Guide to nonlinear potential estimates
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Abstract
One of the basic achievements in nonlinear potential theory is that the typical linear pointwise estimates via fundamental solutions find a precise analog in the case of nonlinear equations. We give a comprehensive account of this fact and prove new unifying families of potential estimates. We also describe new fine properties of solutions to measure data problems.
1 A synopsis
The aim of this paper is to give a rather comprehensive introduction to nonlinear potential estimates, i.e., pointwise estimates for solutions to quasilinear, possibly degenerate elliptic equations via linear and nonlinear potentials. The paper contains both new and old results. They fall into two categories. The first consists of those results that have been proved elsewhere, and that are here given in different and/or streamlined version. The second contains new results that are presented for the first time. These fill some of the gaps that were making the current theory still somewhat incomplete.
To ease the reading, the already known results will be stated together with the reference to the corresponding original paper where they have appeared for the first time. The new ones will be presented pointing at the place of this paper where the proof can be found. In general, the first part of the paper is devoted to the presentation of the general setting, with the statements of the main theorems; this goes up to Sect. 8. The remaining parts are instead devoted to the proofs.
We shall start from a presentation of the classical results valid for linear elliptic equations in Sect. 2, and this will serve to give the general guideline to the topics we are going to cover in the nonlinear case. The first potential estimates for nonlinear equations will be introduced in Sect. 3. There the classical pointwise estimates will be presented. In the subsequent Sects. 4 and 5 we shall instead give a class of estimates aimed at unifying the theory. These allow to frame the classical pointwise potential inequalities in a more general setting, allowing for estimates of both size and oscillations of solutions and their derivatives, including fractional derivates. At this stage we shall use “intermediate” Wolff and Riesz potentials and various fractional maximal operators. Especially, we prove Theorem 10, whose role is to unify estimates for $u$ and $Du$ via a family of estimates for suitable fractional operators of $u$. We shall eventually add a few remarks on the case of equations with coefficients in Sect. 6. All the results up to Sect. 6 are presented for energy solutions, that is for solutions belonging to the natural energy space ${W}^{1,p}$ associated to the problems considered. In Sect. 7 we then turn to the case of general measure data problems, that is when very weak solutions come into the play. This will give us the opportunity, in Sect. 8, to present a few interesting consequences of potential estimates. Specifically, we shall prove two theorems about the possibility of describing the fine behaviour of solutions to nonlinear measure data problems and of their gradient via potentials. These theorems are the nonlinear analogs of classical results about fine properties of solutions to linear equations, which are usually derived via linear potential; see Remark 3 below. In Sect. 9 we then gather a series of regularity results that follow as a corollary of the theory presented. Some of these results are wellknown and they are now framed in the general and unifying context of nonlinear potential estimates. The remaining sections are devoted to the proofs of the results introduced in the preceding ones, and their titles are selfexplaining. In this paper we are not going to deal with parabolic problems, for which we refer to [42, 43, 44].
2 From linear to nonlinear
Definition 1
Definition 2
This is just another way to say that $v$ has “fractional derivatives”. The advantage is that nonlocality is reduced to a minimal status: only two points are considered in (11). Moreover, when $\mathrm{\Omega}$ is a suitably regular domain, Calderón spaces are closely related to the usual fractional Sobolev spaces ${W}^{\mathit{\alpha},q}$; see [17]. The function $m$ plays in fact the role of a fractional derivative of $v$ of order $\mathit{\alpha}$ in the ${L}^{q}$sense. Definition 2 is implicit in the work of DeVore and Sharpley [17], where the authors fix the canonical choice $m={M}_{\mathit{\alpha}}^{\#}(v)$ that is indeed always possible in (11) (see Proposition 1 below). The symbol ${M}_{\mathit{\alpha}}^{\#}(v)$ denotes the standard fractional sharp maximal operator introduced in the following (take $\mathrm{\Omega}={\mathbb{R}}^{n}$ and $R=\infty $):
Definition 3
One of the aims of this paper is to show that completely similar estimates actually hold for a large class of quasilinear, possibly degenerate equations. A potential theory which is completely analogous to the linear one can be constructed in the nonlinear case too.
3 Basic nonlinear estimates
Since we are going to deal with local results, we shall use in a standard way the truncated version of the classical Riesz potentials.
