1 Introduction
The present paper deals with continuity properties of local solutions to the p-Laplacian system in open subsets of the Euclidean space ℝn with 2≤p≤n. Our main results provide minimal integrability conditions on the right-hand side of the relevant system for every solution to be continuous or uniformly continuous. Moreover, they yield sharp information on their modulus of continuity. These results, in their general formulation, are stated in Section 3.
In this section, we enucleate some special cases of our conclusions for the n-Laplace system, which exhibits certain peculiar interesting features due to its borderline character. In recent years, the regularity properties of its solutions have been the subject of various contributions, most of which confined to the case of a single equation, such as [1, 20, 21, 23, 24, 25, 31]. The study of optimal continuity properties of its solutions was, in fact, our original motivation for the present research, which, in particular, complements and improves earlier results in [20, 21].
Let Ω be an open subset of ℝn and let 𝒇:Ω→ℝN. Since we are concerned with local solutions to the n-Laplace system
(1.1)-div(|∇𝒖|n-2∇𝒖)=𝒇(x) in Ω,
we may suppose that Ω is bounded, with Lebesgue measure |Ω|=1. Our first result tells us that membership of 𝒇 to the Lorentz space L(1,1n-1)(Ω,ℝN) is an optimal condition for every solution 𝒖 to (1.1) to be continuous.
Recall that L(1,1n-1)(Ω,ℝN) is the space of all measurable functions 𝒈:Ω→ℝN whose quasi-norm
∥𝒈∥L(1,1n-1)(Ω,ℝN)=(∫01|𝒈|**(s)1n-1s-1+1n-1𝑑s)n-1
is finite. Here, |𝒈|* denotes the decreasing rearrangement of the length |𝒈| of 𝒈 and
|𝒈|**(s)=1s∫0s|𝒈|*(r)𝑑r for s>0.
When N=1, the abridged notation L(1,1n-1)(Ω) will be employed for L(1,1n-1)(Ω,ℝ). An analogous simplified notation will be adopted for other function spaces.
Theorem 1.1Let n≥2. Then, L(1,1n-1)(Ω,RN) is the largest rearrangement-invariant quasi-normed space X(Ω,RN) such that any local solution to (1.1) is continuous for every 𝐟∈X(Ω,RN).
In the scalar case (N=1), the fact that every solution to (1.1) is continuous under the assumption that 𝒇∈L(1,1n-1)(Ω) is proved in [21, Theorem 1.1] via a different approach. Theorem 1.1 above not only includes systems, but also characterizes the space L(1,1n-1)(Ω,ℝN) as the optimal rearrangement-invariant quasi-normed space enjoying this property. One important trait of Theorem 1.1 is thus to reduce the question of the continuity of all solutions 𝒖 to (1.1), as 𝒇 ranges in some rearrangement-invariant quasi-normed space X(Ω,ℝN), to the merely functional-analytic problem of the inclusion
X(Ω,ℝN)⊂L(1,1n-1)(Ω,ℝN).
As a consequence, minimal conditions on X(Ω,ℝN) in classes of function spaces for any solution to (1.1) to be continuous for every 𝒇∈X(Ω,ℝN) can be derived.
The following result deals with the situation when X(Ω,ℝN) is an Orlicz spaceLA(Ω,ℝN) associated with a Young function A.
Theorem 1.2Let A be a Young function. Assume that
{lim supt→∞tlog(1+t)A(t)<∞if n=2,∫∞(ttA′(t)-A(t))1n-2dtt<∞if n≥3.
If 𝐟∈LA(Ω,RN), then any local solution 𝐮 to (1.1) is continuous.
Let us now focus on the case when X(Ω,ℝN) is a so-called classical Lorentz spaceΛ1(ν)(Ω,ℝN) associated with a weight ν. Unless otherwise stated, we shall assume that the weight function ν:(0,1)→[0,∞) is continuous. The space Λ1(ν)(Ω,ℝN) consists of all measurable functions 𝒈 on Ω for which the quantity
∥𝒈∥Λ1(ν)(Ω,ℝN)=∫01|𝒈|*(s)ν(s)𝑑s
is finite.
Theorem 1.3Let ν be a non-decreasing weight. Then, any local solution 𝐮 to (1.1) is continuous for every 𝐟∈Λ1(ν)(Ω,RN) if and only if
(1.2){lim supt→0+tlog1t∫0tν(s)𝑑s<∞if n=2,∫0(tlog1t∫0tν(s)𝑑s)1n-2dtt<∞if n≥3.
Theorem 1.3 should be compared with [21, Corollary 4.1], where the scalar case (N=1) is considered and a more implicit, just sufficient condition on the weight ν is given for any solution to (1.1) to be continuous when f∈Λ1(ν)(Ω).
Remark 1.4Theorem 1.3 is a special case of Theorem 3.4 in Section 3, where not only solutions to the p-Laplace system with 2≤p≤n are considered, but weights ν which need not be monotone are also allowed.
Example 1.5Assume that n≥3, and f∈Λ1((1+log1t)n-1ω(t))(Ω,ℝN) for some slowly varying non-increasing function ω:(0,1)→(0,∞) in the sense of Karamata, namely, such that limt→0+ω(λt)ω(t)=1 for every λ>0. As a consequence of [5, Theorems 1.5.11 and 1.6.1], the function ω is slowly varying if and only if
(1.3)∫0t(1+log1s)n-1ω(s)𝑑s=t(1+log1t)n-1ω(t)+higher order terms as t→0+.
Note that, in particular, (1.3) holds if ω is locally absolutely continuous and the (essential) limit of tω′(t)ω(t) as t→0+ equals 0.
By Theorem 1.3 and Remark 1.4, any local solution u to (1.1) is continuous if and only if
∫0dttlog1tω(t)1n-2<∞.
For instance, the choice
(1.4)ω(t)=ℓ2(1t)n-2⋯ℓk-1(1t)n-2ℓk(1t)n-2+ε for t∈(0,1)
and for any k≥2 and ε>0 is admissible, where ℓk is inductively defined for s>0 as
(1.5)ℓ1(s)=max{1,logs},ℓk(s)=max{1,log(ℓk-1(s))} if k≥2.
On the other hand, discontinuous (in fact, locally unbounded) solutions to (1.1) may exist if ω is defined as in (1.4) with ε=0. This recovers, and extends to systems, a result of [21].
Let us emphasize that discontinuous solutions may exist when ε=0 even if an infinite product extended to all k≥2 appears on the right-hand side of (1.4), namely, if
ω(t)=∏k=2∞ℓk(1t)n-2 for t>0.
Indeed, such a function ω can be shown to fulfil (1.3), whereas (1.2) fails. This last example is closely related to a question raised in [21, Remark 5.2] about the case when the right-hand side of (1.1) belongs to an Orlicz space associated with a Young function defined in terms of an infinite product of logarithms.
Let us now turn to the question of the modulus of continuity of solutions to (1.1). Let φ:(0,∞)→(0,∞) be a function equivalent near 0 (up to
multiplicative constants) to a non-decreasing function. We denote by C0,φ(Ω,ℝN) the space of all functions 𝒖 for which the semi-norm
∥𝒖∥C0,φ(Ω,ℝN)=supx,y∈Ωx≠y|𝒖(x)-𝒖(y)|φ(|x-y|)
is finite. Clearly, functions φ which are equivalent (up to multiplicative constants) near 0 yield the same spaces (up to equivalent semi-norms). The space C0,φ(Ω,ℝN) is nontrivial if and only if lim sups→0+sφ(s)<∞. Furthermore, it consists of uniformly continuous functions in Ω, with modulus of continuity not exceeding φ, provided that lims→0+φ(s)=0.
The Hölder space C0,α(Ω,ℝN) with α∈(0,1] agrees with C0,φ(Ω,ℝN) with the choice φ(s)=sα. In particular, C0,1(Ω,ℝN)=Lip(Ω,ℝN), the space of Lipschitz continuous functions in Ω. We denote by Cloc0,φ(Ω,ℝN) the space of those functions which belong to C0,φ(Ω′,ℝN) for every open set Ω′⊂⊂Ω.
Let us emphasize that no uniform modulus of continuity, depending only on n, is guaranteed for all solutions 𝒖 to (1.1) under the mere assumption that 𝒇∈L(1,1n-1)(Ω,ℝN), see Proposition 3.7 in Section 3. However, if right-hand sides 𝒇 from function spaces essentially smaller than L(1,1n-1)(Ω,ℝN) are considered, every solution to (1.1) does admit a modulus of continuity depending only on the relevant function space. The following result is paradigmatic in this connection, in that it involves certain borderline Orlicz spaces of Zygmund type and yields the best possible modulus of continuity. In particular, it improves earlier results for Orlicz spaces from the same family, namely, Orlicz spaces of the form L(logL)α(Ω,ℝN), associated with a Young function equivalent to tℓ1(t)α near infinity. In what follows, we denote by BR a ball in ℝn of radius R>0. The notation BR(x) will be used to specify that the ball is centered at the point x∈ℝn.
Theorem 1.6Let 𝐮 be any local solution to (1.1) with 𝐟∈L(logL)n-1+ε(Ω,RN). Then, 𝐮∈Cloc0,φ(Ω,RN), where
φ(s)=(log1s)-εn-1 near 0,
and there exists a constant C such that
(1.6)∥𝒖∥C0,φ(BR,ℝN)≤C(∥𝒇∥L(logL)n-1+ε(B2R,ℝN)1n-1+∥∇𝒖∥L1(B2R,ℝN×n))
for every ball B2R⊂⊂Ω.
