, } \\ {0, } \\ {u - {u_ * } - k, } \\ \end{array} } \right.\begin{array}{*{20}{c}} {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} u - {u_ * } < - k, } \\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} - k \leq u - {u_ * } \leq k, } \\ {{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} u - {u_ * } > k.} \\ \end{array} \end{equation} $

(3.2)

Using $u \in {u_ * } + W_0^{1, p}(\Omega)$ and (3.2) we have $\psi \in W_0^{1, p}(\Omega)$ and

$ \begin{equation}\label{eq1.10} X\psi = (Xu - X{u_ * }){1_{\left\{ {\left| {u - {u_ * }} \right| > k} \right\}}}, \end{equation} $

(3.3)

$ \begin{equation}\label{eq1.11} {\kern 1pt} \left| \psi \right| = \left( {\left| {u - {u_ * }} \right| - k} \right){1_{\left\{ {\left| {u - {u_ * }} \right| > k} \right\}}}, \end{equation} $

(3.4)

where ${1_A}(x) = 1$ if $x \in A$, ${1_A}(x) = 0$ if $x \notin A$. Let us consider

$\begin{equation}\label{eq1.12} w = u - \psi , \end{equation} $

(3.5)

which implies $w \in {u_ * } + W_0^{1, p}(\Omega)$. By (3.3) and (3.5) we obtain

$ \begin{align}\label{eq1.13} Xw &= Xu-X\psi = Xu - (Xu - X{u_ * }){1_{\left\{ {\left| {u - {u_ * }} \right| > k} \right\}}} = Xu - (Xu){1_{\left\{ {\left| {u - {u_ * }} \right| > k} \right\}}} + (X{u_ * }){1_{\left\{ {\left| {u - {u_ * }} \right| > k} \right\}}} \nonumber\\ &=(Xu){1_{\left\{ {\left| {u - {u_ * }} \right| \leq k} \right\}}} + (Xu){1_{\left\{ {\left| {u - {u_ * }} \right| > k} \right\}}} - (Xu){1_{\left\{ {\left| {u - {u_ * }} \right| > k} \right\}}} + (X{u_ * }){1_{\left\{ {\left| {u - {u_ * }} \right| > k} \right\}}} \nonumber\\ &= (Xu){1_{\left\{ {\left| {u - {u_ * }} \right| \leq k} \right\}}} + (X{u_ * }){1_{\left\{ {\left| {u - {u_ * }} \right| > k} \right\}}}. \end{align} $

(3.6)

Combining (2.2), (3.6) and $\Omega = \left\{ {\left| {u - {u_ * }} \right| \leq k} \right\} \cup \left\{ {\left| {u - {u_ * }} \right| > k} \right\}$, it concludes

$ \begin{eqnarray}\label{eq1.14} &&\int_{\{|{u-{u_*}}|\leq k\}} f(x, Xu )dx +\int_{\{|{u-{u_*}}|> k\}} f(x, Xu )dx \nonumber\\ &\leq &\int_{\{|{u-{u_*}}|\leq k\}} f(x, Xw )dx +\int_{\{|{u-{u_*}}|> k\}} f(x, Xw )dx \nonumber\\ &= &\int_{\{|{u-{u_*}}|\leq k\}} f(x, Xu )dx +\int_{\{|{u-{u_*}}|> k\}} f(x, Xu_{*} )dx, \end{eqnarray} $

(3.7)

and then by (3.7),

$ \begin{equation}\label{eq1.15} \int_{\left\{ {\left| {u - {u_ * }} \right| > k} \right\}} {f(x, Xu)dx} \le \int_{\left\{ {\left| {u - {u_ * }} \right| > k} \right\}} {f(x, X{u_ * })dx}. \end{equation} $

(3.8)

