Gradient estimate of a variable power for nonlinear elliptic equations with Orlicz growth

Definition 1.1

A Orlicz space L^Φ(Ω) is the set of all measurable functions f : Ω → ℝ such that

Therefore, L^Φ(Ω) is a Banach space equipped with the Luxemburg norm

A Orlicz-Sobolev space W^1,Φ(Ω) is the set of all measurable functions f ∈ L^Φ(Ω) for which Df ∈ L^Φ(Ω,ℝⁿ) with the norm ∥f∥_W^1,Φ(Ω) = ∥f∥_L^Φ(Ω) + ∥Df∥_{L^Φ(Ω,ℝⁿ)}. For more details about Orlicz spaces, we refer to [14, 15, 28, 29].

The equation of generalized p-Laplacian type is arising in the fields of fluid dynamics, magnetism and mechanics. Particularly, if φ(s) = s^p−1 for 1 < p < ∞, our problem becomes the p-Laplace equation. For this case, there is a great deal of literature concerning the regularity of weak solutions, for instance, see [4, 9, 11, 13, 24]. While φ(s) = s^p−1 + as^q−1, where 1 < p < q and a is a positive constant, Filippis and Mingione [18] obtained a global estimate in the setting of Lebesgue spaces, which is covered by our problem (1.1). We also refer to [8] for the existence of solutions to the (p, q)-Laplace equations.

In recent years, there have been significant advances of this type of problem in the regularity theory, see for example [12, 14, 34, 35] since starting with a seminal paper of Lieberman [23]. In 2011, Verde [34] studied the elliptic system

where A(x, ξ) = G(x, ξ) = φ(|ξ|)|ξ|ξ with φ(s) defined as in (1.3). He proved the global Calderón-Zygmund estimates over whole domain ℝⁿ for the weak solutions. Very recently, Yao and Zhou [35] obtained local L^q-estimates for weak solutions of (1.1), which implies the fact that

Also, Byun and Cho [12] further extended it to a global gradient estimate in the setting of Orlicz spaces under the assumption that the boundary of underlying domain is Reifenberg flat. We would like to point out that Cho [14] also considered the zero-Dirichlet problem of (1.1) for A defined as in (1.2) and G(x, ξ) = φ(|ξ|)|ξ|ξ , who obtained a global Calderón-Zygmund estimate in the setting of Orlicz spaces. We also refer to [6, 7, 17, 36] for a further study of these problems with Orlicz growth. In particular, Baroni and Lindfors [7], and Yao [36] studied parabolic problems with Orlicz growth.

In this paper we are to prove a global Calderón-Zygmund type estimate in the Lorentz spaces for general nonlinear elliptic equations (1.1). As we know, there are mainly three kinds of different arguments to handle the Calderón-Zygmund theory for elliptic and parabolic problems with VMO or small BMO discontinuous coefficients except for a classical technique by using singular integral operators and their commutators. The first one is the so-called geometric method originally traced from Byun and Wang’s work in [11], which is based on the weak compactness, the Hardy-Littlewood maximal operators and the modified Vitali covering. Here, the so-called modified Vitali covering actually refers to the argument as “crawling of ink spots” as in the early papers by Safonov and Krylov [20, 30]. Indeed, this is also a development from Caffarelli and Peral’s paper in [13]. Secondly, Kim and Krylov [19, 21] gave a unified approach of studying L^p solvability for elliptic and parabolic problems due to the Fefferman-Stein theorem, which is mainly based on the sharp functions. In this present paper, we have to highlight the third technique being called large-M-inequality principle originating from Acerbi and Mingione’s work [1, 2], which is directly based on arguing on certain Calderón-Zygmund-type covering instead of the boundedness of a maximal function operator and the so-called good-λ-inequality used by Byun and Wang [11] and Kim and Krylov’s papers [19, 21].

Here we are revising the so-called large-M-inequality principle and geometric method to get an estimate in the Lorentz spaces for the variable power of the gradients to (1.1). Regarding what we consider, we would like to point out that Byun, Ok and Wang [10] studied the zero-Dirichlet problem of linear elliptic system in divergence form under the assumptions that their coefficients are partially BMO and the variable exponent p(x) is log-Hölder continuous. They first obtained a global Calderón-Zygmund estimate in the variable exponent Lebesgue spaces:

Also, Tian and Zheng in [32] further extended the above result to the global Calderón-Zygmund type estimate in the Lorentz spaces for a variable power of the gradients of weak solutions. Note that the Lorentz space is a two-parameter scale of the Lebesgue space obtained by refining it in the fashion of a second index, and there are a lot of research activities on Lorentz regularity for partial differential equations. For examples, the first estimates in the Lorentz spaces are obtained by Mingione [27]. Later, Mengesha and Phuc [24] derived the weighted Lorentz estimate to quasilinear p-Laplacian type equations based on the geometric approach. In 2014, Zhang and Zhou [39] extended the above result in [24] to that for a general equation of p(x)-Laplacian also using a geometrical argument. Adimurthil and Phuc [3] proved the global Lorentz and Lorentz-Morrey estimates for quasilinear equations below the natural exponent. Meanwhile, Baroni [4, 5] obtained interior Lorentz estimates to evolutionary p-Laplacian systems and obstacle parabolic p-Laplacian with the given obstacle function Dψ ∈ L^y,q locally in Ω_T, respectively, which means that

for y > p and q ∈ (0, ∞], where he just used the so-called large-M-inequality principle. Very recently, Tian and Zheng [31] showed a global weighted Lorentz estimate to linear elliptic equations with lower order items under assumptions of partially BMO coefficients in Reifenberg flat domain. Zhang and Zheng [37, 38] also proved Hessian Lorentz estimates for fully nonlinear parabolic and elliptic equations with small BMO nonlinearities, and weighted Hessian Lorentz estimates of strong solutions for nondivergence linear elliptic equations with partially BMO coefficients, respectively.

To this end, we introduce some related notation and basic facts being useful in the article. The Lorentz space L^t,q(U) for open subset U ⊂ ℝⁿ with parameters 1 ≤ t < ∞ and 0 < q < ∞, is the set of all measurable functions g : U → ℝ requiring

while the Lorentz space L^t,∞ for 1 ≤ t < ∞ and q = ∞ is defined by the Marcinkiewicz space 𝓜^t(U) as usual, which is the set of all measurable functions g with

The local variant of such spaces is defined as usual way. If t = q, then the Lorentz space L^t,t(U) is nothing but a classical Lebesgue space. Indeed, by Fubini’s theorem it yields

which implies L^t(U) = L^t,t(U), see also [4, 5, 24, 37].

