Higher-order Fefferman-Poincaré type inequalities and applications

Abstract

We establish higher-order Fefferman-Poincaré type inequalities with a potential belonging to an appropriate higher-order Stummel-Kato type class introduced in this paper. As an application, we obtain a priori ${L^{p}}$ estimates for solutions of higher-order elliptic equations with discontinuous coefficients of small BMO type and a potential belonging to the higher-order Stummel-Kato type class.

1 Introduction and main results

Fefferman proved in [1] that if a potential V belonging to the classical Morrey space $L^{r,n-2r}(\mathbb{R}^{n})$ with $1< r\leq n/2$, then there exists a positive constant c, independent of u, such that

$$ { \int_{{\mathbb{R}^{n}}} { \bigl\vert {V(x)} \bigr\vert \bigl\vert {u(x)} \bigr\vert } ^{2}}\,dx \leq c{ \int_{{\mathbb{R}^{n}}} { \bigl\vert {\nabla u(x)} \bigr\vert } ^{2}}\,dx, \quad u \in C_{0}^{\infty}\bigl({ \mathbb{R}^{n}} \bigr). $$

The result has been extended to many more general settings and applied to study Harnack’s inequality, unique continuation for nonnegative solutions and regularity of solutions of elliptic equations (cf. [2–8] etc.). Especially, Schechter [7] deduced a similar inequality for $V \in S({\mathbb{R}^{n}})$, here the space $S({\mathbb{R}^{n}})$ is the classical Stummel-Kato class. Let us recall that one says $V \in S({\mathbb{R}^{n}})$ ($n \geq3$) if $V \in L_{\mathrm{loc}}^{1}({\mathbb{R}^{n}})$, and for any $r>0$,

$$ {\varphi_{V}}(r): = \sup_{x \in{\mathbb{R}^{n}}} \int_{B(x,r)} { \bigl\vert {V(y)} \bigr\vert } {\vert {x - y} \vert ^{2 - n}}\,dy $$

is finite and

$$ \lim_{r \to{0^{+} }} {\varphi_{V}}(r) = 0, $$

where $B(x,r)$ is a ball of radius r and center x in $\mathbb{R}^{n}$. Recently, Zamboni [8] introduced a new function space including $S(\mathbb{R}^{n})$ in terms of nonlinear Riesz potentials, and also provided a Fefferman-Poincaré inequality extending the result in [7].

Inspired by Zamboni [8], Definition 2.3, we introduce the following higher-order Stummel-Kato type class $S^{m}_{p}(\mathbb{R}^{n})$.

Definition 1.1

Let $V \in L^{1}_{\mathrm{loc}}(\mathbb{R}^{n})$ ($n \geq3$), and let $p>1$ and m be a positive integer with $1 \leq m < n$. For $r>0$, denote

$$ {\eta_{V}}(r): = \sup_{x \in{\mathbb{R}^{n}}} { \biggl( {{{ \int_{B_{r}(x)} {\frac{1}{{{{\vert {x - y} \vert }^{n - m}}}} \biggl( { \int_{B_{r}(x)} {\frac{{\vert {V(z)} \vert }}{ {{{\vert {y - z} \vert }^{n - m}}}}\,dz} } \biggr)} }^{\frac{1}{{p - 1}}}} \,dy} \biggr)^{p - 1}}. $$

We say that V belongs to the higher-order Stummel-Kato type class $S^{m}_{p}(\mathbb{R}^{n})$, if

$$ \eta_{V}(r)< +\infty,\quad r>0 \quad \mbox{and} \quad \lim _{r \to {0^{+} }} {\eta_{V}}(r) = 0. $$

Here $|\cdot|$ denotes the Euclidean norm and ${B_{r}}(x) = B(x,r)= \{ {y \in {\mathbb{R}^{n}}: \vert {x - y} \vert < r} \}$. Sometimes we call that $\eta_{V}(r)$ is the higher-order Stummel-Kato type modulus of V.

Remark 1.2

If $m=1$, the definition above is identical with Definition 2.3 in [8]; if $m=2$, the definition above is the same as Definition 1 in [6] corresponding to the Euclidean case.

We also need the following definitions.

