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Research article

On substructures of semigroups of inductive terms

  • Received: 28 November 2021 Accepted: 14 March 2022 Published: 18 March 2022
  • MSC : 08A40, 08A70, 20M10

  • An inductive composition is an operation generalizing from a superposition Sn on the set of all n-ary terms of type τ. A binary operation called inductive product is obtainable from such composition. It is a generalization of a tree language product but on the set of all n-ary terms of type τ. Unlike the original one, this inductive product is not associative on the mentioned set. Nonetheless, it turns to be associative on some restricted set. A semigroup arising in this way is the main focus of this paper. We consider its special subsemigroups and semigroup factorizations.

    Citation: Pongsakorn Kitpratyakul, Bundit Pibaljommee. On substructures of semigroups of inductive terms[J]. AIMS Mathematics, 2022, 7(6): 9835-9845. doi: 10.3934/math.2022548

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  • An inductive composition is an operation generalizing from a superposition Sn on the set of all n-ary terms of type τ. A binary operation called inductive product is obtainable from such composition. It is a generalization of a tree language product but on the set of all n-ary terms of type τ. Unlike the original one, this inductive product is not associative on the mentioned set. Nonetheless, it turns to be associative on some restricted set. A semigroup arising in this way is the main focus of this paper. We consider its special subsemigroups and semigroup factorizations.



    This article deals with the nonlinear elliptic equation with measure data of the type

    divA(x,u)=μin Ω. (1.1)

    Here ΩRn (n2) is a bounded domain, the unknown uW1,P(Ω) with an N-function P() which will be introduced in Section 2, and A(x,u)L1loc(Ω,Rn). In (1.1), μ is a Radon measure defined on Ω with finite total mass |μ|(Ω)<. Moreover, if the measure μ is actually an L1-function, then

    |μ|(Q):=Q|μ(x)|dx (1.2)

    for a measurable subset QΩ. We assume that measure μ satisfies the following condition:

    (M) There exists some θ0(0,n1) such that

    |μ|(B(x0,2R))(2R)θ0 (1.3)

    holds for all x0Ω and every R>0.

    In this article, we assume that the Carathéodory vector field A:Ω×RnRn is C1-regular in the gradient variable ξ, and satisfies A(x,0)=0, and the growth, ellipticity, and continuity conditions, i.e., there are constants 0s1, 0<νL, and K1 such that

    (A1) {|A(x,ξ)|+|Aξ(x,ξ)|(|ξ|2+s2)12LP((|ξ|2+s2)12)(|ξ|2+s2)12,ν1P(|ξ2ξ1|)A(x,ξ2)A(x,ξ1),ξ2ξ1,|A(x,ξ)A(y,ξ)|Kω(|xy|)P((|ξ|2+s2)12)(|ξ|2+s2)12

    for any x,yΩ and ξ,ξ1,ξ2Rn. In (A1), the function ω:[0,)[0,) has the following hypothesis.

    (A2)ωis a non-decreasing concave function such thatω(0)=limρ0ω(ρ)=0andω()1.

    Moreover, ω is assumed to satisfy the Dini-continuous condition:

    (A3)  d(R):=R0ω(ρ)dρρ<

    for every 0<R1.

    A significant example of (1.1) is the p-Laplacian type equation, for which s=0, p(1,), P(x)=xp, and A(u)=|u|p2u. Then Eq (1.1) can be expressed as

    div(|u|p2u)=μ.

    The relevant research on the regularities of solutions to elliptic equations starts with Kilpeläinen and Malý [1,2], and extends with a different technique by Trudinger and Wang [3]. Later, Duzaar and Mingione make a further study in [4]. Those results show a standard fact that solutions to non-homogeneous p-Laplacian-type equations with measure data can be pointwise estimated in a natural way by involving the classical nonlinear Wolff potential Wμβ,p(x,R) [5], that is,

    Wμβ,p(x,R):=R0(|μ|(B(x,ρ))ρnβp)1p1dρρ,β(0,np].

    Based on the relationship between Wolff potential and Riesz potential, Mingione et al. [6,7] find out pointwise gradient estimates hold for general quasilinear degenerate equations by applying the Riesz potential

    Iμβ(x,R):=Wμβ2,2(x,R)=R0|μ|(B(x,ρ))ρnβdρρ,β>0. (1.4)

    Baroni proves pointwise gradient bounds for solutions in terms of linear Riesz potentials in [8]. In addition, the caloric Riesz potential serves as a means for pointwise estimation of the spatial gradient of solutions to nonlinear degenerate parabolic equations [9].

    Further, pointwise gradient estimates via the nonlinear Wolff potentials for weak solutions to various quasilinear elliptic equations with measure data are obtained by Mingione [10] and Yao [11]. More generally, pointwise potential estimates for elliptic equations and systems with Orlicz growth are studied in [12,13,14], respectively.

    In recent years, a great deal of effort has gone into investigating nonlinear elliptic equations and systems involving measure data. Chilebicka et al. [15] study estimates including precise continuity and Hölder continuity criteria by the means of potential of a Wolff type; they also provide regularity estimates of the solutions and their gradients in the generalized Marcinkiewicz scale [16]. The existence of solutions in the framework of renormalized solutions is introduced in [17]. There are many interesting results in [18,19,20].

    In this article, a weak solution to (1.1) is a function uW1,P(Ω) such that

    ΩA(x,u),φdζ=Ωφdμ, (1.5)

    whenever φC0(Ω). Inspired by Mingione et al. [4,10,21], the main objective is to present pointwise potential estimates and interior Hölder continuity of weak solutions to (1.1) by using (1.4) in the Sobolev-Orlicz spaces.

    We state our pointwise estimates in Theorems 1.1 and 1.2. It is important to note that θ0 in the following theorems will be introduced in (1.3). The first main result is the gradient pointwise estimates of u as follows.

    Theorem 1.1. Let uC1(Ω)W1,P(Ω) be a weak solution to (1.1) under the assumptions (A1), (A2), (A3), Radon measure μ satisfy μL1(Ω) and (M), and P be an N-function with Δ2(P,˜P)< satisfying Assumption 2.4 and (2.12). There exists a constant CC(n,ν,L,K,s,θ0,CΔ2,C1)>0 and a positive radius ˜R<1 such that the pointwise estimate

    P(|u(x0)|)CB(x0,R)[P(|u|)+P(s)]dx+CIμnθ0(x0,4R) (1.6)

    holds whenever B(x0,R)Ω and 0<R˜R. In (1.6), denotes integral average, Δ2-condition, CΔ2, and C1 will be described in Definition 2.3 and Lemma 2.5, respectively.

    On the basis of Theorem 1.1, we demonstrate the pointwise estimate of u.

    Theorem 1.2. Let uC0(Ω)W1,P(Ω) be a weak solution to (1.1) under the assumptions (A1)1, (A1)2, Radon measure μ satisfy μL1(Ω) and (M), and P be an N-function with Δ2(P,˜P)< satisfying Assumption 2.4, and (2.12). There exists a constant CC(n,ν,L,θ0,CΔ2,C1,diam(Ω))>0 and a positive radius R<1 such that for every rR the pointwise estimate

    P(|u(x0)|)Cr1+ε0B(x0,r)P(|u|r)dx+CrαIμnθ0α(x0,2r) (1.7)

    holds, where ε0 and α will be introduced in Lemmas 2.5 and 3.3, respectively.

    Inspired by Mingione's result [10], the following theorem expounds the Hölder continuity of the solution u.

    Theorem 1.3 (Interior Hölder type estimate). Let uC0(Ω)W1,P(Ω) be a weak solution to (1.1) under the assumptions (A1), (A2), let μ be a Radon measure satisfying μL1(Ω) and (M), and let P be an N-function satisfying Δ2(P,˜P)<, Assumption 2.4, and (2.12). Then there exists constants α[0,1) and 0<R<1 such that for every x,yB(x0,2R)Ω, there holds

    P(|u(x)u(y)|)C[Iμnθ0α(x,2R)+Iμnθ0α(y,2R)+B(x0,2R)P(|u|R)dζ+P(s)]|xy|α,

    where the constant C depends on n,ν,L,θ0,ω(),CΔ2,C1, and diam(Ω).

    Remark 1.4. Drawing upon the Riesz potential, our results present pointwise estimates and Hölder continuity within the more generalized framework of Sobolev-Orlicz spaces. Notably, (1.6) provides an estimate of u, whereas (1.7) estimates u itself. We shall leverage Lemma 2.11 to relate u to u.

    In this article, we adopt several technical tools and methods in Sobolev-Orlicz spaces, and explore the properties of solutions of homogeneous equations to those of inhomogeneous equations with measure data. We first establish Proposition 3.2, which reveals the density of the Riesz potential and serves as a crucial conclusion in facilitating the proof of subsequent comparison lemmas. The proof of the comparison estimate is divided into two intricate steps; the first step requires Sobolev-type embedding, while the second primarily employs scaling changes. Our primary objective is to prove pointwise potential estimates and Hölder continuity, which is shown essentially by oscillation estimates of solutions. By utilizing summation methods and involving the Riesz potential, we proceed from deriving the estimates of the gradient of u to those of u itself. Ultimately, we employ the sharp maximal functions, and achieve the interior Hölder estimates of solutions.

    This article is organized as follows: In Section 2, we state fundamental tools and definitions such as N-functions and maximal functions. Section 3 is devoted to the proof of Lemmas 3.4 and 3.9, and Section 4 presents supporting results, to gather Caccioppoli-type inequality and maximal estimate towards the proof of the main theorems. In the last section, we present the proofs of Theorems 1.1–1.3, respectively.

    In this section, we give the definitions and tools of N-functions, function spaces; classical inequalities, and maximal functions.

    The following definitions and results are standard in the context of N-function; see [22].

    Definition 2.1. A function P:[0,)[0,) is said to be an N-function, if P is convex, differentiable, and its derivative P is a right continuous, non-decreasing function satisfying that P(0)=0 and P(t)>0 for t>0.

    Definition 2.2. The complementary function ˜P:[0,)[0,) is defined by

    ˜P(x)=supt0[xtP(t)].

    Definition 2.3. We say that P satisfied the Δ2-condition, if there exists CΔ2>0 such that

    P(2t)CΔ2P(t)

    for all t0. By Δ2(P) we denote the smallest constant CΔ2. The function P is said to satisfy the 2-condition, if ˜PΔ2. Then we define

    Δ2(P,˜P):=max{Δ2(P),Δ2(˜P)}. (2.1)

    By the Δ2-condition of Definition 2.3, we can easily obtain

    P(x+y)CΔ2[P(x)+P(y)] (2.2)

    for every x,y0.

    Assumption 2.4. Let P be a convex function that satisfies Δ2(P,˜P)< as (2.1), and P is C2 on (0,). Moreover, let P(0)=0, limtP(t)= and uniformly in t0

    P(t)tP(t).

    Assumption 2.4 assures that P is an N-function.

    We let P1:[0,)[0,) be the right-continuous inverse function of P, and (P)1:[0,)[0,) the inverse function of P. Then ~˜P(t)=P(t) and (˜P)(t)=(P)1(t) hold. By [23], one has

    P(t)tP(t),  and  ˜P(P(t))P(t) (2.3)

    hold uniformly in t0. By (2.2) and (2.3), we have

    P(x+y)CΔ2[P(x)+P(y)]. (2.4)

    Lemma 2.5. [23] Let P be an N-function with Δ2(P,˜P)<. Then there exist ε0>0, C1>0 which only depend on Δ2(P,˜P) such that for all t0 and all λ[0,1], one has

    P(λt)C1λ1+ε0P(t). (2.5)

    From Definition 2.2 of N-function ˜P(), it is easy to obtain that (2.5) holds for ˜P, i.e.,

    ˜P(λt)C1λ1+ε0˜P(t). (2.6)

    Using (2.3) and (2.5), it is not difficult to obtain

    P(λt)C1λε0P(t). (2.7)

    In this article, we denote ab by CbaCb for two constants C and C.

    Lemma 2.6. [23] Let P be an N-function under Assumption 2.4, and A satisfies the continuity and growth condition of (A1)1, (A1)2. Then

    (A(x,ξ1)A(x,ξ2))(ξ1ξ2)  P(|ξ1ξ2|)  |ξ1ξ2|P(|ξ1|+|ξ2|).

    Moreover,

    A(x,ξ)ξP(|ξ|)

    uniformly in ξRn and xΩ.