Let us denote

for y ∈ Ω and radius r > 0, and

with briefly B_r = B_r(0), Ω_r = Ω_r(0) and T_r = T_r(0). For a bounded open set U ⊂ ℝⁿ, we write the integral average of g(x) over U of a locally integrable function g in ℝⁿ by

As usual, it is a necessary assumption that the variable exponent p(⋅) is log-Hölder continuous, which ensures that the mollification, the singular integrals and the Hardy-Littlewood maximal operator are all bounded within the framework of generalized Lebesgue space. For this, we are recalling the definition that p(x) is log-Hölder continuous denoted it by p(x) ∈ LH(Ω), if there exist positive constants c₀ and δ such that for all x, y ∈ Ω with ∣x − y∣ < δ it holds

In the following context, we assume that p(x) : Ω → ℝ is a log-Hölder continuous function and there exist positive constants y₁ and y₂ such that

Without loss of generality, let

where ω: [0, ∞) → [0, ∞) is a modulus of continuity of p(x) such that ω is a nondecreasing continuous function with ω(0) = 0 and limsupr→0⁡ω(r)log⁡(1r)<∞. With the above assumptions in hand, it is clear that p(x) ∈ LH(Ω) yields that there exists a positive number K₀ such that

To obtain a global Calderón-Zygmund type estimate for general nonlinear elliptic problem (1.1), it is also necessary to impose some regular assumptions on the nonlinearity A = A(x, ξ) and the rough boundary of underlying domain Ω. Let us set

with

Assumption 1.2

Let R₀ > 0, we say that (A, Ω) is (δ, R₀)-vanishing if

1. sup0<r≤R0supy∈Rn⧸∫Br(y)θ(A,Br(y))(x)dx≤δ. (1.9)

2. For any y ∈ ∂ Ω and for every number r ∈ (0, R₀], there exists a coordinate system depending only on y and r, such that in this new coordinate system, y is the origin and

   Br∩{xn>δr}⊂Br∩Ω⊂Br∩{xn>−δr}. (1.10)

Remark 1.3

In this article, we always assume that δ is a small positive constant with 0 < δ < 18 by a scaling transformation. Note that the rough boundary with the so-called (δ, R₀)-Reifenberg flatness (1.10) yields that the boundary might be locally very rough between two hyperplanes, which may go beyond the boundaries with C¹-smooth or the Lipschitz category with a small Lipschitz constant. However, this is still an A-type domain with the following measure density condition

ensuring some natural properties in geometric analysis to hold, such as Sobolev embedding theorem and Sobolev extension theorem, see [11, 22, 25, 33].

Finally, we state the main result of this article.

Theorem 1.4

Let u ∈ W01,Φ (Ω) be a solution of (1.1) under the assumptions (1.2), (1.3) and (1.4) with the following higher integrability data

If p(⋅) ∈ LH(Ω) with (1.6) and (1.7), and there exists a small constant δ = δ(n, ν, Λ, y₁, y₂, t, q, σ_φ, τ_φ, R₀, K₀, ∣Ω∣) > 0 such that (A, Ω) is (δ, R₀)-vanishing with Assumption 1.2. Then we have (Φ(∣Du∣))^p(x) ∈ L^t,q(Ω) with the estimate

where the constant c depends only on n, ν, Λ, y₁, y₂, t, q, σ_φ, τ_φ, R₀, K₀, ω(⋅) and ∣Ω∣ (except in the case q = ∞, where c depends on n, ν, Λ, y₁, y₂, t, σ_φ, τ_φ, R₀, K₀, ω(⋅) and ∣Ω∣).

This article is devoted to a global Calderón-Zygmund type estimate in the framework of Lorentz spaces for a variable power of the gradients of weak solutions to the zero-Dirichlet problem of general nonlinear elliptic equations (1.1) over Reifenberg domains. As already mentioned, our problem and proof are inspired by work of Acerbi and Mingione [1, 2] and Baroni [4, 5], and recent work from Cho [14]. The key ingredient is to make use of the so-called large-M-inequality principle, Calderón-Zygmund-type covering, approximate estimate and an iteration argument to attain the variable Lorentz estimate (1.12).

The rest of the paper is organized as follows. In Section 2, we introduce notation and some useful lemmas. In Section 3, we focus on proving the main theorem.

Lemma 2.1

Let Ω ⊂ ℝⁿ be a bounded domain, and f ∈ L^Φ(Ω). Then there exists a unique solution u ∈ W01,Φ (Ω) to (1.1) such that the following energy estimate is valid:

where c = c(n, ν, Λ, σ_φ, τ_φ).

Proof

The existence and uniqueness of weak solution to (1.1) has been proved in [14]. Next, we take a test function ϕ = u ∈ W01,Φ (Ω) and use (1.2) and (1.4) to get the energy estimate (2.1).□

In what follows, let us give the scaling invariant property of (1.1), see Lemma 3.1 in [14].

Lemma 2.2

Let u ∈ W01,Φ (Ω) be the weak solution of (1.1) under the assumptions (1.2), (1.3) and (1.4). For fixed x₀ ∈ Ω, ρ > 0 and K > 0, we define

and the set Ω~={z−x0ρ:z∈Ω}, where ξ ∈ ℝⁿ and t ≥ 0. It leads to the following conclusions:

1. ũ ∈ W01,Φ~ (Ω̃) is a weak solution to

   divA~(x,Du~)=divG~(x,f~)inΩ~,u~=0on∂Ω~,

   where

   Φ~(s)=∫0sφ~(r)dr=1KΦ(Ks).

2. Ã and G̃ satisfy assumptions (1.2) and (1.4), respectively, with the same constants ν and L.

3. φ̃ satisfies (1.3).

4. If (A, Ω) is (δ, R₀)-vanishing, then (Ã, Ω̃) is (δ,R0ρ) -vanishing.

Next, we recall a reverse Hölder inequality for Φ(∣ξ∣), see Theorem 9 in [16].

Lemma 2.3

Let u ∈ W01,Φ (Ω) be the weak solution of (1.1) under the assumptions (1.2), (1.3) and (1.4). Suppose that (Φ(∣f∣))^p(x) ∈ L^t for p(x) > y₁ > 1 and t > 1. Then there exists σ₀ = σ₀(n, ν, Λ, σ_φ, τ_φ) ∈ (0, ty₁ − 1) such that for B_r with B_2r ⊂ ⊂ Ω and any σ ∈ (0, σ₀], it holds

where c = c(n, ν, Λ, σ_φ, τ_φ) > 0.

In addition, the following reverse Hölder inequality on the boundary version is also a self-improving result due to the Reifenberg flatness domain belonging to A-type condition as in (1.11).

Lemma 2.4

Let u ∈ W01,Φ (Ω) be the weak solution of (1.1) under the assumptions (1.2), (1.3) and (1.4). Suppose that (Φ(∣f∣))^p(x) ∈ L^t for p(x) > y₁ > 1 and t > 1, and (A, Ω) satisfies (δ, R₀)-vanishing with Assumption 1.2. Then there exists σ₀ = σ₀(n, ν, Λ, σ_φ, τ_φ) ∈ (0, ty₁ − 1) such that for Ω_r = Ω ∩ B_r(x₀) with x₀ ∈ ∂Ω and any σ ∈ (0, σ₀], it holds

where c = c(n, ν, Λ, σ_φ, τ_φ) > 0.