The modulus of continuity φ is optimal, in the sense that if (1.6) holds with φ replaced by some other modulus of continuity ψ for every 𝐟∈L(logL)n-1+ε(Ω,RN) and any local solution 𝐮 to (1.1), then Cloc0,φ(Ω,RN)⊂Cloc0,ψ(Ω,RN).
Theorem 1.6 augments [20, Theorem 1.1], where just the scalar case was considered and the weaker conclusion u∈Cloc0,ψ(Ω) with
ψ(s)=(log1s)-εn near 0
was established under the same assumption that f∈L(logL)n-1+ε(Ω). In fact, one can more generally show that if 𝒇∈L(ℓ1L)n-1(ℓ2L)n-2⋯(ℓk-1L)n-2(ℓkL)n-2+ε(Ω,ℝN), the Orlicz space built upon a Young function equivalent to tℓ1(t)n-1ℓ2(t)n-2⋯ℓk-1(t)n-2ℓk(t)n-2+ε near infinity, then 𝒖∈Cloc0,φ(Ω,ℝN), where
φ(t)=ℓk(1t)-εn-1 near 0.
This follows as a special case of Theorem 3.8 in Section 3, via (2.5) and (2.11).
After recalling the necessary background from the theory of function spaces in the next section, our main results for the p-Laplace system for any p∈[2,n] are stated in Section 3. Specifically, Section 3.1 deals with plain continuity, whereas uniform continuity is the subject of Section 3.2. Proofs of these two sets of results are presented in Sections 4 and 5, respectively. Our approach makes substantial use of precise pointwise estimates for the gradient of solutions to p-Laplacian type elliptic systems which are established in [25] (and refine earlier estimates from [27, 14, 15]), of optimal Sobolev type embeddings into spaces of uniformly continuous functions from [11], and of various Hardy type inequalities and related embeddings between spaces of measurable functions appearing in diverse contributions, including [6, 7, 8, 9, 17, 19, 26].
We note that part of the results of the present paper were announced in [2].
2 Spaces of measurable functions
Let Ω be a measurable subset of ℝn having finite Lebesgue measure, which, without loss of generality, will be assumed to be equal to 1, and let g:Ω→ℝ be a measurable function. The decreasing rearrangement of g is the function g*:[0,∞)→[0,∞] defined as
g*(s)=inf{t≥0:|{x∈Ω:|g(x)|>t}|≤s} for s≥0.
In other words, g* is the (unique) non-increasing, right-continuous function in [0,∞) equidistributed with g. By g** we denote the maximal type function associated with g* as recalled in Section 1. Notice that
(2.1)g*(s)≤g**(s) for s>0
and for every measurable function g, since g* is non-increasing.
A quasi-normed function spaceX(Ω) on Ω is a linear space of measurable functions on Ω equipped with a quasi-norm ∥⋅∥X(Ω) satisfying the following properties.
∥g∥X(Ω)>0 if g≠0;∥λg∥X(Ω)=|λ|∥g∥X(Ω) for every λ∈ℝ and g∈X(Ω);∥g+h∥X(Ω)≤c(∥g∥X(Ω)+∥h∥X(Ω)) for some constant c≥1 and for every g,h∈X(Ω).
0≤|h|≤|g| a.e. in Ω implies ∥h∥X(Ω)≤∥g∥X(Ω).
0≤gk↗g a.e. implies ∥gk∥X(Ω)↗∥g∥X(Ω) as k→∞.
There exists a constant C such that ∫Ω|g|𝑑x≤C∥g∥X(Ω) for every g∈X(Ω).
Given a measurable set G⊂Ω, we define
∥g∥X(G)=∥gχG∥X(Ω)
for every measurable function g on Ω. Here, χG stands for the characteristic function of G. Moreover, we denote by Xloc(Ω) the space of measurable functions g on Ω such that ∥g∥X(G)<∞ for every compact set G⊂Ω.
If assumption (i) holds with c=1, then the functional ∥⋅∥X(Ω) is a norm which makes X(Ω) a Banach space. In this case, X(Ω) will be called a Banach function space.
Given a Banach function space X(Ω) and a quasi-normed space Y(Ω), we have that
(2.2)X(Ω)⊂Y(Ω) if and only if X(Ω)→Y(Ω),
where the arrow “→” means that there exists a constant C such that ∥g∥Y(Ω)≤C∥g∥X(Ω) for every g∈X(Ω). Let us notice that property (2.2) is usually stated under the assumption that both X(Ω) and Y(Ω) are Banach function spaces (see [4, Chapter 1, Theorem 1.8]). An inspection of the proof reveals that it continues to hold even if Y(Ω) is just a quasi-normed space.
A quasi-normed function space (in particular, a Banach function space) X(Ω) is called rearrangement-invariant if
∥g∥X(Ω)=∥h∥X(Ω) whenever g*=h*.
If this is the case, the (quasi)-norm ∥⋅∥X(Ω) is also said to be rearrangement-invariant.
Given a nonnegative functional ∥⋅∥X(Ω) defined on the space of measurable functions (not necessarily a norm nor a quasi-norm), its associate functional ∥⋅∥X′(Ω) is given by
∥h∥X′(Ω)=supg≠0∫Ω|g(x)h(x)|𝑑x∥g∥X(Ω).
If ∥⋅∥X(Ω) is a rearrangement-invariant norm, then ∥⋅∥X′(Ω) is also a rearrangement-invariant norm.
Let σ,ρ∈(0,∞) and let ν be a weight on (0,1). The classical Lorentz spacesΛσ(ν)(Ω) and Γσ(ν)(Ω), and the
weak Lorentz spacesΛρ,∞(ν)(Ω) and Γρ,∞(ν)(Ω), are the families of those measurable functions g in Ω such that the corresponding functional from
(2.3){∥g∥Λσ(ν)(Ω)=(∫01(g*(t))σν(t)𝑑t)1σ,∥g∥Λρ,∞(ν)(Ω)=sup0<t<1g*(t)𝒱(t)1ρ,∥g∥Γσ(ν)(Ω)=(∫01(g**(t))σν(t)𝑑t)1σ,∥g∥Γρ,∞(ν)(Ω)=sup0<t<1g**(t)𝒱(t)1ρ
is finite. Here, 𝒱:(0,1)→[0,∞) denotes the function defined by
(2.4)𝒱(t)=∫0tν(s)𝑑s for t∈(0,1).
Observe that, owing to (2.1), any functional of type Γ is not weaker than the corresponding functional of type Λ associated with the same weight and powers. Inequalities among these functionals have been extensively studied in the literature. In particular, it is easily seen, via Fubini’s theorem, that
(2.5)∥⋅∥Γ1(ν)(Ω)=∥⋅∥Λ1(ν~)(Ω),
where ν~(t)=∫t1ν(s)s𝑑s for t∈(0,1). Moreover, if ρ∈(0,∞), then
{∥⋅∥Λρ,∞(ν)(Ω)=∥⋅∥Λ1,∞(νρ)(Ω),∥⋅∥Γρ,∞(ν)(Ω)=∥⋅∥Γ1,∞(νρ)(Ω),
where νρ(t)=1ρ𝒱(t)1ρ-1ν(t) for t∈(0,1). The quantities defined in (2.3) are not always norms. For example, if σ≥1, then ∥⋅∥Λσ(ν)(Ω) is a norm if and only if ν is non-increasing. If σ∈(1,∞), then ∥⋅∥Λσ(ν)(Ω) is equivalent to a norm if and only if
(2.6)tσ∫t1s-σν(s)𝑑s≤C∫0tν(s)𝑑s for t∈(0,1)
and for some constant C (see [29, Theorem 4]). A necessary and sufficient condition for ∥⋅∥Λ1(ν)(Ω) to be equivalent to a norm, and hence for Λ1(ν)(Ω) to be equivalent to a Banach space, is that
(2.7)𝒱(t)t≤C𝒱(s)s for 0<s≤t≤1
and for some constant C (see [6, Theorem 2.3]).
Given ρ∈(0,∞), the quasi-norm ∥⋅∥Λρ,∞(ν)(Ω) is equivalent to a norm if
1t∫0tds𝒱(s)1ρ≤C𝒱(t)1ρ for t∈(0,1)
and for some constant C. The space Γσ(ν)(Ω) is a Banach space if σ≥1. The space Γρ,∞(ν)(Ω) is a Banach space for every ρ∈(0,∞).
For technical reasons, Lorentz type functionals associated with weights dμ associated with a Borel measure μ, which are not necessarily absolutely continuous with respect to the Lebesgue measure, will come into play in some of our proofs. Their definition parallels (2.3) and reads
{∥g∥Λσ(dμ)(Ω)=(∫01(g*(t))σ)dμ(t))1σ,∥g∥Λρ,∞(dμ)(Ω)=sup0<t<1g*(t)(1t∫0t𝑑μ(s))1ρ,∥g∥Γσ(dμ)(Ω)=(∫01(g**(t))σ𝑑μ(t))1σ,∥g∥Γρ,∞(dμ)(Ω)=sup0<t<1g**(t)(1t∫0t𝑑μ(s))1ρ
for a measurable function g in Ω.
