Annales Academiæ Scientiarum Fennicæ Mathematica Volumen 30, 2005, 407–436

VERY WEAK SOLUTIONS OF NONLINEAR SUBELLIPTIC EQUATIONS

Anna Zatorska-Goldstein

University of Warsaw, Institute of Applied Mathematics and Mechanics Banacha 2, PL-02-097 Warszawa, Poland; azator@mimuw.edu.pl

Abstract. We prove a generalization of a theorem of Iwaniec, Sbordone and Lewis on higher integrability of very weak solutions of the A-harmonic equation onto a case of subelliptic operators defined by a family of vector fields satisfying the Hörmander condition. The main tool is a form of the Gehring Lemma formulated and proved in an arbitrary metric space with a doubling measure. This result might be of special interest, as the Gehring Lemma is a vital tool in many applications.

1. Introduction

Our aim is to study properties of the so-called very weak solutions to nonlinear subelliptic equations in the form

(1.1)  X^{∗}A(x, u, Xu) + B(x, u, Xu) = 0.

Here x belongs to a bounded region Ω ⊂ R^{n} and X = (X_{1}, …, X_{k}) is a family of smooth vector fields in R^{n} defined on a neighborhood of Ω, satisfying the Hörmander condition, and X^{∗} = (X_{1}^{∗}, …, X_{k}^{∗}) is a family of operators formal adjoint to X_{i} in L^{2} . We will call the equation a subelliptic A-harmonic equation. In the classical situation

X = ∇ = (\frac{∂}{∂x_{1}} , \frac{∂}{∂x_{2}} , …, \frac{∂}{∂x_{n}} )

we obtain the familiar A-harmonic equation. The vector fields X_{i} also satisfy some additional assumptions which are described in Section 2. A and B are both Carathéodory functions and satisfy standard growth conditions, i.e. A( · , · , ξ) ≈ |ξ|^{p−2}ξ. The precise statement of the conditions is given in Section 4.

We say that u is a weak solution of the equation (1.1) if for every ϕ ∈ C_{0}^{∞}(Ω)

(1.2)  ∫_{Ω} A(x, u, Xu) · Xϕ(x) dx + ∫_{Ω} B(x, u, Xu)ϕ(x) dx = 0

and the function u belongs to the Sobolev space W^{1,p}. The last assumption comes from the variational formulation of the problem. If the function A satisfies standard growth conditions, i.e. |A(x, s, ξ)| ≈ |ξ|^{p−1} , then the L^{p} -integrability condition on u and its derivatives allows us to take as a test function an appropriate power of u multiplied by a smooth cut-off function (or another local construction of a test function based on u). In such a way we can obtain better properties of solutions (e.g. Hölder continuity).

2000 Mathematics Subject Classification: Primary 35H20, 35J60.

On the other hand, the integrals in (1.2) are well defined for |Xu|^{p−1} ∈ L^{1} . It is natural to ask, if one can work with weaker regularity assumptions for weak solutions. In the classical situation where X = ∇, T. Iwaniec and C. Sbordone [15] proved that if u satisfies (1.2) but its derivatives are à priori integrable with some exponent strictly lower than the natural exponent p, then in fact they are integrable with the exponent p and therefore u belongs to Sobolev space W^{1,p}.

Definition 1.1. A function u is called a very weak solution of (1.1) if u satisfies (1.2) but belongs to the Sobolev space W^{1,r}, where the exponent r is strictly lower than the natural exponent p.

Assume that functions A and B satisfy conditions (4.23) and the set Ω ⊂ R^{n} is open and bounded. Let X_{1}, …, X_{k} be vector fields on a neighborhood of Ω, with real, C^{∞} smooth and globally Lipschitz coefficients satisfying the Hörmander condition.

Theorem 1.2. There exists δ > 0, such that if u is a very weak solution of (1.1), u ∈ W_{X,loc}^{1,p−δ}(Ω), then u ∈ W_{X,loc}^{1,p+δ̃}(Ω) for some δ̃ > 0, and hence it is a classical weak solution of (1.1).

Recently, a similar theorem on very weak solutions for parabolic equations (in case X = ∇) was proved by J. Lewis and J. Kinnunen [18], [19].

The idea of Iwaniec and Sbordone was to use the Hodge decomposition in construction of a test function. Later J. Lewis [17] showed another proof, where a construction of a test function was based on a Hardy–Littlewood maximal function. We follow the idea of Lewis. We also follow the way of Iwaniec, Sbordone and Lewis to show the higher integrability of the gradient by application of the Gehring Lemma. We use it in a version formulated by Giaquinta [10, Chapter V, Proposition 1.1], introducing changes that are necessary to adapt it to arbitrary metric spaces with a doubling measure. To the best of the author’s knowledge, this lemma is not available in the mathematical literature in such generality. We need a metric version of the theorem (see Theorem 3.3 in Section 3) because of the change of a metric in R^{n}. This is a result of working with a differential operator X instead of a classical gradient. The idea of the proof is analogous to that in the euclidean case. We cannot, however, use tools which are strictly connected with the euclidean geometry: decomposition into dyadic cubes, the classical Calderon– Zygmund Theorem etc. In general metric spaces one then has to use different arguments, see e.g. Lemma 3.1 which replaces the classical Calderon–Zygmund decomposition.

The Gehring Lemma is widely used in the theory of quasi-regular mappings and nonlinear p.d.e.’s (see [14], [11], [20]). For the proof in the euclidean case see for instance [1], [10], [23].

In Section 2 we present basic information on Carnot–Carathéodory spaces. Section 3 contains the proof of the metric version of Gehring’s Lemma. Section 4 contains a precise statement of the assumptions on the operator and the proof of Theorem 1.2. As an application of the theorem we have the following compactness theorem in Section 5:

Theorem 1.3. Let F be a compact subset of Ω and δ be a constant defined by Theorem 1.2. Let {u_{i}}_{i∈N} be a family of very weak solutions of (1.1) such that u_{i} ∈ W_{X}^{1,r}(Ω) for some p − δ < r < p. If the family is bounded in W_{X}^{1,r}(Ω), then it is compact in W_{X}^{1,p}(F).

2. Carnot–Carathéodory spaces

Let X_{1}, …, X_{k} be a family of vector fields in R^{n} with real, C^{∞} coefficients. The family satisfies the Hörmander condition if there exists an integer m such that a family of commutators of the vector fields up to the length m, i.e. the family of vector fields

X_{1}, …, X_{k}, [X_{i_{1}}, X_{i_{2}}], …, [X_{i_{1}}, [X_{i_{2}}, […, X_{i_{m}}]] …], i_{j} = 1, 2, …, k,

spans the tangent space T_{x}R^{n} at every point x ∈ R^{n}.

For u ∈ Lip(R^{n}) we define X_{j}u by

X_{j}u(x) = ⟨X_{j}(x), ∇u(x)⟩, j = 1, 2, …, k,

and set Xu = (X_{1}u, …, X_{k}u). Its length is given by

|Xu(x)| = (∑_{j=1}^{k} |X_{j}u(x)|^{2})^{1/2},

where X_{j}^{∗} is a formal adjoint to X_{j} in L^{2} , i.e.

∫_{R^{n}}(X_{j}^{∗}u)v dx = − ∫_{R^{n}} uX_{j}v dx for functions u, v ∈ C_{0}^{∞}(R^{n}).

Given R^{n} with the family of vector fields, we define a distance function ϱ. We say that an absolutely continuous curve γ: [a, b] → R^{n} is admissible, if there exist functions c_{j}: [a, b] → R, j = 1, …, k, such that

γ̇(t) = ∑_{j=1}^{k} c_{j}(t)X_{j}(γ(t)) and ∑_{j=1}^{k} c_{j}(t)^{2} ≤ 1.

Functions c_{j} do not need to be unique, because vector fields X_{j} do not need to be linearly independent. The distance ϱ(x, y) between points x and y is defined as the infimum of those T > 0 for which there exists an admissible curve γ: [0, T] → R^{n} such that γ(0) = x and γ(T) = y. If such a curve does not exist, we set ϱ(x, y) = ∞. The function ϱ is called the Carnot–Carathéodory distance. In general it does not need to be a metric. When the family X_{1}, …, X_{k} satisfies the Hörmander condition, then ϱ is a metric and we say that (R^{n}, ϱ) is a Carnot– Carathéodory space. For more information about such spaces and their geometry see for instance [26], [22], [12].

Here and subsequently all the distances will be with respect to the metric ϱ. In particular all the balls B are balls with respect to the C.-C. metric. If σ > 0 and B = B(x, r) then σB will denote a ball centered in x of radius σ · r. By diam Ω we will denote the diameter of the set Ω.

The metric ϱ is locally Hölder continuous with respect to the euclidean metric. Thus the space (R^{n}, ϱ) is homeomorphic with the euclidean space R^{n}, and every set which is bounded in euclidean metric is also bounded in the metric ϱ. The reverse implication is not true. However, if X_{1}, …, X_{k} have globally Lipschitz coefficients, then Garofalo and Nhieu [8] have shown that every bounded set with respect to ϱ is also bounded in euclidean metric.