Proof

By a similar procedure to Theorem 9 in [16] and using the measure density property (1.11) for Ω, and a zero-extension of u in B_2r, the conclusion is clearly true.□

Note that Φ(s) is an N-function from the definition of Φ(s) in (1.5). Besides, we deduce that

and

We define an auxiliary vector field V : ℝⁿ → ℝⁿ given by

Then there is the following relation between V and Φ, see [14]:

for all ξ, η ∈ ℝⁿ and m₀ > 0 depending only on n, ν, Λ, σ_φ and τ_φ.

Now, we give the comparison estimates with a limiting problem, and show its Lipschitz regularity by following from Proposition 5.5 and 5.11 in [14]. We state the required comparison estimates in the following lemmas.

For the interior case, we assume

and

for η = 1+σ₂ ≤ 1+σ with σ as the same of Lemma 2.3, and δ ∈ (0,18) determined later.

Lemma 2.5

Let u ∈ W01,Φ (Ω) be a weak solution of (1.1) under the assumptions (1.2), (1.3) and (1.4). If for any 0 < ϵ < 1, there exists a constant δ = δ(n, ϵ, ν, Λ, σ_φ, τ_φ) > 0 such that (2.5) and (2.6) hold. Then there exists a weak solution v ∈ W01,Φ (B₃) of

where Ā_B3 = ⧸∫B3 A(x, ξ)dx, such that we have

and

where c₁ = c₁(n, ν, Λ, σ_φ, τ_φ) > 1.

For the boundary case, let Ω₅ satisfy that

and

Lemma 2.6

Let u ∈ W01,Φ (Ω) be a weak solution of (1.1) under the assumptions (1.2), (1.3) and (1.4). If for any 0 < ϵ < 1, there exists a constant δ = δ(n, ϵ, ν, Λ, σ_φ, τ_φ) > 0 such that (2.7), (2.8) and (2.9) hold. Then there exists a weak solution v ∈ W01,Φ(B2+) of

where A¯B2+=⧸∫B2+A(x,ξ)dx, and we have

and

where c₂ = c₂(n, ν, Λ, σ_φ, τ_φ) > 1 and v̄ is zero extension of v from B1+ to Ω₁.

Let us now collect some preliminary results concerning the so-called embedding relations involved in the Lorentz spaces, which will be used in the sequel.

Proposition 2.7

Let U be a bounded measurable subset of ℝⁿ, then the following relations hold:

1. If 0 < q ≤ ∞, and 1 ≤ t₁ < t₂ < ∞, then L^t₂,q(U) ⊂ L^t₁,q(U) with the estimate

   ∥g∥Lt1,q(U)≤c|U|1t1−1t2∥g∥Lt2,q(U). (2.10)

2. If 1 ≤ t < ∞, and 0 < q₁ < q₂ ≤ ∞, then L^t,q₁(U) ⊂ L^t,q₂(U) ⊂ L^t,∞(U) with the estimate

   ∥g∥Lt,q2(U)≤c(t,q1,q2)∥g∥Lt,q1(U). (2.11)

3. If ∣g∣^α ∈ L^t,q(U) for some 0 < α < ∞, then g ∈ L^αt,αq(U) with the estimate

   ∥|g|α∥Lt,q(U)=∥g∥Lαt,αq(U)α. (2.12)

   The following two lemmas will play important roles in our main proof, which are the variants of classical Hardy’s inequality and a reverse Hölder inequality, respectively, see Lemma 3.4 and 3.5 in [4].

Lemma 2.8

Let ψ: [0, +∞) → [0, +∞) be a measurable function such that

Then for any α ≥ 1 and r > 0, there holds

Lemma 2.9

Let h : [0, +∞) → [0, +∞) be a nonincreasing, measurable function. For α₁ ≤ α₂ ≤ ∞ and r > 0, if α₂ < ∞, then we have

with every ε ∈ (0, 1] and λ ≥ 0. If α₂ = ∞, then it holds

where the constant c depends only on α₁,α₂ and r; except in the case α₂ = ∞ with c ≡ c(α₁, r).

Proof

Let σ₀ be the same as in Lemma 2.4, and let σ₂ = min{σ₀,y₁ − 1} > 0 due to y₁ > 1. For a fixed point y ∈ Ω, we take r_y < R250 with

where c^* = c^*(n, y₁, y₂, ν, Λ, σ_φ, τ_φ, ω(⋅), ∣Ω∣) ≥ ∣Ω∣ + 1 determined later.

Here, we only consider the boundary case, saying B_25ry(y) ⊄ Ω. For this, we can find a boundary point ȳ ∈ B_25ry (y) ∩ ∂Ω and for r_y ∈ (0,R250), there exists a coordinate system depending only on ȳ and r_y, such that in this new coordinate system it holds

We select 0 < δ < 18 such that Ω_5ry(z) ⊂ Ω_50ry(0), which implies the fact that

Then we also have

For a fixed point x₀ ∈ Ω, we set

For the weak solution u of original problem, by a scaling argument to the nonhomogeneous terms f and the N-functions φ, Φ, then we write

By the assumption (Φ(∣f∣))^p(x) ∈ L^t,q(Ω), then there holds

Hereafter, for the sake of simplicity, we still use u, f, φ and Φ replacing ū, f̄, φ̄ and Φ̄ in the following.

1. In this step, we give a Calderón-Zygmund type covering on the super-level set E(λ, Ω_r1(x₀)) as below. Let u be the weak solution of (1.1), we define the quantity for any r₁ and r₂ with R ≤ r₁ ≤ r₂ ≤ 2R

   λ0:=⧸∫Ωr2(x0)(Φ(|Du|))p(x)p−dx+1δ(⧸∫Ωr2(x0)(Φ(|f|))p(x)p−ηdx+1)1η, (3.7)

   where δ > 0 and η > 1 will be specified later. We treat the super-level set

   Eλ,Ωr1(x0):={x∈Ωr1(x0),(Φ(|Du|))p(x)p−>λ}

   for any λ > Mλ₀ ≥ 1 with M=(2320r27(r2−r1))n. For y ∈ E(λ, Ω_r1(x₀)) and radii 0 < r ≤ r₂-r₁, we let

   CZ(Ωr(y)):=⧸∫Ωr(y)(Φ(|Du|))p(x)p−dx+1δ(⧸∫Ωr(y)(Φ(|f|))p(x)p−ηdx)1η. (3.8)

   If r2−r1145 ≤ r ≤ r₂-r₁, then we discover that

   CZ(Ωr(y))≤|Ωr2(x0)||Ωr(y)|⧸∫Ωr2(x0)(Φ(|Du|))p(x)p−dx+(|Ωr2(x0)||Ωr(y)|)1η1δ(⧸∫Ωr2(x0)(Φ(|f|))p(x)p−ηdx)1η≤|Ωr2(x0)||Ωr(y)|(⧸∫Ωr2(x0)(Φ(|Du|))p(x)p−dx+1δ(⧸∫Ωr2(x0)(Φ(|f|))2p(x)p−ηdx)1η)≤|Br2(x0)||Br(y)||Br(y)||Ωr(y)|(⧸∫Ωr2(x0)(Φ(|Du|))p(x)p−dx+1δ(⧸∫Ωr2(x0)(Φ(|f|))2p(x)p−ηdx)1η)≤(r2r)n(167)nλ0≤(2320r27(r2−r1))nλ0<λ,

   which means that while r2−r1145 ≤ r ≤ r₂ − r₁ one has CZ(Ω_r(y)) < λ. At the same time, by Lebesgue’s differentiation theorem we find that CZ(Ω_r(y)) > λ for 0 < r ≪ 1. Therefore, by absolute continuity of the integral with respect to the domain we can pick the maximal radius r_y such that