Definition 1.3

([9])

Let Ω be an open set in $\mathbb{R}^{n}$. For $p \geq1$ and k a nonnegative integer, the Sobolev space $W^{k,p}(\Omega)$ consists of all distributions u on Ω such that $D^{\alpha}u \in L^{p}(\Omega)$ for all multi-index α with $\vert \alpha \vert \leq k$. Furthermore, $W^{k,p}(\Omega)$ is a Banach space with the norm

$${\Vert u \Vert _{{W^{k,p}}(\Omega)}} = \biggl( { \int_{\Omega}{\sum_{\vert \alpha \vert \leq k} {{{ \bigl\vert {{D^{\alpha}}u} \bigr\vert }^{p}}\,dx} } } \biggr)^{1/p}. $$

Here, $\alpha = ({\alpha _{1}},\ldots,{\alpha_{n}})$ is a multi-index of order $\vert \alpha \vert : = {\alpha_{1}} + \cdots+{\alpha_{n}}$. The Banach space $W_{0}^{k, p}(\Omega)$ is the closure of $C_{0}^{\infty}(\Omega)$ in $W^{k, p}(\Omega)$.

We denote ${D^{k}}u: = \{ {{D^{\alpha}}u: | \alpha| = k} \}$, and its norm is defined by

$$\bigl\vert {{D^{k}}u} \bigr\vert = \biggl( {\sum _{\vert \alpha \vert = k} {{{ \bigl\vert {{D^{\alpha}}u} \bigr\vert }^{2}}} } \biggr)^{1/2}. $$

For convenience, we denote by $MB(x,r)$ ($M>0$) the ball concentric with $B(x,r)$ of radius M times that of $B(x,r)$.

Definition 1.4

([10])

A domain Ω in $\mathbb{R}^{n}$ is said to be a weak Boman chain domain, or a member of $\mathcal{F}(1,M)$, if there exist a constant $M \geq1$ and a family $\mathcal{F}$ of balls $B\subset \Omega$ such that

• (i) $\Omega = { \bigcup_{B \in\mathcal{F}}}B$;

• (ii) $\sum_{B \in\mathcal{F}} {{\chi}_{B}(x)} \leq M{\chi _{\Omega}}(x)$, $x \in{\mathbb{R}^{n}}$;

• (iii) there is a ‘central ball’ $B_{0} \in\mathcal{F}$ such that for every ball $B \in\mathcal{F}$, there exist a positive integer $k=k(B)$ and a chain $\{ {{B_{j}}} \}_{j = 0}^{k}$ of balls for which and each $B_{j}\cap B_{j+1}$ contains a ball $D_{j}$ (this ball does not need to be a member of $\mathcal{F}$) with $B_{j}\cup B_{j+1}\subset MD_{j}$;

• (iv) $B \subset MB_{j}$ for all $j=0,1,\ldots,k(B)$.

Let us note that if the hypothesis (ii) is replaced by

$$\sum_{B \in\mathcal{F}} {{\chi_{\tau B}}(x)} \leq M{ \chi_{\Omega}}(x),\quad x \in {\mathbb{R}^{n}}, \tau>1\mbox{ is a constant}, $$

then Ω is said to be a Boman chain domain, cf. [11, 12]. The classes $\mathcal{F}(1,M)$ contain many important types of domains in $\mathbb{R}^{n}$, for instance, balls, cubes, and John domains (cf. [11, 13, 14]).

Based on the class $S_{p}^{m}$, we establish the following higher-order Fefferman-Poincaré type inequality with the aid of the method in [4, 8] proving the Fefferman-Poincaré type inequality for the case $m=1$. It is interesting that the mth order derivatives arise in this setting.

Theorem 1.5

Let Ω be a weak Boman domain, $p>1$, and m be a positive integer with $1 \leq m < n$. Assume $V \in S_{p}^{m}(\mathbb{R}^{n})$, then for any $u \in W^{m,p}(\Omega)$, there exists a polynomial $P_{B_{0}}(x)$ of order less than m such that

$$ { \int_{\Omega}{ \bigl\vert {u(x) - {P_{{B_{0}}}}(x)} \bigr\vert } ^{p}} \bigl\vert {V(x)} \bigr\vert \,dx \leq c{ \eta_{V}}(2M{r_{{B_{0}}}}){ \int_{\Omega}{ \bigl\vert {{D^{m}}u(x)} \bigr\vert } ^{p}}\,dx, $$

(1.1)

where the positive constant c is independent of u, $B_{0}$ is the central ball of radius $r_{B_{0}}$ in Ω, and the constant M is in Definition  1.4.

When u is a distribution with compact support in Ω, we have the following.

Theorem 1.6

Under the assumptions of Theorem  1.5, for any $u \in W^{m,p}_{0}(\Omega)$, there exists a positive constant c independent of u such that

$$ { \int_{\Omega}{ \bigl\vert u(x) \bigr\vert } ^{p}} \bigl\vert {V(x)} \bigr\vert \,dx \leq c{\eta_{V}}(2M{r_{{B_{0}}}}){ \int_{\Omega}{ \bigl\vert {{D^{m}}u(x)} \bigr\vert } ^{p}}\,dx. $$

(1.2)

Remark 1.7

• (i) In the special case when $m=1$, similar results were obtained in [4, 8]; when $m=2$ and $\Omega=B(x,r)$, (1.1) and (1.2) have been obtained in [6].