Standard instances of classical Lorentz spaces are the customary Lorentz spaces L(r,q)(Ω) with r,q∈(0,∞], and, more generally, the Lorentz–Zygmund spacesL(r,q)(logL)α(Ω). Given r,q∈(0,∞] and α∈ℝ, they are defined in terms of the quasi-norm
(2.8)∥g∥L(r,q)(logL)α(Ω)=∥s1r-1q(1+log1s)αg**(s)∥Lq(0,1)
for a measurable function g in Ω. The space L(r,q)(Ω) is defined as L(r,q)(logL)0(Ω). If r,q≥1 and α∈ℝ, the space L(r,q)(logL)α(Ω) is a rearrangement-invariant space endowed with the norm (2.8). The quantity obtained on replacing g** by g* in (2.8) will be denoted by ∥g∥Lr,q(logL)α(Ω). Accordingly, we set ∥g∥Lr,q(Ω)=∥g∥Lr,q(logL)0(Ω). If r>1, q∈(0,∞), and α∈ℝ, then
(2.9)∥g∥L(r,q)(logL)α(Ω)≈∥g∥Lr,q(logL)α(Ω)
for every measurable function g in Ω. The notation “≈” means that the two sides are bounded by each other up to multiplicative constants, which in (2.9) depend on r,q,α. In particular, if r>1, then L(r,r)(logL)α(Ω)=Lr,r(logL)α(Ω)) (up to equivalent norms). On the contrary, L(1,1)(Ω)⫋L1(Ω). Actually, ∥⋅∥L(1,1)(Ω) is equivalent to ∥⋅∥L1,1(logL)1(Ω), whereas ∥⋅∥L1(Ω)=∥⋅∥L1,1(logL)0(Ω)=∥⋅∥L(1,∞)(Ω).
Observe that L(r,q)(logL)α(Ω)=Γq(ν)(Ω) if ν(t)=tqr-1(1+log1t)αq and r,q∈(0,∞). Furthermore, L(r,∞)(logL)α(Ω)=Γr,∞(ν)(Ω) if ν(t)=t(1+log1t)αr and r∈(0,∞). Similarly, Lr,q(logL)α(Ω)=Λq(ν)(Ω) and Lr,∞(logL)α(Ω)=Λr,∞(ν)(Ω) with the same choices of the weight ν.
A function A:[0,∞)→[0,∞] is called a Young function if it has the form
(2.10)A(t)=∫0ta(τ)𝑑τ for t≥0
and for some non-decreasing, left-continuous function a:[0,∞)→[0,∞] which is neither identically equal to 0 nor to ∞. Clearly, any convex (nontrivial) function from [0,∞) into [0,∞], which is left-continuous and vanishes at 0, is a Young function.
The Orlicz spaceLA(Ω), associated with a Young function A, is the Banach function space of those measurable functions g in Ω for which the Luxemburg norm
∥g∥LA(Ω)=inf{λ>0:∫ΩA(|g(x)|λ)𝑑x≤1}
is finite.
The Orlicz spaces are rearrangement-invariant spaces, since
∥g∥LA(Ω)=∥g*∥LA(0,1)
for every measurable function g in Ω. We shall also need to make use of the rearrangement-invariant space L(A)(Ω), associated with a Young function A, consisting of those measurable functions g in Ω such that the norm
∥g∥L(A)(Ω)=∥g**∥LA(Ω)
is finite.
A function A of the form (2.10) for some function a which is just nonnegative and locally bounded in (0,∞) (but not necessarily non-decreasing) will be called a generalized Young function. Clearly, a generalized Young function is not necessarily convex. In particular, it may have sublinear growth near infinity.
If A is just a generalized Young function, the set of all measurable functions g in Ω such that the quantity
|g|LA(Ω)=inf{k>0:∫ΩA(|g(x)|k)𝑑x≤k}
is finite will be still denoted by LA(Ω) and is a complete Frechét space, in the sense that |⋅|LA(Ω) satisfies the triangle inequality, |g|LA(Ω)=0 if and only if g=0, and it is a complete metric space endowed with the metric induced by |⋅|LA(Ω). One also has that |λg|LA(Ω)→0 if λ→0. Let us point out that functions from a generalized Orlicz space need not be even locally integrable. Classical examples of generalized Orlicz spaces are the spaces Lp(Ω) with 0<p<1.
The functional |⋅|LA(Ω) is rearrangement-invariant, since
|g|LA(Ω)=|g*|LA(0,1)
for every measurable function g in Ω.
The Frechét space L(A)(Ω), associated with a generalized Young function A, consists of those measurable functions g in Ω such that the functional
∥g∥L(A)(Ω)=|g**|LA(Ω)
is finite.
A generalized Young function A is said to dominate another generalized Young function B near infinity if there exist constants C>0 and t0≥0 such that
B(t)≤A(Ct) for t≥t0.
The functions A and B are called equivalent near infinity if they dominate each other near infinity. If A dominates B near infinity, then LA(Ω)⊂LB(Ω). Hence, LA(Ω)=LB(Ω) if A and B are equivalent near infinity. Analogous inclusion relations hold for the spaces L(A)(Ω) and L(B)(Ω).
Orlicz spaces associated with a (generalized) Young function equivalent to the function tpℓ1(t)α1ℓ2(t)α2⋯ℓk(t)αk near infinity, where ℓk is defined as in (1.5), will be denoted by Lp(ℓ1L)α1(ℓ2L)α2⋯(ℓkL)αk(Ω). The expression ℓ1L will also simply be denoted by logL. One can show that
L(q,q)(logL)α(Ω)=Lq,q(logL)α(Ω)=Lq(logL)αq(Ω),
up to equivalent norms, if q>1 and α∈ℝ. On the other hand, if α≥0,
L1,1(logL)α(Ω)=L1(logL)α(Ω),
but
L(1,1)(logL)α(Ω)=L1(logL)α+1(Ω),
up to equivalent norms. More generally, analogous relations hold among the Orlicz spaces Lp(ℓ1L)α1(ℓ2L)α2⋯(ℓkL)αk(Ω) and the Lorenz spaces Λq(ν) and Γq(ν) associated with weights ν of power-multiple-logarithmic type. For instance,
(2.11)Γ1(ν)(Ω)=L(ℓ1L)α1+1(ℓ2L)α2⋯(ℓkL)αk(Ω),
up to equivalent norms, if ν(s)=ℓ1(1s)α1ℓ2(1s)α2⋯ℓk(1s)αk.
Spaces X(Ω,ℝN) of vector-valued functions 𝒈:Ω→ℝN are defined analogously, on replacing g with |𝒈| in the definitions given above. In particular, a space X(Ω,ℝN) will be called rearrangement-invariant if
∥𝒈∥X(Ω,ℝN)=∥𝒉∥X(Ω,ℝN) whenever |𝒈|*=|𝒉|*.
Spaces X(Ω,ℝN×n) of matrix-valued functions from Ω into ℝN×n are defined accordingly.
3 Main results
A generalized notion of solutions to the p-Laplace system will be adopted, which also applies to the case when the right-hand side is merely an integrable function. In the spirit of [13, 12, 25], the solutions to be considered are required to be approximable, in a suitable sense, by a sequence of solutions to the p-Laplace system corresponding to a sequence of smooth right-hand sides converging to the original one. If the latter enjoys a sufficiently high degree of integrability, then an approximable solution turns out to be a weak solution in the usual sense.
Specifically, let 𝒇∈L1(Ω,ℝN) and p≥2. A function 𝒖 in the Sobolev space W1,p-1(Ω,ℝN) is called a local approximable solution to the system
(3.1)-div(|∇𝒖|p-2∇𝒖)=𝒇 in Ω
if there exists a sequence of smooth functions {𝒇k} and a sequence {𝒖k} of local weak solutions to the systems
-div(|∇𝒖k|p-2∇𝒖k)=𝒇k in Ω
such that 𝒇k→𝒇 in Lloc1(Ω,ℝN) and 𝒖k→𝒖 in Wloc1,p-1(Ω,ℝN) as k→∞.
In particular, any local approximable solution satisfies
∫Ω|∇𝒖|p-2∇𝒖⋅∇ϕdx=∫Ω𝒇ϕ𝑑x
for every ϕ∈C0∞(Ω,ℝN).
3.1 Continuity
The key result of this section is a description of the optimal (largest) rearrangement-invariant quasi-normed space X(Ω,ℝN) such that membership of 𝒇 to X(Ω,ℝN) ensures the continuity of any local solution to (3.1).
Theorem 3.1Theorem 3.1 (Optimal space for continuity)
Let 2≤p≤n. Then, L(np,1p-1)(Ω,RN) is the largest rearrangement-invariant quasi-normed space X(Ω,RN) such that any local solution 𝐮 to (3.1) is continuous for every 𝐟∈X(Ω,RN).
In the case of a single equation, namely, when N=1, the continuity of any solution to (3.1) under the assumption that f∈L(np,1p-1)(Ω) was established in [16] for p<n and, as mentioned above, in [21] for p=n. The special case when p=n=2 goes back to [3]. Theorem 3.1 provides us with the optimality of the space L(np,1p-1)(Ω,ℝN) and tells us that the same conclusion holds for systems as well.
Theorem 3.1 can be exploited, in combination with the characterization of embeddings between function spaces, to provide sharp conditions on the right-hand side 𝒇 in (3.1), in classes of function spaces, for the continuity of solutions. Data 𝒇 in classical Lorentz spaces of type Γσ(ν)(Ω,ℝN) are considered in the following result.
Theorem 3.2Theorem 3.2 (Continuity with f∈Γσ(ν)(Ω,RN))
Assume that 2≤p≤n. Let 0<σ<∞ and let ν be a weight. Let Vσ:(0,1)→[0,∞) be defined as
Vσ(t)=∫0tν(s)𝑑s+tσ∫t1s-σν(s)𝑑s for t∈(0,1).
