J. Math. Pures Appl. 156 (2021) 179–214 Contents lists available at ScienceDirect Journal de Mathématiques Pures et Appliquées www.elsevier.com/locate/matpur Scale invariant regularity estimates for second order elliptic equations with lower order coefficients in optimal spaces Georgios Sakellaris 1 Department of Mathematics, Universitat Autònoma de Barcelona, Barcelona, 08193, Spain a r t i c l e i n f o a b s t r a c t Article history: We show local and global scale invariant regularity estimates for subsolutions Received 30 June 2020 and supersolutions to the equation − div(A∇u + bu) + c∇u + du = − div f + g, Available online 12 October 2021 assuming that A is elliptic and bounded. In the setting of Lorentz spaces, under the n assumptions b, f ∈ Ln,1 , d, g ∈ L 2 ,1 and c ∈ Ln,q for q ≤ ∞, we show that, with the MSC: surprising exception of the reverse Moser estimate, scale invariant estimates with primary 35B45, 35B50, 35B51, 35B65, 35J15 “good” constants (that is, depending only on the norms of the coefficients) do not secondary 35D30, 35J10, 35J20, hold in general. On the other hand, assuming a necessary smallness condition on 35J86 b, d or c, d, we show a maximum principle and Moser’s estimate for subsolutions with “good” constants. We also show the reverse Moser estimate for nonnegative Keywords: supersolutions with “good” constants, under no smallness assumptions when q < ∞, Maximum principle leading to the Harnack inequality for nonnegative solutions and local continuity of Moser estimate solutions. Finally, we show that, in the setting of Lorentz spaces, our assumptions Harnack inequality are the sharp ones to guarantee these estimates. Continuity of solutions © 2021 The Author. Published by Elsevier Masson SAS. This is an open access Lorentz spaces article under the CC BY-NC-ND license Symmetrization (http://creativecommons.org/licenses/by-nc-nd/4.0/). r é s u m é Nous montrons des estimations de régularité invariante à l’échelle locale et globale pour les sous-solutions et les supersolutions de l’équation − div(A∇u + bu) + c∇u + du = − div f + g, en supposant que A est elliptique et borné. Dans le cadre des n espaces de Lorentz, sous les hypothèses b, f ∈ Ln,1 , d, g ∈ L 2 ,1 et c ∈ Ln,q pour q ≤ ∞, nous montrons qu’à l’exception surprenante de l’estimation de Moser inverse, les estimations invariantes d’échelle avec de “bonnes” constantes (c’est-à-dire ne dépendant que des normes des coefficients) ne tiennent pas en général. D’autre part, en supposant une condition de petitesse nécessaire sur b, d ou c, d, nous montrons un principe du maximum et l’estimation de Moser pour les sous-solutions avec de “bonnes” constantes. Nous montrons également l’estimation inverse de Moser pour les supersolutions non négatives avec de “bonnes” constantes, sous des hypothèses de non petitesse lorsque q < ∞, conduisant à l’inégalité de Harnack pour les solutions non négatives et la continuité locale des solutions. Enfin, nous montrons que, dans E-mail address: gsakellaris@mat.uab.cat. 1 The author has received funding from the European Union’s Horizon 2020 research and innovation programme under Marie Skłodowska-Curie grant agreement No. 665919, and is partially supported by MTM-2016-77635-P (MICINN, Spain) and 2017 SGR 395 (Generalitat de Catalunya). https://doi.org/10.1016/j.matpur.2021.10.009 0021-7824/© 2021 The Author. Published by Elsevier Masson SAS. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 180 G. Sakellaris / J. Math. Pures Appl. 156 (2021) 179–214 le cadre des espaces de Lorentz, nos hypothèses sont les plus pointues pour garantir ces estimations. © 2021 The Author. Published by Elsevier Masson SAS. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). 