Definition 4
Definition 5
Wolff potentials, that despite their name were first considered and studied in [28], reduce to Riesz potentials when $p=2$, i.e. ${\mathbf{I}}_{\mathit{\beta}}^{\mathit{\mu}}\equiv {\mathbf{W}}_{\mathit{\beta}/2,2}^{\mathit{\mu}}$. They play a crucial role in nonlinear potential theory and in the description of the fine properties of solutions to nonlinear equations in divergence form [2, 3, 28, 29, 32, 59, 60].
As first shown in the fundamental works of Kilpeläinen and Malý [37, 38] for the case of nonnegative measures, a neat analog of the first estimate in (5) holds using Wolff potentials. Later on, a new and interesting proof has been offered Trudinger and Wang [65, 66], and this allows to cover the case of general subelliptic operators. Yet different proofs can be found in [39], and in [21], where an approach covering the case of general signed measures has been developed. The final outcome is summarized in the following:
Theorem 1
A sketchy proof of this theorem is proposed in Sect. 18 below; extensions to more general operators have been given in [50, 53]. If $p=2$, then ${\mathbf{W}}_{1,p}^{\mathit{\mu}}\equiv {\mathbf{I}}_{2}^{\mathit{\mu}}$ and we retrieve a local analog of the first estimate in (5). A remarkable point here is that estimate (20) is sharp, and the nonlinear potential ${\mathbf{W}}_{1,p}^{\mathit{\mu}}$ cannot be replaced by any other smaller potential. This is in fact reported in the following:
Theorem 2
Theorem 3
The proof of Theorem 3 will be presented in Sect. 15 below, where it will be obtained as a corollary of more general potential estimates. An extension of the previous result to a class of general operators including the $p$Laplacean has been recently given by Baroni in [5].
Remark 1
An obvious ambiguity arises in the statements of Theorems 1–3 when saying that the related estimates hold whenever the right hand side is finite. This might appear obvious. Another point is that estimates in Theorems 1–3 are stated for every $x$, while both $u(x)$ and $Du(x)$ are only defined if $x$ is Lebesgue point, where the so called precise representative can be defined. Both ambiguities are clarified in Sect. 8. There we prove that, both for $u$ and $Du$, the set of Lebesgue points coincides with the one for which inequalities (20) and (24) feature a finite right hand side, respectively.
Theorem 4
(Nonlinear Stein theorem [41]) Let $u\in {W}^{1,p}(\mathrm{\Omega})$ be a solution to the Eq. (13), under the assumptions (16) and such that $\mathit{\mu}\in L(n,1)$ locally in $\mathrm{\Omega}$. Then $Du$ is continuous in $\mathrm{\Omega}$.
See [12] for a global Lipschitz bound. Without appealing to potentials, but by using different means, the result of the previous theorem also holds for systems.
Theorem 5
Finally, let us mention that starting from the techniques developed for the last two theorems, similar results can be proved in the case of fully nonlinear equations [15].
4 Universal potential estimates
Theorem 6
Notice that when $\mathit{\mu}\equiv 0$ estimates (31) and (33) coincide. In general, counterexamples show that in (30) ${\mathit{\alpha}}_{m}\to 0$ when $L/\mathit{\nu}\to \infty $. This prevents estimate (33) to hold in general for the full range $\mathit{\alpha}\in [0,1)$ when $x\to a(x,\xb7)$ is measurable. Things change when the dependence on $x$ is more regular. For instance, when considering Eq. (29), estimate (31) holds for $\mathit{\alpha}=1$. In this case the validity of (32) extends to the whole interval $\mathit{\alpha}\in [0,1]$. Moreover, estimates (20) and (23) can be uniformly obtained as limiting endpoint cases of (32) for $\mathit{\alpha}=0$ and $\mathit{\alpha}=1$, respectively. In fact, it holds the following:
Theorem 7
(Uniform Wolff potential estimate [40]) Let $u\in {W}^{1,p}(\mathrm{\Omega})$ be a weak solution to the Eq. (13) under assumptions (16). Let ${B}_{R}\subset \mathrm{\Omega}$ be such that $x,y\in {B}_{R/2}$; then estimate (32) holds for $\mathit{\alpha}\in [0,1]$, whenever the right hand side is finite. The constant $c$ depends only on $n,p,\mathit{\nu},L$ but is otherwise independent of $\mathit{\alpha}$.