It follows from (1.2), (3.3), (3.8) and Lemma 2.5 that

$ \begin{eqnarray}\label{eq1.16} && { ( {\int_\Omega {{{ | \psi |}^{{p^ * }}}dx} } )^{\frac{p} {{{p^ * }}}}} \leq c\int_\Omega {{{ | {X\psi } |}^p}dx} =c\int_{ \{ { | {u - {u_ * }} | > k} \}} {{{ | {Xu - X{u_ * }} |}^p}dx} \nonumber\\ & \leq&{2^p}c\int_{ \{ { | {u - {u_ * }} | > k} \}} {{{ | {Xu} |}^p}dx} + {2^p}c\int_{ \{ { | {u - {u_ * }} | > k} \}} {{{ | {X{u_ * }} |}^p}dx} \nonumber\\ & \leq&{2^p}c [ {\int_{ \{ { | {u - {u_ * }} | > k} \}} { ( {f(x, Xu)dx + {g_1}} )dx} } ] + {2^p}c [ {\int_{ \{ { | {u - {u_ * }} | > k} \}} { ( {f(x, X{u_ * })dx + {g_1}} )dx} } ] \nonumber\\ &\leq&{2^p}c [ {\int_{ \{ { | {u - {u_ * }} | > k} \}} { ( {f(x, X{u_ * })dx + {g_1}} )dx} } ]{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} + {2^p}c [ {\int_{ \{ { | {u - {u_ * }} | > k} \}} { ( {f(x, X{u_ * })dx + {g_1}} )dx} } ] \nonumber\\ &\leq&c [ {\int_{ \{ { | {u - {u_ * }} | > k} \}} { ( {{{ | {X{u_ * }} |}^p} + {g_1} + {g_2}} )dx} } ]. \end{eqnarray} $

(3.9)

Since $p < q$, there exists a positive number $t$ such that $p < t \leq q$, thus from (3.9), Hölder's inequality and Minkowski's inequality, we can get

$\begin{eqnarray}\label{eq1.17} &&{ ( {\int_\Omega {{{ | \psi |}^{{p^ * }}}dx} } )^{\frac{p} {{{p^ * }}}}} \\ &\leq& c{ ( {\int_{ \{ { | {u - {u_ * }} | > k} \}} {{{ ( {{{ | {X{u_ * }} |}^p} + {g_1} + {g_2}} )}^{\frac{t} {p}}}dx} } )^{\frac{p} {t}}}{ [ {\text{meas} \{ { | {u - {u_ * }} | > k} \}} ]^{\frac{{t - p}} {t}}} \nonumber\\ &\leq& c ( {{{ ( {\int_\Omega {{{ | {X{u_ * }} |}^t}dx} } )}^{\frac{p} {t}}} + {{ ( {\int_\Omega {{{ ( {{g_1}} )}^{\frac{t} {p}}}dx} } )}^{\frac{p} {t}}} + {{ ( {\int_\Omega {{{ ( {{g_2}} )}^{\frac{t} {p}}}dx} } )}^{\frac{p} {t}}}} ) \cdot{ [ {\text{meas} \{ {| {u - {u_ * }} | > k} \}}]^{\frac{{t - p}} {t}}}.\nonumber \end{eqnarray} $

(3.10)

Moreover, we can define a positive number $b = \frac{{t - p}} {t}$ in (3.10), namely we have $0 < b = 1 - \frac{p} {t} \leq 1 - \frac{p} {q} < 1$; and thus from ${u_*} \in {W^{1, q}}(\Omega)$ and ${g_1}, {g_2} \in {L^{\frac{1} {{1 - b}}}}(\Omega)$, we can set

$ \begin{eqnarray}\label{eq1.18} M &=& {\left( {\int_\Omega {{{\left| {X{u_ * }} \right|}^{\frac{p} {{1 - b}}}}dx} } \right)^{1 - b}} + {\left( {\int_\Omega {{{\left( {{g_1}} \right)}^{\frac{1} {{1 - b}}}}dx} } \right)^{1 - b}} + {\left( {\int_\Omega {{{\left( {{g_2}} \right)}^{\frac{1} {{1 - b}}}}dx} } \right)^{1 - b}} < + \infty.\quad \end{eqnarray} $

(3.11)

Finally we insert (3.11) into (3.10), we easily obtain

$\begin{equation}\label{eq1.19} \int_\Omega {{{\left| \psi \right|}^{{p^ * }}}dx} \leq c{M^{\frac{{{p^ * }}} {p}}}{\left[ {meas\left\{ {\left| {u - {u_ * }} \right| > k} \right\}} \right]^{\frac{{b{p^ * }}} {p}}}. \end{equation} $