We will consider the Lebesgue measure in the Carnot–Carathéodory space. As we change the metric, the measure of B(x, r) is no longer equal to the familiar ω_{n}r^{n}. However, the important fact is that the Lebesgue measure in the Carnot–Carathéodory space satisfies the so-called doubling condition (although only locally—see [22]):

Theorem 2.1. Let Ω be an open, bounded subset of R^{n}. There exists a constant C_{d} ≥ 1 such that

(2.3)  |B(x_{0}, 2r)| ≤ C_{d}|B(x_{0}, r)|

provided x_{0} ∈ Ω and r < 5 diam Ω.

The best constant C_{d} is known as the doubling constant and we call a measure satisfying the above condition a doubling measure. Iterating (2.3) we obtain a lower bound on μ(B(x, r)).

Lemma 2.2. Let μ be a Borel measure in a metric space Y , finite on bounded sets. Assume that μ satisfies the doubling condition on an open, bounded set Ω ⊂ Y . Then for every ball B = B(x, r) such that x ∈ Ω and r < diam Ω the following inequality holds:

μ(B) ≥ \frac{μ(Ω)r^{s}}{(2 diam Ω)^{s}}

where s = log_{2} C_{d}.

We say that Q is of homogeneous dimension relative to Ω, if there exists a constant C > 0 such that for every ball B_{0} with a center in Ω and with a radius r_{0} < diam Ω we have

\frac{μ(B)}{μ(B_{0})} ≥ C(\frac{r}{r_{0}} )^{Q}

where B = B(x, r) is any ball such that x ∈ B_{0} and r ≤ r_{0} . If Ω ⊂ R^{n} is open and bounded and a family of vector fields on Ω satisfies the Hörmander condition, then the Carnot–Carathéodory space (Ω, ϱ) with a Lebesgue measure has the homogeneous dimension Q = s = log_{2} C_{d}.

Given a first-order differential operator X = (X_{1}, …, X_{k}), we define the Sobolev space W_{X}^{1,p} in the following way:

W_{X}^{1,p}(Ω) = {u ∈ L^{p}(Ω) : X_{j}u ∈ L^{p}(Ω), j = 1, 2, …, k},

where X_{j}u is distributional derivative. The W_{X}^{1,p} norm is defined by

∥u∥_{1,p} = ∥u∥_{p} + ∥Xu∥_{p}.

Smooth functions are dense in W_{X}^{1,p}(Ω) ([6], [7]). The existence of smooth cut-off functions in C.-C. spaces was shown in [4] and [8]. We have Sobolev and Poincaré type inequalities ([8], [12], [16]):

Theorem 2.3. Let Q be a homogeneous dimension relative to Ω. There exist constants C_{1}, C_{2} > 0, such that for every metric ball B = B(x, r), where x ∈ Ω and r ≤ diam Ω, the following inequalities hold:

(\overline{∫}_{B} |u − u_{B}|^{s^{∗}} dx)^{1/s^{∗}} ≤ C_{1}r(\overline{∫}_{B} |Xu|^{s} dx)^{1/s} for 1 ≤ s < Q,

where s^{∗} = Qs/(Q − s) and

\overline{∫}_{B} |u − u_{B}|^{s} dx ≤ C_{2}r^{s}\overline{∫}_{B} |Xu|^{s} dx for 1 ≤ s < ∞.

We will consider the following maximal functions:

M_{Ω}f(x) := sup_{r>0} \frac{1}{|B(x, r)|} ∫_{Ω∩B(x,r)} |f| dy

and

M_{Ω,R}f(x) := sup_{R≥r>0} \frac{1}{|B(x, r)|} ∫_{Ω∩B(x,r)} |f| dy.

Our setting requires the use of the theory of maximal functions and Muckenhoupt weights in metric spaces equipped with a doubling measure. We refer to [12], [23] and [25] for more details.

Theorem 2.4 (Hardy–Littlewood). Assume Y is a metric space and μ is a doubling measure on an open set Ω ⊂ Y . Let u ∈ L_{loc}^{1}(Ω). Then

|\{x ∈ Ω : M_{Ω}u(x) > t\}| ≤ \frac{C}{t} ∫_{Ω} |u| dμ

for t > 0, where the constant C depends only on the doubling constant (C_{d}) and

∥M_{Ω}u∥_{L^{p}(Ω,μ)} ≤ C∥u∥_{L^{p}(Ω,μ)}

for 1 < p ≤ ∞, where C = C(C_{d}, p).

We will use the above theorem on a bounded and open set Ω in Carnot– Carathéodory space and also on balls σB, such that B ⊂ Ω and σ > 1. Such balls are contained in Ω′ = {x : ϱ(x, ∂Ω) < σ diam Ω} which is open and bounded. Therefore the doubling constant may change and so may the constants in the Hardy–Littlewood Theorem, but this does not affect the final result.

Theorem 2.5. Assume Ω is an open and bounded subset of R^{n} with Carnot– Carathéodory metric and u ∈ L_{loc}^{1}(Ω), s ≥ 1. Then for almost all x, y ∈ Ω we have

|u(x) − u(y)| ≤ Cϱ(x, y)[(M_{Ω,2ϱ}|Xu|^{s}(x))^{1/s} + (M_{Ω,2ϱ}|Xu|^{s}(y))^{1/s}],

and for any metric ball B ⊂ Ω with radius r and for almost every x ∈ B we have

|u(x) − u_{B}| ≤ Cr(M_{Ω}|Xu|^{s}(x))^{1/s}.

We say that a nonnegative, locally integrable function w belongs to the space A_{p} for p > 1, if

sup_{B⊂R^{n}}(\overline{∫}_{B} w dx)(\overline{∫}_{B} w^{1/(1−p)} dx)^{p−1} < ∞.

A function w belongs to the space A_{1} if there exists a constant c ≥ 1 such that for every ball B ⊂ R^{n}

\overline{∫}_{B} w dx ≤ c ess inf_{B} w.

Functions in A_{p} are called Muckenhoupt weights.

Theorem 2.6 (Muckenhoupt Theorem). Assume v ∈ L_{loc}^{1}(R^{n}) is nonnegative and 1 < p < ∞. Then v ∈ A_{p} if and only if there exists a constant C > 0 such that

∫_{R^{n}} |Mf|^{p}v dx ≤ C ∫_{R^{n}} |f|^{p}v dx for all f ∈ L^{p}(R^{n}, v);

i.e., M is a bounded operator from L^{p}(R^{n}, v) into L^{p}(R^{n}, v).

A metric version of this theorem, with some additional assumptions (in fact— unnecessary and easy to remove^{(1)} ) can be found in [25].

(1) The author would like to thank J. Kinnunen for pointing this out.

3. Gehring’s Lemma for metric spaces

In this section we assume (Y, ϱ, μ) to be an arbitrary metric space with a doubling (Borel regular) measure μ, i.e. there exists a constant C_{d}, such that

μ(B(x, 2r)) ≤ C_{d}μ(B(x, r)).

The doubling condition implies the inequality

(3.4)  \frac{μ(B)}{μ(B_{0})} ≥ \frac{1}{4^{Q}} (\frac{r}{r_{0}} )^{Q},

where Q = log_{2} C_{d} and B_{0} has the radius r_{0} , and B = B(x, r) is any ball such that x ∈ B_{0} and r ≤ r_{0} .

Fix σ > 1. Given a ball B_{0} = B_{0}(x_{0}, R) ⊂ Y define a decomposition of a ball σB_{0} into sets C^{k}, k = 0, 1, 2, …, defined by

\begin{aligned}C^{0} & = B_{0}, \\ C^{k} & = \{x ∈ σB_{0} : \frac{(σ − 1)R}{2^{k−1}} ≥ dist (x, ∂(σB_{0})) > \frac{(σ − 1)R}{2^{k}}\} for k ≥ 1. \end{aligned}

The following lemma is a version of the Calderon–Zygmund decomposition for metric spaces:

Lemma 3.1. Assume a function u ∈ L^{1}(σB_{0}, μ) is nonnegative. Let α be such that

\overline{∫}_{σB_{0}} u(x) dμ < α.

Then, for every k = 0, 1, 2, …, there exists a countable family of pairwise disjoint balls ℱ^{k} = {B_{j}^{k}} centered in C^{k} such that

(3.5)  u(x) ≤ α2^{kQ} for almost all x ∈ C^{k}\ ⋃_{j} 5B_{j}^{k}

and

(3.6)  α2^{kQ} <\overline{∫}_{5B_{j}^{k}} u(x) dμ ≤ α2^{kQ}K,

where the constant

K = max\{C_{d}, 8^{Q}(\frac{σ^{2}}{σ − 1})^{Q}\}.

Proof. Define 𝒢_{0}^{k} := ∅ and S_{0}^{k} := C^{k}. Define a family of balls

ℬ_{1}^{k} = \{B(x, r)x ∈ S_{0}^{k}; r = \frac{(σ − 1)R}{5 · 2^{k+1}σ}\}.

Let ℬ̃_{1}^{k} be a subfamily of ℬ_{1}^{k} defined by

ℬ̃_{1}^{k} = \{B ∈ ℬ_{1}^{k} : α2^{kβ} <\overline{∫}_{5B} u(x) dμ\}.

The Vitali covering lemma implies that we can choose from ℬ̃_{1}^{k} a countable subfamily ℱ_{1}^{k} of pairwise disjoint balls such that

⋃_{B∈ℱ_{1}^{k}} 5B ⊃ ⋃_{B∈_{ℬ_{1}^{k}}} B.