   CZ(Ωry(y))=⧸∫Ωry(y)(Φ(|Du|))p(x)p−dx+1δ(⧸∫Ωry(y)(Φ(|f|))p(x)p−ηdx)1η=λ (3.9)

   for each point y ∈ E(λ, Ω_r1(x₀)). Moreover, one has

   CZ(Ωr(y))<λ,for anyr∈(ry,r2−r1]. (3.10)

   From (3.9), we conclude the following alternatives:

   λ2≤⧸∫Ωry(y)(Φ(|Du|))p(x)p−dxor(δλ2)η≤⧸∫Ωry(y)(Φ(|f|))p(x)p−ηdx. (3.11)

   First, we suppose that the first case of (3.11) is valid to have

   ⧸∫Ωry(y)(Φ(|Du|))p(x)p−dx=|Ωry(y)∖E(λ4,Ωr2(x0))||Ωry(y)|⧸∫Ωry(y)∖E(λ4,Ωr2(x0))(Φ(|Du|))p(x)p−dx+1|Ωry(y)|∫Ωry(y)∩E(λ4,Ωr2(x0))(Φ(|Du|))p(x)p−dx≤λ4+c(|Ωry(y)∩E(λ4,Ωr2(x0))||Ωry(y)|)1−11+σ1(⧸∫Ωry(y)(Φ(|Du|))p(x)p−(1+σ1)dx)11+σ1, (3.12)

   where σ₁ > 0 is determined later.

   Considering that ω(4R) < y₁σ₂ in (3.1), it leads to y1(1+σ2)y1+ω(4R)−1>0. Now let us take

   0<σ1≤y1(1+σ2)y1+ω(4R)−1,

   which yields that

   py+p−(1+σ1)=(1+p+−p−p−)(1+σ1)≤(1+ω(4R)p−)(1+σ1)≤(1+σ2),

   where py+:=supΩ125ryp(x) with Ω_125ry ⊂ Ω_2R(x₀), and ω(⋅) is the modulus of continuity for p(x). Then we use the reverse Hölder inequality shown in Lemma 2.4 to obtain that

   (⧸∫Ωry(y)(Φ(|Du|))p(x)p−(1+σ1)dx)11+σ1≤(⧸∫Ωry(y)(Φ(|Du|))py+p−(1+σ1)dx+1)11+σ1≤c(⧸∫Ω2ry(y)(Φ(|Du|))dx)py+p−+c(⧸∫Ω2ry(y)(Φ(|f|))py+p−(1+σ1)dx)11+σ1+c≤c(⧸∫Ω2ry(y)Φ(|Du|)dx)py+p−+c(⧸∫Ω2ry(y)(Φ(|f|))1+σ2dx)py+p−11+σ2+c.

   Taking into account (3.10), we have CZ(Ω_2ry(y)) < λ, then by a similar proof of (3.19) in Step 2, we get that

   ⧸∫Ω2ry(y)Φ(|Du|)dx≤cλp−py+,⧸∫Ω2ry(y)(Φ(|f|))ηdx≤cλp−py+ηδy1y2η.

   Thus, by taking η = 1 + σ₂ we have

   (⧸∫Ωry(y)(Φ(|Du|))p(x)p−(1+σ1)dx)11+σ1≤c(λ+λδy1y2py+p−+1)≤cλ. (3.13)

   Therefore, we combine (3.11), (3.12) and (3.13) to have

   λ4≤c(|Ωry(y)⋂E(λ4,Ωr2(x0))||Ωry(y)|)1−11+σ1λ,

   which implies

   |Ωry(y)|≤c|Ωry(y)∩E(λ4,Ωr2(x0))| (3.14)

   with the positive constant c depending only on n, ν, Λ, y₂, y₂, σ_φ, τ_φ, R₀, K₀, and ∣Ω∣.

   If the case of the second estimate in (3.11) is valid, by taking ζ=δ4 and Fubini’s theorem, we get

   (λδ2)η≤⧸∫Ωry(y)(Φ(|f|))p(x)p−ηdx=η|Ωry(y)|∫0∞μη|{x∈Ωry(y):(Φ(|f|))p(x)p−>μ}|dμμ=η|Ωry(y)|∫0ζλμη|{x∈Ωry(y):(Φ(|f|))p(x)p−>μ}|dμμ+η|Ωry(y)|∫ζλ∞μη|{x∈Ωry(y):(Φ(|f|))p(x)p−>μ}|dμμ≤(ζλ)η+η|Ωry(y)|∫ζλ∞μη|{x∈Ωry(y):(Φ(|f|))p(x)p−>μ}|dμμ.

   Let δ = 4ζ, we derive that

   |Ωry(y)|≤η(ζλ)η∫ζλ∞μη|{x∈Ωry(y):(Φ(|f|))p(x)p−>μ}|dμμ. (3.15)

   Now we put (3.14) and (3.15) together to get that

   |Ωry(y)|≤c|Ωry(y)∩E(λ4,Ωr2(x0))|+cη(ζλ)η∫ζλ∞μη|{x∈Ωry(y):(Φ(|f|))p(x)p−>μ}|dμμ. (3.16)

2. This step is devoted to various comparison estimates with the reference problems and the limiting one. Note that Ω is (δ, R₀)-Reifenberg flat, then it follows from (3.3) and (3.10) that

   ⧸∫Ω125ry(Φ(|Du(z′)|))p(z′)p−dz′+1δ(⧸∫Ω125ry(Φ(|f|))p(z′)p−ηdz′)1η≤2⋅(3425)n(⧸∫Ω170ry(y)(Φ(|Du(z′)|))p(z′)p−dz′+1δ(⧸∫Ω170ry(y)(Φ(|f|))p(z′)p−ηdz′)1η)≤2⋅(3425)nλ, (3.17)

   which implies that

   ⧸∫Ω125ry(Φ(|Du|))p(x)p−dx≤cλ,(⧸∫Ω125ry(Φ(|f|))p(x)p−ηdx)1η≤cλδ, (3.18)

   where we still use variable x for simplicity. Therefore, it suffices to show that

   ⧸∫/Ω125ryΦ(|Du|)dx≤c3λp−py+,⧸∫Ω125ry(Φ(|f|))ηdx≤c3λp−py+ηδy1y2η (3.19)

   for a constant c₃ ≥ 1. We first claim that

   (⧸∫Ω125ryΦ(|Du|)dx)py+−py−≤c4 (3.20)

   with c₄ ≥ 1 a universal constant. In fact, since (Φ(∣f∣))^p(x) ∈ L^t,q(Ω) for t > 1 and 0 < q ≤ ∞, it deduces that

