Universal potential estimates for $1 < p\leq 2-\frac{1}{n}$

1.   Introduction

We are concerned here with the quasilinear elliptic equation with measure data

in a bounded open subset $\Omega$ of $\mathbb{R}^n$, $n\geq 2$. Here $\mu$ is a finite signed measure in $\Omega$ and the nonlinearity $A = (A_1, \dots, A_n):\mathbb{R}^n\times \mathbb{R}^n\to \mathbb{R}^n$ is vector valued function. Throughout the paper, we assume that there exist $\Lambda\geq 1$ and $p > 1$ such that

for every $x\in \mathbb{R}^n$ and every $(\xi, \eta)\in \mathbb{R}^n\times \mathbb{R}^n\backslash\{(0, 0)\}$. More regularity assumptions on function $x\mapsto A(x, \xi)$ will be needed later.

A typical example of (1.1) is the $p$-Laplace equation with measure data

Since the seminal work of Kilpeläinen and Malý ^[7] (see also ^[16] for a different approach), the study of pointwise behaviors of solutions to quasilinear equations with measure data (1.1) has undergone substantial progress. In particular, the series of works ^[4,5,9] (see also ^[12]) provide interesting pointwise bounds for gradients of solutions to the seemingly unwieldy Eq (1.1), at least for $p > 2-\frac{1}{n}$. These pointwise gradient bounds have been extended recently in ^[3,14,15] for the more singular case $1 < p\leq 2- \frac{1}{n}$.

On the other hand, a more unified approach to pointwise bounds for solutions and their gradients was presented in ^[8]. The results of ^[8] give pointwise bounds not only for the size but also for the oscillation of solutions and their derivatives expressed in terms of bounds by linear or nonlinear potentials in certain Calderón spaces. These cover different kinds of pointwise fractional derivative estimates as well as estimates for (sharp) fractional maximal functions of the solutions and their gradients.

However, the treatment of ^[8] is still confined to the range $p > 2-\frac{1}{n}$, and the purpose of this note is to extend it to the singular case $1 < p\leq 2-\frac{1}{n}$. Note that, for $1 < p\leq 2-\frac{1}{n}$, by looking at the fundamental solution we see that in general distributional solutions of (1.4) may not even belong to $W^{1, 1}_{\rm loc}(\Omega)$.

Thus in this paper we shall restrict ourselves only to the case

and note that the main results obtained here also hold in the case $2-\frac{1}{n} < p < 2$ thanks to ^[8]. Moreover, except for the comparison estimates obtained earlier in ^[13,15], the methods used in this paper are very much guided by those of ^[8]. We would also like to point out that there are analogous results in the case $p\ge 2$ that we refer to ^[8] for the precise statements.

In some sense our pointwise regularity for the non-homogeneous equation (1.1) is obtained from perturbation/interpolation arguments involving the associated homogeneous equations. Thus information on the regularity of associated homogeneous equations will play an important role. In this direction, we first recall a quantitative version of the well-known De Giorgi's result that established $C^{\alpha_0}$, $\alpha_0\in (0, 1)$, regularity for solutions of ${\rm div}\left({A(x, \nabla w)} \right) = 0$. Henceforth, by $Q_r(x_0)$ we mean the open cube $Q_r(x_0): = x_0+ (-r, r)^n$ with center $x_0\in \mathbb{R}^n$ and side-length $2r$. In other words,

Lemma 1.1. Under (1.2)–(1.3), let $w\in W^{1, p}({\Omega})$, $p > 1$, be a solution of the equation ${\rm div}\left({A(x, \nabla w)} \right) = 0$ in ${\Omega}$. Then there exists $\alpha_0\in (0, 1)$, depending only on $n, p$ and $\Lambda$, such that for any cubes $Q_\rho(x_0)\subset Q_R(x_0)\subset{\Omega}$, and ${\epsilon}\in (0, 1]$, we have

and

We point out that the proof of (1.5) follows from [6,Chapter 7], whereas the proof of (1.6) follows from (1.5) and the reverse Hölder property of $w$.

In the case the nonlinearity $A(x, \xi)$ is independent of $x$, we actually have $C^{1, \beta_0}$, $\beta_0\in (0, 1)$, regularity the homogeneous equation (see, e.g., ^[2,10,11]). For our purpose, we shall use the following quantitative version of this regularity result (see ^[3,5]).