    Example 2.7. [24] Assume that P is a Young function such that

    P(t)tp1(logt)p2,t1,

    where p1>1 and p2R. The derivative P of P is as follow:

    P(t)tp11(log(1+t))p2nearinfinity.

    The complementary function ˜P satisfies

    ˜P(t)tp1p11(log(t))p2p11nearinfinity.

    It is not difficult to verify that P() is an N-function satisfying Definitions 2.1–2.3, and Assumption 2.4.

    In this article, we need the following definitions of function spaces. The classical Orlicz spaces LP(Rn) with its norm are given via [22]

    LP(Rn)={fL1(Rn)|RnP(|f(x)|)dx<},

    and

    f(x)LP(Rn)=inf{k>0|RnP(f(x)k)dx1}. (2.8)

    If f(x)LP(Rn) is finite, then f(x)LP(Rn). The Sobolev-Orlicz spaces W1,P(Rn) and its norm are given by [25]

    W1,P(Rn)={fLP(Rn)|fLP(Rn)},

    and

    f(x)W1,P(Rn)=fLP(Rn)+fLP(Rn).

    If f(x)W1,P(Rn) is finite, then f(x)W1,P(Rn). Both LP(Rn) and W1,P(Rn) are Banach spaces.

    In this subsection, we recall several classical inequalities.

    Lemma 2.8. (Young's inequality [26]) For all ε>0, there exist Cε, ˜Cε depending on Δ2(P,˜P), such that for all ζ1,ζ20, there holds

    ζ1ζ2εP(ζ1)+Cε˜P(ζ2), (2.9)

    and

    ζ1ζ2ε˜P(ζ1)+˜CεP(ζ2). (2.10)

    Lemma 2.9. Let P be an N-function with the Δ2-condition. Then for all ε>0, there exists a constant C such that

    |P(x)P(y)|εCΔ2P(y)+CP(|xy|)

    for x,y>0, where the constant C depends on CΔ2 and C1.

    Proof. By the mean value theorem, for x>y>0, there exists λ0(0,1) such that

    P(x)P(y)=P[λ0x+(1λ0)y](xy)=P[y+λ0(xy)](xy).

    We use (2.4) and (2.7) to get

    P(x)P(y)CΔ2[P(y)(xy)+C1λε00P(xy)(xy)].

    Then applying (2.3) and (2.10), we obtain that

    |P(x)P(y)|CΔ2[P(y)|xy|+C1λε00P(|xy|)]εCΔ2˜P(P(y))+˜CεP(|xy|)+CP(|xy|)εCΔ2P(y)+CP(|xy|).

    We complete the proof of Lemma 2.9.

    Let B be a measurable set with positive measure, and f:BRn a measurable function. We denote the integral average of f by

    (f)B=Bf(x)dx=1|B|Bf(x)dx.

    Lemma 2.10. (Jesen's inequality [27]) Let P be an N-function with Δ2(P,˜P)<. If fW1,P(B(x,R)), then there exists CC(n) for B(x,R)Ω such that

    P(|B(x,R)fdζ|)CB(x,R)P(|f|)dζ.

    Lemma 2.11. (Sobolev-Poincaré's inequality [27]) Let P be an N-function with Δ2(P,˜P)< and satisfy Assumption 2.4. If fW1,P(B(x,R)), then there exist 0<θ1<1 and C>0 such that

    B(x,R)P(|f(f)B(x,R)|R)dζC(B(x,R)Pθ1(|f|)dζ)1θ1 (2.11)

    holds whenever B(x,R)Ω.

    The following lemma describes an embedding into a space of continuous functions; see Theorem 8.39 in [28] and (2.22) in [24].

    Lemma 2.12. If an N-function P satisfies that

    1P1(x)xn+1ndx<, (2.12)

    then W1,P0(Ω)C0(Ω)L(Ω), that is, there exists a constant CC(n) such that

    fL(Ω)CfW1,P0(Ω)CfLP(Ω)

    for all fW1,P(Ω).

    The following iteration lemma plays an essential role in proving the Caccioppoli-type inequality (4.1).

    Lemma 2.13. (Iteration lemma [29]) Let f:[γR,R][0,) be a bounded function such that the inequality

    f(ϱ)12f(r)+C2(rϱ)κ

    holds for fixed constants C2,κ0, and γRϱrR with 0<γ<1. Then we have

    f(γR)C[(1γ)R]κ

    for a constant C depending only on κ.

    Throughout this subsection, we provide powerful tools for Hölder estimate of Theorem 1.3. Analogously to the definition of the classical maximal operator in [30], we define the generalized maximal operator and sharp maximal function in Sobolev-Orlicz spaces.

    Definition 2.14. [31] Let 1<τ<n, R<dist(x,Ω),xΩ, where Ω is the boundary of Ω. Let f be a function in Orlicz space LP(Ω) or a measure with finite mass, and P be an N-function with Δ2(P,˜P)<. The function defined by

    MPτ,R(f)(x):=sup0<rRrτB(x,r)P(|f|)dζ (2.13)

    is called the restricted fractional τ generalized maximal function of f.

    Definition 2.15. [31] Let β(0,1), xΩ, and R<dist(x,Ω), let fLP(Ω) and P be an N-function with Δ2(P,˜P)<. The function defined by

    M#,Pβ,R(f)(x):=sup0<rRrβB(x,r)P(|f(f)B(x,r)|)dζ

    is called the restricted fractional β generalized sharp maximal function of f. For θ>0, we also denote

    ˜M#,Pθ,R(f)(x):=sup0<rRrθB(x,r)P(|f(f)B(x,r)|r)dζ. (2.14)

    The following note gives us a connection between maximal functions and sharp maximal functions.

    Remark 2.16. Combining the generalized sharp maximal functions and Lemma 2.11, we see that if Assumption 2.4 holds, then it follows that

    ˜M#,Pθ,R(f)(x)Csup0<rRrθ(B(x,r)Pθ1(|f|)dζ)1θ1. (2.15)

    In this section, we collect the relevant difference estimates and decay estimates, and consider the density of the Riesz potential Iμβ explicitly. We define vu+W1,P0(B(x0,2R)) as the unique solution to the homogeneous Dirichlet problem

    {divA(x,v)=0 in B(x0,2R),v=u onB(x0,2R). (3.1)

    The existence of v is guaranteed by a standard monotonicity argument; see [13]. One obtains the following control estimate.

    Lemma 3.1. Let uW1,P(Ω) be as in Theorem 1.1 satisfying the continuity and growth condition of (A1), vu+W1,P0(B(x0,2R)) be a solution to (3.13)1, let P be an N-function with Δ2(P,˜P)< and satisfy Assumption 2.4. Then the following estimate

    B(x0,R)P(|v|)dxCB(x0,2R)P(|u|+s)dx (3.2)

    holds, where C depends on n,ν,s,L.

    We consider the estimate of the difference of a solution to (1.1), and that of the corresponding solution to the Dirichlet problem (3.1).

    Proposition 3.2. Let uW1,P(Ω) be as in Theorem 1.1 satisfying (A1)2, vu+W1,P0(B(x0,2R)) be a solution to (3.1), let P be an N-function with Δ2(P,˜P)<, and satisfy Assumption 2.4, (2.12). Radon measure μ satisfies (M). There exists a constant CC(n,ν,C1) such that

    B(x0,2R)P(|uv|)dxC|μ|(B(x0,2R))(2R)θ0. (3.3)

    Proof. Without loss of generality, we first assume that B(x0,2R)B(0,1) and |μ|(B(0,1))=1. Then we shall remove these two conditions for general situations.

    Step 1. We assume that B(x0,2R)B(0,1) with |μ|(B(0,1))=1. By choosing φuv as a test function in (1.5), and using (1.1) and (3.1)1, we obtain

    B(0,1)A(x,u)A(x,v),uvdx=B(0,1)(uv)dμ. (3.4)

    By using the ellipticity assumption (A1)2, we deduce that

    ν1B(0,1)P(|uv|)dxB(0,1)A(x,u)A(x,v),uvdx.

    By (3.4) and Lemma 2.12, it follows that

    ν1B(0,1)P(|uv|)dxB(0,1)(uv)dμsupB(0,1)|uv||μ|(B(0,1))CuvLP(B(0,1)).

    That is, there exists a constant C3 such that

    B(0,1)P(|uv|)dxC3uvLP(B(0,1)). (3.5)

    We claim that there exists a positive constant CC(n,ν) such that

    B(0,1)P(|uv|)dxC (3.6)

    with 2R=1 and |μ|(B(0,1))=1. According to (2.8), we write uvLP(B(0,1))=k. We shall prove (3.6) in the following two scenarios. On the one hand, if 0<k1, then it follows from (3.5) that

    B(0,1)P(|uv|)dxC3kC.

    On the other hand, if k>1, then we set k=(C1C3)(2+ε0), where C1 and C3 are given in (2.5) and (3.5), respectively. It is obvious that 1C1C3>1, and so

    B(0,1)(C1C3)1+ε0P(|uv|)dx(C1C3)1+ε0C31(C1C3)2+ε0=1C1.

    By involving λ=C1C3<1 in (2.5), we obtain that

    B(0,1)P(C1C3|uv|)dxC1B(0,1)(C1C3)1+ε0P(|uv|)dx1. (3.7)

    By (3.7) and (2.8), one finds that the norm k1C1C3, that is, 1C1C31, which is contradictory to the condition 1C1C3>1. Therefore, the inequality (3.6) holds.

    Step 2. Scaling procedures.

    We assume that |μ|(B(x0,2R))=1, and we shall reduce to the case B(x0,2R)B(0,1) by a standard scaling argument. By letting

    ˜u(y):=u(x0+2Ry)2R,˜v(y):=v(x0+2Ry)2R,˜A(y,ξ):=(2R)nθ01A(x0+2Ry,ξ),˜μ(y):=(2R)nθ0μ(x0+2Ry)

    for yB(0,1), one has the following equations:

    div˜A(y,˜u)=˜μ  on B(0,1),div˜A(y,˜v)=0  on B(0,1).

    With the definition of Radon measure (1.2), we find the following relation between |˜μ|(B(0,1)) and |μ|(B(x0,2R)), namely,

    |˜μ|(B(0,1))=B(0,1)˜μ(y)dy = B(0,1)(2R)nθ0μ(x0+2Ry)dy=1(2R)nB(x0,2R)(2R)nθ0μ(x)dx = |μ|(B(x0,2R))(2R)θ0.

    Next, we shall reduce the general case to the special case |μ|(B(0,1))=1. We define

    M=[|˜μ|(B(0,1))]11+ε0=[|μ|(B(x0,2R))(2R)θ0]11+ε0. (3.8)

    By (1.3), one has M1. Hence the new solution, coefficient, and datum become

    ˉu:=˜uM,ˉv:=˜vM,ˉA(x,ξ):=˜A(x,Mξ)|˜μ|(B(0,1)),ˉμ:=˜μ|˜μ|(B(0,1)).

    Then we find that

    divˉA(x,ˉu)=ˉμ  on B(0,1),divˉA(x,ˉv)=0  on B(0,1)

    hold in the weak sense and |\bar{\mu}|(B(0, 1)) = 1 . Then by applying the result (3.6) in Step 1, one has

    \begin{eqnarray} &&\int_{B(0, 1)}P\left(|\nabla\bar u(x_0+2Ry)-\nabla\bar v(x_0+2Ry)|\right)\, \mathrm{d}y\\ & = &\int_{B(0, 1)}P\left(\frac{|\nabla\tilde u(x_0+2Ry)-\nabla\tilde v(x_0+2Ry)|}{M}\right)\, \mathrm{d}y\ \leq\ C. \end{eqnarray} (3.9)

    Considering (3.9) on B(x_0, 2R) , then

    \begin{equation*} \int_{B(0, 1)}P\left(|\nabla\bar u(x_0+2Ry)-\nabla\bar v(x_0+2Ry)|\right)\, \mathrm{d}y = \frac{1}{(2R)^n}\int_{B(x_0, 2R)}P\left(\frac{|\nabla u(x)-\nabla v(x)|}{M}\right)\, \mathrm{d}x, \end{equation*}

    we apply Lemma 2.5 with \lambda replaced by M to obtain

    \begin{eqnarray} &&\rlap{-} \displaystyle {\int}_{B(x_0, 2R)}P\left(|\nabla u(x)-\nabla v(x)|\right)\, \mathrm{d}x\\ &\leq& C_1M^{1+\varepsilon_0}\rlap{-} \displaystyle {\int}_{B(x_0, 2R)}P\left(\frac{|\nabla u(x)-\nabla v(x)|}{M}\right)\, \mathrm{d}x\ \leq\ C\, \frac{|\mu|(B(x_0, 2R))}{(2R)^{\theta_0}}, \end{eqnarray} (3.10)

    where C depends on n, \, \nu , and C_1 . From (3.10), we complete the proof of Proposition 3.2.