   ∫ΩΦ(|f|)dx≤∫Ω(Φ(|f|)+1)p(x)dx≤c∥(Φ(|f|))p(x)∥Lt,q(Ω)+c|Ω|≤c(1+|Ω|),

   where we have used (3.6). By the energy estimate of Lemma 2.1, it leads to that

   ∫ΩΦ(|Du|)dx≤c(1+|Ω|). (3.21)

   By considering py+−py−≤ω(250ry), we have

   (⧸∫Ω125ryΦ(|Du|)dx)py+−py−=(1|Ω125ry|)py+−py−(∫Ω125ryΦ(|Du|)dx)py+−py−≤c(1|B250ry|)py+−py−(∫Ω125ryΦ(|Du|)dx)py+−py−≤c(1250ry)nω(250ry)(∫Ω125ryΦ(|Du|)dx)py+−py−≤c(∫Ω125ryΦ(|Du|)dx)py+−py−.

   On the other hand, we use (3.21) and 1250ry≥1R≥c∗R0≥|Ω|+1 with (3.1) to find

   (∫Ω125ryΦ(|Du|)dx)py+−py−≤(∫ΩΦ(|Du|)dx)py+−py−≤c(|Ω|+1)py+−py−≤c(1250ry)ω(250ry)≤c,

   where we have used the so-called log-Hölder continuity (1.8) for p(x) in the last inequality, which yields (3.20). Recalling y1≤py+ and (3.20) with λ > 1, we obtain

   ⧸∫Ω125ryΦ(|Du|)dx=(⧸∫Ω125ryΦ(|Du|)dx)py+−py−py+(⧸∫Ω125ryΦ(|Du|)dx)py−py+≤c41y1(⧸∫Ω125ryΦ(|Du|)dx)py−py+≤c(⧸∫Ω125ry(Φ(|Du|))py−p−dx)p−py+≤c(⧸∫Ω125ry(Φ(|Du|))p(x)p−dx+1)p−py+≤cλp−py+.

   Note that (Φ(∣f∣))^p(x) ∈ L^t,q(Ω) for 1 < y₁ ≤ p(x) ≤ y₂ < ∞, and 1 < η ≤ 1+σ with σ as the same of Lemma 2.4, it holds

   ∫Ω(Φ(|f|))ηdx≤∫Ω((Φ(|f|))1+σ+2)dx≤c(1+|Ω|).

   Similarly, recalling δλ₀ ≥ 1 and λ ≥ Mλ₀ we find

   ⧸∫Ω125ry(Φ(|f|))ηdx≤c(⧸∫Ω125ry(Φ(|f|))p(x)p−ηdx+1)p−py+≤c(δλ+1)p−py+η≤c(δλ+δλ0)p−py+η≤cλp−py+ηδy1y2η.

   Next, we define

   A~(x,ξ):=A(25ryx,λp−py+ξ),G~(x,ξ):=G(25ryx,λp−py+ξ),

   u~(x):=u(25ryx)25ryλp−py+,f~(x):=f(25ryx)λp−py+,φ~(s):=φ(λp−py+s),Φ~(s):=Φ(λp−py+s)λp−py+.

   In light of (3.2), (3.4), (3.19) and Lemma 2.2, we find that the hypothesis of Lemma 2.6 is valid, which implies that

   ⧸∫Ω25ry|V(Du)−V(Dv¯)|2dx≤ϵλp−py+,

   and

   ∥Φ(|Dv¯|)∥L∞(Ω25ry)≤c0λp−py+,

   where ϵ > 0 is small and c₀ = max{c₁, c₂} ≥ 1.

   For the case of interior, by Lemma 2.5 similarly we have

   ⧸∫B25ry|V(Du)−V(Dv)|2dx≤ϵλp−py+,

   and

   ∥Φ(|Dv|)∥L∞(B25ry)≤c0λp−py+.

3. We are here to estimate the super level set E(λ, Ω_r2(x₀)). For any fixed point x ∈ Ω, we select a universal constant R with 0 < R ≤ min{R02,R0|Ω|+1,1}, and there exists a constant δ = δ(ϵ) > 0 such that Lemma 2.5 and 2.6 hold. Let A=(2τφ+1c0(m0+1))y2y1, for any x ∈ E(Aλ, Ω_r1(x₀)) we consider the collection 𝓑_λ of all subset Ω_ry(y). By“ crawling of ink spots” argument, we extract a countable subcollection {Ω_ri(y_i)} ∈ 𝓑_λ, such that

   Ωri(yi)∩Ωrj(yj)=∅,wheneveri≠j,andE(Aλ,Ωr1(x0))⊂∪i∈NΩ5ri(yi)∪Nλ

   with ∣𝓝_λ∣ = 0. Let us denote pi+=pyi+, then we derive that

   |E(Aλ,Ωr1(x0))|=|E((2τφ+1c0(m0+1))y2y1λ,Ωr1(x0)|=|{x∈Ωr1(x0):(Φ(|Du|))p(x)p−>(2τφ+1c0(m0+1))y2y1λ}|≤∑i≥1|{x∈Ω5ri(yi):Φ(|Du|)>2τφ+1c0(m0+1)λp−p(x)}|≤∑i≥1|{x∈Ω5ri(yi):Φ(|Du|)>2τφ+1c0(m0+1)λp−pi+}|=∑ interior case|{x∈Ω5ri(yi):Φ(|Du|)>2τφ+1c0(m0+1)λp−pi+}|+∑ boundary case|{x∈Ω5ri(yi):Φ(|Du|)>2τφ+1c0(m0+1)λp−pi+}|. (3.22)

   For the boundary case, we recall (2.2), (2.3) and (2.4) to find that

   Φ(|Du|)≤Φ(|Du−Dv¯|+|Dv¯|)≤12(Φ(2|Du−Dv¯|)+Φ(2|Dv¯|))≤2τφ(Φ(|Du−Dv¯|)+Φ(|Dv¯|))≤2τφ(m0+1)(|V(Du)−V(Dv¯)|2+Φ(|Dv¯|)),

   which implies that

   |{x∈Ω5ri(yi):Φ(|Du|)>2τφ+1c0(m0+1)λp−pi+}|≤|{x∈Ω50ri:|V(Du)−V(Dv¯)|2>c0λp−pi+}|+|{x∈Ω50ri:Φ(|Dv¯|)>c0λp−pi+}|≤1c0λp−pi+∫Ω50ri|V(Du)−V(Dv¯)|2dx≤cϵ|Ω50ri|≤cϵ|B50ri|=cϵ|Bri(yi)|≤cϵ|Bri(yi)||Ωri(yi)||Ωri(yi)|≤cϵ(21−δ)n|Ωri(yi)|≤cϵ|Ωri(yi)|,

   where we used the following weak (1, 1)−type estimate:

   |{x∈U:g(x)>λ}|≤1λ∫Ug(x)dx.