Lemma 1.2. Let $v\in W^{1, p}({\Omega})$, $p > 1$, be a solution of $\operatorname{div}\left({A_0(\nabla v)} \right) = 0$ in ${\Omega}$, where $A_0(\xi)$ satisfies (1.2)–(1.3) and is independent of $x$. Then there exists $\beta_0\in (0, 1)$, depending only on $n, p$ and $\Lambda$, such that for any cubes $Q_\rho(x_0)\subset Q_R(x_0)\subset{\Omega}$ and ${\epsilon}\in (0, 1]$, we have

and

In what follows, we shall use the (maximal) constants $\alpha_0$ in Lemma 1.1 and $\beta_0$ in Lemma 1.2 as certain thresholds in our regularity theory. Also, henceforth, we reserve the letter $\kappa$ for the following constant

Our first result provides a De Giorgi's theory for non-homogeneous equations with measure data, which also includes [15,Theorem 1.4] as an end-point case. For the case $p > 2-1/n$, see [8,Theorem 1.1].

Theorem 1.1. Under (1.2)–(1.3), with $1 < p \leq 2- \frac{1}{n}$, let $\kappa$ be as in (1.8), and suppose that $u\in C^0(\Omega)\cap W^{1, p}_{\rm loc}({\Omega})$ is a solution of (1.1). Let $Q_R(x_0)\subset\Omega$ and $\bar{\alpha} \in (0, \alpha_0)$, where $\alpha_0$ is as in Lemma 1.1. Then for any $x, y\in Q_{R/8}(x_0)$ we have

uniformly in $\alpha\in [0, \bar{\alpha}]$. Here the implicit constant depends only on $n, p, \Lambda$, and $\bar{\alpha}$.

In (1.9), the function ${\bf W}_{1-\alpha (p-1)/p, p}^{R}(|\mu|)(\cdot)$ is a truncated Wolff's potential of $|\mu|$. In general, given a nonnegative measure $\nu$ and $\rho > 0$, the Wolff's potential ${\bf W}^{\rho}_{\alpha, s} \nu$, $\alpha > 0, s > 1$, is defined by

Note that ${\bf W}^{\rho}_{\alpha, 2} (\nu) = {\bf I}^{\rho}_{2\alpha} (\nu)$, where ${\bf I}^{\rho}_{\gamma} (\nu)$, $\gamma > 0$, is a truncated Riesz's potential defined by

We remark that, except for (1.2)–(1.3), no further regularity assumption is needed in Theorem 1.1. However, this will force the constant $\bar{\alpha}$ to be small in general.

On the other hand, it is possible to allow $\bar{\alpha}$ to be arbitrarily close to $1$ as long as we further impose a 'small BMO' condition on the map $x\mapsto A(x, \xi)$. This condition entails the smallness of the limit $\limsup_{\rho\rightarrow 0} {\omega}(\rho)$, where

and for each cube $Q_r(y)$ we set

with $\overline{A}_{Q_r(y)}(\xi) = \mathit{{\rlap{-} \smallint }}_{Q_r(y)}A(x, \xi)dx$. The precise statement is as follows.

Theorem 1.2. Under (1.2)–(1.3), with $1 < p \leq 2- \frac{1}{n}$, let $\kappa$ be as in (1.8), and suppose that $u\in C^0(\Omega)\cap W^{1, p}_{\rm loc}({\Omega})$ is a solution to (1.1). Let $Q_{R}(x_0)\subset{\Omega}$. Then for any positive $\bar{\alpha} < 1$ there exists a small $\delta = \delta(n, p, \Lambda, \bar{\alpha}) > 0$ such that if

then for any $x, y\in Q_{R/8}(x_0)\subset{\Omega}$, we have

uniformly in $\alpha\in [0, \bar{\alpha}]$. Here the implicit constant depends on $n, p, \Lambda, \bar{\alpha}, {\omega}(\cdot)$, and ${\rm diam}({\Omega})$.

Under a certain Dini-VMO condition, we could also allow $\bar{\alpha} = 1$ in the above theorem. However, in this case the Wolff's potential is replaced with a Riesz's potential raised to the power of $\frac{1}{p-1}$.