• (ii) The higher-order Fefferman-Poincaré type inequalities on stratified groups can also be proved, because of the higher-order representation formulas proved by Lu and Wheeden [10, 15].

In the rest of this paper, we will show some applications of the above results to regularity of solutions to the higher-order elliptic equations with a potential. Let us consider the equation

$$ \sum_{\vert \alpha \vert \leq2k} {{a_{\alpha}}(x){D^{\alpha}}u} (x) + V(x)u(x) - \lambda u(x) = f(x),\quad x \in \mathbb{R}^{n}, $$

(1.3)

where k is a positive integer with $1\leq k < n/2$, $V\in S_{p}^{2k}(\mathbb{R}^{n})$ for $p>1$, and the coefficients $a_{\alpha}(x)$ satisfy

$$ {( - 1)^{k - 1}}\sum_{|\alpha| = 2k} {{a_{\alpha}}(x){\xi ^{\alpha}}} \geq{\Lambda_{1}},\quad \xi \in{\mathbb{S}^{n - 1}} \subset{\mathbb{R}^{n}} $$

(1.4)

and

$$ \sum_{\vert \alpha \vert \leq2k} { \bigl\vert {{a_{\alpha}}(x)} \bigr\vert } \leq{\Lambda_{2}} $$

(1.5)

for all $x\in\mathbb{R}^{n}$ and positive constants $\Lambda_{1}$, $\Lambda_{2}$. In addition, we assume that the leading coefficients $a_{\alpha}(x)$ ($| \alpha | = 2k$) are in BMO space and their semi-norms are small enough. More precisely, we recall the following definition.

Definition 1.8

(Small BMO condition, cf. [16])

We say that the coefficients $a_{\alpha}(x)$ satisfy $(\delta, R)$-vanishing condition if for given $\delta>0$, there exists $R>0$ such that

$$ \sup_{0 < r \leq R} \sup_{x \in {\mathbb{R}^{n}}} { \bigl\vert {{B_{r}}(x)} \bigr\vert ^{ - 1}} \int_{{B_{r}}(x)} { \bigl\vert {{a_{\alpha}}(y) - {{ \overline{({a_{\alpha}})} }_{{B_{r}}(x)}}} \bigr\vert } \,dy \leq\delta, $$

where

$${\overline{({a_{\alpha}})} _{{B_{r}}(x)}} = { \bigl\vert {{B_{r}}(x)} \bigr\vert ^{ - 1}} \int_{{B_{r}}(x)} {{a_{\alpha}}(y)} \,dy. $$

When $k=1$ and V in general is not bounded, or $k>1$ and $V\in L^{\infty}(\mathbb{R}^{n})$, regularity for the elliptic equation (1.3) has been studied by many authors, cf. [16–19] and so forth. Here we are concerned with (1.3) for the case $1< k< n/2$ and a singular potential V in $S_{p}^{m}(\mathbb{R}^{n})$.

Our result from (1.3) is the following.

Theorem 1.9

Let $p \in(1,\infty)$, k be a positive integer with $1< k< n/2$. There exist a positive ${\lambda_{0}}={\lambda_{0}}(n,{\Lambda _{1}},{\Lambda_{2}},p)$ and a small $\delta=\delta(n,{\Lambda_{1} },{\Lambda_{2}},p)>0$ such that for the coefficients $a_{\alpha}(x)$ satisfying (1.4)-(1.5) and $(\delta,R)$-vanishing condition with $|\alpha|=2k$, for $V^{p}\in S^{2k}_{p}(\mathbb{R}^{n})$ and $f\in L^{p}(\mathbb{R}^{n})$, if $u\in W^{2k,p}(\mathbb{R}^{n})$ solves equation (1.3), then

$$ \sum_{\vert \alpha \vert \leq2k} {{{ \bigl\Vert {{D^{\alpha}}u} \bigr\Vert }_{{L^{p}}({\mathbb{R}^{n}})}}} + {\Vert {Vu} \Vert _{{L^{p}}({\mathbb{R}^{n}})}} \leq c{\Vert f \Vert _{{L^{p}}({\mathbb{R}^{n}})}}, $$

(1.6)

provided $\lambda\geq\lambda_{0}$, where the positive constant c is independent of u and f.