Estimate (32) does not catch up, when $\mathit{\alpha}\to 1$, the optimal gradient bound in (24). There is a natural reason for such a lack of endpoint property: it catches the other optimal bound (20) as $\mathit{\alpha}\to 0$, and the two cases involve different potentials. In fact, the change in the nature of the estimates when passing from (20) to (24) requires another theorem, parallel to Theorem 7.
Theorem 8
The previous result is here presented for the first time and for the proof we refer to Sect. 5 below. There Theorem 8 is actually obtained as a corollary of a more general estimate for certain maximal operators of $Du$. From the discussion made there it will be clear that the points $x,y$ are Lebesgue points of $u$ when the right hand side in (33) is finite; compare with Remark 1. Theorems 6 and 8 provide altogether the optimal analog of estimate (9) and in fact they both unify with Theorem 7 when $p=2$, when Wolff and Riesz potentials do coincide. We remark that is also possible to quantify the blowup of the constant $c$ in (33) as $\mathit{\alpha}\to 0$; see Remark 7 below.
Theorem 9
The proof of this theorem will be presented in Sect. 16 below. Notice that the limitation in (37) makes the statement of Theorem 9 consistent with the definition of Wolff potential given in Definition 5, where it must be $\mathit{\beta}>0$.
5 Maximalpotential estimates
Definition 6
Using fractional maximal operators allows to get Riesz potential estimates which are uniform in the whole range $[0,1]$, and that provide a rigorous interpretation of previous heuristic arguments. The outcome is summarized in the following theorem, which appears here for the first time. The proof is contained in Sect. 12 below (see Sect. 14.4 for the case of SOLA).
Theorem 10
The turning point to pointwise estimates is now given by the fact that the sharp maximal operator ${M}_{\mathit{\alpha},R}^{\#}(u)(x)$ controls the pointwise behaviour of $u$ provided $\mathit{\alpha}>0$. This fact is expressed in the following Proposition, a first form of which is present in [17]. It in turn relies on some original arguments of Campanato [10].
Proposition 1
5.1 Nonendpoint maximal estimates
Theorem 11
As for gradient oscillations, we instead have the following result, whose proof is contained in Sect. 13 below.
Theorem 12
6 Equations with coefficients
Theorem 13
 [40] For every $\stackrel{~}{\mathit{\alpha}}\in (0,1)$ there exists a number $\mathit{\delta}\in (0,1)$, depending only on $n,p,\mathit{\nu},L,\stackrel{~}{\mathit{\alpha}}$, such thatimplies the validity of estimate (32) whenever $\mathit{\alpha}\in [0,{\mathit{\alpha}}_{m})$. In particular, if the limit in (48) is zero estimate (32) holds for every $\mathit{\alpha}<1$$$\begin{array}{c}\hfill \underset{\mathit{\varrho}\to 0}{lim\; sup}\phantom{\rule{0.166667em}{0ex}}\mathit{\omega}(\mathit{\varrho})\le \mathit{\delta}\end{array}$$(48)
 [6] If the partial map $x\to a(x,\xb7)$ is Dinicontinuous, i.e.then estimates (33) and (40) continue to hold, for $\mathit{\alpha}\in [\stackrel{~}{\mathit{\alpha}},1]$ and $\mathit{\alpha}\in [0,1]$, respectively$$\begin{array}{c}\hfill {\int}_{0}\mathit{\omega}(\mathit{\varrho})\phantom{\rule{0.166667em}{0ex}}\frac{d\mathit{\varrho}}{\mathit{\varrho}}<\infty ,\end{array}$$(49)
 [40] If the partial map $x\to a(x,\xb7)$ is Hölder continuous with exponent $\stackrel{~}{\mathit{\alpha}}<min\{1/(p1),{\mathit{\alpha}}_{M}\}$ in the sense thatthen estimate (36) holds whenever $\mathit{\alpha}<\stackrel{~}{\mathit{\alpha}}$.$$\begin{array}{c}\hfill \underset{r}{sup}{\int}_{0}^{r}\frac{{[\mathit{\omega}(\mathit{\varrho})]}^{2/p}}{{\mathit{\varrho}}^{\stackrel{~}{\mathit{\alpha}}}}\phantom{\rule{0.166667em}{0ex}}\frac{d\mathit{\varrho}}{\mathit{\varrho}}<\infty ,\end{array}$$(50)
Let us now briefly comment on the assumptions made in the last theorem, remarking that all of them are essentially necessary. Assumption (48) allows to catch the case in which, when referring to Eq. (47), the function $c(\xb7)$ is not even continuous, but of class VMO or BMO with small seminorm. This reconnects the theory presented here to the classical Calderón–Zygmund theory, where VMOregularity of coefficients is required to establish integrability estimates related to estimate (32); see for instance [9]. Assumption (149) is necessary too. Indeed, estimate (33) implies gradient boundedness when $\mathit{\mu}$ is good enough, while Dini continuity of coefficients is known to be necessary to get Lipschitz regularity of solutions; see [33]. The proof of the second statement in the last theorem has the techniques developed for Theorem 10 as starting point, and will appear in the forthcoming paper [6]. Finally, assumption (50) seems to be natural in view of the usual Schauder theory. This prescribes that, in order to have Hölder continuity of the gradient, the Hölder continuity of $x\mapsto a(x,\xb7)$ is necessary.