Then we put

𝒢_{1}^{k} = ℱ_{1}^{k}, S_{1}^{k} = C^{k}\ ⋃_{B∈𝒢_{1}^{k}} 5B.

Iteration of this procedure gives in the ith step

\begin{aligned}ℬ_{i}^{k} & = \{B(x, r) : x ∈ S_{i−1}^{k}; r = \frac{(σ − 1)R}{5 · 2^{k+i}σ}\}, \\ ℬ̃_{i}^{k} & = \{B ∈ ℬ_{i}^{k} : α2^{kQ} <\overline{∫}_{5B} u(x) dμ\}, \end{aligned}

ℱ_{i}^{k} being a countable subfamily of pairwise disjoint balls such that

⋃_{B∈ℱ_{i}^{k}} 5B ⊃ ⋃_{B∈_{ℬ_{i}^{k}}} B.

We also have

𝒢_{i}^{k} = 𝒢_{i−1}^{k} ∪ ℱ_{i}^{k}, and S_{i}^{k} = C^{k}\ ⋃_{B∈𝒢_{i}^{k}} 5B.

Define ℱ^{k} = {B_{j}^{k}} = ⋃_{i} 𝒢_{i}^{k} = ⋃_{i} ℱ_{i}^{k}. For every ball belonging to that family we have

α2^{kQ} <\overline{∫}_{5B_{j}^{k}} u(x) dμ,

which gives us the lower estimation of (3.6). We proceed to show the upper estimation of (3.6).

Very weak solutions of nonlinear subelliptic equations 415 Assume B ∈ ℱ_{1}^{k} has radius r. Thus we have

r = \frac{(σ − 1)R}{5 · 2^{k+1}σ}

and applying (3.4) we obtain

\overline{∫}_{5B} u(x) dμ ≤ \frac{μ(B_{0})}{μ(5B)}\overline{∫}_{B_{0}} u(x) dμ < α2^{kQ}8^{Q}(\frac{σ^{2}}{σ − 1})^{Q}.

If B = B(x, r) ∈ ℱ_{i}^{k} for i > 1, then

x ∈ S_{i−1}^{k} ⊂ S_{i−2}^{k} and r = \frac{(σ − 1)R}{5 · 2^{k+i}σ}.

For the ball 2B = B(x, 2R) we have x ∈ S_{i−2}^{k} and

2r = \frac{(σ − 1)R}{5 · 2^{k+i−1}σ};

thus 2B ∈ ℬ_{i−1}^{k}. By construction, 2B does not belong to ℬ̃_{i−1}^{k} , because

x ∈ S_{i−1}^{k} = C^{k}\ ⋃_{B∈𝒢_{i−1}^{k}} 5B ⊂ C^{k}\ ⋃_{B∈ℱ_{i−1}^{k}} 5B ⊂ C^{k}\ ⋃_{B∈_{ℬ_{i−1}^{k}}} B.

Therefore

\overline{∫}_{5·2B} u(x) dμ ≤ α2^{kQ}.

The doubling condition leads to

\overline{∫}_{5B} u(x) dμ < α2^{kQ}C_{d}. To obtain (3.5) assume x ∈ C^{k}\ ⋃_{B∈ℱ^{k}} 5B and let {B_{i}} be a sequence of

balls centered in x with

r_{i} = \frac{(σ − 1)R}{5 · 2^{k+i}σ}.

For every i = 1, 2, …, we have x ∈ S_{i}^{k}. Therefore B_{i} does not belong to ℬ̃_{i}^{k} . Thus

\overline{∫}_{5B_{i}} u(x) dμ ≤ α2^{kQ} for i = 1, 2, ….

The Lebesgue Theorem implies

u(x) ≤ α2^{kQ}

for almost all x ∈ C^{k}\ ⋃_{j} 5B_{j}^{k}. The proof is complete.

The lemma below is standard (see e.g. [9]).

Lemma 3.2. Fix a ball B ⊂ Y . Assume that functions F, G are nonnegative and belong to the space L^{q}(B, μ) for some q > 1. If there exists a constant a > 1 such that for every t ≥ 1 we have

∫_{E(G,t)} G^{q} dμ ≤ a[t^{q−1} ∫_{E(G,t)} G dμ + ∫_{E(F,t)} F^{q} dμ],

where E(G, t) = {x ∈ B : G(x) > t} and E(F, t) = {x ∈ B : F (x) > t}. Then the following inequality holds:

∫_{B} G^{p} dμ ≤ μ_{p}(∫_{B} G^{q} dμ + a ∫_{B} F^{p} dμ)

for p ∈ [q, q + ε), where ε = (q − 1)/(a − 1) and μ_{p} = (p − 1)/(p − 1 − a(p − q)).

The following theorem is a version of the Gehring Lemma for metric spaces with a doubling measure (see e.g. [9], [10]).

Theorem 3.3. Let q ∈ [q_{0}, 2Q], where q_{0} > 1 is fixed. Assume the functions f, g to be nonnegative and such that g ∈ L_{loc}^{q}(Y, μ), f ∈ L_{loc}^{r_{0}}(Y, μ) for some r_{0} > q. Assume that there exist constants b > 1 and θ such that for every ball B ⊂ σB ⊂ Y the following inequality holds

\overline{∫}_{B} g^{q} dμ ≤ b[(\overline{∫}_{σB} g dμ)^{q} +\overline{∫}_{σB} f^{q} dμ] + θ\overline{∫}_{σB} g^{q} dμ.

Then there exist nonnegative constants θ_{0} and ε_{0} , θ_{0} = θ_{0}(q_{0}, Q, C_{d}, σ) and ε_{0} = ε_{0}(b, q_{0}, Q, C_{d}, σ) such that if 0 < θ < θ_{0} then g ∈ L_{loc}^{p}(Y, μ) for p ∈ [q, q+ε_{0}) and moreover

(\overline{∫}_{B} g^{p} dμ)^{1/p} ≤ C[(\overline{∫}_{σB} g^{q} dμ)^{1/q} + (\overline{∫}_{σB} f^{p} dμ)^{1/p}]

for C = C(b, q_{0}, Q, C_{d}, σ).

Remark. For the definitions of the constants θ_{0} , ε_{0} see (3.19), (3.20) and (3.22).

Proof. Fix a ball B ⊂ σB ⊂ Ω. Let u be given by

u(x) = \frac{g^{q}(x)}{\overline{∫}_{σB} g^{q} dx}.

Take s > t ≥ 1 (their precise values shall be fixed later). Let α = s^{q} > 1. By Lemma 3.1 we obtain a decomposition of σB into sets C^{k}, k = 1, 2, …, and for every k a family of pairwise disjoint balls {B_{j}^{k}}_{j=1,2...} ⊂ C^{k} such that

u(x) ≤ α2^{kQ} for a.e. x ∈ C^{k} \ ⋃_{j} 5B_{j}^{k}

and

α2^{kQ} <\overline{∫}_{5B_{j}^{k}} u(x) dx ≤ α2^{kQ}K,

where

K = max\{C_{d}, 8^{Q}(\frac{σ^{2}}{σ − 1})^{Q}\}.

Assume x ∈ C^{k}. Define functions

(3.7)  F (x) := \frac{f(x)}{2^{kQ/q}(\overline{∫}_{σB} g^{q} dμ)^{1/q}}

and

(3.8)  G(x) := \frac{g(x)}{2^{kQ/q}(\overline{∫}_{σB} g^{q} dμ)^{1/q}} .

By the assumptions of the theorem

\overline{∫}_{5B_{j}^{k}} g^{q} dμ ≤ b[(\overline{∫}_{σ5B_{j}^{k}} g dμ)^{q} +\overline{∫}_{σ5B_{j}^{k}} f^{q} dμ] + θ\overline{∫}_{σ5B_{j}^{k}} g^{q} dμ.

Consider a ball 5σB_{j}^{k}. It is centered in C^{k} with radius r ≤ (σ − 1)R/2^{k+1}σ; hence 5σB_{j}^{k} ⊂ ⋃_{i=(k−1)^{+}}^{k+1} C^{i}. Therefore for any x ∈ 5σB_{j}^{k} we have

f(x) ≤ F (x)(2^{(k+1)q}\overline{∫}_{σB} g^{q} dμ)^{1/q}

and

g(x) ≤ G(x)(2^{(k+1)q}\overline{∫}_{σB} g^{q} dμ)^{1/q}.

It follows that

(3.9) \overline{∫}_{5B_{j}^{k}} G^{q} dμ ≤ b · 2^{q}[(\overline{∫}_{σ5B_{j}^{k}} G dμ)^{q} +\overline{∫}_{σ5B_{j}^{k}} F^{q} dμ] + θ · 2^{q}\overline{∫}_{σ5B_{j}^{k}} G^{q} dμ.

By the definition of G, for every ball B_{j}^{k} we have

(3.10)  s^{q} <\overline{∫}_{5B_{j}^{k}} G^{q} dμ ≤ s^{q}K.

Let us now set

(3.11)  s := 2^{Q/q} b^{1/(q−1)} \frac{2q}{q − 1}t > t.