This paper is organized as follows. In Section 2 we first give an example showing the class $S_{p}^{m}(\mathbb{R}^{n})$ contains some nontrivial singular functions, and then prove Theorems 1.5 and 1.6 using the higher-order representation formulas by Lu and Wheeden in [10, 15]. The proof of Theorem 1.9 is given in Section 3 based on results in previous section and $L^{p}$ estimates for the higher-order elliptic equations without potentials in [16, 18].

Dependence of constants

Throughout this paper, the letter c denotes a positive constant which may vary from line to line.

2 Proofs of Theorems 1.5 and 1.6

Before giving the proofs of Theorems 1.5 and 1.6, we first show that the higher-order Stummel-Kato type class $S_{p}^{m}(\mathbb{R}^{n})$ ($p>1$, $1\leq m< n$) is nonempty. Clearly, ${L^{\infty}}({\mathbb{R}^{n}}) \subset S_{p}^{m}({\mathbb{R}^{n}})$ and, in general, the class $S_{p}^{m}({\mathbb{R}^{n}})$ involves some singular potentials. For example, the function

$$ V(x) = \frac{1}{{{{\vert x \vert }^{m}}{{\vert {\log \vert x \vert } \vert }^{2m}}}}{\chi_{B(0,{e^{ - 2}})}}(x)\quad \mbox{for }p>1\mbox{ and }1\leq m< n, $$

where ${\chi_{B(0, {e^{ - 2}})}}$ is the characteristic function of $B(0,e^{-2})$, belongs to $S_{p}^{m}({\mathbb{R}^{n}})$. To illustrate it, we need to show that

$$ {\eta_{V}}(r) = \sup_{x \in{\mathbb{R}^{n}}} { \biggl( { \int_{{B_{r}}(x)} {\frac{1}{{{{\vert {x - y} \vert }^{n - m}}}}} {{ \biggl( { \int_{{B_{r}}(x)} {\frac{{{{\vert z \vert }^{ - m}}{{\vert {\log \vert z \vert } \vert }^{ - 2m}}}}{{{{\vert {y - z} \vert }^{n - m}}}}{\chi_{{B_{{e^{ - 2}}}}(0)}}(z)\,dz} } \biggr)}^{\frac{1}{{p - 1}}}}\,dy} \biggr)^{p - 1}} $$

satisfies

$$\begin{aligned} (\mathrm{i})&\quad \eta_{V}(r)< \infty, \quad r>0; \\ (\mathrm{ii})&\quad \lim_{r \to{0^{+} }} {\eta_{V}}(r) = 0. \end{aligned}$$

In fact, for $x\in\mathbb{R}^{n}$ and $r>0$, one has

$$\begin{aligned} \Phi(x,r): &= \int_{B(x,r)} {\frac{1}{{{{\vert {x - y} \vert }^{n - m}}}}} { \biggl( { \int_{B(x,r)} {\frac{{{{\vert z \vert }^{ - m}}{{\vert {\log \vert z \vert } \vert }^{ - 2m}}}}{{{{\vert {y - z} \vert }^{n - m}}}}{\chi_{B(0, {e^{ - 2}})}}(z)\,dz} } \biggr)^{\frac{1}{p-1}}}\,dy \\ &\equiv \int_{B(x,r)} {{{\vert {x - y} \vert }^{m - n}}} {J^{\frac{1}{{p - 1}}}}\,dy, \end{aligned}$$

and one has

$$\begin{aligned} J \leq& \int_{B(y,2r)} {\frac{{{{\vert z \vert }^{ - m}}{{\vert {\log \vert z \vert } \vert }^{ - 2m}}}}{{{{\vert {y - z} \vert }^{n - m}}}}{\chi_{B(0, {e^{ - 2}})}}(z)\,dz} \\ \leq& \int_{\{ z: \vert z \vert < \vert {y - z} \vert < 2r\} } {\frac{{{{\vert z \vert }^{ - m}}{{\vert {\log \vert z \vert } \vert }^{ - 2m}}}}{{{{\vert {y - z} \vert }^{n - m}}}}{\chi_{B(0, {e^{ - 2}})}}(z)\,dz} \\ &{} + \int_{\{ z: \vert z \vert \geq \vert {y - z} \vert \} \cap B(y,2r)} {\frac{{{{\vert z \vert }^{ - m}}{{\vert {\log \vert z \vert } \vert }^{ - 2m}}}}{{{{\vert {y - z} \vert }^{n - m}}}}{\chi_{B(0, {e^{ - 2}})}}(z)\,dz} \\ \equiv&{J_{1}} + {J_{2}}. \end{aligned}$$