   Similarly, for the interior case, we discover that

   |{x∈Ω5ri(yi):Φ(|Du|)>2τφ+1c0(m0+1)λp−pi+}|=|{x∈B5ri(yi):Φ(|Du|)>2τφ+1c0(m0+1)λp−pi+}|≤cϵ|Bri(yi)|.

   Therefore, it follows from (3.22) that

   |E(Aλ,Ωr1(x0))|≤cϵ∑i≥1|Ωri(yi)|. (3.23)

   Using“ crawling of ink spots” argument again and (3.16), we conclude that

   |E(Aλ,Ωr1(x0))|≤cϵ∑i≥1|Ωri(yi)∩E(λ4,Ωr2(x0))|+cϵη(ζλ)η∑i≥1∫ζλ∞μη|{x∈Ωri(yi):(Φ(|f|))p(x)p−>μ}|dμμ≤cϵ|E(λ4,Ωr2(x0))|+cϵη(ζλ)η∫ζλ∞μη|{x∈Ωr2(x0):(Φ(|f|))p(x)p−>μ}|dμμ. (3.24)

4. This step is devoted to the estimate of ∥(Φ(∣Du∣))^p(x)∥_{L^t,q(Ωr1(x0))} for 0 < q < ∞. Since t > 1, we multiply the inequality (3.24) by (tp−)tq(Aλ)tp−, and integrate the resulting expression with a power qt in the measure dλAλ from Mλ₀ to ∞, which yields that

   tp−∫Mλ0∞((Aλ)tp−|{x∈Ωr1(x0):(Φ(|Du|))p(x)p−>Aλ}|)qtdλAλ≤c⋅tp−ϵqt∫0∞(λtp−|{x∈Ωr2(x0):(Φ(|Du|))p(x)p−>λ4}|)qtdλλ+c⋅tp−ϵqt∫0∞λq(p−−ηt)(∫ζλ∞μη|{x∈Ωr2(x0):(Φ(|f|))p(x)p−>μ}|dμμ)qtdλλ=cϵqt(I1+I2), (3.25)

   where c depends on n, ν, Λ, y₁, y₂, t, q, σ_φ, τ_φ, R₀, K₀,∣Ω∣, and ω(⋅). Thanks to (2.12) in Proposition 2.7, we have

   ∥(Φ(|Du|))p(x)∥Lt,q(Ωr1(x0))q=∥(Φ(|Du|))p(x)p−∥Ltp−,qp−(Ωr1(x0))qp−=tp−∫0∞(μtp−|{x∈Ωr1(x0):(Φ(|Du|))p(x)p−>μ}|)qtdμμ. (3.26)

   By a simple change of variable and (3.26) it leads to that

   I1=c(q)∥(Φ(|Du|))p(x)∥Lt,q(Ωr2(x0))q.

   We are now to estimate I₂. For this we part it in two cases.

5. If q ≥ t, noticing that (2.13) is satisfied since (Φ(|f|))p(x)p−∈Lη(Ωr2). By making the change of variables λ̄ = ζλ and ζ=δ4, then we employ Lemma 2.8 with ψ(μ)=μη−1|{x∈Ωr2(x0):(Φ(|f|))p(x)P−>μ}|,α=qt≥1, r=q(p−ηt)>0 and (3.26) to infer

   I2=c⋅tp−∫0∞λ¯q(p−−ηt)(∫λ¯∞μη|{x∈Ωr2(x0):(Φ(|f|))p(x)p−>μ}|dμμ)qtdλ¯λ¯≤c⋅tp−∫0∞λ¯qp−|{x∈Ωr2(x0):(Φ(|f|))p(x)p−>λ¯}|qtdλ¯λ¯=c∥(Φ(|f|))p(x)∥Lt,q(Ωr2(x0))q,

   where c = c(y₁, y₂, t, q).

6. If 0 < q < t, we use Lemma 2.9 with h(μ)=|{x∈Ωr2(x0):(Φ(|f|))p(x)p−>μ}|qt,r=ηqt,α1=1<tq=α2 and ε = 1, which yields the following

   (∫λ∞μη|{x∈Ωr2(x0):(Φ(|f|))p(x)p−>μ}|dμμ)qt≤ληqt|{x∈Ωr2(x0):(Φ(|f|))p(x)p−>λ}|qt+c∫λ∞μηqt|{x∈Ωr2(x0):(Φ(|f|))p(x)p−>μ}|qtdμμ.

   After a change variable ζ λ → λ, (3.26) and Fubini’s theorem we get

   I2≤c⋅tp−∫0∞λq(p−−ηt)ληqt|{x∈Ωr2(x0):(Φ(|f|))p(x)p−>λ}|qtdλλ+c⋅tp−∫0∞λq(p−−ηt)∫λ∞μηqt−1|{x∈Ωr2(x0):(Φ(|f|))2p(x)p−>μ}|qtdμdλλ≤c∥(Φ(|f|))p(x)∥Lt,q(Ωr2(x0))q+c⋅tp−∫0∞λq(p−−ηt)(∫λ∞μηqt−1|{x∈Ωr2(x0):(Φ(|f|))p(x)p−>μ}|qtdμ)dλλ≤c∥(Φ(|f|))p(x)∥Lt,q(Ωr2(x0))q,

   where c = c(y₁, y₂, t, q).

   We are now in a position to put the estimates of I₁ and I₂ into (3.25), and after simple manipulation, then for t > 1 it follows that

   ∥(Φ(|Du|))p(x)∥Lt,q(Ωr1(x0))≤c(tp−∫Mλ0∞((Aλ)tp−|{x∈Ωr1(x0):(Φ(|Du|))p(x)p−>Aλ}|)qtd(Aλ)Aλ)1q+c(tp−∫0Mλ0((Aλ)tp−|{x∈Ωr1(x0):(Φ(|Du|))p(x)p−>Aλ}|)qtd(Aλ)Aλ)1q≤c(tp−∫0Mλ0((Aλ)tp−|{x∈Ωr1(x0):(Φ(|Du|))p(x)p−>Aλ}|)qtd(Aλ)Aλ)1q+c¯ϵ1t(∥(Φ(|Du|))p(x)∥Lt,q(Ωr2(x0))+∥(Φ(|f|))p(x)∥Lt,q(Ωr2(x0)))≤cλ0p−|Ωr2(x0)|1t+c¯ϵ1t(∥(Φ(|Du|))p(x)∥Lt,q(Ωr2(x0))+∥(Φ(|f|))p(x)∥Lt,q(Ωr2(x0))), (3.27)

   where c̄ = c̄(n, ν, Λ, y₁, y₂, t, q, σ_φ, τ_φ, R₀, K₀, ∣Ω∣, ω(⋅)) > 0. Now we choose ϵ > 0 small enough such that c¯ϵ1t≤12, then we can find a corresponding positive constant δ = δ(n, ϵ, ν, Λ, y₁, y₂, t, q, σ_φ, τ_φ, R₀, K₀, ∣Ω∣) such that we deduce

   ∥(Φ(|Du|))p(x)∥Lt,q(Ωr1(x0))≤cλ0p−|Ωr2(x0)|1t+12∥(Φ(|Du|))p(x)∥Lt,q(Ωr2(x0))+c∥(Φ(|f|))p(x)∥Lt,q(Ωr2(x0)). (3.28)

7. This step is devoted to proving that ∥(Φ(∣Du∣))^p(x)∥_{L^t,q(Ωr2(x0))} < ∞. To this end, we first refine the estimate of (Φ(∣Du∣))^p(x) in the scale of Lorentz spaces. Consider the following truncated function

   |(Φ(|Du|))p(x)|k=min{(Φ(|Du|))p(x),k}forx∈Ωandk∈N∩[Mλ0,∞).