    Via a classical approach, we have the following estimate inspired by [10].

    Lemma 3.3. Let v\in W^{1, P}(\Omega) be a weak solution to the Dirichlet problem (3.1) under the assumptions (A1), let P be an N -function satisfying \Delta_2({P, \widetilde{P}}) < \infty , Assumption 2.4 and (2.12). There exist constants \alpha\in (0, 1] and C_4\equiv C_4(n, \nu, L)\geq 1 such that the estimate

    \begin{equation} \rlap{-} \displaystyle {\int}_{B(x_0, \rho)}P(|\nabla v|)\, \mathrm{d}x\leq C_4\left(\frac{\rho}{R}\right)^{-1+\alpha}\rlap{-} \displaystyle {\int}_{B(x_0, R)}P(|\nabla v|)\, \mathrm{d}x \end{equation} (3.11)

    holds whenever B(x_0, \rho)\subset B(x_0, R)\subset\Omega .

    We note that the estimate (3.11) is tenable by imitating the proof for the p -Laplacian equations.

    Lemma 3.4. Let u\in W^{1, P}(\Omega) be a weak solution to (1.1) under the assumptions {\text{(A1)}}_2 . Let P be an N -function satisfying \Delta_2({P, \widetilde{P}}) < \infty , Assumption 2.4, and (2.12). Then there exist constants \ell\equiv \ell(n, \, \nu, \, L, \, C_{\Delta_2})\geq 1 and C\equiv C(n, \, \nu, \, L, \, C_{\Delta_2}, \, C_1)\geq 1 such that

    \begin{equation*} \rlap{-} \displaystyle {\int}_{B(x_0, \rho)}P(|\nabla u|)\, \mathrm{d}x\leq \ell\left(\frac{\rho}{R}\right)^{-1+\alpha}\rlap{-} \displaystyle {\int}_{B(x_0, R)}P(|\nabla u|)\, \mathrm{d}x+C\left(\frac{R}{\rho}\right)^{n}\frac{|\mu|(B(x_0, 2R))}{(2R)^{\theta_0}} \end{equation*}

    holds whenever B(x_0, \rho)\subset B(x_0, R)\subset\Omega .

    Proof. From the triangle inequality and (2.2), we see that

    \begin{eqnarray*} \rlap{-} \displaystyle {\int}_{B(x_0, \rho)}P(|\nabla u|)\, \mathrm{d}x&\leq&C_{\Delta_2}\left(\rlap{-} \displaystyle {\int}_{B(x_0, \rho)}P(|\nabla u-\nabla v|)\, \mathrm{d}x+\rlap{-} \displaystyle {\int}_{B(x_0, \rho)}P(|\nabla v|)\, \mathrm{d}x\right)\nonumber\\ &\leq&C_{\Delta_2}\left(\frac{R}{\rho}\right)^{n}\rlap{-} \displaystyle {\int}_{B(x_0, R)}P(|\nabla u-\nabla v|)\, \mathrm{d}x+C_{\Delta_2}\rlap{-} \displaystyle {\int}_{B(x_0, \rho)}P(|\nabla v|)\, \mathrm{d}x. \end{eqnarray*}

    By applying (3.11), one carries out

    \begin{eqnarray} \rlap{-} \displaystyle {\int}_{B(x_0, \rho)}P(|\nabla u|)\, \mathrm{d}x&\leq& C_{\Delta_2}\left(\frac{R}{\rho}\right)^{n}\rlap{-} \displaystyle {\int}_{B(x_0, R)}P(|\nabla u-\nabla v|)\, \mathrm{d}x\\ &&+\ C_{\Delta_2}\, C_4\, \left(\frac{\rho}{R}\right)^{-1+\alpha}\rlap{-} \displaystyle {\int}_{B(x_0, R)}P(|\nabla v|)\, \mathrm{d}x. \end{eqnarray} (3.12)

    Using the triangle inequality |\nabla v|\leq|\nabla u-\nabla v|+|\nabla u| again, the inequality (3.12) leads to

    \begin{eqnarray*} \rlap{-} \displaystyle {\int}_{B(x_0, \rho)}P(|\nabla u|)\, \mathrm{d}x&\leq&C_{\Delta_2}\left[\left(\frac{R}{\rho}\right)^{n}+C_{\Delta_2}C_4\, \left(\frac{\rho}{R}\right)^{-1+\alpha}\right]\rlap{-} \displaystyle {\int}_{B(x_0, R)}P\left(|\nabla u-\nabla v|\right)\, \mathrm{d}x\\ &&+\ \ell\, \left(\frac{\rho}{R}\right)^{-1+\alpha}\rlap{-} \displaystyle {\int}_{B(x_0, R)}P(|\nabla u|)\, \mathrm{d}x \end{eqnarray*}

    with \ell = C_{\Delta_2}^{2}C_4 . It follows from (3.3) that

    \begin{eqnarray*} \rlap{-} \displaystyle {\int}_{B(x_0, \rho)}P(|\nabla u|)\, \mathrm{d}x&\leq& C\left[\left(\frac{R}{\rho}\right)^{n}+C_{\Delta_2}C_4\, \left(\frac{\rho}{R}\right)^{-1+\alpha}\right]\, \frac{|\mu|(B(x_0, 2R))}{(2R)^{\theta_0}}\\ &&+\ \ell\, \left(\frac{\rho}{R}\right)^{-1+\alpha}\rlap{-} \displaystyle {\int}_{B(x_0, R)}P(|\nabla u|)\, \mathrm{d}x. \end{eqnarray*}

    Notice that \rho\leq R with

    \begin{equation*} C_{\Delta_2}C_4\, \left(\frac{\rho}{R}\right)^{-1+\alpha} < C\left(\frac{R}{\rho}\right)^{n}, \end{equation*}

    we complete the proof of Lemma 3.4.

    We also define w\in v+W_{0}^{1, P}(B(x_0, R)) as the unique solution to the homogeneous Dirichlet problem with frozen coefficients

    \begin{equation} \left\{ \begin{array}{rcll} \operatorname{div}\mathcal{A}(x_0, \nabla w)& = &0 & \ \mathrm{in}\ B(x_0, R), \\ w& = &v & \ \mathrm{on}\ \partial B(x_0, R).\\ \end{array} \right. \end{equation} (3.13)

    We have the following decay estimate.

    Lemma 3.5. Let w\in W^{1, P}(\Omega) be a weak solution to (3.13) under the assumption (A1). Then there exist constants \tilde{\alpha}\in(0, 1] and C\geq 1 , both depending on n, \, \nu, \, L , such that

    \begin{equation} \rlap{-} \displaystyle {\int}_{B(x_0, \rho)}P\left(|\nabla w-(\nabla w)_{B(x_0, \rho)}|\right)\, \mathrm{d}x\leq C\left(\frac{\rho}{R}\right)^{\tilde{\alpha}}\rlap{-} \displaystyle {\int}_{B(x_0, R)}P\left(|\nabla w-(\nabla w)_{B(x_0, R)}|\right)\, \mathrm{d}x \end{equation} (3.14)

    holds whenever B(x_0, \rho)\subset B(x_0, R)\subset\Omega .

    Notice that the conclusion (3.14) is inspired by [32].

    Lemma 3.6. Under the assumptions (A1) and (A2) of Theorem 1.1, with v as in (3.1) and w as in (3.13), there exists a constant C\equiv C(n, \, \nu, \, L) such that

    \begin{equation} \rlap{-} \displaystyle {\int}_{B(x_0, R)}P(|\nabla v-\nabla w|)\, \mathrm{d}x\leq CK\omega(R)\rlap{-} \displaystyle {\int}_{B(x_0, R)}P(|\nabla v|+s)\, \mathrm{d}x \end{equation} (3.15)

    for B(x_{0}, R)\subset\Omega , where K and \omega(R) are given in the assumption (A1).

    Proof. We test Eq (3.13)_1 with v-w . Since both v and w are weak solutions, then the assumption {\text{(A1)}}_2 gives us that

    \begin{eqnarray*} C\nu^{-1}\int_{B(x_0, R)}P(|\nabla v-\nabla w|)\, \mathrm{d}x&\leq& \int_{B(x_0, R)}{\langle \mathcal{A}(x_0, \nabla v)-\mathcal{A}(x_0, \nabla w), \nabla v-\nabla w\rangle} \, \mathrm{d}x\\ & = &\int_{B(x_0, R)}{\langle \mathcal{A}(x_0, \nabla v)-\mathcal{A}(x, \nabla v), \nabla v-\nabla w\rangle} \, \mathrm{d}x. \end{eqnarray*}

    By using {\text{(A1)}}_3 with |x-x_{0}|\leq R and Young's inequality (2.9), we derive that

    \begin{eqnarray} C\nu^{-1}\int_{B(x_0, R)}P(|\nabla v-\nabla w|)\, \mathrm{d}x &\leq& K\omega(R)\int_{B(x_0, R)}\frac{P\left((|\nabla v|^2+s^2)^{\frac{1}{2}}\right)}{(|\nabla v|^2+s^2)^{\frac{1}{2}}}|\nabla v-\nabla w|\, \mathrm{d}x\\ &\leq& \varepsilon\, K\omega(R)\int_{B(x_0, R)}P(|\nabla v-\nabla w|)\, \mathrm{d}x \\ &&+\ C_{\varepsilon}K\omega(R)\int_{B(x_0, R)}\widetilde{P}\left[\frac{P\left((|\nabla v|^2+s^2)^{\frac{1}{2}}\right)}{(|\nabla v|^2+s^2)^{\frac{1}{2}}}\right]\, \mathrm{d}x. \end{eqnarray} (3.16)

    Finally, Lemma 3.6 is proved by using the assumptions (A2), (3.16), and (2.3).

    Lemma 3.7. Assume that u\in W^{1, P}(\Omega) is a weak solution to (1.1) satisfying (A1), (A2), and P is an N -function with \Delta_2({P, \widetilde{P}}) < \infty , and satisfies Assumption 2.4, (2.12). Let w be defined in (3.13), and \mu be a Radon measure that satisfies {\text{(M)}} . There exists a constant C\equiv C(n, \, \nu, \, L, \, s, \, C_{\Delta_2}, \, C_1) such that

    \begin{equation} \rlap{-} \displaystyle {\int}_{B(x_0, R)}P(|\nabla u-\nabla w|)\, \mathrm{d}x\leq C\, \frac{|\mu|(B(x_0, 2R))}{(2R)^{\theta_0}}+CK\omega(R)\rlap{-} \displaystyle {\int}_{B(x_0, 2R)}P(|\nabla u|+s)\, \mathrm{d}x. \end{equation} (3.17)

    The key to the proof of Lemma 3.7 is the triangle inequality as follows:

    \begin{equation*} P(|\nabla u-\nabla w|)\leq C_{\Delta_2}\bigl[P\left(|\nabla u-\nabla v|\right)+P\left(|\nabla v-\nabla w|\right)\bigr] \end{equation*}

    with (3.3), (3.15), and (3.2).

    Corollary 3.8. Let u\in W^{1, P}(\Omega) be a weak solution to (1.1) under the assumptions {\text{(A1), (A2)}} . Let P be an N -function satisfying \Delta_2({P, \widetilde{P}}) < \infty , Assumption 2.4, and (2.12). Then there exists a constant C\equiv C(n, \, \nu, \, L, \, C_{\Delta_2}, \, C_1)\geq 1 such that

    \begin{equation*} \rlap{-} \displaystyle {\int}_{B(x_0, \rho)}P(|\nabla u|)\, \mathrm{d}x\, \leq\, C\, \rlap{-} \displaystyle {\int}_{B(x_0, R)}P(|\nabla u|)\, \mathrm{d}x\, +\, C\, \left(\frac{R}{\rho}\right)^{n}\left[\frac{|\mu|(B(x_0, 2R))}{(2R)^{\theta_0}}+K\omega(R)\rlap{-} \displaystyle {\int}_{B(x_0, 2R)}P(|\nabla u|)\, \mathrm{d}x\right] \end{equation*}

    holds whenever B(x_0, \rho)\subset B(x_0, R)\subset B(x_0, 2R)\subset\Omega .