1. Introduction In this article we are interested in local and global regularity for subsolutions and supersolutions to the equation Lu = − div f + g, in domains Ω ⊆ Rn , where L is of the form Lu = − div(A∇u + bu) + c∇u + du. In particular, we investigate the validity of the maximum principle, Moser’s estimate, the Harnack inequality and continuity of solutions, in a scale invariant setting; that is, we want our estimates to not depend on the size of Ω. We will also assume throughout this article that n ≥ 3. In this work A will be bounded and uniformly elliptic in Ω: for some λ > 0, A(x)ξ, ξ ≥ λξ2 , ∀x ∈ Ω, ∀ξ ∈ Rn . For the lower order coefficients and the terms on the right hand side, we consider Lorentz spaces that are scale invariant under the natural scaling for the equation. That is, we assume that n b, f ∈ Ln,1 (Ω), c ∈ Ln,q (Ω), d, g ∈ L 2 ,1 (Ω), q ≤ ∞. In the case that q = ∞, it is also necessary to assume that the norm of c is small for our results to hold. As explained in Section 6, these assumptions are the optimal ones to imply our estimates in the setting of Lorentz spaces. Note also that there will be no size assumption on Ω and no regularity assumption on ∂Ω. The main inspiration for this work comes from the local and global pointwise estimates for subsolutions to the fore mentioned operator in [1], where it is also assumed that d ≥ div c in the sense of distributions. Focusing on the case when c, d ≡ 0 for simplicity, and assuming that b ∈ Ln,1 , a maximum principle for subsolutions to − div(A∇u + bu) ≤ − div f + g is shown in [1, Proposition 7.5], while a Moser type estimate is the context of [1, Proposition 7.8]. The main feature of these estimates is their scale invariance, with constants that depend only on the ellipticity of A and the Ln,1 norm of b, as well as the L∞ norm of A for the Moser estimate. Following this line of thought, it could be expected that the consideration of all the lower order coefficients in the definition of L should yield the same type of scale invariant estimates, with constants being “good”; that is, depending only on n, q, the ellipticity of A, and the norms of the coefficients involved (as well as A∞ in some cases). However, it turns out that this does not hold. In particular, if B1 is the unit ball in Rn , n in Proposition 6.3 we construct a bounded sequence (dN ) in L 2 ,1 (B1 ) and a sequence (uN ) of nonnegative W01,2 (B1 ) solutions to the equation −ΔuN + dN uN = 0 in B1 , such that uN W01,2 (B1 ) ≤ C, while uN L∞ (B1/2 ) −−−−→ ∞. N →∞ We also show in Remark 6.4 that the equation −Δu − div(bu) + c∇u = 0 has the same feature, which implies that the constants in Moser’s local boundedness estimate, as well as the Harnack inequality, cannot be “good” without any further assumptions. Since scale invariant estimates with “good” constants do not hold in such generality, we first prove estimates where the constants are allowed to depend on the coefficients themselves. This is the context of G. Sakellaris / J. Math. Pures Appl. 156 (2021) 179–214 181 the global bound in Proposition 3.2, where it is shown that, if Ω ⊆ Rn is a domain and u ∈ Y 1,2 (Ω) (see (2.1)) is a subsolution to Lu ≤ − div f + g, then, for any p > 0, ⎛ ⎞ p1 ˆ sup u+ ≤ C sup u+ + C  ⎝ |u+ |p ⎠ + Cf n,1 + Cg n2 ,1 , (1.1) Ω ∂Ω Ω where C is a “good” constant, while C  depends on the coefficients themselves and p. In the subcritical setting, an analogous estimate is the context of [2, Theorem 8.15]; however, despite considering the optimal scale invariant setting for such an estimate to hold, the constant C  in (1.1) is not “good”. Note also the appearance of a constant in front of the term sup∂Ω u+ ; such a constant can be greater than 1, and this follows from the fact that constants are not necessarily subsolutions to our equation in the generality of our assumptions. Having proven the previous estimate, we then turn to show various scale invariant estimates with “good” constants, assuming an extra condition on the lower order coefficients, which is necessary in view of the fore mentioned discussion. Such a condition is some type of smallness: in particular, we either assume that the norms of b, d are small, or that the norms of c, d are small. Under these smallness assumptions, we