Theorem 1.3. Under (1.2)–(1.3), with $1 < p \leq 2- \frac{1}{n}$, let $\kappa$ be as in (1.8), and suppose that $u\in C^1({\Omega})$ is a solution to (1.1). Let $Q_{R}(x_0)\subset{\Omega}$. If for some $\sigma_1\in (0, 1)$ such that ${\omega}(\cdot)^{\sigma_1}$ is Dini-VMO, i.e.,

then for any $x, y\in Q_{R/8}(x_0)\subset{\Omega}$, we have

uniformly in $\alpha \in [0, 1]$. Here the implicit constant depends on $n, p, \Lambda, \bar{\alpha}, \sigma_1, {\omega}(\cdot)$, and ${\rm diam}({\Omega})$.

We remark that, when $\alpha = 1$, Theorem 1.3 recovers the pointwise gradient estimates of ^[3] and ^[15] that were obtained under a slightly different Dini condition.

Finally, under a stronger Dini-Hölder condition we can also bound solution gradients in appropriate Calderón spaces.

Theorem 1.4. Under (1.2)–(1.3), with $1 < p \leq 2- \frac{1}{n}$, let $\kappa$ be as in (1.8), and suppose that $u\in C^1({\Omega})$ is a solution to (1.1). Let $Q_{R}(x_0)\subset{\Omega}$. If for some $\sigma_1\in (0, 1)$ such that ${\omega}(\cdot)^{\sigma_1}$ is Dini-Hölder of order $\bar{\alpha}$, i.e.,

for some $\bar{\alpha}\in [0, \beta_0)$, then for any $x, y\in Q_{R/4}(x_0)\subset{\Omega}$, we have

uniformly in $\alpha \in [0, \bar{\alpha}]$. Here $\beta_0$ is as in Lemma 1.2, and the implicit constant depends on $n, p, \Lambda, \bar{\alpha}, \sigma_1, {\omega}(\cdot)$, and ${\rm diam}({\Omega})$.

2.   Comparison and Poincaré type inequalities

The study of regularity problems for Eq (1.1) is based on the following comparison estimate that connects the solution of measure datum problem to a solution of a homogeneous problem.

To describe it, we let $u\in W_{\rm loc}^{1, p}(\Omega)$ be a solution of (1.1). Then for a cube $Q_{2R} = Q_{2R}(x_0) \Subset\Omega$, we consider the unique solution $w\in W_0^{1, p}(Q_{2R}(x_0))+u$ to the local interior problem

Lemma 2.1. Suppose that $Q_{3 R}(x_0)\subset{\Omega}$ for some $R > 0$. Let $u$ and $w$ be as in (2.1) and let $\kappa$ be as in (1.8), where $1 < p \leq 2-\frac{1}{n}$. Then it holds that

Proof. For $1 < p\leq \frac{3n-2}{2n-1}$, inequality (2.2) was obtained in [15,Theorem 1.2]. For $\frac{3n-2}{2n-1} < p\leq 2-\frac{1}{n}$, by [13,Lemma 2.2], we have

for some $\gamma_0 \in [\frac{2-p}{2}, \frac{n(p-1)}{n-1})$. In fact, an inspection of the proof of [13,Lemma 2.2] reveals that we can take any $\gamma_0 \in (\frac{n}{2n-1}, \frac{n(p-1)}{n-1})$. Thus we may assume that $\kappa = (p-1)^2/2 < \gamma_0$. To conclude the proof, it is therefore enough to show that

To this end, let $\gamma_1\in (\gamma_0, \frac{n(p-1)}{n-1})$. By [15,Corollay 2.4], we have

for any $\lambda\in\mathbb{R}$ and any cube $Q_\rho(x)$ such that $Q_{9\rho/8}(x)\subset{\Omega}$.

Now, with $Q_{8\rho/7}(x)\subset{\Omega}$, let $w_1$ be the unique solution $w_1\in W_0^{1, p}(Q_{8\rho/7}(x))+u$ to the problem

Then from the proof of [13,Lemma 2.2] (using (2.8) and (2.18) in ^[13]), we can deduce that

By Young's inequality, this yields

Thus by quasi-triangle and Hölder's inequalities we get

where we choose $\lambda = \mathit{{\rlap{-} \smallint }}_{Q_{9\rho/8}(x)} w_1 dz$.