    Our goal is to derive an oscillation decay estimate of \nabla u . Based on Lemmas 3.5 and 3.7, we first involve the corresponding oscillation decay estimate (3.14) of \nabla w , and then compare \nabla u and \nabla w by (3.17). We note that Lemma 2.10 and the triangle inequality play an essential role in the following lemma.

    Lemma 3.9. Let u be a weak solution to (1.1) under the assumptions (A1), (A2), (M), \Delta_2({P, \widetilde{P}}) < \infty , Assumption 2.4, and (2.12). Then there exists C\equiv C(n, \, \nu, \, L, \, s, \, C_{\Delta_2}, \, C_1) > 0 such that

    \begin{eqnarray*} \rlap{-} \displaystyle {\int}_{B(x_0, \rho)}P\left(|\nabla u-(\nabla u)_{B(x_0, \rho)}|\right)\, \mathrm{d}x&\leq& C\left(\frac{\rho}{R}\right)^{\tilde{\alpha}}\rlap{-} \displaystyle {\int}_{B(x_0, 2R)}P\left(|\nabla u-(\nabla u)_{B(x_0, 2R)}|\right)\, \mathrm{d}x\nonumber\\ &&+\ CK\left(\frac{R}{\rho}\right)^{n}\omega(R)\rlap{-} \displaystyle {\int}_{B(x_0, 2R)}P\left(|\nabla u|+s\right)\, \mathrm{d}x\nonumber\\ &&+\ C\left(\frac{R}{\rho}\right)^{n}\frac{|\mu|(B(x_0, 2R))}{(2R)^{\theta_0}} \end{eqnarray*}

    for B(x_{0}, \rho)\subset B(x_{0}, 2R)\subset\Omega . Here the constant \tilde{\alpha} is introduced in Lemma 3.5.

    In this section, we use the estimate established in Proposition 3.2 to derive the Caccioppoli-type inequality and the maximal estimate. First, the following Caccioppoli-type inequality gives a connection between \nabla u and u .

    Proposition 4.1. (Caccioppoli-type inequality) Let u\in W^{1, P}(\Omega) be a weak solution to (1.1) with measurable coefficients and satisfy {\text{(A1)}}_1 , {\text{(A1)}}_2 . Let \mu be a Radon measure with (M). Suppose P is an N -function satisfying Assumption 2.4. Then there exists a constant C\equiv C(n, \, \nu, \, L, \, C_1, \, C_{\Delta_2}) such that

    \begin{equation} \rlap{-} \displaystyle {\int}_{B(x_0, R)}P(|\nabla u|)\, \mathrm{d}x\leq C\rlap{-} \displaystyle {\int}_{B(x_0, 2R)}P\left(\frac{\left|u-(u)_{B(x_0, 2R)}\right|}{R}\right)\, \mathrm{d}x+C\, \frac{|\mu|(B(x_0, 2R))}{(2R)^{\theta_0}}, \end{equation} (4.1)

    where B(x_0, 2R)\subset\Omega .

    Proof. We may assume that (u)_{B(x_0, 2R)} = 0 as if u solves (1.1) also u-(u)_{B(x_0, 2R)} dose. Let \eta\in C_{0}^{\infty}(B(x_0, 2R)) such that 0\leq \eta\leq 1 , and

    \begin{equation} \left\{ \begin{array}{r@{\ }c@{\ }ll} \eta& = &1, & \mathrm{in}\: B(x_0, R)\, , \\[0.1cm] |\nabla\eta|&\leq& \frac{1}{R}, & \mathrm{in}\: B(x_0, 2R)\setminus B(x_0, R)\, , \\[0.2cm] \eta& = &0\, , & \mathrm{otherwise}.\\ \end{array} \right. \end{equation} (4.2)

    Let v\in u+W_0^{1, P}(B(x_{0}, 2R)) be the weak solution to (3.1). We choose a test function \varphi: = v\eta to (3.1), and obtain

    \begin{equation*} \label{eq21} \int_{B(x_0, 2R)}\langle \mathcal{A}(x, \nabla v), \nabla \varphi\rangle\mathrm{d}x = 0. \end{equation*}

    It is clear that

    \begin{equation} \int_{B(x_0, 2R)}\langle\mathcal{A}(x, \nabla v), \eta\nabla v\rangle\mathrm{d}x = -\int_{B(x_0, 2R)}\langle\mathcal{A}(x, \nabla v), v\nabla \eta\rangle\mathrm{d}x. \end{equation} (4.3)

    By Lemma 2.6 and (4.2), we deduce that

    \begin{equation} \int_{B(x_0, R)}P(|\nabla v|)\, \mathrm{d}x\, \leq\, C\int_{B(x_0, R)}\langle \mathcal{A}(x, \nabla v), \nabla v\rangle\, \mathrm{d}x\, \leq\, C\int_{B(x_0, 2R)}\langle \mathcal{A}(x, \nabla v), \eta\nabla v\rangle\, \mathrm{d}x. \end{equation} (4.4)

    By (4.3), (4.4), Cauchy-Schwartz inequality, and Lemma 2.6, one has

    \begin{equation*} \int_{B(x_0, R)}P(|\nabla v|)\, \mathrm{d}x\leq\int_{B(x_0, 2R)} P'(|\nabla v|)\cdot |v|\, \left|\nabla\eta\right|\, \mathrm{d}x. \end{equation*}

    According to Young's inequality (2.10), we derive that for \varepsilon > 0 , there exists \widetilde{C}_{\varepsilon} such that

    \begin{equation*} \int_{B(x_0, R)}P(|\nabla v|)\, \mathrm{d}x\leq \varepsilon\int_{B(x_0, 2R)}\widetilde{P}\left(P'(|\nabla v|)\right)\, \mathrm{d}x+\widetilde{C}_{\varepsilon} \int_{B(x_0, 2R)}P\left(|v|\left|\nabla\eta\right|\right)\, \mathrm{d}x. \end{equation*}

    By (2.3) and Lemma 2.13, we deduce that

    \begin{equation*} \int_{B(x_0, R)}P(|\nabla v|)\, \mathrm{d}x\leq \widetilde{C}_{\varepsilon} \int_{B(x_0, 2R)\setminus B(x_0, R)}P\left(\frac{|v|}{R}\right)\, \mathrm{d}x. \end{equation*}

    Dividing by |B(x_0, R)| , one gives

    \begin{equation} \rlap{-} \displaystyle {\int}_{B(x_0, R)}P(|\nabla v|)\, \mathrm{d}x\leq 2^{n}\, \widetilde{C}_\varepsilon \rlap{-} \displaystyle {\int}_{B(x_0, 2R)}P\left(\frac{|v|}{R}\right)\, \mathrm{d}x. \end{equation} (4.5)

    Applying the triangle inequality with P(|\nabla u|)\leq C_{\Delta_2}\left(P(|\nabla u-\nabla v|)+P(|\nabla v|)\right) and (4.5), one has

    \begin{equation} \rlap{-} \displaystyle {\int}_{B(x_0, R)}P(|\nabla u|)\, \mathrm{d}x\, \leq\, 2^nC_{\Delta_2}\rlap{-} \displaystyle {\int}_{B(x_0, 2R)}P(|\nabla u-\nabla v|)\, \mathrm{d}x\, +\, 2^n\, C_{\Delta_2}\widetilde{C}_\varepsilon\rlap{-} \displaystyle {\int}_{B(x_0, 2R)}P\left(\frac{|v|}{R}\right)\, \mathrm{d}x. \end{equation} (4.6)

    In order to estimate the last term of (4.6), we use the triangle inequality again to obtain

    \begin{equation} \rlap{-} \displaystyle {\int}_{B(x_0, 2R)}P\left(\frac{|v|}{R}\right)\, \mathrm{d}x\leq C_{\Delta_2}\rlap{-} \displaystyle {\int}_{B(x_0, 2R)}P\left(\frac{|u-v|}{R}\right)\, \mathrm{d}x+C_{\Delta_2}\rlap{-} \displaystyle {\int}_{B(x_0, 2R)}P\left(\frac{|u|}{R}\right)\, \mathrm{d}x. \end{equation} (4.7)

    Lemma 2.11 and the classical Hölder's inequality give us that

    \begin{eqnarray} \rlap{-} \displaystyle {\int}_{B(x_0, 2R)}P\left(\frac{|u-v|}{R}\right)\, \mathrm{d}x&\leq&C\left(\rlap{-} \displaystyle {\int}_{B(x_0, 2R)}P^{\theta_1}(|\nabla u-\nabla v|)\, \mathrm{d}x\right)^{\frac{1}{\theta_1}}\\ &\leq&C\rlap{-} \displaystyle {\int}_{B(x_0, 2R)}P(|\nabla u-\nabla v|)\, \mathrm{d}x. \end{eqnarray} (4.8)

    Thus we combine (3.3) and (4.6)–(4.8), and conclude that

    \begin{equation*} \rlap{-} \displaystyle {\int}_{B(x_0, R)}P(|\nabla u|)\, \mathrm{d}x\leq C\, \frac{|\mu|(B(x_0, 2R))}{(2R)^{\theta_0}}+C\rlap{-} \displaystyle {\int}_{B(x_0, 2R)}P\left(\frac{|u|}{R}\right)\, \mathrm{d}x. \end{equation*}

    This establishes the Caccioppoli-type inequality (4.1).

    Based on the definition of maximal functions in Section 2.4 and the control estimate in Lemma 3.4, we present the following pointwise estimate involving the maximal functions.

    Proposition 4.2. (Maximal estimate) Let u\in W^{1, P}(\Omega) be a weak solution to (1.1) under (A1), (A2). Let P be an N -function with \Delta_2({P, \widetilde{P}}) < \infty , Assumption 2.4, and (2.12). Let Radon measure \mu satisfy (M). Then there exists a constant C\equiv C(n, \, \nu, \, L, \, \theta_0, \, \omega(\cdot), \, C_{\Delta_2}, \, C_1) such that

    \begin{eqnarray} && \widetilde{M}_{1+\varepsilon_0-\alpha, \, R}^{\#, \, P}(u)(x_0)+\left[\, M_{(1+\varepsilon_0-\alpha)\theta_1, \, R}^{P^{\theta_1}}\left(\nabla u\right)(x_0)\, \right]^{\frac{1}{\theta_1}} \\ &\leq& C\, R^{1+\varepsilon_0}\, \mathbf{I}_{n-\theta_0-\alpha} ^\mu(x_0, 2R)+C, R^{1+\varepsilon_0-\alpha}\rlap{-} \displaystyle {\int}_{B(x_0, R)}P(|\nabla u|+s)\, \mathrm{d}x. \end{eqnarray} (4.9)

    Here \mathbf{I}_{n-\theta_0-\alpha} ^\mu(x_0, 2R) is a Riesz potential that is introduced in (1.4). In (4.9), the constants \varepsilon_{0} , \alpha\in(0, 1] , \theta_1 and \theta_{0} are given in Lemmas 2.5, 3.3, 2.11, and (1.3), respectively.

    Proof. The key of the proof is to consider the radii R satisfying that R\leq R_0 , where the quantity R_0 > 0 is in dependence of the data n, \, \nu, \, L, \, \alpha , and \omega(\cdot) . More precisely, by (A2), we shall choose R_0 so that

    \begin{equation*} \omega(R_0)\leq\delta, \end{equation*}

    where \delta will be a small quantity that will be reduced at several stages, as a decreasing function of the quantities n, \, \nu, \, L , and also \alpha . The proof of Proposition 4.2 is accomplished through two steps, to which the following content is devoted.