   By considering E_k(λ, Ω_ρ) = {x ∈ Ω_ρ : ∣(Φ(∣Du∣))^p(x)∣_k > λ} in line with (3.24), we discover that

   Ek(Aλ,Ωr1(x0))≤cϵ|Ek(λ4,Ωr2(x0))|+cϵη(ζλ)η∫ζλ∞μη|{x∈Ωr2(x0):(Φ(|f|))p(x)p−>μ}|dμμ. (3.29)

   For 0 < k ≤ Aλ we have E_k(Aλ, Ω_r1(x₀)) = ∅, which implies that the above estimate holds trivially. For k > Aλ, the estimate is also valid since

   Ek(Aλ,Ωr1(x0))=E(Aλ,Ωr1(x0))={x∈Ωr1(x0),(Φ(|Du|))p(x)>Aλ}andEk(λ4,Ωr2(x0))=E(λ4,Ωr2(x0)).

   Then working exactly as in Step 4, we get that (3.28) holds with ∣(Φ(∣Du∣))^p(x)∣_k in place of (Φ(∣Du∣)^p(x).

   Now we let L=cλ0p−|Ω2R(x0)|1t+c∥(Φ(|f|))p(x)∥Lt,q(Ω2R(x0)) for a fixed small radius R > 0, Θ(ρ) = ∥∣(Φ(∣Du∣))^p(x)∣_k∥_{L^t,q(Ωρ(x0))} for any R ≤ ρ < 2R, and take a small ϵ > 0 such that cϵ=12 in (3.28). Then for any R ≤ ρ₁ < ρ₂ < ρ₃ < ⋯ < 2R with ρ_d → 2R, d → ∞,

   Θ(ρ1)≤12Θ(ρ2)+L,Θ(ρ2)≤12Θ(ρ3)+L,⋯

   by the iteration we get that

   Θ(R)≤12dΘ(ρd)+L∑l=0d−112l.

   Note that

   Θ(2R)=∥|(Φ(|Du|))p(x)|k∥Lt,q(Ω2R(x0))<∞,

   we let d → ∞ to get

   ∥|(Φ(|Du|))p(x)|k∥Lt,q(ΩR(x0))≤cλ0p−|Ω2R(x0)|1t+c∥(Φ(|f|))p(x)∥Lt,q(Ω2R(x0)).

   In what follows, we use a standard finite covering argument to realize our global estimate. Note that Ω is a bounded domain in ℝⁿ and let us now take x₀ as every point in Ω. Then there exist N = N(n, ∣Ω∣) ∈ ℕ and x_j ∈ Ω for j = 1, 2, ⋯, N, where we replace the point x₀ by each x_j, such that

   Ω¯⊂∪j=1NΩR(xj).

   Therefore, we deduce that

   ∥|(Φ(|Du|))p(x)|k∥Lt,q(Ω)≤∑j=1N∥|(Φ(|Du|))p(x)|k∥Lt,q(ΩR(xj))≤c∑j=1N(λ0p−|Ω2R(xj)|1t+∥(Φ(|f|))p(x)∥Lt,q(Ω2R(xj)))≤c∑j=1N|Ω2R(xj)|1t(⧸∫Ωr2(xj)(Φ(|Du|))p(x)p−dx+(⧸∫Ωr2(xj)((Φ(|f|))p(x)p−η+1)dx)1η)p−+c∑j=1N∥(Φ(|f|))p(x)∥Lt,q(Ω2R(xj))≤c∑j=1N|Ω|1t(⧸∫Ωr2(xj)(Φ(|Du|))p(x)p−dx+(⧸∫Ωr2(xj)((Φ(|f|))p(x)p−η+1)dx)1η)p−+cN∥(Φ(|f|))p(x)∥Lt,q(Ω), (3.30)

   where we use the definition of λ₀. By (3.1), we notice that

   p+p−=1+p+−p−p−≤1+ω(4R)y1≤1+σ,

   where σ is the same as in Lemma 2.4. Then, it yields that

   ⧸∫Ωr2(xj)(Φ(|Du|))p(x)p−dx≤⧸∫Ωr2(xj)(Φ(|Du|))p+p−dx+1≤c(⧸∫Ω2r2(xj)(Φ(|Du|))dx+1)p+p−+c⧸∫Ω2r2(xj)(Φ(|f|))p+p−dx, (3.31)

   where we employed the reverse Höder inequality of Lemma 2.4 in the last inequality. Using (2.1) and Höder inequality, we obtain that

   (⧸∫Ω2r2(xj)Φ(|Du|)dx)p+p−≤(1|Ω2r2(xj)|)p+p−(∫Ω(Φ(|Du|))dx)p+p−≤c(1|Ω2r2(xj)|)p+p−(∫ΩΦ(|f|)dx)p+p−≤c(1|Ω2r2(xj)|)p+p−|Ω|1−p−p+∫Ω(Φ(|f|))p+p−dx≤c(1|Ω2r2(xj)|)p+p−∫Ω((Φ(|f|))p(x)p−p+p−+1)dx. (3.32)

   We now combine (3.30), (3.31) and (3.32) to get that

   ∥|(Φ(|Du|))p(x)|k∥Lt,q(Ω)≤c∑j=1N((1|Ω2r2(xj)|)p+p−∫Ω((Φ(|f|))p(x)p−p+p−+1)dx+(1|Ωr2(xj)|)1η(∫Ω((Φ(|f|))p(x)p−η+1)dx)1η)p−+cN∥(Φ(|f|))p(x)∥Lt,q(Ω). (3.33)

   Using a standard Hardy’s inequality in the Marcinkiewicz spaces (cf. Lemma 2.8 in [26]) and the reverse Hölder inequality of Lemma 2.9, we conclude that

   ∫Ω(Φ(|f|))p(x)p−p+p−dx≤t(p−)2t(p−)2−p+|Ω|1−p+t(p−)2∥(Φ(|f|))p(x)p−∥Mtp−(Ω)p+p−=t(p−)2t(p−)2−p+|Ω|1−p+t(p−)2(suph>0(htp−|{x∈Ω:(Φ(|f|))p(x)p−>h}|)1tp−)p+p−≤c|Ω|1−p+t(p−)2∥(Φ(|f|))p(x)p−∥Ltp−,qp−(Ω)p+p−≤c|Ω|1−p+t(p−)2∥(Φ(|f|))p(x)∥Lt,q(Ω)p+(p−)2. (3.34)

   Similarly, we also show that

   (∫Ω(|Φ(|f|))p(x)p−ηdx)1η≤c(y1,y2,t,q)|Ω|1η−1tp−∥(Φ(|f|))p(x)p−∥Ltp−,qp−(Ω)≤c(y1,y2,t,q)|Ω|1η−1tp−∥(Φ(|f|))p(x)∥Lt,q(Ω)1p−.