Assume that
(3.2){lim supt→0+tpnVσ(t)1σ<∞if 0<σ≤1p-1, 2≤p<n,lim supt→0+t(log1t)n-1Vσ(t)1σ<∞if 0<σ≤1n-1, 2≤p=n,∫0tpσn[σ(p-1)-1]-1Vσ(t)1σ(n-1)-1𝑑t<∞if 1p-1<σ≤1, 2<p≤n, or σ>1, 2≤p<n,∫0t(log1t)1σ-1Vσ(t)1σ-1𝑑t<∞if σ>1, 2=p=n,∫0tσσ(n-1)-1-1(log1t)1σ(n-1)-1Vσ(t)1σ(n-1)-1𝑑t<∞if σ≥1, 2<p=n.
If 𝐟∈Γσ(ν)(Ω,RN), then every local solution 𝐮 to (3.1) is continuous.
Conversely, if σ≥1 and for every 𝐟∈Γσ(ν)(Ω,RN) any local solution to (3.1) is continuous, then one of the alternatives in (3.2) holds.
Right-hand sides 𝒇 in weak type spaces Γρ,∞(ν)(Ω,ℝN) are the subject of the next theorem.
Theorem 3.3Theorem 3.3 (Continuity with f∈Γρ,∞(ν)(Ω,RN))
Assume that 2≤p≤n. Let 0<ρ<∞ and let ν be a weight. Let V be defined as in (2.4). Assume that
(3.3)∫0t1p-1(pn-1)-1sups≥t(s-1𝒱(s)1ρ)𝑑t<∞.
If 𝐟∈Γρ,∞(ν)(Ω,RN), then every local solution 𝐮 to (3.1) is continuous.
Conversely, if ρ≥1 and for every 𝐟∈Γρ,∞(ν)(Ω,RN) any local solution to (3.1) is continuous, then (3.3) holds.
In Theorems 3.4 and 3.5 below, functions 𝒇 from the Lorentz spaces Λσ(ν)(Ω,ℝN) and the weak Lorentz spaces Λρ,∞(ν)(Ω,ℝN), respectively, are taken into account.
Theorem 3.4Theorem 3.4 (Continuity with f∈Λσ(ν)(Ω,RN))
Assume that 2≤p≤n. Let 0<σ<∞ and let ν be a weight. Let V be defined as in (2.4). Assume that
(3.4){lim supt→0+tpn𝒱(t)1σ<∞if 0<σ≤1p-1, 2≤p<n,lim supt→0+t(log1t)n-1𝒱(t)1σ<∞if 0<σ≤1n-1, 2≤p=n,∫0tpσn(σ(p-1)-1)𝒱(t)1σ(p-1)-1dtt<∞if σ>1p-1, 2≤p<n,∫0sup0<s≤t(sσσ(n-1)-1𝒱(s)1σ(n-1)-1)(log1t)1σ(n-1)-1dtt<∞if 1n-1<σ≤1, 2<p=n,∫0(∫0t(s𝒱(s))1σ-1𝑑s)σ-1σ(n-1)-1(log1t)1σ(n-1)-1dtt<∞if σ>1, 2≤p=n.
If 𝐟∈Λσ(ν)(Ω,RN), then every local solution 𝐮 to (3.1) is continuous.
Conversely, if either σ>1 and (2.6) holds or σ=1 and (2.7) holds, and for every 𝐟∈Λσ(ν)(Ω,RN) any local solution to (3.1) is continuous, then one of the alternatives in (3.4) holds.
Theorem 3.5Theorem 3.5 (Continuity with f∈Λρ,∞(ν)(Ω,RN))
Assume that 2≤p≤n. Let 0<ρ<∞ and let ν be a weight. Let V be defined as in (2.4). Assume that
(3.5)∫0(1t∫0t𝒱(s)-1ρ𝑑s)1p-1tpn(p-1)-1𝑑t<∞.
If 𝐟∈Λρ,∞(ν)(Ω,RN), then every local solution 𝐮 to (3.1) is continuous.
Conversely, if (2.6) holds and for every 𝐟∈Λρ,∞(ν)(Ω,RN) any local solution to (3.1) is continuous, then (3.5) holds.
We conclude this subsection with the case when 𝒇 belongs to an Orlicz space.
Theorem 3.6Theorem 3.6 (Continuity with f∈LA(Ω,RN))
Assume that 2≤p≤n. Let A be a Young function. Assume that
{∫∞(tA(t))pnp-n-pdttn(p-2)np-n-p<∞if 2≤p<n,∫∞(ttA′(t)-A(t))1n-2dtt<∞if 2<p=n,lim supt→∞tlog(1+t)A(t)<∞if 2=p=n.
If 𝐟∈LA(Ω,RN), then any local solution 𝐮 to (3.1) is continuous.
3.2 Modulus of continuity
This subsection is devoted to the modulus of continuity of solutions to (3.1). Loosely speaking, the results to be presented tell us that if a sufficiently strong rearrangement-invariant (quasi)-norm of 𝒇 is finite, then the modulus of continuity of any solution 𝒖 to (3.1) is locally bounded by the relevant norm of 𝒇 and the L1 norm of ∇𝒖.
In the following proposition, we preliminarily note that no uniform estimate of this kind holds for the modulus of continuity of solutions 𝒖 to the p-Laplace system (3.1) (even for N=1) when 𝒇 just belongs to the space L(np,1p-1)(Ω,ℝN). Recall from Theorem 3.1 that such a space is the largest ensuring the mere continuity of 𝒖.
Proposition 3.7Proposition 3.7 (Failure of uniform continuity for f∈L(np,1p-1)(Ω,RN))
Let 2≤p≤n. Assume that there exist a non-decreasing function φ and a constant C such that
(3.6)∥𝒖∥C0,φ(BR,ℝN)≤C(∥𝒇∥L(np,1p-1)(B2R,ℝN)1p-1+∥∇𝒖∥L1(B2R,ℝN×n))
for every ball B2R⊂⊂Ω, every 𝐟∈Lloc(np,1p-1)(Ω,RN), and every local solution 𝐮 to (3.1). Then, lims→0+φ(s)>0.
Although our approach applies, in principle, to a variety of rearrangement-invariant spaces, we focus on the case when 𝒇 belongs to Lorentz spaces of type Γσ(ν)(Ω,ℝN) or Γσ,∞(ν)(Ω,ℝN). These include, as special instances, most customary function spaces, such as the usual Lorentz spaces L(r,q)(Ω,ℝN) and certain borderline Orlicz spaces of Zygmund type which are of crucial use in the analysis of the n-Laplace system, as already shown in Theorem 1.1 in Section 1. The modulus of continuity established in these instances can also be shown to be sharp in most situations, see Proposition 3.12 below.
In what follows, given a number σ∈[1,∞], we set σ′=σσ-1, the Hölder conjugate of σ, with the usual conventions if either σ=1 or σ=∞.
Theorem 3.8Theorem 3.8 (Modulus of continuity with f∈Γσ(ν)(Ω,RN))
Assume that 2≤p≤n. Let 𝐟∈Γσ(ν)(Ω,RN) for some σ∈[1,∞). Let 𝐮 be a local solution to (3.1).
Part I. Assume that the function φ, defined as
(3.7)φ(s)={(∫0snt(σ(p-1))′n-1(∫t1τ-σ′n′ν(τ)-1σ-1𝑑τ)σ-1σ(p-1)-1𝑑t)1(σ(p-1))′if σ>1, 2≤p≤n,(∫0snt(p-1)′n-11(inft<τ<1(τ1n′ν(τ)))(p-1)′-1𝑑t)1(p-1)′if σ=1, 2<p≤n,sup0<t<1min{t12,s}inft<τ<1(τ12ν(τ))if σ=1, 2=p=n,
for s near 0, is finite and lims→0+φ(s)=0 (of course, the latter assumption is relevant only in the last case of (3.7)). Then, 𝐮∈Cloc0,φ(Ω,RN) and there exists a constant C such that
∥𝒖∥C0,φ(BR,ℝN)≤C(∥𝒇∥Γσ(ν)(B2R,ℝN)1p-1+∥∇𝒖∥L1(B2R,ℝN×n))
for every ball B2R⊂⊂Ω.
Part II. In particular, if
(3.8)∫0t-σ′n′ν(t)-1σ-1𝑑t<∞,
then 𝐮∈Liploc(Ω,RN) and there exists a constant C such that
∥𝒖∥Lip(BR,ℝN)≤C(∥𝒇∥Γσ(ν)(B2R,ℝN)1p-1+∥∇𝒖∥L1(B2R,ℝN×n))
for every ball B2R⊂⊂Ω.
Theorem 3.9Theorem 3.9 (Modulus of continuity with f∈Γρ,∞(ν)(Ω,RN))
Assume that 2≤p≤n. Let 𝐟∈Γρ,∞(ν)(Ω,RN) for some ρ∈(0,∞). Let 𝐮 be a local solution to (3.1). Assume that the function φ, defined as
φ(s)=∫0snt-1n′(∫t1τ-1n′(∫0τν(ϱ)𝑑ϱ)-1ρ𝑑τ)1p-1𝑑t
for s near 0, is finite. Then, 𝐮∈Cloc0,φ(Ω,RN) and there exists a constant C such that
∥𝒖∥C0,φ(BR,ℝN)≤C(∥𝒇∥Γρ,∞(ν)(B2R,ℝN)1p-1+∥∇𝒖∥L1(B2R,ℝN×n))
for every ball B2R⊂⊂Ω.
Specializing the previous theorems to the case when the weight ν is a power times a logarithm yields the following result.