Combining (3.9) with (3.10) and applying (3.11) we obtain, after simple computations,

(3.12) \begin{aligned}\frac{2q}{q − 1}tμ(σ5B_{j}^{k}) & ≤ ∫_{σ5B_{j}^{k}} G dμ + (μ(σ5B_{j}^{k}))^{1−(1/q)}(∫_{σ5B_{j}^{k}} F^{q} dμ)^{1/q} \\ & + (\frac{θ}{b} )^{1/q}(μ(σ5B_{j}^{k}))^{1−(1/q)}(∫_{σ5B_{j}^{k}} G^{q} dμ)^{1/q}. \end{aligned}

Let E(G, s) = {x ∈ σB : G(x) > s}. Since

G(x) = (\frac{u(x)}{2^{kQ}})^{1/q}

for almost all x ∈ B \ ⋃_{j} 5B_{j}^{k}, we have G ≤ s. Thus

μ(E(G, s)) = μ(E(G, s) ∩ (⋃_{j,k} 5B_{j}^{k}))

and

∫_{E(G,s)} G^{q} dμ = ∫_{E(G,s)∩{∪_{j,k}5B_{j}^{k}}} G^{q} dμ ≤ ∑_{j,k} ∫_{5B_{j}^{k}} G^{q} dμ.

Combining this with (3.10) and (3.11) and applying the doubling condition we obtain

(3.13) ∫_{E(G,s)} G^{q} dμ ≤ s^{q}K ∑_{j,k} μ(5B_{j}^{k}) ≤ b 2^{Q} K C_{d}^{3} t^{q} (\frac{2q}{q − 1})^{q} ∑_{j,k} μ(B_{j}^{k}).

By the definitions of E(F, t) and E(G, t) we have

∫_{σ5B_{j}^{k}} F dμ ≤ ∫_{σ5B_{j}^{k}∩E(F,t)} F dμ + tμ(σ5B_{j}^{k})

Very weak solutions of nonlinear subelliptic equations 419

and

∫_{σ5B_{j}^{k}} G dμ ≤ ∫_{σ5B_{j}^{k}∩E(G,t)} G dμ + tμ(σ5B_{j}^{k}).

Applying Young’s inequality we obtain

(t^{q−1}μ(B′)^{1−(1/q)})(∫_{B′}F^{q} dμ)^{1/q}≤ \frac{q − 1}{q}(t^{q−1}μ(B′)^{(q−1)/q})^{q/(q−1)}+ \frac{1}{q} ∫_{B′}F^{q} dμ

≤ \frac{q − 1}{q}t^{q}μ(B′) + \frac{1}{q} ∫_{B′} F^{q} dμ

≤ ∫_{B′∩E(F,t)} F^{q} dμ + t^{q}μ(B′).

Hence

(μ(σ5B_{j}^{k}))^{1−(1/q)}(∫_{σ5B_{j}^{k}} F^{q} dμ)^{1/q} ≤ t^{1−q} ∫_{σ5B_{j}^{k}∩E(F,t)} F^{q} dμ + tμ(σ5B_{j}^{k}). In the same manner we check that

(\frac{θ}{b} )^{1/q}(μ(σ5B_{j}^{k}))^{1−(1/q)}(∫_{σ5B_{j}^{k}}G^{q} dμ)^{1/q}≤ \frac{θt^{1−q}}{b} ∫_{σ5B_{j}^{k}∩E(G,t)}G^{q} dμ + tμ(σ5B_{j}^{k}).

Substituting the last two inequalities into (3.12) yields

\begin{aligned}\frac{2q}{q − 1}μ(σ5B_{j}^{k}) & ≤ t^{−1} ∫_{σ5B_{j}^{k}∩E(G,t)} G dμ + t^{−q} ∫_{σ5B_{j}^{k}∩E(F,t)} F^{q} dμ \\ & + \frac{θ}{b} t^{−q} ∫_{σ5B_{j}^{k}∩E(G,t)} G^{q} dμ + 2μ(σ5B_{j}^{k}), \end{aligned}

and therefore

(3.14)  \begin{aligned}μ(σ5B_{j}^{k}) & ≤ \frac{q − 1}{2t^{q}}[t^{q−1} ∫_{σ5B_{j}^{k}∩E(G,t)} G dμ \\ & + ∫_{σ5B_{j}^{k}∩E(F,t)} F^{q} dμ + \frac{θ}{b} ∫_{σ5B_{j}^{k}∩E(G,t)} G^{q} dμ]. \end{aligned}

Let D^{k} = ⋃_{j} σ5B_{j}^{k}. There exists a countable subfamily of pairwise disjoint balls (σ5B_{j(h)}^{k})_{j(1),j(2),...} such that D^{k} ⊂ ⋃_{h} σ25B_{j(h)}^{k}. Hence

μ(D^{k}) ≤ C_{d}^{3} ∑_{h} μ(σ5B_{j(h)}^{k}).

From (3.14) it follows that

\begin{aligned}μ(D^{k}) & ≤ \frac{C_{d}^{3}(q − 1)}{2t^{q}} ∑_{h}[t^{q−1} ∫_{σ5B_{j(h)}^{k}∩E(G,t)} G dμ \\ & + ∫_{σ5B_{j(h)}^{k}∩E(F,t)} F^{q} dμ + \frac{θ}{b} ∫_{σ5B_{j(h)}^{k}∩E(G,t)} G^{q} dμ]. \end{aligned}

The balls σ5B_{j(h)}^{k} are pairwise disjoint and contained in ⋃_{i=(k−1)^{+}}^{k+1} C^{i}. Thus

\begin{aligned}μ(D^{k}) & ≤ \frac{C_{d}^{3}(q − 1)}{2t^{q}} ∑_{i=(k−1)^{+}}^{k+1}[t^{q−1} ∫_{C^{i}∩E(G,t)} G dμ \\ & + ∫_{C^{i}∩E(F,t)} F^{q} dμ + \frac{θ}{b} ∫_{C^{i}∩E(G,t)} G^{q} dμ]. \end{aligned}

By summing over k = 1, 2, … we obtain (note that each C^{k} can appear at most 3 times)

\begin{aligned}∑_{k} μ(D^{k}) & ≤ \frac{3 · C_{d}^{3}(q − 1)}{2t^{q}} ∑_{k}[t^{q−1} ∫_{C^{k}∩E(G,t)} G dμ \\ & + ∫_{C^{k}∩E(F,t)} F^{q} dμ + \frac{θ}{b} ∫_{C^{k}∩E(G,t)} G^{q} dμ]. \end{aligned}

Therefore we have (3.15)

∑_{k} μ(D^{k}) ≤ \frac{3 · C_{d}^{3}(q − 1)}{2t^{q}}[t^{q−1} ∫_{E(G,t)} G dμ + ∫_{E(F,t)} F^{q} dμ + \frac{θ}{b} ∫_{E(G,t)} G^{q} dμ].

By the definition of D_{k} we also have

(3.16)  ∑_{j,k} μ(B_{j}^{k}) ≤ ∑_{k} μ(D^{k}).

Combining (3.13) with (3.15) and (3.16) we obtain

(3.17)  \begin{aligned}∫_{E(G,s)} G^{q} dμ & ≤ \frac{3 K 2^{Q} C_{d}^{6} (2q)^{q}}{2(q − 1)^{q−1}} [t^{q−1}b ∫_{E(G,t)} G dμ \\ & + b ∫_{E(F,t)} F^{q} dμ + θ ∫_{E(G,t)} G^{q} dμ]. \end{aligned}

We also have

(3.18)  \begin{aligned}∫_{E(G,t)\E(G,s)} G^{q} dμ & ≤ s^{q−1} ∫_{E(G,t)} G dμ \\ & ≤ 2^{Q(q−1)/q}(\frac{2q}{q − 1})^{q−1}t^{q−1}b ∫_{E(G,t)} G dμ. \end{aligned}

Adding both sides of (3.17) and (3.18) we conclude that

\begin{aligned}∫_{E(G,t)} G^{q} dμ & ≤ (a_{1} + a_{2}) · t^{q−1}b ∫_{E(G,t)} G dμ \\ & + a_{1}[b ∫_{E(F,t)} F^{q} dμ + θ ∫_{E(G,t)} G^{q} dμ], \end{aligned}

where the constants

a_{1} = \frac{3 · 2^{Q} K C_{d}^{6} (2q)^{q}}{2(q − 1)^{q−1}} , a_{2} = \frac{2^{Q(1−(1/q))} (2q)^{q}}{2q(q − 1)^{q−1}} .

Assume q ∈ [q_{0}, 2Q]. Then a_{1} , a_{2} < a_{0} , where

(3.19)  a_{0} = 2 K C_{d}^{6} 32^{Q} Q^{2Q} for K = max\{C_{d}; 8^{Q}(\frac{σ^{2}}{σ − 1})^{Q}\}.

Define

(3.20)  θ_{0} := \frac{1}{2a_{0}} .

Then for θ < θ_{0} we have a_{1}θ < \frac{1}{2} and therefore

∫_{E(G,t)} G^{q} dμ ≤ 4a_{0}b[t^{q−1} ∫_{E(G,t)} G dμ + ∫_{E(F,t)} F^{q} dμ]. Since t ≥ 1 was arbitrary and the constants a_{1} , a_{2} do not depend on t, by

Lemma 3.2 we obtain

(3.21)  ∫_{B} G^{p} dμ ≤ μ_{p}(∫_{B} G^{q} dμ + 4a_{0}b ∫_{B} F^{p} dμ),

where

μ_{p} = \frac{p − 1}{p − 1 − 4a_{0}b(p − q)}

and p ∈ [q, q + ε_{0}) for

(3.22)  ε_{0} = \frac{q_{0} − 1}{4a_{0}b}.