    By (2.15) and (2.13) with \theta = 1+\varepsilon_0-\alpha and \tau = (1+\varepsilon_0-\alpha)\theta_1 , there holds

    \begin{equation} \widetilde{M}_{1+\varepsilon_0-\alpha, \, R}^{\#, \, P}(u)(x)\leq C\left[M_{(1+\varepsilon_0-\alpha)\theta_1, \, R}^{P^{\theta_1}}\left(\nabla u\right)(x)\right]^{\frac{1}{\theta_1}}. \end{equation} (4.10)

    By using Hölder's inequality, we obtain

    \begin{eqnarray} \left[M_{(1+\varepsilon_0-\alpha)\theta_1, \, R}^{P^{\theta_1}}\left(\nabla u\right)(x)\right]^{\frac{1}{\theta_1}} & = & \sup\limits_{0 < r\leq R}\left(r^{(1+\varepsilon_0-\alpha)\theta_1}\rlap{-} \displaystyle {\int}_{B(x_0, r)}P^{\theta_1}(|\nabla u|)\, \mathrm{d}x\right)^{\frac{1}{\theta_1}} \\ &\leq& \sup\limits_{0 < r\leq R}\left(r^{1+\varepsilon_0-\alpha}\rlap{-} \displaystyle {\int}_{B(x_0, r)}P(|\nabla u|)\, \mathrm{d}x\right) \\ & = & M_{1+\varepsilon_0-\alpha, \, R}^{P}(\nabla u)(x_0). \end{eqnarray} (4.11)

    Then the inequality (4.9) will follow if we are able to show that

    \begin{equation} M_{1+\varepsilon_0-\alpha, \, R}^{P}(\nabla u)(x_0) \leq C\, R^{1+\varepsilon_0}\, \mathbf{I}_{n-\theta_0-\alpha} ^\mu(x_0, 2R)+C\, R^{1+\varepsilon_0-\alpha}\rlap{-} \displaystyle {\int}_{B(x_0, R)}P(|\nabla u|+s)\, \mathrm{d}x. \end{equation} (4.12)

    Step 1. The case for small radii R\leq R_0 .

    We take 0 < \rho\leq r/2\leq r \leq R , and adopt the estimate in Lemma 3.4 with two radii \rho and r/2 . There exists a constant C_5\equiv C_5(n, \, \nu, \, L, \, C_{\Delta_2}, \, C_1) such that

    \begin{equation} \rlap{-} \displaystyle {\int}_{B(x_0, \rho)}P(|\nabla u|)\, \mathrm{d}x\leq C_5\left(\frac{\rho}{r}\right)^{-1+\alpha}\, \rlap{-} \displaystyle {\int}_{B(x_0, r)}P(|\nabla u|)\, \mathrm{d}x+\, C\left(\frac{r}{\rho}\right)^{n}\frac{|\mu|(B(x_0, r))}{r^{\theta_0}}. \end{equation} (4.13)

    Multiplying both sides of (4.13) by \rho^{1+\varepsilon_0-\alpha} , and taking S = r/\rho , it follows that

    \begin{eqnarray*} \rho^{1+\varepsilon_0-\alpha}\rlap{-} \displaystyle {\int}_{B(x_0, \rho)}P(|\nabla u|)\, \mathrm{d}x &\leq&C_5\, S^{-\varepsilon_0}\, r^{1+\varepsilon_0-\alpha}\rlap{-} \displaystyle {\int}_{B(x_0, r)}P(|\nabla u|)\, \mathrm{d}x\nonumber\\ &&+\ C\, S^{n+\alpha-\varepsilon_0-1}\, r^{1+\varepsilon_0}\, \frac{|\mu|(B(x_0, r))}{r^{\theta_0+\alpha}} \end{eqnarray*}

    for \rho\leq r/2\leq R/2 . We choose the constant S\geq2 large enough, which satisfies that

    \begin{equation*} \frac{C_5}{S^{\varepsilon_0}}\leq \frac{1}{2}\, , \end{equation*}

    and take the supremum with 0 < r\leq R such that the following estimate holds

    \begin{eqnarray} \sup\limits_{0 < r\leq R}\left(\rho^{1+\varepsilon_0-\alpha}\rlap{-} \displaystyle {\int}_{B(x_0, \rho)}P(|\nabla u|)\, \mathrm{d}x\right)&\leq& \frac{1}{2}\sup\limits_{0 < r\leq R}\left(r^{1+\varepsilon_0-\alpha}\rlap{-} \displaystyle {\int}_{B(x_0, r)}P(|\nabla u|)\, \mathrm{d}x\right)\\ &&+\ C\, R^{1+\varepsilon_0}\, \sup\limits_{0 < r\leq R}\frac{|\mu|(B(x_0, r))}{r^{\theta_0+\alpha}}, \end{eqnarray} (4.14)

    where 0 < r\leq R is equivalent to 0 < \rho\leq R/S . By (2.13) and (4.14), we obtain

    \begin{equation} \sup\limits_{\rho\leq R/S}\left(\rho^{1+\varepsilon_0-\alpha}\rlap{-} \displaystyle {\int}_{B(x_0, \rho)}P(|\nabla u|)\, \mathrm{d}x\right)\, \leq\, \frac{1}{2}\, M_{1+\varepsilon_0-\alpha, \, R}^{P}(\nabla u)(x_0)\, +\, C\, R^{1+\varepsilon_0}\sup\limits_{0 < r\leq R}\frac{|\mu|(B(x_0, r))}{r^{\theta_0+\alpha}} \end{equation} (4.15)

    with a constant C depending on n, \, \nu, \, L, \, C_{\Delta_2}, \, C_1, \, S , and \alpha .

    On the other hand, we notice that

    \begin{equation} \sup\limits_{R/S\leq\rho\leq R}\left(\rho^{1+\varepsilon_0-\alpha}\rlap{-} \displaystyle {\int}_{B(x_0, \rho)}P(|\nabla u|)\, \mathrm{d}x\right)\leq C\, S^{n}\, R^{1+\varepsilon_0-\alpha}\rlap{-} \displaystyle {\int}_{B(x_0, R)}P(|\nabla u|+s)\, \mathrm{d}x. \end{equation} (4.16)

    Recalling the constant S , and putting (4.15) and (4.16) together, we obtain the following:

    \begin{eqnarray} M_{1+\varepsilon_0-\alpha, \, R}^{P}(\nabla u)(x_0)&\leq&\frac{1}{2}\, M_{1+\varepsilon_0-\alpha, \, R}^{P}(\nabla u)(x_0)+C\, R^{1+\varepsilon_0}\sup\limits_{0 < r\leq R}\frac{|\mu|(B(x_0, r))}{r^{\theta_0+\alpha}} \\ &&+\ C\, R^{1+\varepsilon_0-\alpha}\rlap{-} \displaystyle {\int}_{B(x_0, R)}P(|\nabla u|+s)\, \mathrm{d}x. \end{eqnarray} (4.17)

    The definition of the supremum shows that for any \varepsilon > 0 , there is r\in (0, R] such that

    \begin{equation} \sup\limits_{0 < r\leq R}\frac{|\mu|(B(x_0, r))}{r^{\theta_0+\alpha}}\leq \frac{|\mu|(B(x_0, r))}{r^{\theta_0+\alpha}}+\varepsilon. \end{equation} (4.18)

    This leads to

    \begin{equation} \frac{|\mu|(B(x_0, r))}{r^{\theta_0+\alpha}} = \frac{|\mu|(B(x_0, r))}{r^{\theta_0+\alpha}}\frac{1}{\ln 2}\int_{r}^{2r}\, \frac{\mathrm{d}\rho}{\rho}\leq\frac{2^{\theta_0+\alpha}}{\ln 2}\int_{r}^{2r}\frac{|\mu|(B(x, \rho))}{\rho^{\theta_0+\alpha}}\frac{\mathrm{d}\rho}{\rho}. \end{equation} (4.19)

    Since \varepsilon is arbitrary, and 0 < r < 2r\leq 2R , the preceding estimates (4.18) and (4.19) show that there exists a constant C\equiv C(n, \, \theta_0) such that

    \begin{equation} \sup\limits_{0 < r\leq R}\frac{|\mu|(B(x_0, r))}{r^{\theta_0+\alpha}}\leq C\int_{0}^{2R}\frac{|\mu|(B(x_0, \rho))}{\rho^{\theta_0+\alpha}}\frac{\mathrm{d}\rho}{\rho} = C\, \mathbf{I}_{n-\theta_0-\alpha} ^\mu(x_0, 2R). \end{equation} (4.20)

    Combining (4.17) and (4.20), we deduce the desired estimate (4.12) with R\leq R_0 .

    Step 2. Removing the condition R\leq R_0 .

    Our goal is to prove (4.12) without the restriction R\leq R_0 . Taking R > R_0 and recalling Definition (2.13), it is clear that

    \begin{eqnarray} &&M_{1+\varepsilon_0-\alpha, \, R}^{P}\left(\nabla u\right)(x_0)\\ & = &\sup\limits_{0 < r\leq R}\left(r^{1+\varepsilon_0-\alpha}\rlap{-} \displaystyle {\int}_{B(x_0, r)}P(|\nabla u|)\, \mathrm{d}x\right)\\ &\leq&\sup\limits_{0 < r\leq R_0}\left(r^{1+\varepsilon_0-\alpha}\rlap{-} \displaystyle {\int}_{B(x_0, r)}P(|\nabla u|)\, \mathrm{d}x\right)+\sup\limits_{R_0 < r\leq R}\left(r^{1+\varepsilon_0-\alpha}\rlap{-} \displaystyle {\int}_{B(x_0, r)}P(|\nabla u|)\, \mathrm{d}x\right)\\ &\leq&M_{1+\varepsilon_0-\alpha, \, R_0}^{P}\left(\nabla u\right)(x_0) +\, \left(\frac{R}{R_0}\right)^n\, R^{1+\varepsilon_0-\alpha}\rlap{-} \displaystyle {\int}_{B(x_0, R)}P(|\nabla u|+s)\, \mathrm{d}x. \end{eqnarray} (4.21)

    We apply (4.12) with radius R_0 , i.e.,

    \begin{eqnarray} M_{1+\varepsilon_0-\alpha, \, R_0}^{P}\left(\nabla u\right)(x_0)&\leq& C\, R_0^{1+\varepsilon_0}\, \mathbf{I}_{n-\theta_0-\alpha} ^\mu(x_0, 2R_0)\\ &&+\ C\, R_0^{1+\varepsilon_0-\alpha}\rlap{-} \displaystyle {\int}_{B(x_0, R_0)}P(|\nabla u|+s)\, \mathrm{d}x. \end{eqnarray} (4.22)

    By the definition of Riesz potential (1.4), one has

    \begin{eqnarray} &&R_0^{1+\varepsilon_0}\, \mathbf{I}_{n-\theta_0-\alpha} ^\mu(x_0, 2R_0)\ = \ R_0^{1+\varepsilon_0}\int_{0}^{2R_0}\frac{|\mu|(B(x_0, \rho))}{\rho^{\theta_0+\alpha }}\frac{\mathrm{d}\rho}{\rho}\\ &\leq&R^{1+\varepsilon_0}\int_{0}^{2R}\frac{|\mu|(B(x_0, \rho))}{\rho^{\theta_0+\alpha }}\frac{\mathrm{d}\rho}{\rho}\ = \ R^{1+\varepsilon_0}\, \mathbf{I}_{n-\theta_0-\alpha} ^\mu(x_0, 2R). \end{eqnarray} (4.23)

    It is apparent to enlarge the integral by

    \begin{equation} R_0^{1+\varepsilon_0-\alpha}\rlap{-} \displaystyle {\int}_{B(x_0, R_0)}P(|\nabla u|+s)\, \mathrm{d}x\leq\left(\frac{R}{R_0}\right)^nR^{1+\varepsilon_0-\alpha}\rlap{-} \displaystyle {\int}_{B(x_0, R)}P(|\nabla u|+s)\, \mathrm{d}x. \end{equation} (4.24)

    By using (4.21)–(4.24), we derive

    \begin{equation*} M_{1+\varepsilon_0-\alpha, \, R}^{P}\left(\nabla u\right)(x_0)\leq R^{1+\varepsilon_0}\, \mathbf{I}_{n-\theta_0-\alpha} ^\mu(x_0, 2R) +\left(\frac{R}{R_0}\right)^n\, R^{1+\varepsilon_0-\alpha}\rlap{-} \displaystyle {\int}_{B(x_0, R)}P(|\nabla u|+s)\, \mathrm{d}x. \end{equation*}

    Since \Omega is bounded, then (4.12) holds.

    Combining (4.10)–(4.12), we obtain (4.9), which means the proof of Proposition 4.2 is completed.

    By establishing the preceding technical tools and lemmas, we are in a position to present the proofs of the main theorems. We first have the following proof.