We now use Poincaré and the reverse Hölder's inequalities for $\nabla w_1$ to obtain that

where we used [13,Lemma 2.2] and Young's inequality in the last bound.

Thus combining this result with (2.7) we find

At this point, plugging this into (2.4) we arrive at

which holds for any cube $Q_{ \rho}(x)$ such that $Q_{ 8\rho/7}(x)\subset{\Omega}$. Recall that $\gamma_1 > \gamma_0$, and thus by a covering/iteration argument as in [6,Remark 6.12], we have

for any $\epsilon > 0$. This obviously yields (2.3) as desired and the proof is complete.

Remark 2.1. Using the above argument, in particular (2.5), we can also show the following comparison estimate for the functions $u$ and $w$: for any $\frac{3n-2}{2n-1} < p\leq 2-\frac{1}{n}$,

and

for any $\lambda\in\mathbb{R}$. For $1 < p\leq \frac{3n-2}{2n-1}$, these inequalities have been obtained in [15,Theorem 1.2].

The following Poincaré type inequality was obtained in the case $1 < p\leq \frac{3n-2}{2n-1}$ in [15,Corollary 1.3]. A similar proof using Lemma 2.1 and inequalities of the form (2.6) and (2.8) also yields the result in the case $\frac{3n-2}{2n-1} < p\leq 2-\frac{1}{n}.$

Corollary 2.1. Suppose that $Q_{3r/2}(x_0)\subset{\Omega}$ for some $r > 0$. Let $u\in W^{1, p}_{\rm loc}({\Omega})$, $1 < p\leq 2- \frac{1}{n}$, be a solution of (1.1). Then for any $\epsilon > 0$ we have

With $u$ and $w$ as in (2.1), we now consider another auxiliary function $v$ such that $v\in W^{1, p}_{0}(Q_R(x_0))+w$ is the unique solution to the equation

where $\overline{A}_{Q_R(x_0)}(\xi) = \mathit{{\rlap{-} \smallint }}_{Q_R(x_0)}A(x, \xi)dx$.

The following result can be deduced from [8,Lemma 2.3] and an appropriate reverse Hölder's inequality.

Lemma 2.2. Let $p > 1$, $0 < {\epsilon}\leq p$, and $u, w$, and $v$ be as in (2.1) and (2.9), where $Q_{2R}(x_0)\Subset{\Omega}$. Then there exists a small positive constant $\sigma_0 > 0$ such that

where ${\omega}(\cdot)$ is as defined in (1.10).

Likewise, following lemma follows from [8,Lemma 2.5].

Lemma 2.3. Let $1 < p < 2$, $0 < {\epsilon}\leq p$, and $u, w$, and $v$ be as in (2.1) and (2.9), where $Q_{2R}(x_0)\Subset{\Omega}$. Then for any $\sigma_1\in (0, 1)$ such that ${\omega}(\cdot)^{\sigma_1}$ is Dini-VMO, i.e., (1.13) holds, it follows that

3.   Pointwise fractional maximal function bounds

As in ^[8], our proofs of Theorems 1.1–1.4 are based on the corresponding pointwise estimates for the associate fractional and sharp fractional maximal functions, which are interesting in their own right. This section is devoted to such pointwise fractional maximal function bounds.

Given $R > 0$ and $q > 0$, following ^[1], we define the following truncated sharp fractional maximal function of a function $f\in L^q_{\rm loc}(\mathbb{R}^n)$:

Also, we define a truncated fractional maximal function by

In the case $q = 1$, we usually drop the index $q$ in the above notation, i.e., we set ${\bf M}^{\#, R}_{\alpha, 1}(f) = {\bf M}^{\#, R}_{\alpha}(f)$ and ${\bf M}^{R}_{\beta, 1}(f) = {\bf M}^{R}_{\beta}(f)$. Moreover, the definition of ${\bf M}^{R}_{\beta}(f)$ can also be naturally extended to the case where $f = \mu$ is a locally finite signed measure in $\mathbb{R}^n$:

Note that by Poincaré inequality we have

for any $f\in W^{1, 1}_{\rm loc}(\mathbb{R}^n)$.