   For the case of q < ∞, from (3.33) we then infer the following relations

   ∥|(Φ(|Du|))p(x)|k∥Lt,q(Ω)≤c∑j=1N((1|Ω2r2(xj)|)p+(∥(Φ(|f|))p(x)∥Lt,q(Ω)p+p−+1)+(1|Ωr2(xj)|)p−η(∥(Φ(|f|))p(x)∥Lt,q(Ω)+1))≤c∑j=1N(1|B2r2(xj)||B2r2(xj)||Ω2r2(xj)|)p+(∥(Φ(|f|))p(x)∥Lt,q(Ω)p+p−+1)+c∑j=1N(1|Br2(xj)||Br2(xj)||Ωr2(xj)|)p−η(∥(Φ(|f|))p(x)∥Lt,q(Ω)+1)≤c∑j=1N(1|B2R(xj)|(21−δ)n)p+(∥(Φ(|f|))p(x)∥Lt,q(Ω)p+p−+1)+c∑j=1N(1|BR(xj)|(21−δ)n)p−η(∥(Φ(|f|))p(x)∥Lt,q(Ω)+1)≤cN(∥(Φ(|f|))p(x)∥Lt,q(Ω)+1)y2y1≤c,

   where we used the uniformly estimate (3.6). Now let us take k → ∞. By the lower semi-continuity of Lorentz quasi-norm we have

   ∥(Φ(|Du|))p(x)∥Lt,q(Ω)≤c,

   where c depends only on n, ν, Λ, y₁, y₂, t, q, σ_φ, τ_φ, R₀, K₀, ω(⋅), and ∣Ω∣. And recalling the definition in (3.5), we get the desired result (1.12)

8. Finally, for the case of q = ∞, we come back to the second inequality in (3.11) and split it into two parts with a small ι > 0 determined later:

   (λ2)η≤1δη⧸∫Ωry(y)(Φ(|f|))p(x)p−ηdx≤(ιλ)ηδη+1δη|Ωry(y)|∫{x∈Ωry(y):(Φ(|f|))p(x)p−>ιλ}(Φ(|f|))p(x)p−ηdx.

   We set

   Ψ(ιλ,Ωry(y))={x∈Ωry(y):(Φ(|f|))p(x)p−>ιλ},Ψ(μ,Ωry(y))={x∈Ωry(y):(Φ(|f|))p(x)p−>μ}.

   Similar to the estimate (3.34), by using Hölder inequality we get

   (λ2)η−(ιλδ)η≤1δη|Ωry(y)|∫{x∈Ωry(y):(Φ(|f|))p(x)p−>ιλ}(Φ(|f|))p(x)p−ηdx≤t(t−η)δη|Ψ(ιλ,Ωry(y))|1−ηt|Ωry(y)|supμ>0μη|{x∈Ψ(ιλ,Ωry(y)):(Φ(|f|))p(x)p−η≥μ}|ηt≤t|Ψ(ιλ,Ωry(y))|1−ηt(t−η)δη|Ωry(y)|((ιλ)η|Ψ(ιλ,Ωry(y))|ηt+supμ>ιλμη|Ψ(μ,Ωry(y))|ηt)=t(t−η)δη((ιλ)η+|Ψ(ιλ,Ωry(y))|1−ηt|Ωry(y)|supμ>ιλμη|Ψ(μ,Ωry(y))|ηt).

   Now we choose ι > 0 sufficiently small so as to satisfy

   (λ2)η−(ιλδ)η−tt−η(ιλδ)η=(λ2)η−(ιλδ)η(1+tt−η)≥(λ4)η,

   and there exists a positive constant c(t) depending only on t such that ι ≤ c(t)δ. Therefore, it follows that

   |Ωry(y)|≤ctt−η|Ψ(ιλ,Ωry(y))|1−ηt(ιλ)η(supμ>ιλμt|Ψ(μ,Ωry(y))|)ηt≤ct(ιλ)−tt−η((ιλ)t|Ψ(ιλ,Ωry(y))|)1−ηt(supμ>ιλμt|Ψ(μ,Ωry(y))|)ηt≤ct(ιλ)−tt−ηsupμ>ιλμt|Ψ(μ,Ωry(y))|. (3.35)

   Next we put the estimates of two cases in (3.11) together into (3.23), that is, we insert the formulas (3.14) and (3.35) into (3.23) to get

   |E(Aλ,Ωr1(x0))|≤cϵ|E(λ4,Ωr2(x0))|+cϵ(ιλ)−tsupμ>ιλμt|Ψ(μ,Ωr2(x0))|, (3.36)

   then we multiply (3.36) by (Aλ)^tp−, and take the supremum with respect to λ over (Mλ₀, ∞) to show

   supλ>Mλ0(Aλ)tp−|{x∈Ωr2(x0):(Φ(|Du|))p(x)p−>Aλ}|≤cϵAtp−(supλ>Mλ0λtp−|{x∈Ωr2(x0):(Φ(|Du|))p(x)p−>λ4}|+supλ>Mιλ0λtp−−t(supμ>λμt|Ψ(μ,Ωr2(x0))|))≤cϵ(supλ>Mλ0λtp−|{x∈Ωr2(x0):(Φ(|Du|))p(x)p−>λ4}|+supλ>Mιλ0(supμ>λμtp−|Ψ(μ,Ωr2(x0))|)).

   Note that

   supλ>Mιλ0supμ>λμtp−|Ψ(μ,Ωr2(x0))|≤∥(Φ(|f|))p(x)∥Mt(Ωr2(x0))t.

   If we take ϵ > 0 so small that it ensures cϵ1t≤12, it follows that

   ∥(Φ(|Du|))p(x)∥Mt(Ωr1(x0))≤cϵ1t(∥(Φ(|Du|))p(x)∥Mt(Ωr2(x0))+∥(Φ(|f|))p(x)∥Mt(Ωr2(x0)))+cMλ0p−|Ωr2(x0)|1t≤12∥(Φ(|Du|))p(x)∥Mt(Ωr2(x0))+c∥(Φ(|f|))p(x)∥Mt(Ωr2(x0))+c|Ωr2(x0)|1t(⧸∫Ωr2(x0)(Φ(|Du|))p(x)p−dx+(⧸∫Ωr2(x0)((Φ(|f|))p(x)p−+1)ηdx)1η)p−.

   In the remainder we use a similar way of the argument in Step 5, and it leads to the desired result for the case q = ∞.□