    Proof of Theorem 1.1. We set a sequence of balls \{B_{i}\}_{i = 0}^{\infty} by

    \begin{equation} B_i: = B(x_0, R_i) = B\left(x_0, \frac{R}{(2\Lambda)^i}\right), \end{equation} (5.1)

    where 2\Lambda > 1 will be chosen later. It is clear that B_{i+1}\subset B_i for every i\geq 0 . We set two sequences \{K_{i}\}_{i = 0}^{\infty} and \{k_{i}\}_{i = 0}^{\infty} by

    \begin{equation} K_i: = \rlap{-} \displaystyle {\int}_{{B}_i}P\left(\left|\nabla u-(\nabla u)_{B_i}\right|\right)\, \mathrm{d}x, \ \ k_i: = P\left(\left|(\nabla u)_{B_i}\right|\right)+P(s). \end{equation} (5.2)

    We also introduce \widetilde{k}_{0} by

    \begin{equation} \widetilde{k}_{0}: = \rlap{-} \displaystyle {\int}_{B(x_0, R)}\left[P(|\nabla u|)+P(s)\right]\, \mathrm{d}x. \end{equation} (5.3)

    By (2.10), it is obvious that

    \begin{equation} k_{0} = P\left(\left|(\nabla u)_{B_{0}}\right|\right)+P(s) = P\left(\left|\rlap{-} \displaystyle {\int}_{B(x_0, R)}\nabla u\, \mathrm{d}x\right|\right)+P(s)\leq \widetilde{k}_{0}, \end{equation} (5.4)

    as well as

    \begin{equation} K_0 = \rlap{-} \displaystyle {\int}_{B_{0}}P\left(\left|\nabla u-(\nabla u)_{B_{0}}\right|\right)\, \mathrm{d}x\leq C\rlap{-} \displaystyle {\int}_{B(x_0, R)}P(|\nabla u|)\, \mathrm{d}x\leq\widetilde{k}_{0}. \end{equation} (5.5)

    Step 1. An estimate of the summation of K_i .

    An application of Lemma 3.9 with B(x_0, \rho)\equiv B\left(x_0, \frac{R}{2\Lambda}\right)\subset B(x_0, R) shows that

    \begin{eqnarray*} &&\rlap{-} \displaystyle {\int}_{B\left(x_0, \frac{R}{2\Lambda}\right)}P\left(\left|\nabla u-(\nabla u)_{B\left(x_0, \frac{R}{2\Lambda}\right)}\right|\right)\, \mathrm{d}x \\ &\leq& C\left(\frac{1}{2\Lambda}\right)^{\tilde{\alpha}}\rlap{-} \displaystyle {\int}_{B(x_0, R)}P\left(\left|\nabla u-(\nabla u)_{B(x_0, R)}\right|\right)\, \mathrm{d}x\\ &&+\ C(2\Lambda)^{n}\, \frac{|\mu|(B(x_0, 2R))}{(2R)^{\theta_0}}\, +\, C(2\Lambda)^{n}K\omega(R)\rlap{-} \displaystyle {\int}_{B(x_0, R)}P(|\nabla u|+s)\, \mathrm{d}x, \end{eqnarray*}

    where the constants C depend on n, \, \nu, \, L, \, s, \, C_{\Delta_2} , and C_1 . Using (5.1), we choose \Lambda\equiv \Lambda(n, \, \nu, \, L, \, s, \, C_{\Delta_2}, \, C_1) > 1 large enough such that

    \begin{equation*} C\left(\frac{1}{2\Lambda}\right)^{\tilde{\alpha}}\leq\frac{1}{4}, \end{equation*}

    where \tilde{\alpha}\in(0, 1] is given in Lemma 3.5. By (2.2), it is clear that

    \begin{equation*} \rlap{-} \displaystyle {\int}_{B(x_0, R)}P(|\nabla u|+s)\, \mathrm{d}x\leq C\left[\rlap{-} \displaystyle {\int}_{B(x_0, R)}P\left(\left|\nabla u-(\nabla u)_{B(x_0, R)}\right|\right)\, \mathrm{d}x+P\left(\left|(\nabla u)_{B(x_0, R)}\right|\right)+P(s)\right]. \end{equation*}

    Hence there exists a constant C_{6} > 0 depending on n, \, \nu, \, L, \, K, \, s, \, C_{\Delta_2} , and C_1 such that

    \begin{eqnarray} && \rlap{-} \displaystyle {\int}_{B\left(x_0, \frac{R}{2\Lambda}\right)}P\left(\left|\nabla u-(\nabla u)_{B\left(x_0, \frac{R}{2\Lambda}\right)}\right|\right)\, \mathrm{d}x\\ &\leq&\left(\frac{1}{4}+C_{6}\, \omega(R)\right)\rlap{-} \displaystyle {\int}_{B(x_0, R)}P\left(\left|\nabla u-(\nabla u)_{B(x_0, R)}\right|\right)\, \mathrm{d}x\\ &&+\ C\, \frac{|\mu|(B(x_0, 2R))}{(2R)^{\theta_0}}+C_{6}\, \omega(R)\left[P\left(\left|(\nabla u)_{B(x_0, R)}\right|\right)+P(s)\right]. \end{eqnarray} (5.6)

    Using (A2), we take \widetilde{R} small enough to obtain

    \begin{equation*} C_{6}\, \omega\left(\widetilde{R}\right)\leq\frac{1}{4}. \end{equation*}

    It follows that if R\leq\widetilde{R} , then all R_{i}\leq\widetilde{R} . Applying the estimate (5.6) with R\equiv R_{i-1} , and noting that \omega(\cdot) is non-decreasing, it yields that

    \begin{eqnarray*} &&\rlap{-} \displaystyle {\int}_{B_{i}}P\left(\left|\nabla u-(\nabla u)_{B_i}\right|\right)\, \mathrm{d}x\\ &\leq&\frac{1}{2}\rlap{-} \displaystyle {\int}_{B_{i-1}}P\left(\left|\nabla u-(\nabla u)_{B_{i-1}}\right|\right)\, \mathrm{d}x+C\, \frac{|\mu|(2B_{i-1})}{(2R_{i-1})^{\theta_0}}+C\omega(R_{i-1})\left[P\left(\left|(\nabla u)_{B_{i-1}}\right|\right)+P(s)\right], \end{eqnarray*}

    which can be simplified as

    \begin{equation} K_i\, \leq\, \frac{1}{2}K_{i-1}\, +C\, \frac{|\mu|(2B_{i-1})}{(2R_{i-1})^{\theta_0}}\, +C\omega(R_{i-1})\, k_{i-1}. \end{equation} (5.7)

    Via a summation, one deduces that

    \begin{equation*} \sum\limits_{i = 1}^{m}K_i\, \leq\, \frac{1}{2}\sum\limits_{i = 0}^{m-1}K_{i}\, +\, C\sum\limits_{i = 0}^{m-1}\frac{|\mu|(2B_{i})}{(2R_{i})^{\theta_0}}\, +\, C\sum\limits_{i = 0}^{m-1}\omega(R_{i})\, k_{i} \end{equation*}

    for C\equiv C(n, \, \nu, \, L, \, K, \, s, \, C_{\Delta_2}, \, C_1) and for every integer m . This implies that

    \begin{equation} \sum\limits_{i = 1}^{m}K_i\, \leq\, K_0\, +\, 2C\sum\limits_{i = 0}^{m-1}\frac{|\mu|(2B_{i})}{(2R_{i})^{\theta_0}}\, +\, 2C\sum\limits_{i = 0}^{m-1}\omega(R_{i})\, k_{i} \end{equation} (5.8)

    holds for every m\in\mathbb{N} .

    Step 2. An estimate of k_{m+1} .

    Using (5.2), we have

    \begin{equation} k_{m+1}\ : = \ \sum\limits_{i = 0}^{m}(k_{i+1}-k_i)+k_0\ \leq\ \sum\limits_{i = 0}^{m}\, \left|\, P\left(\left|(\nabla u)_{B_{i+1}}\right|\right)-P\left(\left|(\nabla u)_{B_{i}}\right|\right)\, \right|+k_0. \end{equation} (5.9)

    By using Lemma 2.9 with x = \left|(\nabla u)_{B_{i+1}}\right| , y = \left|(\nabla u)_{B_{i}}\right| , we estimate the difference as

    \begin{equation*} \left|\, P\left(\left|(\nabla u)_{B_{i+1}}\right|\right)-P\left(\left|(\nabla u)_{B_{i}}\right|\right)\, \right|\leq \varepsilon\, C_{\Delta_2}P\left(\left|(\nabla u)_{B_{i}}\right|\right)+CP\left(\left|\, \left|(\nabla u)_{B_{i+1}}\right|-\left|(\nabla u)_{B_{i}}\right|\, \right|\right). \end{equation*}

    Considering the triangle inequality \left|\, |x|-|y|\, \right|\leq |x-y| and Lemma 2.10, we have

    \begin{eqnarray*} P\left(\left|\, \left|(\nabla u)_{B_{i+1}}\right|-\left|(\nabla u)_{B_{i}}\right|\, \right|\right)&\leq& CP\left(\rlap{-} \displaystyle {\int}_{B_{i+1}}\left|\nabla u-(\nabla u)_{B_i}\right|\, \mathrm{d}x\right)\\ &\leq& C(2\Lambda)^n\rlap{-} \displaystyle {\int}_{B_i}P\left(\left|\nabla u-(\nabla u)_{B_i}\right|\right)\, \mathrm{d}x. \end{eqnarray*}

    Hence there exists a constant C\equiv C(n, \, \nu, \, L, \, s, \, C_{\Delta_2}, \, C_1) such that

    \begin{equation} \left|\, P\left(\left|(\nabla u)_{B_{i+1}}\right|\right)-P\left(\left|(\nabla u)_{B_{i}}\right|\right)\, \right|\leq \varepsilon\, C_{\Delta_2}P\left(\left|(\nabla u)_{B_{i}}\right|\right)+C\rlap{-} \displaystyle {\int}_{B_i}P\left(\left|\nabla u-(\nabla u)_{B_i}\right|\right)\, \mathrm{d}x. \end{equation} (5.10)

    For each i , we choose \varepsilon = \varepsilon(i, n, C_{\Delta_2}, \Lambda) small enough such that

    \begin{equation*} \varepsilon\, C_{\Delta_2}\leq\frac{1}{(2\Lambda)^{i(n+1)}}. \end{equation*}

    Then we have

    \begin{eqnarray} \varepsilon\, C_{\Delta_2}P\left(\left|(\nabla u)_{B_{i}}\right|\right) &\leq& \varepsilon\, C_{\Delta_2}\, k_i \ \leq\ \varepsilon\, C_{\Delta_2}\rlap{-} \displaystyle {\int}_{B_{i}}\left[P\left(\left|(\nabla u)\right|\right)+P(s)\right]\, \mathrm{d}x \\ &\leq&\frac{1}{(2\Lambda)^{i(n+1)}}\, \left[(2\Lambda)^{i}\right]^{n}\rlap{-} \displaystyle {\int}_{B(x_{0}, R)}\left[P\left(\left|(\nabla u)\right|\right)+P(s)\right]\, \mathrm{d}x\\ & = & \frac{1}{(2\Lambda)^i}\, \widetilde{k}_{0}. \end{eqnarray} (5.11)

    Here we use the fact that the sum of geometric series is finite, i.e.,

    \begin{equation} \sum\limits_{i = 0}^{m}\left(\frac{1}{2\Lambda}\right)^{i}\leq \sum\limits_{i = 0}^{\infty}\left(\frac{1}{2\Lambda}\right)^{i} = \frac{1}{1-\frac{1}{2\Lambda}}. \end{equation} (5.12)

    By combining (5.9)–(5.12), and (5.4), there exists a constant C such that

    \begin{equation} k_{m+1}\, \leq\, C\sum\limits_{i = 0}^{m}\left[K_i+\frac{1}{(2\Lambda)^i}\, \widetilde{k}_{0}\right]\, +\, k_0\, \leq\, C\sum\limits_{i = 0}^{m}K_i\, +\, C\, \widetilde{k}_{0}. \end{equation} (5.13)

    Making use of (5.13), (5.8), and (5.4), one derives that for every integer m\geq 1 , there holds

    \begin{eqnarray} k_{m+1}&\leq& C\left[K_{0}+\sum\limits_{i = 0}^{m-1}\frac{|\mu|(2B_{i})}{\left(2R_{i}\right)^{\theta_0}}+\sum\limits_{i = 0}^{m-1}\omega(R_{i})\, k_{i}+\widetilde{k}_{0}\right]\\ &\leq& C\left[\widetilde{k}_{0}+\sum\limits_{i = 0}^{m-1}\frac{|\mu|(2B_{i})}{\left(2R_{i}\right)^{\theta_0}}+\sum\limits_{i = 0}^{m-1}\omega(R_{i})\, k_{i}\right]. \end{eqnarray} (5.14)