On the other hand, if $u\in W^{1, p}_{\rm loc}({\Omega})$, $1 < p\leq 2-\frac{1}{n}$, then it follows from Corollary 2.1 that

for any ${\epsilon} \in (0, 1)$ and any cube $Q_{3R/2}(x)\subset{\Omega}$.

The following fractional maximal function bound will be needed in the proof of Theorem 1.1.

Theorem 3.1. Under (1.2)–(1.3), let $1 < p \leq 2-\frac{1}{n}$, and suppose that $u\in W^{1, p}_{\rm loc}({\Omega})$ is a solution of (1.1). Let $Q_{3R}(x)\subset\Omega$ and $\bar{\alpha} \in (0, \alpha_0)$, where $\alpha_0\in (0, 1)$ is as in Lemma 1.1. Then we have

uniformly in $\alpha\in [0, \bar{\alpha}]$. Here the implicit constant depends on $n, p, \Lambda$, and $\bar{\alpha}$.

Proof. The main idea of the proof of (3.2) lies the proof of [8,Proposition 3.1] that treated the case $p > 2-\frac{1}{n}$. Note that by (3.1) it is enough to show

for some ${\epsilon} = {\epsilon}_1(n, p, \Lambda, \bar{\alpha})\in (0, 1)$.

Let $0 < \rho\leq r\leq R$, and choose $w$ as in (2.1) with $Q_{2r}(x)$ in place of $Q_{2R}(x_0)$. We have

where we used the inequality

which is a modified version of [8,Theorem 2.2], in the second inequality.

Thus by Lemma 2.1 we get

Let $\epsilon \in (0, 1)$, and choose $\rho = \epsilon r$. Then by Young's inequality we have

Multiplying both sides by $(\epsilon r)^{1-\alpha}$, $0 < \alpha\leq \bar{\alpha} < \alpha_0$, and taking the supremum with respect to $r\in(0, R]$, we find

We now choose $\epsilon \in (0, 1)$ such that

to deduce that

This is (3.3) and the proof is complete.

The following result will be needed for the proof of Theorem 1.2.

Theorem 3.2. Let $1 < p\leq 2-\frac{1}{n}$ and $u\in C^0({\Omega})$ be a solution to (1.1). Suppose that $Q_{3R}(x)\subset{\Omega}$. Then for any positive $\bar{\alpha} < 1$ there exists a small $\delta = \delta(n, p, \Lambda, \bar{\alpha}) > 0$ such that if (1.11) holds, then the estimate

holds uniformly in $\alpha\in[0, \bar{\alpha}]$. Here the implicit constant depends on $n, p, \Lambda, \bar{\alpha}, {\omega}(\cdot)$, and ${\rm diam}({\Omega})$.

Proof. The proof is similar to that of Theorem 3.1, but this time we need to use Lemma 2.2. As above, by (3.1) it is enough to show (3.3) for some ${\epsilon} = {\epsilon}_1(n, p, \Lambda, \bar{\alpha})\in (0, 1)$. Let $0 < \rho\leq r\leq R$, and choose $w$ as in (2.1) with $Q_{2r}(x)$ in place of $Q_{2R}(x_0)$. Then choose $v$ as in (2.9) with $Q_{r}(x)$ in place of $Q_{R}(x_0)$. This time we have

Here we used

which is a a modified version of (2.6) in [8,Theorem 2.1] in the second inequality.

Then by Lemma 2.2 we get

for a small constant $\sigma_0 > 0$. Thus using Lemma 2.1 and the fact that ${\omega}(r)\leq 2 \Lambda$, we find

Let $\epsilon \in (0, 1)$, and choose $\rho = \epsilon r$. Then by Young's inequality we have

Multiplying both sides by $(\epsilon r)^{1-\alpha}$, $0 < \alpha\leq \bar{\alpha} < 1$, and taking the supremum with respect to $r\in(0, R]$, we find

We now choose $\epsilon \in (0, 1)$ such that

and then choose $\bar{R} = \bar{R}(n, p, \Lambda, \bar{\alpha}, {\omega}(\cdot)) > 0$ and a small $\delta = \delta(n, p, \Lambda, \bar{\alpha}) > 0$ in (1.11) such that

Then it follows that

provided $R\leq \bar{R}$. Hence, for $R\leq \bar{R}$, we deduce that

This proves (3.3) in the case $R\leq \bar{R}$. For $R > \bar{R}$, we observe that

Thus we also obtain (3.3) in the case $R > \bar{R}$ as long as we allow the implicit constant to depend on ${\rm diam}(\Omega)$, and $n, p, \Lambda, \bar{\alpha}, {\omega}(\cdot)$.