(3.12)

For any $h > k \geq {k_0}$, it follows from (3.4) that

$ \begin{align}\label{eq1.20} {\left( {h - k} \right)^{{p^ * }}}\left[ {meas\left\{ {\left| {u - {u_ * }} \right| > h} \right\}} \right]& = \int_{\left\{ {\left| {u - {u_ * }} \right| > h} \right\}} {{{\left( {h - k} \right)}^{{p^ * }}}dx} \leq \int_{\left\{ {\left| {u - {u_ * }} \right| > h} \right\}} {{{\left( {\left| {u - {u_ * }} \right| - k} \right)}^{{p^ * }}}dx} \nonumber\\ & \leq \int_{\left\{ {\left| {u - {u_ * }} \right| > k} \right\}} {{{\left( {\left| {u - {u_ * }} \right| - k} \right)}^{{p^ * }}}dx} =\int_\Omega {{{\left| \psi \right|}^{{p^ * }}}dx} . \end{align} $

(3.13)

Combining (3.12) and (3.13), it yields

$ \begin{align}\label{eq1.21} meas\left\{ {\left| {u - {u_ * }} \right| > h} \right\} \leq \frac{{c{M^{\frac{{{p^ * }}} {p}}}}} {{{{\left( {h - k} \right)}^{{p^ * }}}}}{\left[ {meas\left\{ {\left| {u - {u_ * }} \right| > k} \right\}} \right]^{\frac{{b{p^ * }}} {p}}}. \end{align} $

(3.14)

In (3.14), setting

$ \begin{align}\label{eq1.22} \varphi (h) = meas\left\{ {\left| {u - {u_ * }} \right| > h} \right\}, \varphi (k) = meas\left\{ {\left| {u - {u_ * }} \right| > k} \right\}, \alpha = {p^ * }, \beta = \frac{{b{p^ * }}} {p}. \end{align} $

(3.15)

We now apply Lemma 2.6 to (3.15). We can prove, respectively.

(ⅰ) If $b < \frac{p} {{{p^ * }}} = \frac{p} {{\frac{{Qp}} {{Q - p}}}} = \frac{{Q - p}} {Q}$, then $\beta < 1$. For any ${k_0} > 0$, it follows from (2.3) that

$ \begin{align}\label{eq1.23} meas\left\{ {\left| {u - {u_ * }} \right| > h} \right\} &\leq \left[ {{c^{\frac{1} {{1 - \beta }}}} + {{\left( {2{k_0}} \right)}^{\frac{\alpha } {{1 - \beta }}}}meas\left\{ {\left| {u - {u_ * }} \right| > {k_0}} \right\}} \right]{2^{\frac{\alpha } {{{{\left( {1 - \beta } \right)}^2}}}}}{\left( {\frac{1} {h}} \right)^{\frac{\alpha } {{1 - \beta }}}} \nonumber\\ &\leq \left[ {{c^{\frac{1} {{1 - \beta }}}} + {{\left( {2{k_0}} \right)}^{\frac{\alpha } {{1 - \beta }}}}meas{\kern 1pt} {\kern 1pt} {\kern 1pt} \Omega } \right]{2^{\frac{\alpha } {{{{\left( {1 - \beta } \right)}^2}}}}}{\left( {\frac{1} {h}} \right)^{\frac{\alpha } {{1 - \beta }}}}, \end{align} $

(3.16)

by (3.16) and (2.1), $u - {u_ * } \in L_{weak}^{\frac{\alpha } {{1 - \beta }}}(\Omega), $ where $\frac{\alpha } {{1 - \beta }} = \frac{{{p^ * }}} {{1 - \frac{{b{p^ * }}} {p}}} = \frac{{Qp}} {{Q - p - bQ}} > {p^ * } = \frac{{Qp}} {{Q - p}}$.