Using the polar coordinate transformation leads to

$$\begin{aligned} {J_{1}} &\leq \int_{\{ z: \vert z \vert < 2r\} } {{{\vert z \vert }^{ - n}} {{ \bigl\vert {\log \vert z \vert } \bigr\vert }^{ - 2m}} { \chi_{B(0, {e^{ - 2}})}}(z)\,dz } \\ &\leq \int_{B(0,2r) \cap B(0, {e^{ - 2}})} {{{\vert z \vert }^{ - n}} {{ \bigl\vert {\log \vert z \vert } \bigr\vert }^{ - 2m}}\,dz } \\ & = c(n) \int_{0}^{\sigma}{\frac{{{s^{n - 1}}}}{{{s^{n}}{{( - \log s)}^{2m}}}}}\,ds \\ &= c(n,m){( - \log\sigma)^{1 - 2m}}, \end{aligned}$$

where $\sigma = \min\{ 2r, {e^{ - 2}}\}$. Since the function ${t^{ - m}}{( - \log t)^{ - 2m}}$ is decreasing in $(0, e^{-2})$, one infers

$$\begin{aligned} {J_{2}} &= \int_{\{ z: \vert {y - z} \vert \leq \vert z \vert < {e^{ - 2}}\} \cap B(y,2r)} {{{\vert z \vert }^{ - m}} {{ \bigl\vert {\log \vert z \vert } \bigr\vert }^{ - 2m}} {{\vert {y - z} \vert }^{m - n}}\,dz} \\ &\leq \int_{B(y,{e^{ - 2}}) \cap B(y,2r)} {{{\vert {y - z} \vert }^{ - n}} {{ \bigl\vert {\log \vert {y - z} \vert } \bigr\vert }^{ - 2m}}\,dz} \\ &= \int_{B(0,\sigma)} {{{\vert z \vert }^{ - n}} {{ \bigl\vert {\log \vert z \vert } \bigr\vert }^{ - 2m}}\,dz} \\ &= c(n,m){( - \log\sigma)^{1 - 2m}}. \end{aligned}$$

Combining $J_{1}$ and $J_{2}$, we have

$$\begin{aligned} \Phi(x,r) &\leq c(n,m)\frac{1}{{{{( - \log\sigma)}^{{{(2m - 1)} / {(p - 1)}}}}}} \int_{B(x,r)} {\frac{1}{ {{{\vert {x - y} \vert }^{n - m}}}}} \,dy \\ &\leq c(n,m)\frac{1}{{{{( - \log\sigma)}^{{{(2m - 1)} / {(p - 1)}}}}}} \int_{0}^{r} {\frac{{{s^{n - 1}}}}{ {{s^{n - m}}}}} \,ds \\ &\leq c(n,m)\frac{1}{{{{( - \log\sigma)}^{{{(2m - 1)} / {(p - 1)}}}}}}{r^{m}}. \end{aligned}$$

Therefore,

$$\begin{aligned} \eta_{V} (r) =& \sup_{x \in{\mathbb{R}^{n}}} {\Phi(x,r)^{p-1}} \\ \leq& c(n,m,p){( - \log\sigma)^{1 - 2m}} {r^{m(p - 1)}} \\ =& \left \{ \textstyle\begin{array}{l@{\quad}l} c(n,m,p)2^{1-2m}r^{m(p-1)},& r \geq e^{-2}/2, \\ c(n,m,p)(-log(2r))^{1-2m}r^{m(p-1)},& r < e^{-2}/2, \end{array}\displaystyle \right . \end{aligned}$$

and it proves (i) and (ii).

Now we devote ourselves to proving Theorems 1.5 and 1.6. Lu and Wheeden [10, 15] derived various higher-order integral representation formulas on Carnot groups and applied them to prove some Sobolev type embedding theorems. Since the Euclidean space $\mathbb{R}^{n}$ is a special case of Carnot groups, the results in [10, 15] are also true in $\mathbb{R}^{n}$. Here, we state these formulas in [10, 15] corresponding to $\mathbb{R}^{n}$ and apply to prove (1.1) and (1.2).