In order to prove Theorem 1.3, we need the following pointwise fractional maximal function bound.

Theorem 3.3. Let $1 < p\leq 2-\frac{1}{n}$ and $u\in C^1({\Omega})$ be a solution to (1.1). Suppose that $Q_{3R}(x)\subset{\Omega}$. If for some $\sigma_1\in (0, 1)$ such that ${\omega}(\cdot)^{\sigma_1}$ is Dini-VMO, i.e., (1.13) holds, then the estimate

holds uniformly in $\alpha\in[0, 1]$. Here the implicit constant depends on $n, p, \Lambda, \bar{\alpha}, {\omega}(\cdot)$, $\sigma_1$, and ${\rm diam}({\Omega})$.

Proof. As in the proof of Theorem 3.2, it is enough to show

Moreover, we may assume that $R\leq \bar{R}$, where $\bar{R} = \bar{R}(n, p, \Lambda, \sigma_1, {\omega}(\cdot)) > 0$ is to be determined.

Arguing as in the proof of (3.5), but this time using (1.7) (in Lemma 1.2) instead of (3.4) and Lemma 2.3 instead of Lemma 2.2, we have for $Q_\rho(x)\subset Q_r(x) \subset Q_{3r}(x)\subset{\Omega}$,

Here ${\bf q}_{Q_\rho(x)}\in\mathbb{R}^n$ is defined by

That is, ${\bf q}_{Q_\rho(x)}$ is a vector such that

Note that for $Q_\rho(x)\subset Q_s(x) \Subset{\Omega}$, one has

and also

For brevity, for any $j = 0, 1, 2, \dots,$ and $Q_{3R}(x)\subset{\Omega}$, we now define

where ${\epsilon}\in (0, 1/3)$ is to be determined, and

Then applying (3.6) with $\rho = {\epsilon} R_j < r = R_j/3$ we have

By quasi-triangle inequality, this yields

We now choose ${\epsilon}$ sufficiently small so that $c_1 {\epsilon}^{\beta_0 }\leq 1/4$ and then restrict $R\leq \bar{R}$, where $\bar{R} = \bar{R}(n, p, \Lambda, \sigma_1, {\omega}(\cdot)) > 0$ is such that

Then we have

Summing this up over $j\in\{0, 1, , \dots, m-1\}$, $m\in \mathbb{N}$, we get

Hence,

On the other hand, for any $m\in\mathbb{N}$, by (3.8) we can write

and therefore in view of (3.7),

At this point, multiplying both sides of the above inequality by $R_{m+1}^{1-\alpha}$, $m\in \mathbb{N}$, we deduce that

Thus,

We next further restrict $\bar{R}$ so that for any $R\leq \bar{R}$,

This is possible because we have

where we used the fact that ${\omega}(\rho_1)\leq c\, {\omega}(\rho_2)$ provided $\rho_1\leq \rho_2\leq C \rho_1$, $C > 1$.

Then by an induction argument we deduce from (3.11) that

for every integer $m\geq 0$.

Let us call the right-hand side of (3.13) by ${\bf Q}$. Then from (3.10) and simple manipulations we obtain

which by (3.13) yields

As $R_{0}^{1-{\alpha}} A_{0}\leq c {\bf Q}$, by iteration we get

for every integer $m\geq 0$.

To conclude the proof, we observe that

where we used (3.13) and (3.14) in the last inequality. Then recalling the definition of ${\bf Q}$ and using Young's inequality we obtain

This completes the proof of the theorem. The following pointwise sharp fractional maximal function bound will be used in the proof of Theorem 1.4.

Theorem 3.4. Let $1 < p\leq 2-\frac{1}{n}$ and $u\in C^1({\Omega})$ be a solution to (1.1). Suppose that $Q_{3R}(x)\subset{\Omega}$. If for some $\sigma_1\in (0, 1)$ such that

for some $\bar{\alpha}\in [0, \beta_0)$, then the estimate

holds uniformly in $\alpha\in[0, \bar{\alpha}]$. Here $\beta_0$ is as in Lemma 1.2, and the implicit constant depends on $n, p, \Lambda, \bar{\alpha}, {\omega}(\cdot), \sigma_1, K$, and ${\rm diam}({\Omega})$.