By inequality (3.21) and definitions of F and G we get

\begin{aligned}∑_{k} \frac{∫_{C^{k}} g^{p} dμ}{2^{kQp/q}(\overline{∫}_{σB} g^{q} dμ)^{p/q}} & ≤ μ_{p} ∑_{k} \frac{∫_{C^{k}} g^{q} dμ}{2^{kQ}(\overline{∫}_{σB} g^{q} dμ)} \\ & + 4μ_{p}a_{0}b ∑_{k} \frac{∫_{C^{k}} f^{p} dμ}{2^{kQp/q}(\overline{∫}_{σB} g^{q} dμ)^{p/q}} . \end{aligned}

Since C^{0} = B we obtain after simple computations

∫_{B} g^{p} dμ ≤ μ_{p}2^{Q}(\overline{∫}_{σB} g^{q} dμ)^{p/q} + 4μ_{p}a_{0}b2^{Q}\overline{∫}_{σB} f^{p} dμ.

Taking C = (4μ_{p}a_{0}b2^{Q})^{1/p} completes the proof.

4. Proof of the main theorem

Throughout this section we assume that Ω ⊂ R^{n} is open and bounded and that vector fields X_{1}, …, X_{k}, defined on a neighborhood of Ω, have smooth (C^{∞}), globally Lipschitz coefficients and satisfy the Hörmander condition.

The functions A = (A_{1}, …, A_{k}): R^{n} ×R×R^{k} → R^{k} and B: R^{n} ×R×R^{k} → R are both Carathéodory functions, i.e. they are measurable in x and continuous in v, ξ. Moreover, there exist constants α, β > 0 such that

(4.23)  \begin{aligned}|A(x, v, ξ)| & ≤ α(|v|^{p−1} + |ξ|^{p−1}), \\ |B(x, v, ξ)| & ≤ α(|v|^{p−1} + |ξ|^{p−1}), \\ ⟨A(x, v, ξ)|ξ⟩ & ≥ β|ξ|^{p} \end{aligned}

for some p ≥ 2.

We consider the following equation in Ω:

(1.1)  X^{∗}A(x, u, Xu) + B(x, u, Xu) = 0.

Theorem (1.2). There exists δ > 0, such that if a function u is a very weak solution of (1.1), i.e. u ∈ W_{X,loc}^{1,p−δ}(Ω) and it satisfies the equation

∫_{Ω}⟨A(x, u, Xu)|Xϕ(x)⟩ dx + ∫_{Ω} B(x, u, Xu)ϕ(x) dx = 0

for every function ϕ ∈ C_{0}^{∞}(Ω), then u ∈ W_{X,loc}^{1,p+δ}(Ω), and hence it is a weak solution of (1.1).

Assume the function u ∈ W_{loc}^{1,p−δ}(Ω) is a very weak solution of the equation (1.1). We can assume also that δ < \frac{1}{2} . Let B ⊂ Ω be a ball with a radius r. Define

s := \frac{(p − δ)Q}{Q + 1} < p − δ.

Let ϕ be a smooth cut-off function, i.e. ϕ ∈ C_{0}^{∞}(2B) such that 0 ≤ ϕ ≤ 1, ϕ = 1 on B and |Xϕ| ≤ c/r. Define

ũ = (u − u_{2B})ϕ

and

E_{λ} = {(M_{Ω}|Xũ|^{s})^{1/s} ≤ λ} for λ > 0.

Then the function ũ is a Lipschitz function on E_{λ} with the Lipschitz constant cλ (see Theorem 2.5). By the Kirszbraun theorem we can prolong ũ to the Lipschitz function v_{λ} defined on the whole R^{n} with the same Lipschitz constant (see e.g. [5]). Moreover, there exists λ_{0} such that for every λ ≥ λ_{0} the function v_{λ} has a compact support. Indeed, if x ∈ R^{n} \ 3B, then

(M_{Ω}|Xũ(x)|^{s})^{1/s} = sup_{B′∋x, B′∩2B≠∅}(\overline{∫}_{B′} |Xũ|^{s} dx)^{1/s} ≤ (C_{d}\overline{∫}_{2B} |Xũ|^{s} dx)^{1/s}

because |B′| ≥ |B|. Define λ_{0} := (C_{d} \overline{∫}_{2B} |Xũ|^{s} dx)^{1/s}. Then we have

(4.24)  (M_{Ω}|Xũ(x)|^{s})^{1/s} < λ for λ ≥ λ_{0} ,

and that implies v_{λ}(x) = ũ(x) = 0. We will take the function v_{λ} as a test function in equation (1.2).

Lemma 4.1. Let ũ be defined as above. Then the function (M_{Ω}|Xũ|^{s})^{−δ/s} belongs to the space A_{r}, where r = p/s.

Proof. Fix a ball B ⊂ R^{n}. Define w(x) = (M_{Ω}|Xũ(x)|^{s})^{−δ/s}. Then we have

\overline{∫}_{B} w dx ≤ (inf_{B} M_{Ω}|Xũ|^{s})^{−δ/s}

and

(\overline{∫}_{B} w^{1/(1−r)} dx)^{r−1} = (\overline{∫}_{B}(M_{Ω}|Xũ|^{s})^{δ/(p−s)} dx)^{r−1}.

Since δ < p − s it follows that (M_{Ω}|Xũ|^{s})^{δ/(p−s)} ∈ A_{1} . Hence

(\overline{∫}_{B} w^{1/(1−r)} dx)^{r−1} ≤ (c inf_{B}(M_{Ω}|Xũ|^{s})^{δ/(p−s)})^{(p−s)/s} = c(inf_{B} M_{Ω}|Xũ|^{s})^{δ/s}.

It follows immediately that

(\overline{∫}_{B} w dx)(\overline{∫}_{B} w^{1/(1−r)} dx)^{r−1} ≤ C,

and the proof is complete.

Lemma 4.2. Let B ⊂ Ω be a metric ball with radius r, and let 0 < σ ≤ 5. The following inequality holds:

\begin{aligned}∫_{σB} |u|^{p−1}(M_{σB}|Xu|^{s})^{(1−δ)/s} & ≤ c_{1} ∫_{σB} |u|^{p−δ} dx \\ & + c_{2}|σB|(\overline{∫}_{σB} |Xu|^{(p−δ)Q/(Q+1)} dx)^{(Q+1)/Q}, \end{aligned}

where the constants c_{1} = c_{1}(p) and c_{2} = c_{2}(p, r).

Proof. By Hölder’s inequality we have

\begin{aligned}\overline{∫}_{σB} |u|^{p−1}(M_{σB}|Xu|^{s})^{(1−δ)/s} & ≤ (\overline{∫}_{σB} |u|^{(p−1)s_{1}}dx)^{1/s_{1}} \\ & × (\overline{∫}_{σB}(M_{σB}|Xu|^{s})^{(1−δ)s_{2}/s} dx)^{1/s_{2}}, \end{aligned}

where

s_{1} = \frac{(p − δ)Q}{(p − 1)Q − (1 − δ)}, s_{2} = \frac{(p − δ)Q}{(1 − δ)(Q + 1)}.

To the right-hand side of the above inequality we apply first the Hardy–Littlewood Theorem (for the maximal function M_{σB}f; all the balls σB, where B ⊂ Ω, are contained in some open and bounded set). Then by Young inequality with the exponents (p − δ)/(p − 1) and (p − δ)/(1 − δ) we obtain

\overline{∫}_{σB} |u|^{p−1}(M_{σB}|Xu|^{s})^{(1−δ)/s} dx ≤ c(\overline{∫}_{σB} |u|^{(p−1)s_{1}} dx)^{1/s_{1}}

× (\overline{∫}_{σB} |Xu|^{(1−δ)s_{2}} dx)^{1/s_{2}}

≤ c(\overline{∫}_{σB} |u|^{(p−1)s_{1}} dx)^{(p−δ)/(s_{1}(p−1))} (4.25)

+ c(\overline{∫}_{σB} |Xu|^{(p−δ)Q/(Q+1)} dx)^{(Q+1)/Q}.

For the first integral on the right-hand side we have

(\overline{∫}_{σB} |u|^{(p−1)s_{1}} dx)^{1/s_{1}(p−1)} ≤ (\overline{∫}_{σB} |u − u_{σB}|^{(p−1)s_{1}} dx)^{1/s_{1}(p−1)} + |u_{σB}|.

Applying Hölder’s inequality and then Sobolev’s inequality we obtain

c(\overline{∫}_{σB}|u|^{(p−1)s_{1}} dx)^{(p−δ)/s_{1}(p−1)}

≤ cr^{p−δ}2^{p}(\overline{∫}_{σB}|Xu|^{(p−δ)Q/(Q+1)(p−1)} dx)^{(p−1)(Q+1)/Q} + 2^{p}\overline{∫}_{σB}|u|^{p−δ} dx (4.26)

Then (4.25), (4.26) and Hölder’s inequality (as p ≥ 2) imply part (i) of the lemma.