7 Interlude on measure data problems
Definition 7
(Very weak solutions) A function $u\in {W}_{\mathrm{loc}}^{1,1}(\mathrm{\Omega})$ is called a very weak solution $u$ to the Eq. (17) in $\mathrm{\Omega}$ if $a(x,Du)\in {L}_{\mathrm{loc}}^{1}(\mathrm{\Omega},{\mathbb{R}}^{n})$ and (19) holds for every $\mathit{\phi}\in {C}_{c}^{\infty}(\mathrm{\Omega})$.
Definition 8
Theorem 14
7.1 Potential estimates and fundamental solutions
8 Fine properties of solutions via potentials
We have seen that linear and nonlinear potentials locally control the behaviour of solutions. It is at this point not surprising to discover that potentials also control their so called fine properties. In this respect we present two theorems. The former is concerned with the pointwise behaviour of gradients of SOLA, and in particular with their Lebesgue points. This result employs Riesz potentials. The latter in instead concerned with Lebesgue points of solutions and uses Wolff potentials.
Theorem 15
The proof of this last theorem is included in Sect. 17 below. The analogous statement for solutions is the following:
Theorem 16
We notice that the assertion about the $p$capacity follows directly from the fact that set of points where the Wolff potential ${\mathbf{W}}_{1,p}^{\mathit{\mu}}$ blowsup has zero $p$capacity (see for instance [30]).
Remark 2
(Hausdorff dimension of singular sets of SOLA) Theorem 16 allows to define SOLA—which are initially defined only almost everywhere via convergence—outside a singular set (i.e. set of nonLebesgue points) of Hausdorff dimension not larger than $np$, when $p\le n$. This establishes a connection with another class of solutions to measure data problem. These are called $p$superharmonic functions and are defined in the case the measure $\mathit{\mu}$ is nonnegative; see [30, 38]. For such solutions every point is a Lebesgue point, by construction. The connection is now given by the fact that, in view of [36], every SOLA has a superharmonic representative (that is, they coincide almost everywhere) whenever the measure is nonnegative. In the case of Theorem 15 we instead can conclude that the Hausdorff dimension of the singular set of $Du$ is not larger than $n1$. The last estimate follows on the other hand also by (57). Indeed, the Hausdorff dimension of the set of nonLebesgue points of a general ${W}^{s,\mathit{\gamma}}$map, with $s\mathit{\gamma}<n$, has Hausdorff dimension not larger that $ns\mathit{\gamma}$; see [54].