Theorem 3.10Theorem 3.10 (Modulus of continuity with f∈L(r,q)(logL)α(Ω,RN))
Assume that 2≤p≤n and also let 𝐟∈L(r,q)(logL)α(Ω,RN) with np≤r≤n, 1≤q≤∞, and α∈R. Let 𝐮 be a local solution to (3.1).
Part I. Let φ be the function defined as
(3.9)φ(s)={(log1s)-α+p-1-1qp-1if r=np, 1≤q≤∞,α>p-1-1q, 2≤p≤n,srp-nr(p-1)(log1s)-αp-1if np<r<n, 1≤q≤∞,α∈ℝ, 2≤p≤n,s(log1s)-α+1-1qp-1if r=n, 1≤q≤∞,α<1q′, 2≤p≤n,s(log(log1s))1-1qp-1if r=n, 1<q≤∞,α=1q′, 2≤p≤n,
for s near 0. Then, 𝐮∈Cloc0,φ(Ω,RN) and there exists a constant C such that
(3.10)∥𝒖∥C0,φ(BR,ℝN)≤C(∥𝒇∥L(r,q)(logL)α(B2R,ℝN)1p-1+∥∇𝒖∥L1(B2R,ℝN×n))
for every ball B2R⊂⊂Ω.
Part II. In particular, if one of the conditions
{r=n,q=1,α≥0, 2≤p≤n;r=n, 1<q≤∞,α>1q′, 2≤p≤n;r>n, 1≤q≤∞,α∈ℝ, 2≤p≤n,
is satisfied, then 𝐮∈Liploc(Ω,RN) and there exists a constant
C such that
∥𝒖∥Lip(BR,ℝN)≤C(∥𝒇∥L(r,q)(logL)α(B2R,ℝN)1p-1+∥∇𝒖∥L1(B2R,ℝN×n))
for every ball B2R⊂⊂Ω.
Example 3.11Theorem 3.8 allows one to deal also with data 𝒇 from unconventional functions spaces. Suppose, for instance, that p=n and 𝒇∈Γ1(ν)(Ω,ℝN), where ν(t)=eγ(log1t)δ(log1t)δ-1 near 0 for some δ∈(0,1) and γ>0. The behavior near 0 of this weight is intermediate between a logarithm and a power. If 𝒖 is any local solution to (3.1), then 𝒖∈Cloc0,φ(Ω,ℝN), where φ(s)=e-γn-1(log1s)δ(log1s)1-δn-1 near 0.
The result of Theorem 3.10 concerning the space Lr,∞(Ω) with np<r<n and N=1 overlaps with [23, Corollary 8.11] and [31, Theorem 4.2]. In fact, we are also in a position to prove the sharpness of the modulus of continuity provided by Theorem 3.10. This is a special instance of our last result.
Proposition 3.12Proposition 3.12 (Sharpness of moduli of continuity)
Let 𝐟∈L(r,q)(logL)α(Ω,RN). Assume that either of the alternatives
(3.11){r=np, 1≤q≤∞,α>p-1-1q, 2≤p≤n,n>2;r=1, 1<q≤∞,α>1-1q,n=p=2;np<r<n, 1≤q≤∞,α∈ℝ, 2≤p≤n;r=n,q=1,α<0, 2≤p≤n,
holds. Then, the modulus of continuity φ, defined as in (3.9), is optimal, in the sense that if ψ is another modulus of continuity such that (3.10) holds for every 𝐟∈L(r,q)(logL)α(Ω,RN) and every local solution 𝐮 to (3.1), then Cloc0,φ(Ω,RN)⊆Cloc0,ψ(Ω,RN).
4 Proofs of the results of Section 3.1
Our point of departure is an estimate, in rearrangement form, for the gradient of local solutions to (3.1). Such an estimate in turn relies upon a pointwise gradient bound from [25].
Proposition 4.1Assume that 2≤p≤n. Let 𝐮 be a local solution to (3.1) for some 𝐟∈L1(Ω,RN). Assume that |Ω|=1 and Ω′ is any open set such that Ω′⊂⊂Ω. Then, there exists a constant C=C(n,N,p,Ω,Ω′) such that
(|∇𝒖|⌊Ω′)*(s)p-1≤(|∇𝒖|⌊Ω′)**(s)p-1≤1s∫0s(|∇𝒖|⌊Ω′)*(t)p-1𝑑t
(4.1)≤C∫s1|𝒇|**(t)t-1n′𝑑t+C∥∇𝒖∥L1(Ω,ℝN×n)p-1 for s∈(0,1).
Proof.
By [25, Theorem 1.1], there exists a constant C=C(n,N,p) such that
(4.2)|∇𝒖|(x)p-1≤C∫0R∫Bϱ(x)|𝒇(y)|𝑑ydϱϱn+C(⨍BR(x)|∇𝒖|𝑑y)p-1
for a.e. x∈Ω and every R>0 such that BR(x)⊂⊂Ω. Fubini’s theorem entails that
∫0R∫Bϱ(x)|𝒇(y)|𝑑ydϱϱn≤∫0∞∫Bϱ(x)|𝒇(y)|χBR(x)(y)𝑑ydϱϱn
=∫ℝn|𝒇(y)|χBR(x)(y)∫0∞χBϱ(x)(y)dϱϱn𝑑y
(4.3)=1n-1∫ℝn|𝒇(y)||x-y|n-1χBR(x)(y)𝑑y
for any such x and R. Owing to (4.2) and (4.3),
(4.4)|∇𝒖|(x)p-1≤C∫ℝn|𝒇(y)||x-y|n-1χBR(x)(y)𝑑y+C(⨍BR(x)|∇𝒖|𝑑y)p-1.
Owing to O’Neil’s rearrangement inequality for convolutions [28], we infer from (4.4) that there exist constants C, C′, and C′′, depending on n,N,p,Ω, and Ω′, such that
1s∫0s(|∇𝒖|⌊Ω′)*(t)p-1𝑑t≤C(∫ℝn|𝒇(y)||⋅-y|n-1χBR(x)(y)𝑑y)*(s)+C∥∇𝒖∥L1(Ω,ℝN×n)p-1
≤Cs|𝒇|**(s)(1|⋅-y|n-1)**(s)+C∫s∞|𝒇|*(t)(1|⋅-y|n-1)*(t)𝑑t+C∥∇𝒖∥L1(Ω,ℝN×n)p-1
≤C′s-1n′∫0s|𝒇|*(t)𝑑t+C′∫s1|𝒇|*(t)t-1n′𝑑t+C∥∇𝒖∥L1(Ω,ℝN×n)p-1
(4.5)≤C∫s1′′|𝒇|**(t)t-1n′𝑑t+C∥∇𝒖∥L1(Ω,ℝN×n)p-1 for s∈(0,1).
This establishes the last inequality in (4.1). The second inequality is a consequence of Hölder’s inequality, since p≥2. The first inequality holds by (2.1). The proof is complete.
∎
We are now in a position to prove our first main result.
Proof of Theorem 3.1.
We claim that, if Ω′⊂⊂Ω, then there exists a constant C such that
(4.6)∥∇𝒖∥Ln,1(Ω′,ℝN×n)≤C∥𝒇∥L(np,1p-1)(Ω,ℝN)1p-1+C∥∇𝒖∥L1(Ω,ℝN×n),
namely,
(4.7)∫01s-1+1n(|∇𝒖|⌊Ω′)*(s)𝑑s≤C∫01(spn|𝒇|**(s))1p-1dss+C∥∇𝒖∥L1(Ω,ℝN×n).
By (4.1), inequality (4.7) will follow if we show that
(4.8)∫01t-1+1n(∫t1|𝒇|**(s)s-1n′𝑑s)1p-1𝑑t≤C∫01(tpn|𝒇|**(t))1p-1dtt
for some constant C and for every measurable function 𝒇 in Ω.
Inequality (4.8) is in turn a consequence of [9, Proposition 2.5]. Having (4.6) at disposal, the continuity of 𝒖 follows, since, by a result of [30], any weakly differentiable function 𝒖 with |∇𝒖|∈Llocn,1(Ω,ℝN) is continuous.
In order to prove the optimality of the space L(np,1p-1)(Ω,ℝN), assume, without loss of generality, that N=1, 0∈Ω, and let R>0 be such that BR(0)⊂⊂Ω. Denote by d the diameter of Ω. Let X(Ω) be a rearrangement-invariant quasi-normed space. Given any nonnegative function h∈X(Ω) which is radially decreasing about 0 and vanishes outside BR(0), the (radially decreasing) function v:Ω→ℝ defined as
v(x)=(nωn1n)-p′∫ωn|x|nωndn(spnh**(s))1p-1dss for x∈Ω
is a local solution to the equation
(4.9)-div(|∇v|p-2∇v))=h(x) in Ω.
One has that
(4.10)∥v∥L∞(BR(0))≥v(0)=c∥h∥L(np,1p-1)(BR(0))1p-1
for a suitable constant c. If, by contradiction, X(Ω)⊈L(np,1p-1)(Ω), then a function h as above can be chosen in such a way that h∈X(Ω)∖L(np,1p-1)(Ω), and hence
∥h∥L(np,1p-1)(BR(0))1p-1=∞.
By (4.10), v∉Lloc∞(Ω), and, a fortiori, v is not continuous in Ω.
∎
The remaining part of this section is devoted to the proofs of Theorems 3.2–3.6.
Proof of Theorem 3.2.
By Theorem 3.1, any local solution 𝒖 to (3.1) is continuous provided that
(4.11)Γσ(ν)(Ω,ℝN)→L(np,1p-1)(Ω,ℝN).