(ⅱ) If $b = \frac{{Q - p}}{Q}$, then $\beta = 1$. It follows from (2.4) that

$ \begin{align}\label{eq1.24} meas\left\{ {\left| {u - {u_ * }} \right| > h} \right\} \leq {\kern 1pt} {\kern 1pt} {e^{1 - \tau \left( {h - {k_0}} \right)}}meas\left\{ {\left| {u - {u_ * }} \right| > {k_0}} \right\}, \end{align} $

(3.17)

where $\tau = {\left({ec} \right)^{ - {\kern 1pt} {\kern 1pt} \frac{1} {{{p^ * }}}}} > 0$. When ${k_0} \leq 0$, we have

$\begin{align}\label{eq1.25} {e^{1 - \tau \left( {h - {k_0}} \right)}} = e{e^{ - \tau \left( {h - {k_0}} \right)}} \leq e{e^{ - \tau h}} \end{align} $

(3.18)

and

$ \begin{align}\label{eq1.26} meas\left\{ {\left| {u - {u_ * }} \right| > {k_0}} \right\} = meas{\kern 1pt} {\kern 1pt} \Omega , \end{align} $

(3.19)

Substituting (3.18) and (3.19) into (3.17),

$ \begin{align}\label{eq1.26aa} meas\left\{ {\left| {u - {u_ * }} \right| > h} \right\} \leq e{e^{ - \tau h}}meas{\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \Omega . \end{align} $

(3.20)

It is easy to see that there exists a positive constant $\theta < \tau $ satisfying

$\begin{align} {e^{\theta \left| {u - {u_ * }} \right|}} - 1 =\int_0^{\left| {u - {u_ * }} \right|} {\theta {e^{\theta h}}} dh = \int_0^\infty {\theta {e^{\theta h}}} {1_{\left\{ {\left| {u - {u_ * }} \right| > h} \right\}}}dh. \label{b50} \end{align} $

(3.21)

It follows from (3.20) and (3.21) that

$ \begin{align*} \int_\Omega {\left( {{e^{\theta \left| {u - {u_ * }} \right|}} - 1} \right)} dx &= \int_\Omega {\int_0^\infty {\theta {e^{\theta h}}} {1_{\left\{ {\left| {u - {u_ * }} \right| > h} \right\}}}dh} dx =\int_\Omega {{1_{\left\{ {\left| {u - {u_ * }} \right| > h} \right\}}}dx\int_0^\infty {\theta {e^{\theta h}}} dh} \nonumber\\ &=\int_{\left\{ {\left| {u - {u_ * }} \right| > h} \right\}} {1dx\int_0^\infty {\theta {e^{\theta h}}} dh} =meas\left\{ {\left| {u - {u_ * }} \right| > h} \right\}\int_0^\infty {\theta {e^{\theta h}}dh} \nonumber\\ &= \int_0^\infty {\theta {e^{\theta h}}meas\left\{ {\left| {u - {u_ * }} \right| > h} \right\}dh} \leq \int_0^\infty {\theta {e^{\theta h}}e{e^{ - \tau h}}meas{\kern 1pt} {\kern 1pt} \Omega }{\kern 1pt} dh \nonumber\\ &= \frac{{\theta e}}{{\tau - \theta }}{\kern 1pt} meas{\kern 1pt} {\kern 1pt} \Omega, \label{b51} \end{align*} $

which implies

$ {{e^{\theta \left| {u - {u_*}} \right|}} \in {L^1}(\Omega )}. $

(ⅲ) If $b > \frac{{Q - p}}{Q}$, then $\beta > 1$. It follows from (2.5) that

$\begin{align}\label{eq1.27} \varphi \left( {{k_0} + d} \right) = meas\left\{ {\left| {u - {u_ * }} \right| > {k_0} + d} \right\} = 0, \end{align} $

(3.22)

where $d = c{\left({meas\left\{ {\left| {u - {u_ * }} \right| > {k_0}} \right\}} \right)^{\frac{{bQ - Q + b}} {{Qp}}}}{2^{\frac{{bQ}} {{bQ - Q + p}}}}$, hence from (3.22) and (2.1), we obtain $u - {u_ * } \in L_{weak}^\infty (\Omega)$. Also since $L_{weak}^\infty (\Omega) = {L^\infty }(\Omega)$, then $u - {u_ * } \in {L^\infty }(\Omega)$.