Lemma 2.1

([10, 15])

Let Ω be a weak Boman chain domain in $\mathbb{R}^{n}$ with a central ball $B_{0}$, and m be a positive integer with $1\leq m < n$. If $u \in W^{m,1}(\Omega)$, then there exists a polynomial $P_{B_{0}}(x)$ of order less than m such that for a.e. $x \in \Omega$,

$$ \bigl\vert {u(x) - {P_{{B_{0}}}}(x)} \bigr\vert \leq c \int_{\Omega}{{{ \vert {x - y} \vert }^{m - n}} \bigl\vert {{D^{m}}u(y)} \bigr\vert } \,dy, $$

(2.1)

where the positive constant c is independent of u, x, and Ω. Moreover, if $u \in W_{0}^{m,1}(\Omega)$, then

$$ \bigl\vert {u(x)} \bigr\vert \leq c \int_{\Omega}{{{ \vert {x - y} \vert }^{m-n}} \bigl\vert {{D^{m}}u(y)} \bigr\vert } \,dy. $$

(2.2)

Proof of Theorem 1.5

Applying (2.1), Fubini’s theorem, and Hölder’s inequality, we obtain

$$\begin{aligned}& { \int_{\Omega}{ \bigl\vert {u(x) - {P_{{B_{0}}}}(x)} \bigr\vert } ^{p}} \bigl\vert {V(x)} \bigr\vert \,dx \\& \quad \leq c \int_{\Omega}{ \bigl\vert {V(x)} \bigr\vert } { \bigl\vert {u(x) - {P_{{B_{0}}}}(x)} \bigr\vert ^{p - 1}} \biggl( { \int_{\Omega}{ \bigl\vert {{D^{m}}u(y)} \bigr\vert } {{\vert {x - y} \vert }^{m - n}}\,dy} \biggr)\,dx \\& \quad = c \int_{\Omega}{ \bigl\vert {{D^{m}}u(y)} \bigr\vert \biggl( { \int_{\Omega}{ \bigl\vert {V(x)} \bigr\vert } {{ \bigl\vert {u(x) - {P_{{B_{0}}}}(x)} \bigr\vert }^{p - 1}} {{\vert {x - y} \vert }^{m - n}}\,dx} \biggr)} \,dy \\& \quad \leq c{ \biggl( { \int_{\Omega}{{{ \bigl\vert {{D^{m}}u(y)} \bigr\vert }^{p}}} \,dy} \biggr)^{\frac{1}{p}}} \\& \qquad {}\times{ \biggl( { \int_{\Omega}{{{ \biggl( { \int_{\Omega}{\frac{{\vert {V(x)} \vert {{\vert {u(x) - {P_{{B_{0}}}}(x)} \vert }^{p - 1}}}}{{{{\vert {x - y} \vert }^{n - m}}}}} \,dx} \biggr)}^{\frac{p}{{p - 1}}}} \,dy} } \biggr)^{\frac{{p - 1}}{p}}} \\& \quad \equiv c{ \biggl( { \int_{\Omega}{{{ \bigl\vert {{D^{m}}u(y)} \bigr\vert }^{p}}} \,dy} \biggr)^{\frac{1}{p}}} \cdot I . \end{aligned}$$

Observing

$$\begin{aligned}& \int_{\Omega}{ \bigl\vert {V(x)} \bigr\vert } { \bigl\vert {u(x) - {P_{{B_{0}}}}(x)} \bigr\vert ^{p - 1}} {\vert {x - y} \vert ^{m - n}}\,dx \\& \quad \leq{ \biggl( { \int_{\Omega}{\frac{|V(z)|}{|z-y|^{n-m}}\,dz} } \biggr)^{\frac{1}{p}}} { \biggl( { \int_{\Omega}{{{ \bigl\vert {u(x) - {P_{{B_{0}}}}(x)} \bigr\vert }^{p}}\frac{|V(x)|}{|x-y|^{n-m}}\,dx} } \biggr)^{\frac{{p - 1}}{p}}}, \end{aligned}$$