Remark 3.1. Condition (3.15) implies the Dini-VMO condition (1.13). In turns, (1.13) implies (1.11), whereas (3.15) is implied by the Dini-Hölder condition (1.14).

Proof. It suffices to show

for $R\leq 1$, where the implicit constant depends on $n, p, \Lambda, \bar{\alpha}, {\omega}(\cdot), \sigma_1, K$, and ${\rm diam}({\Omega})$.

With the notation used in proof of Theorem 3.3, multiplying both sides of (3.9) by $R_{j+1}^{-{\alpha}}$, $j\geq 0$, we have

This time we choose ${\epsilon}\in (0, 1/3)$ such that

and employ (3.15) together with the restriction $R_j\leq 1$, to deduce

On the other hand, applying Theorem 3.3 in the case ${\alpha} = 1$, we can bound

for every integer $j\geq 0$. Thus, using (3.16) and Young's inequality we get

Iterating this inequality, we find for any $m\in \mathbb{N}$,

In view of the fact that

this completes the proof of the theorem.

4.   Proof of Theorem 1.1

Proof of Theorem 1.1. For any cube $Q_\rho(x)\Subset{\Omega}$, let $q_{Q_\rho(x)}\in\mathbb{R}$ be defined by

i.e., $q_{Q_\rho(x)}$ is a real number such that

Then using quasi-triangle inequality a few times and Lemma 1.1, we have for $Q_\rho(x)\subset Q_{r}(x) \subset Q_{3r}(x) \subset{\Omega}$,

Here we choose $w$ as in (2.1) with $Q_{2r}(x)$ in place of $Q_{2R}(x_0)$.

We now apply Remark 2.1 to bound the second term on the right-hand side of the above inequality. This yields that

Letting $\rho = {\epsilon} r$, ${\epsilon} \in (0, 1)$, and using Young's inequality we find

Next, we choose ${\epsilon} \in (0, 1/3)$ small enough so that $C\epsilon^{\alpha_0}\leq \frac{1}{2}$, where $C$ is the constant in (4.1). Let $Q_{R}(x_0)\subset{\Omega}$ be as given in the theorem. Then for any cube $Q_\delta(x)\subset Q_{R}(x_0)$ we set $\delta_j = \epsilon^{j} \delta$, $Q_j = Q_{\delta_j}(x)$, $q_j = q_{Q_j}$, $j\geq 0$, and define

Applying (4.1) with $r = \delta_j/3$ yields

Summing this up over $j\in \{1, 3, ..., m-1\}$, we obtain

As in (3.8), we have

for all integers $j\geq 1$, and thus

holds for every integer $m\geq 2$. Here we use the simple fact (see (3.7)) that

Now for $x, y\in Q_{R/8}(x_0)$ we choose

Note that $\delta < R/8$ and $Q_{\delta}(y)\subset Q_{3\delta}(x)\subset Q_{R/2}(x_0)$. Then applying (4.2), we have

Sending $m\rightarrow \infty$ and using [1,Lemma 4.1], we get

Since $u-m, m\in\mathbb{R},$ is also a solution of (1.1), it follows that

Likewise, we have

Now choosing $m = q_{Q_{3\delta}(x)}$ we find

On the other hand, by Theorem 3.1 and the fact that $3\delta < 3R/8,$ we have

where we used a Caccioppoli type inequality of [15,Corollary 2.4] in the last bound.

Combining inequalities (4.3) and (4.4), we complete the proof of the theorem.

5.   Proof of Theorems 1.2 and 1.3

Proof of Theorems 1.2. The main idea of the proof of Theorem 1.2 lies in the proof of [8,Theorem 1.2]. First, in view of Theorem 1.1, it suffices to prove (1.12) uniformly in ${\alpha}\in [{\alpha}_0/2, \bar{\alpha}]$, $\bar{\alpha} < 1$, for all $x, y\in Q_{R/8}(x_0)$.