The embedding (4.11) is equivalent to the inequality
(4.12)(∫01ϕ**(t)1p-1tpn(p-1)-1𝑑t)p-1≤C(∫01ϕ**(t)σν(t)𝑑t)1σ
for every measurable function ϕ on (0,1). The weighted Hardy type inequality for non-increasing functions (4.12) holds if one of the conditions in (3.2) is fulfilled. This follows from [17, Theorem 5.1 (i)] in the first two cases of (3.2) and from [17, Theorem 5.1 (ii)] in the last three cases.
Conversely, assume that σ≥1 and for every 𝒇∈Γσ(ν)(Ω,ℝN) any local solution 𝒖 to (3.1) is continuous. Then, by Theorem 3.1,
(4.13)Γσ(ν)(Ω,ℝN)⊂L(np,1p-1)(Ω,ℝN).
Since σ≥1, Γσ(ν)(Ω,ℝN) is a rearrangement-invariant space, and hence, owing to (2.2), the inclusion (4.13) is equivalent to the embedding (4.11). Thereby, (4.12) holds. The necessity of the conditions of [17, Theorem 5.1] again tell us that either of the conditions in (3.2) must be fulfilled.
∎
Proof of Theorem 3.3.
By Theorem 3.1, any local solution 𝒖 to (3.1) is continuous provided that
(4.14)Γρ,∞(ν)(Ω,ℝN)→L(np,1p-1)(Ω,ℝN)
or, equivalently,
(4.15)(∫01ϕ**(t)1p-1tpn(p-1)-1𝑑t)p-1≤Csup0<t<1ϕ**(t)𝒱(t)1ρ
for every measurable function ϕ on (0,1). By [8, Theorem 6.4], (4.15) holds if and only if (3.3) holds.
Conversely, assume that ρ≥1 and for every 𝒇∈Γρ,∞(ν)(Ω,ℝN) any local solution 𝒖 to (3.1) is continuous. Then, by Theorem 3.1, Γρ,∞(ν)(Ω,ℝN)⊂L(np,1p-1)(Ω,ℝN). Since ρ≥1, Γρ,∞(ν)(Ω,ℝN) is a rearrangement-invariant space, and, by (2.2), the last inclusion is in fact equivalent to the embedding (4.14), namely, to (4.15). Owing to (the necessity part of) [8, Theorem 6.4], we conclude that (3.3) must hold.
∎
Proof of Theorem 3.4.
By Theorem 3.1, any local solution 𝒖 to (3.1) is continuous provided that
(4.16)Λσ(ν)(Ω,ℝN)→L(np,1p-1)(Ω,ℝN).
If p=n, the embedding (4.16) is equivalent to the inequality
(4.17)(∫01ϕ**(t)1n-1t1n-1-1𝑑t)n-1≤C(∫01ϕ*(t)σν(t)𝑑t)1σ
for every measurable function ϕ on (0,1). On the other hand, if 2≤p<n, then L(np,1p-1)(Ω,ℝN)=Lnp,1p-1(Ω,ℝN) and the embedding (4.16) is equivalent to
(4.18)(∫01ϕ*(t)1p-1tpn(p-1)-1𝑑t)p-1≤C(∫01ϕ*(t)σν(t)𝑑t)1σ
for every measurable function ϕ on (0,1). We claim that either (4.17) or (4.18) holds if the appropriate condition from (3.4) is satisfied. This can be verified by distinguishing into several cases.
If 0<σ≤1p-1 and 2≤p<n, inequality (4.18) is equivalent to the corresponding condition in (3.4) by [8, Theorem 3.1 (i)].
If 0<σ≤1n-1 and 2≤p=n, inequality (4.17) is equivalent to the corresponding condition in (3.4) by [8, Theorem 4.1 (ii)].
If σ>1p-1 and 2≤p<n, inequality (4.18) is equivalent to the corresponding condition in (3.4) by [8, Theorem 3.1 (ii)].
If 1n-1<σ≤1 and 2≤p=n, inequality (4.17) is equivalent to the corresponding condition in (3.4) by [7, Theorem 3.1].
If σ>1 and 2=p=n, by [8, Theorem 4.1 (iv)], inequality (4.17) is equivalent to
(4.19)∫0(t𝒱(t))1σ-1(log1t)σσ-1𝑑t<∞.
An application of Fubini’s theorem shows that (4.19) is equivalent to the condition on the last line of (3.4).
If σ>1 and 2<p=n, by [8, Theorem 4.1 (iii)], inequality (4.17) holds if and only if
(4.20)(∫01(tσ∫0tν(s)𝑑s)1σ(n-1)-1dtt)σ(n-1)-1σ<∞
and
(4.21)∫01(∫0tν(s)sσ′(∫0sν(τ)𝑑τ)σ′𝑑s)σ-1σ(n-1)-1(log1t)1σ(n-1)-1dtt<∞.
An integration by parts in the inner integral on the left-hand side of (4.21) shows that the latter equals
∫01(σ′σ′-1∫0tsσ′-1(∫0sν(τ)𝑑τ)σ′-1𝑑s-tσ′σ′-1(∫0tν(τ)𝑑τ)1-σ′)σ-1σ(n-1)-1(log1t)1σ(n-1)-1dtt.
Thus, the condition appearing on the last line of (3.4), namely,
(4.22)∫01(∫0tsσ′-1(∫0sν(τ)𝑑τ)σ′-1𝑑s)σ-1σ(n-1)-1(log1t)1σ(n-1)-1dtt<∞,
implies (4.21). We claim that the reverse implication is also true, and hence (4.21) and (4.22) are actually equivalent. To verify this claim, it suffices to show that there exists a positive constant C such that
∫01(σ′σ′-1∫0tsσ′-1(∫0sν(τ)𝑑τ)σ′-1𝑑s-tσ′σ′-1(∫0tν(τ)𝑑τ)1-σ′)σ-1σ(n-1)-1(log1t)1σ(n-1)-1dtt
(4.23)≥C∫01(∫0tsσ′-1(∫0sν(τ)𝑑τ)σ′-1𝑑s)σ-1σ(n-1)-1(log1t)1σ(n-1)-1dtt
for every continuous function ν:(0,1)→[0,∞). On setting
ψ*(s)=(∫0s1σ′ν(t)𝑑t)1-σ′ and w(s)=sσ-1σ(n-1)-1-1(log1s)1σ(n-1)-1
for s∈(0,1) and changing variables, (4.23) takes the form
(4.24)∫01(ψ**(s)-ψ*(s))σ-1σ(n-1)-1w(s)𝑑s≥C∫01ψ**(s)σ-1σ(n-1)-1w(s)𝑑s
for some positive constant C. Inequality (4.24), and hence (4.23), follows from [7, Theorem 6.2 (ii)]. The equivalence of (4.22) and (4.21) is fully proved.
Next, observe that, since
∫01ψ**(s)σ-1σ(n-1)-1w(s)𝑑s≥∫01ψ*(s)σ-1σ(n-1)-1w(s)𝑑s,
we have that
(4.25)∫01(∫0tsσ′-1(∫0sν(τ)𝑑τ)σ′-1ds)σ-1σ(n-1)-1(log1t)1σ(n-1)-1dtt≥C∫01(tσ(∫0tν(s)𝑑s))1σ(n-1)-1(log1t)1σ(n-1)-1dtt
for some positive constant C. Thus, (4.22) implies that the right-hand side of (4.25) is finite. On the other hand, the latter condition trivially implies (4.20). Altogether, we have that (4.22) implies both (4.20) and (4.21), and hence (4.17). This concludes the proof of (4.16) and the proof of the embedding is thus complete in all possible cases.
Conversely, assume now that either σ>1 and (2.6) holds or σ=1 and (2.7) holds, and that for every 𝒇∈Λσ(ν)(Ω,ℝN) any local solution 𝒖 to (3.1) is continuous. Then, by Theorem 3.1,
(4.26)Λσ(ν)(Ω,ℝN)⊂L(np,1p-1)(Ω,ℝN).
Under the present assumptions, Λσ(ν)(Ω,ℝN) is a rearrangement-invariant space, and owing to (2.2), the inclusion (4.26) in fact implies the embedding (4.16), namely (4.17) or (4.18). Distinguishing into the same cases as above, and making use of the necessity of the conditions for (4.17) or (4.18), tells us that the corresponding condition in (3.4) must be in force.
∎
Proof of Theorem 3.5.
By Theorem 3.1, any local solution 𝒖 to (3.1) is continuous provided that
(4.27)Λρ,∞(ν)(Ω,ℝN)→L(np,1p-1)(Ω,ℝN),
namely, if
(4.28)(∫01ϕ**(t)1p-1tpn(p-1)-1𝑑t)p-1≤Csup0<t<1ϕ*(t)𝒱(t)1ρ
for every measurable function ϕ on (0,1). By [8, Theorem 4.4], (4.28) holds if (and only if) (3.5) holds.
Conversely, assume that either ρ>1 and (2.6) holds or ρ=1 and (2.7) holds, and that for every 𝒇∈Λρ,∞(ν)(Ω,ℝN) any local solution 𝒖 to (3.1) is continuous. Theorem 3.1 then yields the inclusion
Λρ,∞(ν)(Ω,ℝN)⊂L(np,1p-1)(Ω,ℝN).
Our present assumptions ensure that Λρ,∞(ν)(Ω,ℝN) is a rearrangement-invariant space, and owing to (2.2) this inclusion in fact implies the embedding (4.27), namely (4.28). Making use of the necessity part of [8, Theorem 4.4] tells us that (3.5) must hold.