and noting $\Omega\subset MB_{0}$ from (iv) of Definition 1.4, we deduce

$$\begin{aligned} I^{p/(p-1)} \leq& \int_{\Omega}{{{ \biggl( { \int_{\Omega}{\frac{{\vert {V(z)} \vert }}{{{{\vert {z - y} \vert }^{n - m}}}}} \,dz} \biggr)}^{\frac{1}{{p - 1}}}} \biggl( { \int_{\Omega}{{{ \bigl\vert {u(x) - {P_{{B_{0}}}}(x)} \bigr\vert }^{p}}\frac{{\vert {V(x)} \vert }}{ {{{\vert {x - y} \vert }^{n - m}}}}\,dx} } \biggr)} \,dy \\ =& { \int_{\Omega}{ \bigl\vert {u(x) - {P_{{B_{0}}}}(x)} \bigr\vert } ^{p}} \bigl\vert {V(x)} \bigr\vert \\ &{}\times \biggl( { \int_{\Omega}{\frac{1}{{{{\vert {x - y} \vert }^{n - m}}}}{{ \biggl( { \int_{\Omega}{\frac{{\vert {V(z)} \vert }}{{{{\vert {z - y} \vert }^{n - m}}}}} \,dz} \biggr)}^{\frac{1}{{p - 1}}}} \,dy} } \biggr)\,dx \\ \leq&{ \int_{\Omega}{ \bigl\vert {u(x) - {P_{{B_{0}}}}(x)} \bigr\vert } ^{p}} \bigl\vert {V(x)} \bigr\vert \\ &{}\times \biggl( { \int_{M{B_{0}}} {\frac{1}{{{{\vert {x - y} \vert }^{n - m}}}}{{ \biggl( { \int_{M{B_{0}}} {\frac{{\vert {V(z)} \vert }}{{{{\vert {z - y} \vert }^{n - m}}}}} \,dz} \biggr)}^{\frac{1}{{p - 1}}}} \,dy} } \biggr)\,dx \\ \leq&{ \int_{\Omega}{ \bigl\vert {u(x) - {P_{{B_{0}}}}(x)} \bigr\vert } ^{p}} \bigl\vert {V(x)} \bigr\vert \\ &{}\times \biggl( { \int_{B(x,2M{r_{{B_{0}}}})} {\frac{1}{ {{{\vert {x - y} \vert }^{n - m}}}}{{ \biggl( { \int_{B(x,2M{r_{{B_{0}}}})} {\frac{{\vert {V(z)} \vert }}{ {{{\vert {z - y} \vert }^{n - m}}}}} \,dz} \biggr)}^{\frac{1}{{p - 1}}}} \,dy} } \biggr)\,dx \\ \leq&{ \bigl({\eta_{V}}(2M{r_{{B_{0}}}}) \bigr)^{\frac{1}{{p - 1}}}} { \int_{\Omega}{ \bigl\vert {u(x) - {P_{{B_{0}}}}(x)} \bigr\vert } ^{p}} \bigl\vert {V(x)} \bigr\vert \,dx. \end{aligned}$$

Thus,

$$\begin{aligned}& { \int_{\Omega}{ \bigl\vert {u(x) - {P_{{B_{0}}}}(x)} \bigr\vert } ^{p}} \bigl\vert {V(x)} \bigr\vert \,dx \\& \quad \leq c{ \bigl({\eta _{V}}(2M{r_{{B_{0}}}}) \bigr)^{\frac{1}{p}}} { \biggl( {{{ \int_{\Omega}{ \bigl\vert {u(x) - {P_{{B_{0}}}}(x)} \bigr\vert } ^{p}}} \bigl\vert {V(x)} \bigr\vert \,dx} \biggr)^{\frac{{p - 1}}{p}}} { \biggl( {{{ \int_{\Omega}{ \bigl\vert {{D^{m}}u(y)} \bigr\vert } ^{p}}}\,dy} \biggr)^{\frac{1}{p}}}. \end{aligned}$$

It implies (1.1). □

Proof of Theorem 1.6

By using (2.2) and repeating the argument for (1.1), it immediately get (1.2). □

3 Proof of Theorem 1.9

The following $L^{p}$ estimates for the higher-order elliptic equations without potentials are well known, cf. [16, 18].

Lemma 3.1

Consider the equation

$$ \sum_{\vert \alpha \vert \leq2k} {{a_{\alpha}}(x){D^{\alpha}}u} - \lambda u = f \quad \textit{in }\mathbb{R}^{n}. $$

(3.1)

Let $p>1$. There exist a positive ${\lambda_{0}} = {\lambda _{0}}(n,{\Lambda_{1}},{\Lambda_{2}},p)$ and a small $\delta=\delta(n,\Lambda_{1}, \Lambda_{2},p)>0$ so that for the coefficients $a_{\alpha}(x)$ satisfying (1.4)-(1.5) and $(\delta,R)$-vanishing condition for $|\alpha|=2k$, and for $f\in L^{p}(\mathbb{R}^{n})$, if $u\in W^{2k,p}(\mathbb{R}^{n})$ solves equation (3.1), then

$$ \sum_{\vert \alpha \vert \leq2k} {{{ \bigl\Vert {{D^{\alpha}}u} \bigr\Vert }_{{L^{p}}({\mathbb{R}^{n}})}}} \leq c{\Vert f \Vert _{{L^{p}}({\mathbb{R}^{n}})}}, $$

(3.2)

provided $\lambda\geq\lambda_{0}$, where the positive constant c is independent of u and f.