On the other hand, for a.e. $x, y\in Q_{R/8}(x_0)$ and $f\in L^{\kappa}(Q_R(x_0))$, we have the inequality

provided $\alpha\in (0, 1]$. See inequalities (4.9) and (4.10) in ^[1]. Applying this with $f = u$ and ${\alpha}\in [{\alpha}_0/2, \bar{\alpha}]$, and using Theorem 3.2, we obtain

Then invoking the Caccioppoli type inequality of [15,Corollary 2.4] we obtain (1.12) uniformly in ${\alpha}\in [{\alpha}_0/2, \bar{\alpha}]$ as desired.

Proof. (Proof of Theorem 1.3) The proof of Theorem 1.3 is similar to that of Theorems 1.2, but this time we use Theorem 3.3 instead of Theorem 3.2.

6.   Proof of Theorem 1.4

Proof of Theorem 1.4. Let $Q_R(x_0)\subset{\Omega}$ be as given in the theorem. For any $x, y\in Q_{R/4}(x_0)$, we set $\delta = \frac{1}{2}|x-y|_\infty$. Note that ${\delta} < R/4$ and $Q_\delta(y)\subset Q_{3\delta}(x)\subset Q_{R}(x_0)$. We shall keep the notation in the proof of Theorem 3.3 except that we replace $R$ with ${\delta}$ so that $R_j = {\delta}_j = {\epsilon}^j {\delta}$, $Q_j = Q_{{\epsilon}^j {\delta}}(x)$, ${\bf q}_j = {\bf q}_{Q_{{\epsilon}^j {\delta}}(x)}$, and

for all integers $j\geq 0$.

Then by choosing ${\epsilon}$ in (3.9) such that $c_1 {\epsilon}^{\beta_0 }\leq 1/2$, we have

Summing this up over $j\in\{0, 1, , \dots, m-1\}$, $m\in \mathbb{N}$, and then simplifying, we get

On the other hand, by (3.8),

which holds for any ${\bf m}\in\mathbb{R}^n$ and integer $m\geq 0$.

Hence, it follows that

Then using

which can be proved as in (3.7), we get

We next set

Then applying Theorem 3.3 with ${\alpha} = 1$ and $3R = {\delta}$, we have

Plugging this into the last bound for $|{\bf q}_{m+1}-{\bf m}|$ we deduce that

Also, note that as in (3.12) we have

At this point, using the Dini-Hölder condition (1.14), we obtain, after some simple manipulations,

Here we also used that ${\delta} < R/4 < {\rm diam}({\Omega})$ and the implicit constants are allowed to depend on ${\rm diam}({\Omega})$.

Thus letting $m\rightarrow \infty$ and using Young's inequality we obtain

Likewise, we also have

where we used that $Q_{\delta}(y)\subset Q_{3{\delta}}(x)$.

Combining these two estimates and choosing ${\bf m} = {\bf q}_{Q_{3{\delta}}(x)}$, we find

As ${\delta} < R/4$ and $Q_{R/4}(x)\cup Q_{R/4}(y) \subset Q_{R}(x_0)$, we can apply Theorem 3.3 with ${\alpha} = 1$ to have the bound

Similarly, we can use Theorem 3.4 to bound the last term on the right-hand of (6.1) as follows:

We now plug estimates (6.2) and (6.3) into (6.1) to arrive at

This completes the proof because $\delta\leq \frac{1}{2}|x-y|$.

Remark 6.1. In Theorems 1.3–1.4 and 3.3–3.4 we may take $\sigma_1 = 1$ in (1.13), (1.14) and (3.15), provided we replace ${\omega}$ with a non-decreasing function $\tilde{{\omega}}:[0, 1]\rightarrow [0, \infty)$ such that

for all $x, y, \xi\in \mathbb{R}^n$, $|x-y|\leq 1$. The reason is that in this case solutions to (2.1) are locally Lipschitz, and we can also take $\sigma_1 = 1$ in Lemma 2.3; see [8,Section 8].

Acknowledgments

Q. H. N. is supported by the Academy of Mathematics and Systems Science, Chinese Academy of Sciences startup fund, and the National Natural Science Foundation of China (No. 12050410257 and No. 12288201) and the National Key R & D Program of China under grant 2021YFA1000800. N. C. P. is supported in part by Simons Foundation (award number 426071).

Conflict of interest

The authors declare no conflict of interest.