Remark 3
9 Regularity and corollaries
Corollary 1
 (C1) if $\mathit{\alpha}\in (0,1)$ and $q>n(p1)/[np+\mathit{\alpha}(p1)]$, then(estimate (64))$$\begin{array}{c}\hfill \mathit{\mu}\in {L}^{\frac{nq}{n(p1)+[p\mathit{\alpha}(p1)]q}}\u27f9u\in {C}_{q}^{\mathit{\alpha}}\end{array}$$
 (C2) if $p<n$, thenand$$\begin{array}{c}\hfill \mathit{\mu}\in {\mathcal{M}}^{n/p}\u27f9\mathit{\mu}({B}_{\mathit{\varrho}})\lesssim {\mathit{\varrho}}^{np}\u27f9u\in \text{BMO}\end{array}$$(Theorem 11 with $\mathit{\alpha}=0)$$$\begin{array}{c}\hfill p=n\u27f9u\in \text{BMO}\end{array}$$

(C3) ${\mathbf{W}}_{1,p}^{\mathit{\mu}}\in {L}^{\infty}\u27f9u\in {L}^{\infty}$ (Theorem 1)

(C4) if $p<n$, then $\mathit{\mu}\in L(n/p,1/(p1))\u27f9u\in {C}^{0}$ (Theorem 1)
 (C5) if $\mathit{\alpha}\in (0,1)$ and $p\le n$, then(estimate (64))$$\begin{array}{c}\hfill \mathit{\mu}\in {\mathcal{M}}^{\frac{n}{p\mathit{\alpha}(p1)}}\u27f9\mathit{\mu}({B}_{\mathit{\varrho}})\lesssim {\mathit{\varrho}}^{np+\mathit{\alpha}(p1)}\u27f9u\in {C}^{0,\mathit{\alpha}}\end{array}$$

(C6) if $p\le n$, then $\mathit{\mu}\in {\mathcal{M}}_{\mathrm{b}}\u27f9Du\in {\mathcal{M}}^{n(p1)/(n1)}$ (Theorem 3)
 (C7) if $1<q<n$, then(Theorem 3)$$\begin{array}{c}\hfill \mathit{\mu}\in L(q,\mathit{\gamma})\u27f9Du\in L\left(\frac{nq(p1)}{nq},\mathit{\gamma}(p1)\right)\end{array}$$

(C8) if $1<q<\mathit{\theta}$, then ${Du}^{p1}\in {L}^{\mathit{\theta}q/(\mathit{\theta}q),\mathit{\theta}}$ (Theorem 3)

(C9) $\mathit{\mu}\in {\mathcal{M}}^{n}\u27f9\mathit{\mu}({B}_{\mathit{\varrho}})\lesssim {\mathit{\varrho}}^{n1}\u27f9Du\in $ BMO (Theorem 12)

(C10) ${\mathbf{I}}_{1}^{\mathit{\mu}}\in {L}^{\infty}\u27f9Du\in {L}^{\infty}$ (Theorem 8)

(C11) $\mathit{\mu}\in L(n,1)\u27f9Du\in {C}^{0}$ (Theorem 3)
 (C12) if $0<\mathit{\alpha}<min\{1/(p1),{\mathit{\alpha}}_{M}\}$, then(estimate (65)).$$\begin{array}{c}\hfill \mathit{\mu}\in {\mathcal{M}}^{\frac{n}{1\mathit{\alpha}(p1)}}\u27f9\mathit{\mu}({B}_{\mathit{\varrho}})\lesssim {\mathit{\varrho}}^{n1+\mathit{\alpha}(p1)}\u27f9Du\in {C}^{0,\mathit{\alpha}}\end{array}$$
Proof
The statement in (C3) is simply an obvious consequence of Theorem 1.
As for (C4), a direct computation using certain characterisation of Lorentz spaces shows that if $\mathit{\mu}\in L(n/p,1/(p1))$ for $p<n$, then (21) holds (see for instance [23] for related computations). At this stage (C4) follows by Theorem 1.
In (C5) the first implication follows again by the Hölder type inequality in (69). On the other hand, we observe that the inequality $\mathit{\mu}({B}_{\mathit{\varrho}})\lesssim {\mathit{\varrho}}^{np+\mathit{\alpha}(p1)}$ implies that ${M}_{p\mathit{\alpha}(p1),R}(\mathit{\mu})\in {L}^{\infty}$, locally, by the very definition of fractional maximal operator in Definition 6, so that (C5) follows by estimate (64).
The first implication in (C9) is a consequence of (69) while the second follows applying Theorem 12 with $\mathit{\alpha}=0$. Indeed, assuming that $\mathit{\mu}({B}_{\mathit{\varrho}})\lesssim {\mathit{\varrho}}^{n1}$ implies that ${M}_{1}(\mathit{\mu})$ is locally bounded, and therefore so is ${M}_{0,R}^{\#}(Du)\equiv {M}_{R}^{\#}(Du)$. By definition of sharp maximal function this implies that the gradient belongs to BMO, locally; see (68).