Corollary 4.3. We have from Poincaré’s inequality that

\begin{aligned}∫_{σB} |u|^{p−1}(M_{σB}|Xũ|^{s})^{(1−δ)/s} & ≤ c_{1} ∫_{σB} |u|^{p−δ} dx \\ & + c_{2}|σB|(\overline{∫}_{σB} |Xu|^{(p−δ)Q/(Q+1)} dx)^{(Q+1)/Q}. \end{aligned}

Proof of Theorem 1.2. We first show that |Xu| ∈ L_{loc}^{p+δ̃} for some δ̃ > 0. Let λ ≥ λ_{0} . Take v_{λ} as a test function in (1.2):

∫_{3B} A(x, u, Xu) · Xv_{λ} dx + ∫_{3B} B(x, u, Xu) · v_{λ} dx = 0.

We will show that the assumptions of Theorem 3.3 are satisfied.

By definitions of E_{λ}, v_{λ} and by the growth conditions on A and B we have

∫2B∩E_{λ}A(x, u, Xu) · Xũ dx + ∫_{2B∩E_{λ}} B(x, u, Xu) · ũ dx
≤ ∫_{3B\E_{λ}} |A(x, u, Xu)| · |Xv_{λ}| dx + ∫_{3B\E_{λ}} |B(x, u, Xu)| · |v_{λ}| dx
≤ c ∫_{3B\E_{λ}} λ|Xu|^{p−1} dx + c ∫_{3B\E_{λ}} λ|u|^{p−1} dx.

The last inequality holds because vector fields X_{j} are Lipschitz continuous and there exists a constant c such that |Xv_{λ}| ≤ cλ and |v_{λ}| ≤ crλ, where r is the radius of B.

Multiplying both sides of the last inequality by λ^{−(1+δ)} and integrating over (λ_{0}, +∞) we obtain

(4.27)  \begin{aligned}L & = ∫_{λ_{0}}^{∞} ∫_{2B∩E_{λ}} λ^{−(1+δ)}(A(x, u, Xu) · Xũ + B(x, u, Xu)ũ) dx dλ \\ & ≤ c ∫_{λ_{0}}^{∞} ∫_{3B\E_{λ}} λ^{−δ}(|u|^{p−1} + |Xu|^{p−1}) dx dλ = P. \end{aligned}

Estimation of P. Changing the order of integration and using (4.24) we obtain

\begin{aligned}P & ≤ \frac{c}{1 − δ} ∫_{3B\E_{λ_{0}}} (M_{Ω}|Xũ|^{s})^{1−δ/s}(|u|^{p−1} + |Xu|^{p−1}) dx \\ & ≤ c ∫_{3B}(M_{Ω}|Xũ|^{s})^{1−δ/s}|u|^{p−1} dx + c ∫_{3B}(M_{Ω}|Xũ|^{s})^{1−δ/s}|Xu|^{p−1} dx. \end{aligned}

To estimate the first component of the right-hand side we apply Lemma 4.2. To estimate the second component we apply the Hölder inequality and then the Hardy– Littlewood theorem. It follows that (4.28)

P ≤ c ∫_{3B} |u|^{p−δ} dx + c(∫_{3B} |Xu|^{(p−δ)Q/(Q+1)} dx)^{(Q+1)/Q} + c ∫_{3B} |Xu|^{p−δ} dx

Estimation of L. By changing the order of integration we obtain

\begin{aligned}L & = \frac{1}{δ} ∫_{2B\E_{λ_{0}}} (A(x, u, Xu) · Xũ + B(x, u, Xu)ũ)(M_{Ω}|Xũ|^{s})^{−δ/s} dx \\ & + \frac{1}{δ} ∫_{2B∩E_{λ_{0}}} (A(x, u, Xu) · Xũ + B(x, u, Xu)ũ)λ_{0}^{−δ} dx. \end{aligned}

Since 2B \ E_{λ_{0}} = 2B \ (2B ∩ E_{λ_{0}}), the growth conditions on A and B imply

(4.29)  \begin{aligned}L & ≥ \frac{1}{δ} ∫_{2B}(A(x, u, Xu) · Xũ)(M_{Ω}|Xũ|^{s})^{−δ/s} dx \\ & − \frac{2α}{δ} ∫_{2B∩E_{λ_{0}}} (|u|^{p−1} + |Xu|^{p−1})|Xũ|(M_{Ω}|Xũ|^{s})^{−δ/s} dx \\ & − \frac{3α}{δ} ∫_{2B}(|u|^{p−1} + |Xu|^{p−1})|ũ|(M_{Ω}|Xũ|^{s})^{−δ/s} dx \\ & = \frac{1}{δ}(I_{1} − 2αI_{2} − 3αI_{3}), \end{aligned}

where

\begin{aligned}I_{1} & = ∫_{2B}(A(x, u, Xu) · Xũ)(M_{Ω}|Xũ|^{s})^{−δ/s} dx, \\ I_{2} & = ∫_{2B∩E_{λ_{0}}} (|u|^{p−1} + |Xu|^{p−1})|Xũ|(M_{Ω}|Xũ|^{s})^{−δ/s} dx, \\ I_{3} & = ∫_{2B}(|u|^{p−1} + |Xu|^{p−1})|ũ|(M_{Ω}|Xũ|^{s})^{−δ/s} dx. \end{aligned}

Estimation of I_{1} . Define sets

D_{1} = {x ∈ 2B\B : (M_{Ω}|Xũ|^{s})^{1/s} ≤ δ(M_{2B}|Xu|^{s})^{1/s}}

Very weak solutions of nonlinear subelliptic equations 427

and

D_{2} = {x ∈ 2B\B : (M_{Ω}|Xũ|^{s})^{1/s} > δ(M_{2B}|Xu|^{s})^{1/s}}.

Hence

I_{1} ≥ ∫_{B∪D_{2}} A(x, u, Xu) · Xu(M_{Ω}|Xũ|^{s})^{−δ/s} dx

+ ∫_{D_{2}} A(x, u, Xu)(u − u_{2B})Xϕ(M_{Ω}|Xũ|^{s})^{−δ/s} dx − α ∫_{D_{1}}(|u|^{p−1} + |Xu|^{p−1})|Xũ|(M_{Ω}|Xũ|^{s})^{−δ/s} dx

≥ β ∫_{B} |Xu|^{p}(M_{Ω}|Xũ|^{s})^{−δ/s} dx

− \frac{cα}{r} ∫_{D_{2}}(|u|^{p−1} + |Xu|^{p−1})|u − u_{2B}|(M_{Ω}|Xũ|^{s})^{−δ/s} dx

− α ∫_{D_{1}}(|u|^{p−1} + |Xu|^{p−1})|Xũ|(M_{Ω}|Xũ|^{s})^{−δ/s} dx.

Lemma 4.1 yields

I_{1} ≥ cβ ∫_{B}(M_{B}|Xu|^{s})^{p/s}(M_{Ω}|Xũ|^{s})^{−δ/s} dx

− \frac{cα}{r} ∫_{D_{2}}(|u|^{p−1} + |Xu|^{p−1})|u − u_{2B}|(M_{Ω}|Xũ|^{s})^{−δ/s} dx

− α ∫_{D_{1}}(|u|^{p−1} + |Xu|^{p−1})|Xũ|(M_{Ω}|Xũ|^{s})^{−δ/s} dx =: I_{1,1} − I_{1,2} − I_{1,3}.

We will estimate each integral I_{1,k}, for k = 1, 2, 3. If x ∈ \frac{1}{2}B then we have

(M_{Ω}|Xũ|^{s})^{1/s}(x) ≤ sup_{B′∋x, B′⊂B}(\overline{∫}_{B′} |Xũ|^{s})^{1/s} + sup_{B′∋x, B′∩∂B≠∅}(\overline{∫}_{B′} |Xũ|^{s})^{1/s}

≤ (M_{B}|Xu|^{s})^{1/s} + c(\overline{∫}_{2B} |Xu|^{s} dx)^{1/s}

+ \frac{c}{r} (\overline{∫}_{2B} |u − u_{2B}|^{s} dx)^{1/s}

≤ (M_{B}|Xu|^{s})^{1/s} + c(\overline{∫}_{2B} |Xu|^{s} dx)^{1/s}.

The second inequality comes from the doubling condition and the last one from Poincaré’s inequality. Let G ⊂ \frac{1}{2}B be such that if x ∈ G then

(M_{B}|Xu|^{s})^{1/s} ≥ c(\overline{∫}_{2B} |Xu|^{s} dx)^{1/s}.

Then we have

\begin{aligned}I_{1,1} & ≥ cβ ∫_{G}(M_{B}|Xu|^{s})^{p/s}(M_{B}|Xu|^{s})^{−δ/s} dx \\ & = c ∫_{B/2}(M_{B}|Xu|^{s})^{(p−δ)/s} dx − c(\overline{∫}_{2B} |Xu|^{s} dx)^{(p−δ)/s} ∫_{B/2\G} dx. \end{aligned}

Hence

(4.30)  I_{1,1} ≥ c ∫_{B/2} |Xu|^{p−δ} dx − c|B|(\overline{∫}_{2B} |Xu|^{s} dx)^{(p−δ)/s}.