Proof of Theorem 1.9

Let $r_{0}$ be a positive constant which will be chosen later. By the theorem of the partition of unity (e.g., cf. [17] or [20], p.66), there is a sequence of nonnegative functions $\{ {\varphi_{i}}\} _{i = 1}^{\infty}$ in $\mathbb{R}^{n}$ such that

$$ 0 \leq{\varphi_{i}}(x) \leq1; \qquad {\varphi_{i}} \in C_{0}^{\infty}\bigl(B({z_{i}},{r_{0}}) \bigr); \qquad \sum_{i = 1}^{\infty}{ { \varphi_{i}}(x) = 1, \quad x \in{\mathbb{R}^{n}}} $$

and the family of balls $B({z_{i}},{r_{0}})$ has the finite overlapping property. One may obviously note ${\varphi_{i}}u \in {W^{2k, p}}({\mathbb{R}^{n}})$ and $\operatorname{supp}({\varphi_{i}}u) \subset B({z_{i}},{r_{0}})$ and the fact that the ball is a special weak Boman domain. Hence (1.2) also holds for ${\varphi_{i}}u \in W_{0}^{2k, p}(B({z_{i}},{r_{0}}))$. Since $f \in L^{p}(\mathbb{R}^{n})$ and $\sum_{\vert \alpha \vert \leq2k} {{a_{\alpha}}(x){D^{\alpha}}u} \in{L^{p}}({\mathbb{R}^{n}})$ from the boundedness of $a_{\alpha}(x)$, it follows from (1.3) that $Vu \in {L^{p}}({\mathbb{R}^{n}})$. Thus,

$$\begin{aligned} \int_{{\mathbb{R}^{n}}} {{{\vert {Vu} \vert }^{p}}} \,dx &= \int_{{\mathbb{R}^{n}}} {{{ \biggl\vert {\sum_{i} {(V{\varphi_{i}}u)} } \biggr\vert }^{p}}} \,dx \\ &\leq c\sum_{i} { \int_{B({z_{i}},{r_{0}})} {{{ \bigl\vert {V({\varphi_{i}}u)} \bigr\vert }^{p}}} \,dx} \\ &\leq c{\eta_{{V^{p}}}}(2{r_{0}})\sum _{i} { \int_{B({z_{i}},{r_{0}})} {{{ \bigl\vert {{D^{2k}}({ \varphi_{i}}u)} \bigr\vert }^{p}}} \,dx} \\ &\leq c{\eta_{{V^{p}}}}(2{r_{0}})\sum _{i} { \int_{B({z_{i}},{r_{0}})} {{{ \biggl( {\sum_{\vert \alpha \vert = 2k} { \bigl\vert {{D^{\alpha}}({\varphi_{i}}u)} \bigr\vert } } \biggr)}^{p}}} \,dx} \\ &\leq c{\eta_{{V^{p}}}}(2{r_{0}})\sum _{i} { \int_{B({z_{i}},{r_{0}})} {\sum_{\vert \alpha \vert \leq2k} {{{ \bigl\vert {{D^{\alpha}}u} \bigr\vert }^{p}}} } \,dx} \\ &\leq c{\eta_{{V^{p}}}}(2{r_{0}}) \int_{{\mathbb{R}^{n}}} {\sum_{\vert \alpha \vert \leq2k} {{{ \bigl\vert {{D^{\alpha}}u} \bigr\vert }^{p}}} } \,dx. \end{aligned}$$

(3.3)

By Lemma 3.1 and (3.3), we have

$$\begin{aligned}& \sum_{\vert \alpha \vert \leq2k} {{{ \bigl\Vert {{D^{\alpha}}u} \bigr\Vert }_{{L^{p}}({\mathbb{R}^{n}})}}} + {\Vert {Vu} \Vert _{{L^{p}}({\mathbb{R}^{n}})}} \\& \quad \leq c \bigl( {{{\Vert {f - Vu} \Vert }_{{L^{p}}({\mathbb{R}^{n}})}} + {{\Vert {Vu} \Vert }_{{L^{p}}({\mathbb{R}^{n}})}}} \bigr) \\& \quad \leq c \bigl( {{{\Vert f \Vert }_{{L^{p}}({\mathbb{R}^{n}})}} + {{\Vert {Vu} \Vert }_{{L^{p}}({\mathbb{R}^{n}})}}} \bigr) \\& \quad \leq c{\Vert f \Vert _{{L^{p}}({\mathbb{R}^{n}})}} + c{\eta _{{V^{p}}}} {(2{r_{0}})^{\frac{1}{p}}}\sum_{\vert \alpha \vert \leq2k} {{{ \bigl\Vert {{D^{\alpha}}u} \bigr\Vert }_{{L^{p}}({\mathbb{R}^{n}})}}}. \end{aligned}$$

Choosing $r_{0}>0$ such that $c\eta_{V^{p}}(2r_{0})^{1/p}\leq1/2$, (1.6) is obtained. □