(C10) follows from Theorem 8 in an obvious way.
(C11) is again a consequence of (25) and of some basic computation involving the definition of Lorentz norm (see [22, 41]). Indeed, the condition $\mathit{\mu}\in L(n,1)$ allows to conclude that (25) holds, again by basic manipulations on Lorentz equivalent norms.
Finally, exactly as for the proof of (C5), (C12) follows again from (69) and estimate (65). $\square $
Some of the points in the previous corollary are well known results when considering for instance the model equation (15). The theory above allows to extend and embed them in a more general context where results follow in a unified way. Specifically, for (C2) see [55, 64], for (C6) see [7, 8, 20], for (C7) see [19, 31, 56], for (C8) see [56]. We also remark that the previous corollary is just a sample of what is possible to have using the nonlinear potential estimates approach; further spaces, as for instance Lorentz–Morrey or Besov–Morrey spaces are considerable as well (see [55, 56] for relevant definitions). More rearrangement invariant function spaces regularity results via potentials are contained in [11].
10 A basic comparison estimate
The lemmas in this section are already scattered in [21, 41, 55]. The proofs proposed here are anyway different and shorter. We start with a basic, weighted type energy estimate.
Lemma 1
Proof
Lemma 1 and Sobolev embedding theorem in turn imply a first comparison estimate.
Lemma 2
Proof
Remark 4
11 A sequence of comparison estimates
Lemma 3
Proof
12 Proof of Theorems 8 and 10
The proof of Theorem 10 falls in six steps, going through Sects. 12.1–12.6 respectively. It requires the use of several different tools, and ultimately relies on a delicate iteration technique. Finally, in Sect. 12.7, we shall briefly show how to get Theorem 8 from Theorem 10.
12.1 A density property of $a$harmonic functions
Proposition 2
Proof
Theorem 17
Remark 5
The exponent $\mathit{\beta}$ appearing in (105) can be chosen arbitrarily close to the exponent ${\mathit{\alpha}}_{M}$ appearing in (34). As a matter of fact, estimate (35) is in fact obtained as a corollary of the one in (105) via the standard Campanato’s integral characterisation of Hölder continuous functions [10]. The exponent ${\mathit{\alpha}}_{M}$ considered in Theorem 9 is indeed nothing but the sup of the numbers $\mathit{\beta}$ considerable in Theorem 17. Explicit, though not optimal, estimates on these numbers via the structure parameters $n,p,\mathit{\nu},L$, are retrievable tracking the dependence of the constants in the proofs in [19, 21, 47].
Theorem 18
Proof
Remark 6
Exactly as in Remark 5, the exponent $\stackrel{~}{\mathit{\beta}}$ appearing in Theorem 18 can be actually taken as close to ${\mathit{\alpha}}_{m}$—appearing in (30)–(31)—as we please. As a matter of fact estimates (31) and (106) are equivalent by noticing that if $v$ solves then so does $vk$, whenever $k$ is a real number. Again, the exponent ${\mathit{\alpha}}_{m}$ used in Theorem 6 is actually defined as the sup of the numbers $\mathit{\beta}$ for which (106) works. Estimates for such numbers are available in terms on the structure parameters $n,p,\mathit{\nu},L$.
12.2 Setting of the constants
12.3 Iterating quantities
12.4 Iteration chains
12.5 Density and decay properties along iteration chains
The idea is now that along iteration chains, a condition of the type ${C}_{j}\ge {\mathit{\lambda}}_{M}/100$ guarantees that the equation becomes nondegenerate and can be in a sense linearized. The consequence is that the excess functional of the gradient decays in a way that resembles the one of solutions to the Poisson equation. To implement this we need to get a few density estimates. Let us preliminary prove a cheap decay estimate, that actually holds also outside the context of the present proof, but whose assumptions are obviously satisfied here by (111).
Lemma 4
Proof
The following lemma is an easy consequence of the previous one:
Lemma 5
Proof
The cheap decay estimate for the excess functional of Lemma 4 is not sufficient to get gradient estimates via linear Riesz potentials. Indeed, it only implies Wolff potential estimates as in (23), as shown in [21]. Anyway, by using it together with the density properties of Proposition 2 and Lemma 3, we deduce another, better decay estimate, which is this time implying linear potentials estimates.