By the definition of D_{2} , Theorem 2.5 and the properties of maximal function we have

I_{1,2} ≤ \frac{cαδ^{−δ}}{r} ∫_{2B}(|u|^{p−1} + |Xu|^{p−1})|u − u_{2B}|(M_{2B}|Xu|^{s})^{−δ/s} dx

≤ cαδ^{−δ}[∫_{2B} |u|^{p−1}(M_{2B}|Xu|^{s})^{(1−δ)/s} dx

+ \frac{1}{r} ∫_{2B} |u − u_{2B}|(M_{2B}|Xu|^{s})^{(p−1−δ)/s} dx].

The first component of the right-hand side is estimated, by Lemma 4.2,

c ∫_{2B} |u|^{p−δ} dx + c(\overline{∫}_{2B} |Xu|^{(p−δ)Q/(Q+1)} dx)^{(Q+1)/Q}.

To the second component of the right-hand side we apply Hölder’s inequality with exponents

\frac{(p − δ)Q}{Q + 1} and \frac{p − δ}{p − 1 − δ} \frac{Q}{Q + 1}.

Next, by Poincaré’s inequality and the Hardy–Littlewood Theorem, we have

\frac{1}{r} ∫_{2B}|u − u_{2B}|(M_{2B}|Xu|^{s})^{(p−1−δ)/s} dx

≤ \frac{|2B|}{r}(\overline{∫}_{2B} |u − u_{2B}|^{(p−δ)Q/(Q+1)} dx)^{(Q+1)/(p−δ)Q}

× (\overline{∫}_{2B}(M_{2B}|Xu|^{s})^{(p−δ)Q/s(Q+1)} dx)^{(p−1−δ)(Q+1)/(p−δ)Q}

≤ |2B|(\overline{∫}_{2B} |Xu|^{(p−δ)Q/(Q+1)})^{(Q+1)/Q}.

Thus

(4.31) I_{1,2} ≤ cαδ^{−δ}[∫_{2B} |u|^{p−δ} dx + |2B|(\overline{∫}_{2B} |Xu|^{(p−δ)Q/(Q+1)})^{(Q+1)/Q}].

For the integral I_{1,3} we have

I_{1,3} ≤ α ∫_{D_{1}}(|u|^{p−1} + |Xu|^{p−1})(M_{Ω}|Xũ|^{s})^{(1−δ)/s} dx,

and, using the definition of D_{1} ,

I_{1,3} ≤ αδ^{1−δ} ∫_{2B}(|u|^{p−1} + |Xu|^{p−1})(M_{2B}|Xu|^{s})^{(1−δ)/s} dx
≤ αδ^{1−δ} ∫_{2B} |u|^{p−1}(M_{2B}|Xu|^{s})^{(1−δ)/s} dx
+ αδ^{1−δ} ∫_{2B}(M_{2B}|Xu|^{s})^{(p−δ)/s} dx.

To the first component of the right-hand side we apply Lemma 4.2. Because of the coefficient δ · δ^{−δ} it will be consumed in the inequality (4.31). The second component, by the Hardy–Littlewood Theorem, is estimated by

cαδ^{1−δ} ∫_{2B} |Xu|^{p−δ} dx.

Combining (4.30) and (4.31) with the estimation of I_{1,3} , we obtain finally

(4.32)  \begin{aligned}I_{1} & ≥ c ∫_{B/2} |Xu|^{p−δ} dx − c ∫_{2B} |u|^{p−δ} dx − cδ ∫_{2B} |Xu|^{p−δ} dx \\ & − c|2B|(\overline{∫}_{2B} |Xu|^{(p−δ)Q/(Q+1)})^{(Q+1)/Q}. \end{aligned}

Estimation of I_{2} . We have (4.33)

I_{2} ≤ ∫_{2B} |u|^{p−1}(M_{Ω}|Xũ|^{s})^{1−δ/s} dx + ∫_{2B∩E_{λ_{0}}} |Xu|^{p−1}|Xũ|(M_{Ω}|Xũ|^{s})^{−δ/s} dx.

Estimation of the first component follows from Lemma 4.2. We will work with the second one. Fix a constant γ > 0. Assume that y ∈ 2B ∩E_{λ_{0}} . If |Xu(y)| ≥ λ_{0}/γ, we have

\begin{aligned}∫_{2B∩E_{λ_{0}}} |Xu|^{p−1}|Xũ|(M_{Ω}|Xũ|^{s})^{−δ/s} dx & ≤ λ_{0}^{1−δ} ∫_{2B} |Xu|^{p−1} dx \\ & ≤ γ^{1−δ} ∫_{2B} |Xu|^{p−δ} dx. \end{aligned}

If |Xu(y)| ≤ λ_{0}/γ then, since

λ_{0} := c(\overline{∫}_{2B} |Xũ|^{s} dx)^{1/s} and λ_{0} ≤ inf_{2B}(M_{Ω}|Xũ|^{s})^{1/s},

we obtain

∫2B∩E_{λ_{0}} |Xu|^{p−1}|Xũ|(M_{Ω}|Xũ|^{s})^{−δ/s} dx ≤ λ_{0}^{p−1}γ^{1−p}λ_{0}^{−δ} ∫_{2B} |Xũ| dx

≤ cγ^{1−p}|2B|(\overline{∫}_{2B} |Xũ|^{s}dx)^{p−1−δ/s}\overline{∫}_{2B} |Xũ| dx

≤ cγ^{1−p}|2B|(\overline{∫}_{2B} |Xu|^{s} dx)^{p−δ/s},

where the last inequality follows from Poincaré’s inequality. Thus the second component of the right-hand side in (4.33) is estimated by

γ^{1−δ} ∫_{2B} |Xu|^{p−δ} dx + cγ^{1−p}|2B|(\overline{∫}_{2B} |Xu|^{s} dx)^{p−δ/s}.

Since s = (p − δ)Q/(Q + 1), the integral I_{2} satisfies

(4.34) \begin{aligned}I_{2} & ≤ c ∫_{2B} |u|^{p−δ} dx + (c + γ^{1−p})|2B|(\overline{∫}_{2B} |Xu|^{(p−δ)Q/(Q+1)})^{(Q+1)/Q} \\ & + γ^{1−δ} ∫_{2B} |Xu|^{p−δ} dx. \end{aligned}

Estimation of I_{3} . For the integral I_{3} we have

I_{3} ≤ ∫_{2B} |u|^{p−1}|ũ|(M_{Ω}|Xũ|^{s})^{−δ/s} dx + ∫_{2B} |Xu|^{p−1}|ũ|(M_{Ω}|Xũ|^{s})^{−δ/s} dx.

By Corollary 4.3 the first component on the right-hand side gives the parts which were considered earlier in the integral I_{1,2} . We might estimate the second component

(4.35) \begin{aligned}∫_{2B} |Xu|^{p−1}|ũ|(M_{Ω}|Xũ|^{s})^{−δ/s} dx & ≤ ∫_{2B} |Xu|^{p−1}|ũ| |Xũ|^{−δ} dx \\ & ≤ ∫_{2B} |Xu|^{p−1−δ}|u − u_{2B}| dx \\ & + cr^{δ}\overline{∫}_{2B} |Xu|^{p−1}|u − u_{2B}|^{1−δ}dx, \end{aligned}

because |Xũ| ≤ |Xu| + (c/r)|u − u_{2B}|. To the last two integrals we apply Hölder’s inequality choosing for the first integral

s_{1} = \frac{Q(p − δ)}{(Q + 1)(p − 1 − δ)}, s_{2} = \frac{Q(p − δ)}{(Q + 1) − (p − δ)}

and for the second integral

t_{1} = \frac{(p − δ)Q}{(p − 1)(Q + 1)}, t_{2} = \frac{(p − δ)Q}{Q(1 − δ) − (p − 1)}.

Then, by the Sobolev inequality, the right-hand side of (4.35) does not exceed

c|2B|(\overline{∫}_{2B} |Xu|^{Q(p−δ)/(Q+1)} dx)^{(Q+1)/Q}.

Thus the integral I_{3} can be estimated by the same expression as I_{1,2} , i.e.

(4.36)  I_{3} ≤ c ∫_{2B} |u|^{p−δ} dx + c|2B|(\overline{∫}_{2B} |Xu|^{(p−δ)Q/(Q+1)})^{(Q+1)/Q}.

Combining (4.28), (4.29) with (4.32), (4.34) and (4.36) we obtain

\frac{1}{δ}[c ∫_{B/2} |Xu|^{p−δ} dx − c ∫_{2B} |u|^{p−δ} dx − (cδ + γ^{1−δ}) ∫_{2B} |Xu|^{p−δ} dx

− (c + γ^{1−p})|2B|(\overline{∫}_{2B} |Xu|^{(p−δ)Q/(Q+1)})^{(Q+1)/Q}]
≤ c ∫_{3B} |Xu|^{p−δ} dx.

Therefore u satisfies a reverse Hölder inequality:

\overline{∫}_{B/2} |Xu|^{p−δ} dx ≤ cγ^{1−p}[(\overline{∫}_{3B} |Xu|^{(p−δ)Q/(Q+1)})^{(Q+1)/Q} +\overline{∫}_{3B} |u|^{p−δ} dx]

+ c(γ^{1−δ} + δ)\overline{∫}_{3B} |Xu|^{p−δ} dx. (4.37)

We can apply Theorem 3.3 with

g = |Xu|^{(p−δ)Q/(Q+1)}, f = |u|^{(p−δ)Q/(Q+1)},

and

θ = c(γ^{1−δ} + δ), b = cγ^{1−p}, q = \frac{Q + 1}{Q}.