∎
In view of the proof of Theorem 3.6, we preliminarily establish a lemma on inclusions between Lorentz and Orlicz type spaces. In fact, just one of the relevant inclusions will be needed in what follows. However, we believe that its full content can be of use for future applications, and we thus provide a complete statement and proof.
Lemma 4.2Let 0<p,q<∞ and let A be a generalized Young function. Let Ω be a measurable subset of Rn with |Ω|<∞.
Part I. Assume that
(4.29){lim supt→∞tpA(t)<∞if p≤q,∫∞(tA(t))qp-qtp(q-1)p-q𝑑t<∞if p>q.
Then,
(4.30)LA(Ω,ℝN)⊂Lp,q(Ω,ℝN)
and
(4.31)L(A)(Ω,ℝN)⊂L(p,q)(Ω,ℝN).
Part II. Assume that
(4.32){lim supt→∞A(t)tp<∞if p≥q,∫∞(A(t)t)qq-ptq-1𝑑t<∞if q>p.
Then,
(4.33)Lp,q(Ω,ℝN)⊂LA(Ω,ℝN)
and
(4.34)L(p,q)(Ω,ℝN)⊂L(A)(Ω,ℝN).
Proof.
Let 𝒈:Ω→ℝN be a measurable function. An application of Fubini’s theorem tells us that
(4.35)∫0|Ω||𝒈|*(s)qsqp-1ds=p∫0∞|{|𝒈|*>t}|qptq-1dt
and
(4.36)∫0|Ω|A(|𝒈|*(s))ds=∫0∞|{|𝒈|*>t}|a(t)dt,
where a is the nonnegative function appearing in (2.10). Thus, (4.30) holds, provided that the inequality
(4.37)(∫0∞ϕ(t)qptq-1𝑑t)pq≤C∫0∞ϕ(t)a(t)𝑑t
holds for some constant C and for every non-increasing function ϕ:(0,∞)→[0,∞). Inequality (4.37) follows from (4.29), owing to [8, Theorem 3.1]. Here, one also makes use of the fact that, since |Ω|<∞, the behavior of A near 0 is irrelevant in the definition of the space LA(Ω,ℝN).
Similarly, (4.33) follows from the inequality
(4.38)∫0∞ϕ(t)a(t)𝑑t≤C(∫0∞ϕ(t)qptq-1𝑑t)pq
for some constant C and for every non-increasing function ϕ:(0,∞)→[0,∞). Inequality (4.38) is a consequence of (4.32), thanks to [8, Theorem 3.1] again.
The proof of (4.31) and (4.34) is completely analogous: one has just to replace |𝒈|* with |𝒈|** in (4.35) and (4.36).
∎
Proof of Theorem 3.6.
Assume that 2≤p<n. By Theorem 3.1, it suffices to show that
LA(Ω,ℝN)⊂L(np,1p-1)(Ω,ℝN).
Since L(np,1p-1)(Ω,ℝN)=Lnp,1p-1(Ω,ℝN) if 2≤p<n, the conclusion follows via Lemma 4.2.
Assume next that 2<p=n. On replacing, if necessary, A with an equivalent Young function, we may suppose, without loss of generality, that A∈C2(0,∞). By Theorem 3.1, it suffices to show that
(4.39)LA(Ω,ℝN)⊂L(1,1n-1)(Ω,ℝN).
Define the generalized Young function B as B(t)=tA′(t)-A(t) for t≥0. Observe that B is non-decreasing, since B′(t)=tA′′(t)≥0 for t≥0. By [10, 18, 22], LA(Ω,ℝN)⊂L(B)(Ω,ℝN). Hence, (4.39) holds if, in particular, L(B)(Ω,ℝN)⊂L(1,1n-1)(Ω,ℝN), and the latter inclusion follows from (4.31).
Finally, assume that 2=p=n. Now, we have to show that LA(Ω,ℝN)⊂L(1,1)(Ω,ℝN). Since L(1,1)(Ω,ℝN)=LlogL(Ω,ℝN), the
conclusion follows, inasmuch as A(t) dominates tlog(1+t) near infinity, owing to our assumption on A.
∎
5 Proofs of the results of Section 3.2
We begin by establishing Theorem 3.8.
Proof of Theorem 3.8.
For the proof of Part I, we distinguish the following cases.
If σ>1, 2≤p≤n, we determine a weight m, optimal in a sense, such that
(5.1)∥∇𝒖∥Γσ(p-1)(m)(BR,ℝN×n)≤C∥𝒇∥Γσ(ν)(B2R,ℝN)1(p-1)+C∥∇𝒖∥L1(B2R,ℝN×n)
for every ball B2R⊂⊂Ω. By Proposition (4.1), inequality (5.1) will follow if we show that
(5.2)(∫01(∫s1|𝒇|**(t)t-1n′𝑑t)σm(s)𝑑s)1σ≤C(∫01|𝒇|**(s)σν(s)𝑑s)1σ
for every measurable function 𝒇:Ω→ℝN. Inequality (5.2) holds, in particular, if
(5.3)(∫01(∫s1ϕ(t)𝑑t)σm(s)𝑑s)1σ≤C(∫01ϕ(s)σs1n′ν(s)𝑑s)1σ
for every nonnegative measurable function ϕ in (0,1). A classical criterion for weighted Hardy type inequalities [26, Theorem 1.3.2/2] ensures that the weight m, defined by
m(s)=dds(∫s1(t1n′ν(t)1σ)-σ′𝑑t)1-σ if 0<s≤12
and m(s)=0 if 12<s<1, renders (5.3) true. In conclusion, inequality (5.1) is fulfilled with this choice of m.
By [11, Theorem 1.3],
∥𝒖∥C0,φ(BR,ℝN)≤C∥∇𝒖∥Γσ(p-1)(m)(BR,ℝN×n),
where
φ(s)=∥t-1n′χ(0,sn)(t)∥(Γσ(p-1)(m))′(0,1) for s near 0.
Inasmuch as ∥⋅∥(Λσ(p-1)(m)(0,1)≤∥⋅∥Γσ(p-1)(m)(0,1), one has that
∥⋅∥(Γσ(p-1)(m))′(0,1)≤∥⋅∥(Λσ(p-1)(m))′(0,1).
Since 1<σ(p-1)<∞, by [8, Theorem 9.1 (i)],
∥ϕ∥(Λσ(p-1)(m))′(0,1)≈(∫01ϕ**(t)(σ(p-1))′t(σ(p-1))′m(t)M(t)(σ(p-1))′𝑑t)1(σ(p-1))′
for every measurable function ϕ in (0,1), where the function M is given by
M(t)=∫0tm(s)𝑑s=(∫t1s-σ′n′ν(s)-1(σ-1)𝑑s)1-σ if 0<t≤12
and M(t)=M(12) if 12<t<1. Therefore, there exists a constant C such that
φ(s)≤C{∫01(((⋅)-1n′χ(0,sn)(⋅))**(t))(σ(p-1))′t(σ(p-1))′m(t)M(t)(σ(p-1))′𝑑t}1(σ(p-1))′
≈{∫0snt(σ(p-1))′nm(t)M(t)(σ(p-1))′𝑑t+s(σ(p-1))′∫sn1m(t)M(t)(σ(p-1))′𝑑t}1(σ(p-1))′
≈{limt→0+t(σ(p-1))′nM(t)1-(σ(p-1))′(σ(p-1))′-1+σ(p-1)n∫0snt(σ(p-1))′n-1M(t)1-(σ(p-1))′dt
(5.4)-s(σ(p-1))′M(1)1-(σ(p-1))′(σ(p-1))′-1}1(σ(p-1))′ for s near 0.
An application of de L’Hôpital’s rule tells us that the second addend in braces on the rightmost side of (5.4) decays to 0 as s→0+ more slowly than the third addend. On the other hand,
(5.5)∫0snt(σ(p-1))′n-1M(t)1-(σ(p-1))′𝑑t≥1M(sn)(σ(p-1))′-1∫0snt(σ(p-1))′n-1𝑑t≈s(σ(p-1))′M(sn)(σ(p-1))′-1 for s∈(0,1),
where the first inequality is a consequence of the fact that M(sn)1-(σ(p-1))′ is a non-increasing function. Equation (5.5) implies that the limit in (5.4) equals 0. Altogether, we have shown that
φ(s)≈{∫0snt(σ(p-1))′n-1M(t)1-(σ(p-1))′𝑑t}1(σ(p-1))′
(5.6)≈{∫0snt(σ(p-1))′n-1(∫t1τ-σ′n′ν(τ)-1σ-1𝑑τ)(σ-1)((σ(p-1))′-1)𝑑t}1(σ(p-1))′ for s near 0.
If σ=1, 2<p≤n, let us determine a measure μ in (0,1) such that
(5.7)∥∇𝒖∥Γp-1(dμ)(BR,ℝN×n)≤C(∥𝒇∥Γ1(ν)(B2R,ℝN)1p-1+∥∇𝒖∥L1(B2R,ℝN×n))
for every B2R⊂⊂Ω. Owing to estimate (4.5), inequality (5.7) is a consequence of the inequality
(5.8)∫01(∫t1s-1n′ϕ(s)𝑑s)𝑑μ(t)≤c∫01ϕ(s)ν(s)𝑑s
for every nonnegative measurable function ϕ in (0,1). By [26, Theorem 1.3.2/4], inequality (5.8) holds provided that μ obeys
μ([0,s])=infs<t<1(t1n′ν(t)) if 0<s≤12
and μ([0,s])=μ([0,12]) if 12<s<1. By [11, Theorem 1.3],
∥𝒖∥C0,φ(BR,ℝN)≤C∥∇𝒖∥Γp-1(dμ)(B2R,ℝN×n),
where
(5.9)φ(s)=∥t-1n′χ(0,sn)(t)∥(Γp-1(dμ))′(0,1)≤∥t-1n′χ(0,sn)(t)∥(Λp-1(dμ))′(0,1) for s near 0.