    For the second term on the right side of (5.14), one has

    \begin{eqnarray*} \sum\limits_{i = 0}^{m-1}\frac{|\mu|(2B_{i})}{\left(2R_{i}\right)^{\theta_0}}\leq\sum\limits_{i = 0}^{\infty}\frac{|\mu|(2B_{i})}{\left(2R_{i}\right)^{\theta_0}}\leq\frac{|\mu|(B(x_0, 2R))}{(2R)^{\theta_0}}+\sum\limits_{i = 0}^{\infty}\frac{|\mu|(2B_{i+1})}{(2R_{i+1})^{\theta_0}}. \end{eqnarray*}

    Using \Lambda > 1 , the method adopted in (4.20), and (1.4), we obtain that

    \begin{eqnarray} \sum\limits_{i = 0}^{m-1}\frac{|\mu|(2B_{i})}{(2R_i)^{\theta_0}}&\leq&\frac{2^{\theta_0}}{\ln 2}\int_{2R}^{4R}\frac{|\mu|(B(x_0, \rho))}{\rho^{\theta_0}}\frac{\mathrm{d}\rho}{\rho}+\frac{(2\Lambda)^{\theta_0}}{\ln(2\Lambda)}\sum\limits_{i = 0}^{\infty}\int_{2R_{i+1}}^{2R_i}\frac{|\mu|(B(x_0, \rho))}{\rho^{\theta_0}}\frac{\mathrm{d}\rho}{\rho}\\ &\leq&C\, \mathbf{I}_{n-\theta_0} ^\mu(x_0, 4R) \end{eqnarray} (5.15)

    holds with a constant C depending on n and \theta_0 . Inserting (5.15) in (5.14), we obtain the following inequality:

    \begin{equation} k_{m+1}\leq C\left(\widetilde{k}_{0}+\mathbf{I}_{n-\theta_0} ^\mu(x_0, 4R)\right)+C\, \sum\limits_{i = 0}^{m-1}\omega(R_{i})\, k_{i}. \end{equation} (5.16)

    Step 3. An induction approach.

    By setting

    \begin{equation*} J: = \widetilde{k}_{0}+\mathbf{I}_{n-\theta_0} ^\mu(x_0, 4R) = \rlap{-} \displaystyle {\int}_{B(x_0, R)}\left[P(|\nabla u|)+P(s)\right]\, \mathrm{d}x+\mathbf{I}_{n-\theta_0} ^\mu(x_0, 4R), \end{equation*}

    we shall use the mathematical induction to prove

    \begin{equation} k_{m+1}\leq C\, J. \end{equation} (5.17)

    Initial Step. If m = -1 , then by (5.4), we see that (5.17) is trivial. For the case m = 0 , (5.17) holds by using (5.13).

    Inductive Step. Assuming that (5.17) is valid for any \tilde{m} < m , we shall prove it for m+1 . By (5.16), we have

    \begin{equation*} k_{m+1}\ \leq\ C\, J+C\, \sum\limits_{i = 0}^{m-1}\omega(R_{i})\, k_{i}\ \leq\ C\, J+C\, J\, \sum\limits_{i = 0}^{m-1}\omega(R_{i}). \end{equation*}

    Due to the fact that \omega(\cdot) is non-decreasing, we estimate

    \begin{eqnarray*} \sum\limits_{i = 0}^{m-1}\omega(R_{i})&\leq&\omega(R_{0})+\sum\limits_{i = 0}^{\infty}\omega(R_{i+1})\\ &\leq&\frac{1}{\ln 2}\int_R^{2R}\omega(\rho)\frac{\mathrm{d}\rho}{\rho}+\sum\limits_{i = 0}^{\infty}\omega(R_{i+1})\\ &\leq&\frac{1}{\ln 2}\int_R^{2R}\omega(\rho)\frac{\mathrm{d}\rho}{\rho}+\frac{1}{\ln(2\Lambda)}\sum\limits_{i = 0}^{\infty}\int_{R_{i+1}}^{R_i}\omega(\rho)\frac{\mathrm{d}\rho}{\rho}\\ &\leq&\left(\frac{1}{\ln 2}+\frac{1}{\ln(2\Lambda)}\right)\int_0^{2R}\omega(\rho)\frac{\mathrm{d}\rho}{\rho}. \end{eqnarray*}

    Considering the fact that \Lambda > 1 and the definition of d(\cdot) in (A3), we have

    \begin{equation} \sum\limits_{i = 0}^{m-1}\omega(R_{i})\leq \frac{2\, d(2R)}{\ln 2}. \end{equation} (5.18)

    By applying (5.18), we complete the proof of the inequality (5.17).

    For every Lebesgue point x_{0} of P(|(\nabla u)|) , we let m\rightarrow \infty , and show that

    \begin{eqnarray*} P(|\nabla u(x_0)|)+P(s)& = &\lim\limits_{m\rightarrow \infty}k_{m+1}\\ &\leq& C\, \rlap{-} \displaystyle {\int}_{B(x_0, R)}\left[P(|\nabla u|)+P(s)\right]\, \mathrm{d}x+C\, \mathbf{I}_{n-\theta_0} ^\mu(x_0, 4R), \end{eqnarray*}

    where C depends on n, \, \nu, \, L, \, K, \, s, \, \theta_0, \, C_{\Delta_2} and C_1 . Therefore, (1.6) has been proved.

    Via a similar approach to the previous proof, we are in a position to prove Theorem 1.2.

    Proof of Theorem 1.2. We introduce a sequence of concentric balls \{\widetilde{B}_{i}\}_{i = 0}^{\infty} by

    \begin{equation*} \widetilde{B}_i: = B(x_0, r_i) = B\left(x_0, \frac{r}{(2H)^i}\right), \end{equation*}

    where H > 1 is a constant determined later, and r\leq R < 1 . Hence \widetilde{B}_{i+1}\subset B\left(x_0, \frac{r_i}{2}\right)\subset \widetilde{B}_i for every i\ge 0 . We define

    \begin{equation*} A_i: = r_i^{1+\varepsilon_0}\rlap{-} \displaystyle {\int}_{\widetilde{B}_i}P\left(\frac{\big|u-(u)_{\widetilde{B}_i}\big|}{r_i}\right)\, \mathrm{d}x, \ \ \mbox{and}\ \ a_i: = P\left(\, \big|(u)_{\widetilde{B}_i}\big|\, \right). \end{equation*}

    We also introduce \widetilde{a}_{0} by

    \begin{equation*} \widetilde{a}_{0}: = r^{1+\varepsilon_0}\rlap{-} \displaystyle {\int}_{B(x_0, r)}P\left(\frac{|u|}{r}\right)\, \mathrm{d}x. \end{equation*}

    By (2.10) and (2.5), it is obvious that

    \begin{equation} a_0 = P\left(\left|\rlap{-} \displaystyle {\int}_{\widetilde{B}_0}u\, \mathrm{d}x\right|\right)\leq\rlap{-} \displaystyle {\int}_{B(x_0, r)}P(|u|)\, \mathrm{d}x\leq C_1\, r^{1+\varepsilon_0}\rlap{-} \displaystyle {\int}_{B(x_0, r)}P\left(\frac{|u|}{r}\right)\mathrm{d}x = C\, \widetilde{a}_{0}, \end{equation} (5.19)

    as well as

    \begin{eqnarray} A_0& = &r^{1+\varepsilon_0}\rlap{-} \displaystyle {\int}_{\widetilde{B}_0}P\left(\frac{\big|u-(u)_{\widetilde{B}_0}\big|}{r}\right)\, \mathrm{d}x\\ &\leq& C_{\Delta_2}\, r^{1+\varepsilon_0}\left[\rlap{-} \displaystyle {\int}_{\widetilde{B}_0}P\left(\frac{|u|}{r}\right)\mathrm{d}x+P\left(\frac{\big|(u)_{\widetilde{B}_0}\big|}{r}\right)\right]\, \leq\, C\, \widetilde{a}_{0}. \end{eqnarray} (5.20)

    One applies Lemma 2.11 and Hölder's inequality to obtain

    \begin{equation*} A_{i+1}\, \leq\, Cr_{i+1}^{1+\varepsilon_0}\left[\rlap{-} \displaystyle {\int}_{\widetilde{B}_{i+1}}P^{\theta_1}(|\nabla u|)\, \mathrm{d}x\right]^{\frac{1}{\theta_1}} \leq\, C\, r_{i+1}^{1+\varepsilon_0}\rlap{-} \displaystyle {\int}_{\widetilde{B}_{i+1}}P(|\nabla u|)\, \mathrm{d}x. \end{equation*}

    Applying Lemma 3.4 with \rho\equiv r_{i+1} , R\equiv \frac{r_i}{2} , one has

    \begin{equation} A_{i+1}\leq C\left[r_i\, r_{i+1}^{\varepsilon_0}\left(\frac{2r_{i+1}}{r_i}\right)^{\alpha}\rlap{-} \displaystyle {\int}_{B\left(x_0, \frac{r_i}{2}\right)}P(|\nabla u|)\, \mathrm{d}x+r_{i+1}^{1+\varepsilon_0}\left(\frac{r_i}{2r_{i+1}}\right)^n\frac{|\mu|(\widetilde{B}_{i})}{r_i^{\theta_0}}\right], \end{equation} (5.21)

    where the constant C depends on n, \, \nu, \, L, \, C_{\Delta_2} , and C_1 . By Caccioppoli-type inequality (4.1) and Definition 2.3, we obtain

    \begin{equation} \rlap{-} \displaystyle {\int}_{B\left(x_0, \frac{r_i}{2}\right)}P(|\nabla u|)\, \mathrm{d}x\leq C\rlap{-} \displaystyle {\int}_{\widetilde{B}_i}P\left(\frac{\big|u-(u)_{\widetilde{B}_i}\big|}{r_i}\right)\, \mathrm{d}x+C\, \frac{|\mu|(\widetilde{B}_{i})}{r_i^{\theta_0}}. \end{equation} (5.22)

    Combining (5.21) and (5.22) with r_{i+1}\leq r_{i} , it follows that

    \begin{eqnarray*} A_{i+1}&\leq& C\left\{\left(\frac{1}{H}\right)^{\alpha}r_i^{1+\varepsilon_0}\rlap{-} \displaystyle {\int}_{\widetilde{B}_i}P\left(\frac{\big|u-(u)_{\widetilde{B}_i}\big|}{r_i}\right)\, \mathrm{d}x+\left[r_i^{1+\varepsilon_0}\left(\frac{1}{H}\right)^{\alpha}+r_{i+1}^{1+\varepsilon_0}H^n\right]\frac{|\mu|(\widetilde{B}_{i})}{r_i^{\theta_0}}\right\}\\ &\leq&C_{7}\left(\frac{1}{H}\right)^{\alpha}A_i+C\, \frac{|\mu|(\widetilde{B}_{i})}{r_i^{\theta_0}}, \end{eqnarray*}

    where C_{7} depends on n, \, \nu, \, L, \, C_{\Delta_2}, \, C_1 , and R . By choosing H\equiv H(n, \, \nu, \, L, \, C_{\Delta_2}, \, C_1, \, R) large enough, one has

    \begin{equation*} \left(\frac{1}{H}\right)^{\alpha}\leq\frac{1}{2\, C_{7}}, \end{equation*}

    which implies immediately that

    \begin{equation*} A_{i+1}\leq\frac{1}{2}\, A_i+C\, \frac{|\mu|(\widetilde{B}_{i})}{r_i^{\theta_0}}. \end{equation*}

    We consider a summation with respect to i from 0 to m-1 , and deduce that

    \begin{equation} \sum\limits_{i = 1}^{m}A_i \leq A_0+2\, C\sum\limits_{i = 0}^{m-1}\frac{|\mu|(\widetilde{B}_{i})}{r_i^{\theta_0}}. \end{equation} (5.23)

    For every non-negative integer m , one writes

    \begin{equation*} a_{m+1}\ : = \ \sum\limits_{i = 0}^{m}(a_{i+1}-a_i)+a_0\ \leq\ \sum\limits_{i = 0}^{m}\left|\, P\left(\, \big|(u)_{\widetilde{B}_{i+1}}\big|\, \right)-P\left(\, \big|(u)_{\widetilde{B}_i}\big|\, \right)\, \right|+a_0. \end{equation*}