By Sobolev’s inequality the function f is in L^{r_{0}} , where r_{0} = (Q + 1)/(Q − p + δ) > q. The constants γ and δ can be chosen sufficiently small such that θ < θ_{0} . Then we obtain

Lemma 4.4. If the function u is a very weak solution of (1.1) and |Xu| ∈ L^{p−δ}, where δ ∈ (−δ_{0}, δ_{0}) for some δ_{0} > 0, then there exists ε_{0} > 0 such that |Xu| ∈ L^{p−δ+ε_{0}} .

By iteration of the procedure we prove that |Xu| ∈ L^{p+δ̃}, for some δ̃ > 0, say δ̃ = \frac{1}{2}ε_{0} , where ε_{0} is given by Theorem 3.3. It is important to control the constant b and thus the constant ε_{0} . They both depend on α, β, C_{d}, constants in Poincaré’s and Sobolev’s inequalities and in the Hardy–Littlewood Theorem. The last one depends formally on the exponent p−δ, but by the Riesz–Thorin Theorem (Riesz Convexity Theorem, see e.g. [24]), if p ∈ [2, p_{0}], then the Hardy–Littlewood constant may be chosen independently of p.

Now we show that u ∈ L_{loc}^{p+δ̃}. Assume that Ω′ is a compact subset of Ω. Let ϕ ∈ C_{0}^{∞}(Ω) be a cut-off function, such that ϕ = 1 onto Ω′. Let v = u · ϕ. Then v ∈ W^{1,p−δ}(R^{n}) and |Xv| ∈ L^{p+δ̃}(R^{n}). By Poincaré’s inequality

(\overline{∫}_{B} |v − v_{B}|^{p+δ̃} dx)^{1/p+δ̃} ≤ cr(\overline{∫}_{B} |Xv|^{p+δ̃} dx)^{1/p+δ̃}.

Define

E_{λ} = {x : (M_{Ω}|Xv|^{p+δ̃}(x))^{1/p+δ̃} ≤ λ}.

Then

|R^{n} \ E_{λ}| ≤ cλ^{−p} ∫_{R^{n}} |Xv|^{p+δ̃} dx.

For every λ > 0 there exists a Lipschitz function v_{λ} on R^{n} such that v = v_{λ} for a.e. x ∈ E_{λ}. Then

∫_{R^{n}} |Xv_{λ}|^{p+δ̃} dx ≤ ∫_{E_{λ}} |Xv|^{p+δ̃} dx + cλ^{p+δ̃}|R^{n} \ E_{λ}| ≤ ∫_{R^{n}} |Xv|^{p+δ̃} dx,

and this implies that |Xv_{λ}| are uniformly bounded. It follows from the same argument as previously that there exists λ_{0} such that for λ > λ_{0} we have supp v_{λ} = B ⊃ Ω′. Thus the set {v_{λ} : λ > λ_{0}} is bounded in W_{0}^{1,p+δ̃}(B). Since v_{λ} → v for λ → ∞ we have v ∈ L^{p+δ̃}(B) and u ∈ W^{1,p+δ̃}(B).

5. Compactness

In this section we additionally assume that for all v, w ∈ R the following inequalities hold:

|A(x, v, ξ) − A(x, w, ζ)| ≤ α|ξ − ζ|(|ξ| + |ζ|)^{p−2}

and

⟨A(x, v, ξ) − A(x, w, ζ)|ξ − ζ⟩ ≥ β|ξ − ζ|^{2}(|ξ| + |ζ|)^{p−2}.

Theorem 5.3 below is a classical compactness theorem for weak solutions. It follows from Caccioppoli type estimates with the natural exponent p. Together with Theorem 1.2 they imply Theorem 5.4; i.e., the compactness result where it is enough to assume that very weak solutions are bounded in W_{X}^{1,r} for some r < p. In this way, for the nonlinear elliptic case, Theorem 5.4 was proved by Iwaniec and Sbordone [15]. The alternative approach to prove the result is to use the Caccioppoli type inequalities below the natural exponent.

Theorem 5.1. Let {u_{i}}_{i∈N} , u_{i} ∈ W_{X,loc}^{1,p}(Ω), be a family of weak local solutions of the equation (1.1). Then for j, k ∈ N we have

\begin{aligned}∥ϕX(u_{j} − u_{k})∥_{p}^{p} & ≤ c∥(u_{j} − u_{k})Xϕ∥_{p}^{2} · (∥ϕXu_{j}∥_{p}^{p−2} + ∥ϕXu_{k}∥_{p}^{p−2}) \\ & + c sup_{i∈N} ∥u_{i}∥_{W_{X}^{1,p}}^{p−1} · ∥ϕ(u_{j} − u_{k})∥_{p} \end{aligned}

for every function ϕ ∈ C_{0}^{∞}(Ω), where the constant c depends only on p, α, β.

Proof. Assume ϕ ∈ C_{0}^{∞}(Ω) and let η = ϕ^{p}(u_{j} − u_{k}) be a test function in the equation (1.1). By conditions on A and B we obtain

(5.38)  β ∫_{Ω} |ϕ|^{p}|Xu_{j} − Xu_{k}|^{2}(|Xu_{j}| + |Xu_{k}|)^{p−2} dx ≤ I_{1} + I_{2},

where

I_{1} := pα ∫_{Ω} |Xϕ| |ϕ|^{p−1}|u_{j} − u_{k}| |Xu_{j} − Xu_{k}|(|Xu_{j}| + |Xu_{k}|)^{p−2} dx

and

I_{2} := α ∫_{Ω} |ϕ|^{p}|u_{j} − u_{k}|(|u_{j}|^{p−1} + |u_{k}|^{p−1} + |Xu_{j}|^{p−1} + |Xu_{k}|^{p−1}) dx.

Applying to I_{1} Young’s inequality with exponents 2, 2 we can estimate the integral by sum of expressions:

(5.39)  pαθ^{2} ∫_{Ω} |ϕ|^{p}|Xu_{j} − Xu_{k}|^{2}(|Xu_{j}| + |Xu_{k}|)^{p−2} dx

and

(5.40)  \frac{pαc}{θ^{2}}∥(u_{j} − u_{k})Xϕ∥_{p}^{2}(∥ϕXu_{j}∥_{p}^{p−2} + ∥ϕXu_{k}∥_{p}^{p−2}),

where the constant c depends only on p (to obtain the second component we used Hölder’s inequality with exponents \frac{1}{2}p and p/(p − 2)). We put the expression (5.39) on the left-hand side of (5.38). The constant θ > 0 is such that β−pαθ^{2} > 0.

To estimate the integral I_{2} we apply Hölder’s inequality with exponents p, p/(p − 1). We obtain the estimation of I_{2} by

(5.41)  α∥ϕ(u_{j} − u_{k})∥_{p} · sup_{i∈N} ∥ϕu_{i}∥_{W_{loc}^{1,p}}^{p−1}.

Combining (5.38)–(5.41) we obtain an inequality

\begin{aligned}∥ϕX(u_{j} − u_{k})∥_{p}^{p} & ≤ \frac{pαc}{θ^{2}(β − pαθ^{2})}∥(u_{j} − u_{k})Xϕ∥_{p}^{2} · (∥ϕXu_{j}∥_{p}^{p−2} + ∥ϕXu_{k}∥_{p}^{p−2}) \\ & + \frac{α}{β − pαθ^{2}} sup_{i∈N} ∥u_{i}∥_{W_{X}^{1,p}}^{p−1} · ∥ϕ(u_{j} − u_{k})∥_{p}, \end{aligned}

and the proof is complete.

Theorem 5.2. If a function u ∈ W_{X,loc}^{1,p}(Ω) is a weak local solution of the equation (1.1), then it satisfies a Caccioppoli type inequality

∥ϕXu∥_{p} ≤ c∥uXϕ∥_{p} + c∥uϕ∥_{p}

for every function ϕ ∈ C_{0}^{∞}(Ω), where the constant c depends only on constants p, α and β.

Proof. The procedure is similar to the previous one. As a test function we take η = ϕ^{p}u, where ϕ ∈ C_{0}^{∞}. By the conditions on A and B we obtain an inequality

\begin{aligned}β∥ϕXu∥_{p}^{p} & ≤ (pα∥uXϕ∥_{p} + α∥uϕ∥_{p}) · ∥uϕ∥_{p}^{p−1} \\ & + (pα∥uXϕ∥_{p} + α∥uϕ∥_{p}) · ∥ϕXu∥_{p}^{p−1}, \end{aligned}

which implies the theorem.

The above theorems imply

Theorem 5.3. If a family of weak solutions {u_{i}} ⊂ W_{X,loc}^{1,p}(Ω) of the equation (1.1) is bounded in L^{p}(Ω), then it is compact in W_{X}^{1,p}(Ω).

Combining the theorem on the higher integrability of very weak solutions and the above compactness theorem we obtain the following result:

Theorem 5.4. Let F be any compact subset of Ω and let δ be the constant from Theorem 1.2. Then if a family {u_{i}}_{i∈N} of very weak solutions of the equation (1.1), u_{i} ∈ W_{X}^{1,r}(Ω), is bounded in W_{X}^{1,r}(Ω), where p−δ < r < p, then this family compact in W_{X}^{1,p}(F).

Very weak solutions of nonlinear subelliptic equations 435

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Received 14 September 2004