Since we are assuming that 1<p-1<∞,
(5.10)∥ϕ∥(Λp-1(dμ))′(0,1)≈(∫01ϕ**(t)(p-1)′t(p-1)′dμ(t)M(t)(p-1)′)1(p-1)′
for every measurable function ϕ in (0,1), where now we have set M(t)=μ([0,t]) for t∈(0,1). Equation (5.10) follows from [8, Theorem 9.1 (i)] in the case when μ is absolutely continuous, but the same proof applies to the present slightly more general situation.
By (5.9) and (5.10), there exists a constant C such that
φ(s)≤C{∫01(((⋅)-1n′χ(0,sn)(⋅))**(t))(p-1)′t(p-1)′dμ(t)M(t)(p-1)′}1(p-1)′
≈{∫0snt(p-1)′ndμ(t)M(t)(p-1)′+s(p-1)′∫sn1dμ(t)M(t)(p-1)′}1(p-1)′
≈{limt→0+t(p-1)′nM(t)1-(p-1)′(p-1)′-1+(p-1)n∫0snt(p-1)′n-1M(t)1-(p-1)′𝑑t-s(p-1)′M(1)1-(p-1)′(p-1)′-1}1(p-1)′
≈{∫0snt(p-1)′n-1(inft<τ<1(τ1n′ν(τ)))1-(p-1)′𝑑t}1(p-1)′ for s near 0,
where the last equivalence follows via an argument analogous to that employed in the derivation of (5.6) from (5.4).
If σ=1, 2=p=n, we exhibit a measure μ such that
(5.11)∥∇𝒖∥Γ1(dμ)(BR,ℝN×n)≤C(∥𝒇∥Γ1(ν)(B2R,ℝN)+∥∇𝒖∥L1(B2R,ℝN×n))
for every B2R⊂⊂Ω. By Proposition 4.1, inequality (5.11) holds if the measure μ supports the inequality
(5.12)∫01(∫t1s-12ϕ(s)𝑑s)𝑑μ(t)≤C∫01ϕ(s)ν(s)𝑑s
for every nonnegative measurable function ϕ in (0,1). Owing to [26, Theorem 1.3.2/4], inequality (5.12) is satisfied, provided that μ obeys
μ([0,s])=infs<t<1(t12ν(t)) if 0<s≤12
and μ([0,s])=μ([0,12]) if 12<s<1. Thanks to [11, Theorem 1.3],
∥𝒖∥C0,φ(BR,ℝN)≤C∥∇𝒖∥Γ1(dμ)(BR,ℝN×n),
where
φ(s)=∥t-12χ(0,s2)(t)∥(Γ1(dμ))′(0,1)≤∥t-12χ(0,s2)(t)∥(Λ1(dμ))′(0,1) for s near 0.
Moreover, by [8, Theorem 9.1 (i)],
∥ϕ∥(Λ(dμ))′(0,1)≈sup0<t<1ϕ**(t)tμ([0,t])
for every measurable function ϕ in (0,1). Let us notice that in [11] and [8], absolutely continuous measures μ are considered, but analogous proofs apply to the present slightly more general setting. Thereby, there exists a constant C such that
φ(s)≤Csup0<t<1(((⋅)-12χ(0,s2)(⋅))**(t)tμ([0,t]))≈sup0<t<1(min{t12,s}inft<τ<1(τ12ν(τ))) for s near 0.
For Part II, we assume that (3.8) is in force. We claim that |∇𝒖|∈L∞(BR,ℝN×n), and hence 𝒖∈Lip(BR,ℝN). In order to prove our claim, it suffices to show that
(5.13)∥∇𝒖∥L∞(BR,ℝN×n)≤C(∥𝒇∥Γσ(ν)(B2R,ℝN)1p-1+∥∇𝒖∥L1(B2R,ℝN×n))
for every ball B2R⊂⊂Ω. Inequality (5.13) is a consequence of Proposition 4.1 and Hölder’s inequality, since
∥∫s1|𝒇|**(t)t-1n′𝑑t∥L∞(0,1)=∫01|𝒇|**(t)t-1n′𝑑t≤(∫01|𝒇|**(t)σν(t)𝑑t)1σ(∫01(t-1n′ν(t)-1σ)σ′𝑑t)1σ′.∎
Proof of Theorem 3.9.
We follow the outline of the proof of Theorem 3.8 and find a weight m such that
(5.14)∥∇𝒖∥Γρ(p-1),∞(m)(BR,ℝN×n)≤C(∥𝒇∥Γρ,∞(ν)(B2R,ℝN)1(p-1)+∥∇𝒖∥L1(B2R,ℝN×n))
for every ball B2R⊂⊂Ω. By Proposition 4.1, inequality (5.14) is implied by the inequality
(5.15)∥(∫s1|𝒇|**(t)t-1n′dt)M(s)1ρ∥L∞(0,1)≤C∥|𝒇|**(s)𝒱(s)1ρ∥L∞(0,1)
for every measurable function 𝒇:Ω→ℝN, where M(s)=∫0sm(t)𝑑t and 𝒱(s)=∫0sν(t)𝑑t for s∈(0,1). Owing to [26, Theorem 1.3.2/2], the weight m, defined as
m(s)=dds(∫s1t-1n′𝒱(t)-1ρ𝑑t)-ρ if 0<s≤12
and m(s)=0 if 12<s<1, supports (5.15), and hence (5.14).
An application of [11, Theorem 1.3] now tells us that
∥𝒖∥C0,φ(BR,ℝN)≤C∥∇𝒖∥Γρ(p-1),∞(m)(BR,ℝN×n),
where
φ(s)=∥t-1n′χ(0,sn)(t)∥(Γρ(p-1),∞(m))′(0,1)≤∥t-1n′χ(0,sn)(t)∥(Λρ(p-1),∞(m))′(0,1) for s near 0.
On the other hand, [8, Theorem 9.1 (ii)] ensures that
∥ϕ∥(Λρ(p-1),∞(m))′(0,1)≈∫01ϕ*(t)M(t)-1ρ(p-1)𝑑t
for every measurable function ϕ in (0,1). As a consequence, there exists a constant C such that
φ(s)≤C∫01((⋅)-1n′χ(0,sn)(⋅))*(t)M(t)-1ρ(p-1)𝑑t=∫0snt-1n′(∫t1τ-1n′𝒱(τ)-1ρ𝑑τ)1p-1𝑑t for s near 0.
The conclusion follows.
∎
Proof of Theorem 3.10.
The conclusions follow on applying Theorem 3.8 with appropriate choices of the power σ and the weight ν, via standard computations. The
details are omitted for brevity.
∎
Proof of Proposition 3.12.
Let BR(0), h∈L(r,q)(logL)α(Ω), and v be as in the proof of Theorem 3.1, so that v is a (radially decreasing) solution to the p-Laplace equation (4.9). One has that
(5.16)∥∇v∥L1(BR(0),ℝn)=C∥h∥L(nnp-p+1,1p-1)(BR(0))≤C′∥h∥L(r,q)(logL)α(Ω)
for some constants C,C′ and for every r,q,α as in the statement. Observe that the inequality holds, by well-known inclusion relations between Lorentz–Zygmund spaces, since r>nnp-p+1 for every r as in (3.11). Now, assume that (3.10) holds with φ replaced with some other modulus of continuity ψ, namely,
(5.17)supx,y∈BR2(0)|v(x)-v(y)|ψ(|x-y|)≤C(∥f∥L(r,q)(logL)α(BR(0))1p-1+∥∇v∥L1(BR(0))).
Choosing x=0 and y such that |y|=s<R2 in (5.17) and making use of (5.16) yield
(5.18)suph∈L(r,q)(logL)α(BR(0))sup0<s<R2∫0ωnsn(ϱpnh**(ϱ))1p-1dϱϱψ(s)(∥h∥L(r,q)(logL)α(BR(0))1p-1)≤C.
Exchanging the order of suprema in (5.18) tells us that
(5.19)Cψ(s)≥suph∈L(r,q)(logL)α(BR(0))∫0ωnsn(ϱpnh**(ϱ))1p-1dϱϱ(∫01(ϱ1r(1+log1ϱ)αh**(ϱ))qdϱϱ)1q(p-1) for 0<s<R2.
We claim that the supremum on the right-hand side of (5.19) is equivalent for s near 0 to the function φ(s) defined by (3.9). This equivalence follows from an application of [17, Theorem 5.1 (i)] if p=2, q=1, and n and α obey (3.11). If instead q=∞ and n,p,α satisfy (3.11), then our claim can be derived via [8, Theorem 6.4]. All the remaining cases can be dealt by [17, Theorem 5.1 (ii)]. The details are omitted for brevity.
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Proof of Proposition 3.7.
By the same argument as in the proof of Proposition 3.12, inequality (3.6) implies (5.19) with r=np, q=1p-1, and α=0, namely,
Cφ(s)≥suph∈L(np,1p-1)(logL)α(BR(0))∫0ωnsn(ϱpnh**(ϱ))1p-1dϱϱ∫0ωnsn(ϱpnh**(ϱ))1p-1dϱϱ=1 for 0<s<R2.
Hence, the conclusion follows.
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