    Adopting a similar approach as in the proof of (5.10), we have

    \begin{eqnarray*} && \left|\, P\left(\, \big|(u)_{\widetilde{B}_{i+1}}\big|\, \right)-P\left(\, \big|(u)_{\widetilde{B}_i}\big|\, \right)\, \right|\\ &\leq& \varepsilon\, C_{\Delta_2}P\left(\, \big|(u)_{\widetilde{B}_i}\big|\, \right)+C\rlap{-} \displaystyle {\int}_{{\widetilde{B}}_i}P\left(\big|u-(u)_{\widetilde{B}_i}\big|\right)\, \mathrm{d}x\\ &\leq& \varepsilon\, C_{\Delta_2}P\left(\, \big|(u)_{\widetilde{B}_i}\big|\, \right)+C\, r_{i}^{1+\varepsilon_{0}}\rlap{-} \displaystyle {\int}_{{\widetilde{B}}_i}P\left(\frac{\big|u-(u)_{\widetilde{B}_i}\big|}{r_{i}}\right)\, \mathrm{d}x\\ & = & \varepsilon\, C_{\Delta_2}\, a_{i}+C\, A_{i}, \end{eqnarray*}

    and choose \varepsilon = (2H)^{-i(n+1)} sufficiently small. With the help of (5.19), it follows that there exists a constant C\equiv C(n, \, \nu, \, L, \, C_{\Delta_2}, \, C_1, \, R) such that

    \begin{equation} a_{m+1}\, \leq\, C\sum\limits_{i = 0}^{m}\left[\frac{1}{(2H)^i}\, \widetilde{a}_0+A_i\right]+a_0\, \leq\, C\sum\limits_{i = 0}^{m}A_i+C\, \widetilde{a}_0. \end{equation} (5.24)

    Analogous to (5.15), we obtain

    \begin{equation} \sum\limits_{i = 0}^{m-1}\frac{|\mu|(\widetilde{B}_{i})}{r_i^{\theta_0}}\leq r^{\alpha}\, \sum\limits_{i = 0}^{m-1}\frac{|\mu|(\widetilde{B}_{i})}{r_i^{\theta_0+\alpha}}\leq C\, r^{\alpha}\, \mathbf{I}_{n-\theta_0-\alpha} ^\mu(x_0, 2r). \end{equation} (5.25)

    Applying (5.23)–(5.25) and (5.20), one gets that for every integer m\geq 1 , there holds

    \begin{eqnarray*} a_{m+1}&\leq& C\left[A_{0}+\widetilde{a}_0+\sum\limits_{i = 0}^{m-1}\frac{|\mu|(\widetilde{B}_{i})}{r_i^{\theta_0}}\right]\\ &\leq& C\, r^{1+\varepsilon_0}\rlap{-} \displaystyle {\int}_{B(x_0, r)}P\left(\frac{|u|}{r}\right)\, \mathrm{d}x+C\, r^{\alpha}\, \mathbf{I}_{n-\theta_0-\alpha} ^\mu(x_0, 2r). \end{eqnarray*}

    By the dominated convergence theorem, for every Lebesgue point x_{0}\in P(|u|) , there holds

    \begin{equation} P(|u(x_0)|) = \lim\limits_{m\rightarrow \infty}a_{m+1}\leq C\, r^{1+\varepsilon_0}\rlap{-} \displaystyle {\int}_{B(x_0, r)}P\left(\frac{|u|}{r}\right)\, \mathrm{d}x+C\, r^{\alpha}\, \mathbf{I}_{n-\theta_0-\alpha} ^\mu(x_0, 2r), \end{equation} (5.26)

    where constant C depends on n, \, \nu, \, L, \, \theta_0, \, C_{\Delta_2}, \, C_1 , and \mathrm{diam}(\Omega) .

    By establishing (5.26), we are ready to prove the result of Theorem 1.3.

    Proof of Theorem 1.3. For any real number g , we observe that if u is a weak solution to (1.1), then u-g is still a solution to (1.1). Let B(x_{0}, 2R)\subset\Omega . We consider x, \, y\in B\left(x_{0}, \frac{R}{2}\right) satisfying that r: = |x-y| < \frac{R}{4} . By (5.26), it follows that

    \begin{eqnarray*} P\left(|u(x)-g|\right)&\leq& C\, r^{1+\varepsilon_0}\rlap{-} \displaystyle {\int}_{B(x, r)}P\left(\frac{|u-g|}{r}\right)\, \mathrm{d}\zeta+C\, r^{\alpha}\, \mathbf{I}_{n-\theta_0-\alpha} ^\mu(x, 2r), \\ P\left(|u(y)-g|\right)&\leq& C\, r^{1+\varepsilon_0}\rlap{-} \displaystyle {\int}_{B(y, r)}P\left(\frac{|u-g|}{r}\right)\, \mathrm{d}\zeta+C\, r^{\alpha}\, \mathbf{I}_{n-\theta_0-\alpha} ^\mu(y, 2r). \end{eqnarray*}

    By (1.4), one has the following monotone property:

    \begin{equation*} \mathbf{I}_{n-\theta_0-\alpha} ^\mu(x, 2r)\leq \mathbf{I}_{n-\theta_0-\alpha} ^\mu(x, 2R), \ \mbox{and}\ \mathbf{I}_{n-\theta_0-\alpha} ^\mu(y, 2r)\leq \mathbf{I}_{n-\theta_0-\alpha} ^\mu(y, 2R). \end{equation*}

    Via (2.2) and a direct computation, one has

    \begin{eqnarray} P\left(\left|u(x)-u(y)\right|\right) &\leq& C_{\Delta_2}\left[P\left(\left|u(x)-g\right|\right)+P\left(\left|u(y)-g\right|\right)\right]\\[0.2cm] &\leq&C\, r^{1+\varepsilon_0}\left[\rlap{-} \displaystyle {\int}_{B(x, r)}P\left(\frac{|u-g|}{r}\right)\, \mathrm{d}\zeta+\rlap{-} \displaystyle {\int}_{B(y, r)}P\left(\frac{|u-g|}{r}\right)\, \mathrm{d}\zeta\right]\\[0.1cm] &&+\ C\, r^{\alpha}\left[\mathbf{I}_{n-\theta_0-\alpha} ^\mu(x, 2R)+\mathbf{I}_{n-\theta_0-\alpha} ^\mu(y, 2R)\right]. \end{eqnarray} (5.27)

    We take g: = (u)_{B(x, 2r)} , and observe that B(x, r)\cup B(y, r)\subset B(x, 2r)\subset B\left(x, \frac{R}{2}\right) . Using Definition 2.3, we deduce that

    \begin{equation*} \rlap{-} \displaystyle {\int}_{B(x, r)}P\left(\frac{|u-g|}{r}\right)\, \mathrm{d}\zeta+\rlap{-} \displaystyle {\int}_{B(y, r)}P\left(\frac{|u-g|}{r}\right)\, \mathrm{d}\zeta\leq C\, C_{\Delta_2}\rlap{-} \displaystyle {\int}_{B(x, 2r)}P\left(\frac{|u-(u)_{B(x, 2r)}|}{2r}\right)\, \mathrm{d}\zeta. \end{equation*}

    Since 2r < \frac{R}{2} , then (2.14) gives us that

    \begin{equation*} \rlap{-} \displaystyle {\int}_{B(x, 2r)}P\left(\frac{|u-(u)_{B(x, 2r)}|}{2r}\right)\, \mathrm{d}\zeta \leq (2r)^{-(1+\varepsilon_0-\alpha)}\cdot\widetilde{M}_{1+\varepsilon_0-\alpha, \, \frac{R}{2}}^{\#, \, P}(u)(x). \end{equation*}

    Then it follows from Proposition 4.2 that

    \begin{eqnarray} &&\!\rlap{-} \displaystyle {\int}_{B(x, r)}P\left(\frac{|u-g|}{r}\right)\, \mathrm{d}\zeta+\rlap{-} \displaystyle {\int}_{B(y, r)}P\left(\frac{|u-g|}{r}\right)\, \mathrm{d}\zeta\\[0.1cm] &\leq&\! C r^{-(1+\varepsilon_0-\alpha)} R^{1+\varepsilon_0}\left\{\mathbf{I}_{n-\theta_0-\alpha} ^\mu(x, 2R)+R^{-\alpha}\!\left[\rlap{-} \displaystyle {\int}_{B\left(x, \frac{R}{2}\right)}P(|\nabla u|)\, \mathrm{d}\zeta+P(s)\right]\right\}. \end{eqnarray} (5.28)

    To estimate the last integral, we use Caccioppoli-type inequality (4.1) and (4.19) to obtain that

    \begin{eqnarray} \rlap{-} \displaystyle {\int}_{B\left(x, \frac{R}{2}\right)}P\left(|\nabla u|\right)\mathrm{d}\zeta&\leq& C\, C_{\Delta_2}\rlap{-} \displaystyle {\int}_{B(x, R)}P\left(\frac{| u|}{R}\right)\mathrm{d}\zeta+C\, R^{\alpha}\, \frac{|\mu|(B(x, R))}{R^{\theta_0+\alpha}}\\ &\leq&C\rlap{-} \displaystyle {\int}_{B(x, R)}P\left(\frac{| u|}{R}\right)\mathrm{d}\zeta+C\, R^{\alpha}\, \mathbf{I}_{n-\theta_0-\alpha} ^\mu(x, 2R). \end{eqnarray} (5.29)

    Substituting (5.29) into (5.28) and considering 0\leq\alpha < 1 , it yields that

    \begin{eqnarray} &&\!\rlap{-} \displaystyle {\int}_{B(x, r)}P\left(\frac{|u-g|}{r}\right)\, \mathrm{d}\zeta+\rlap{-} \displaystyle {\int}_{B(y, r)}P\left(\frac{|u-g|}{r}\right)\, \mathrm{d}\zeta\\ &\leq&\! C r^{-(1+\varepsilon_0-\alpha)} R^{1+\varepsilon_0}\left\{\mathbf{I}_{n-\theta_0-\alpha} ^\mu(x, 2R)+R^{-\alpha}\left[\rlap{-} \displaystyle {\int}_{B(x, R)}P\left(\frac{| u|}{R}\right)\, \mathrm{d}\zeta+P(s)\right]\right\}. \end{eqnarray} (5.30)

    Combining (5.27)–(5.30) together with B(x, R)\subset B(x_0, 2R) , there is a constant C such that

    \begin{eqnarray} && P\left(|u(x)-u(y)|\right) \\[0.1cm] &\leq&C\, r^{\alpha}\left[\mathbf{I}_{n-\theta_0-\alpha} ^\mu(x, 2R)+\mathbf{I}_{n-\theta_0-\alpha} ^\mu(y, 2R)\right]+C\, r^{\alpha}\, R^{1+\varepsilon_0}\, \mathbf{I}_{n-\theta_0-\alpha} ^\mu(x, 2R)\\ &&+\ C\, r^{\alpha}R^{1+\varepsilon_0-\alpha}\left[\rlap{-} \displaystyle {\int}_{B(x_{0}, 2R)}P\left(\frac{|u|}{R}\right)\, \mathrm{d}\zeta+P(s)\right], \end{eqnarray} (5.31)

    where the constant C depends on n, \, \nu, \, L, \, \theta_0, \, \omega(\cdot), \, C_{\Delta_2}, \, C_1 , and \mathrm{diam}(\Omega) . Since R < 1 and \alpha\in[0, 1) , then the estimate (5.31) is the desired interior Hölder estimate of Theorem 1.3.

    In this work, we establish pointwise potential estimates of weak solutions to a class of elliptic equations in divergence form with measure data. Our primary result is to employ the Riesz potential to prove the pointwise estimates of the solutions. The key innovation of this paper manifests in the proof of Proposition 3.2, which enables the relationship between measure data and the Riesz potential in the Sobolev-Orlicz spaces. Furthermore, we obtain Hölder continuity estimates for the solutions by establishing the Caccioppoli-type inequality and the maximal estimate. This systematic approach extends the potential estimates of regularity for nonlinear elliptic equations in the existing literature.

    Zhaoyue Sui: Conceptualization, methodology, writing–original draft preparation; Feng Zhou: Supervision, funding acquisition, project administration, writing–review and editing. All authors have read and approved the final version of the manuscript for publication.

    The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

    The authors would like to thank the anonymous referee for their careful reading and valuable comments. The authors are supported by the National Natural Science Foundation of China (NNSF Grant No. 12001333) and the Shandong Provincial Natural Science Foundation (Grant No. ZR2020QA005).

    The authors declare no conflicts of interest regarding the publication of this article.



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