show in Propositions 3.3 and 3.4 that we can take C  = 0 in (1.1), leading to a maximum principle, and the Moser estimate for subsolutions to Lu ≤ − div f + g is shown in Propositions 4.4 and 4.6; that is, in the case when b, d are small, or c, d are small, then for any p > 0, ⎛ ⎞ p1 sup u ≤ C ⎝ |u+ |p ⎠ + Cf Ln,1 (B2r ) + CgL n2 ,1 (B , (1.2) 2r ) Br B2r where the constant C is “good”, and also depends on p. In addition, the analogous estimate close to the boundary is deduced in Propositions 4.9 and 4.10. On the other hand, somewhat surprisingly, we discover that even if the scale invariant Moser estimate with “good” constants requires some type of smallness, it turns out that the scale invariant reverse Moser estimate with “good” constants holds in the full generality of our initial assumptions. That is, in Proposition 5.3, we show that if u ∈ W 1,2 (B2r ) is a nonnegative supersolution to Lu ≥ − div f + g, and under no smallness assumptions (when q < ∞), then for some α = αn , ⎛ ⎞ α1 ⎝ uα ⎠ ≤ C inf u + Cf Ln,1 (B2r ) + CgL n2 ,1 (B , (1.3) Br/2 2r ) Br where C is a “good” constant. Moreover, the analogue of this estimate close to the boundary is deduced in Proposition 5.5. Then, the Harnack inequality (Theorems 5.6 and 5.7) and continuity of solutions (Theo- rems 5.8 and 5.9) are shown combining (1.2) and (1.3); for those, in order to obtain estimates with “good” constants, it is again necessary to assume a smallness condition. Finally, having shown the previous es- timates, we also obtain their analogues in the generality of our initial assumptions, with constants that depend on the coefficients themselves (Remarks 4.7, 4.11 and 5.10). As a special case, we remark that all the scale invariant estimates above hold, with “good” constants, in the case of the operators L1 u = − div(A∇u) + c∇u, L2 u = − div(A∇u + bu), under no smallness assumptions when b ∈ Ln,1 and c ∈ Ln,q , q < ∞. 182 G. Sakellaris / J. Math. Pures Appl. 156 (2021) 179–214 The techniques. The assumption that the coefficients b, d lie in scale invariant spaces is reflected in the fact that the classical method of Moser iteration does not seem to work in this setting. More specifically, an assumption of the form b ∈ Ln,q for some q > 1 does not necessarily guarantee pointwise upper bounds (see Remark 6.1), and it is necessary to assume that b ∈ Ln,1 in order to deduce these bounds. However, Moser’s method does not seem to be “sensitive” enough to distinguish between the cases b ∈ Ln,1 and b ∈ Ln,q for q > 1. Thus, a procedure more closely related to Lorentz spaces has to be followed, and the first results in this article (Section 3) are based on a symmetrization technique, leading to estimates for decreasing rearrangements. This technique involves a specific choice of test functions and has been used in the past by many authors, going back to Talenti’s article [3]; here we use a slightly different choice, utilized by Cianchi and Mazya in [4]. However, since all the lower order coefficients are present, our estimates are more complicated, and we have to rely on an argument using Grönwall’s inequality (as in [5], for example) to give a bound on the decreasing rearrangement of our subsolution. On the other hand, the main drawback of the symmetrization technique is that it does not seem to work well when we combine it with cutoff functions; thus, we are not able to suitably modify it in order to directly show local estimates like (1.2). The idea to overcome this obstacle is to pass from small to large norms using a two-step procedure (in Section 4), utilizing the maximum principle. Thus, relying on Moser’s estimate for the operator L0 u = − div(A∇u) + c∇u when the norm of c is small, the first step is a perturbation argument based on the maximum principle that allows us to pass to the operator L when all the lower order terms have small norms. Then, the second step is an induction argument relying on the maximum principle (similar to the proofs of [1, Propositions 3.4 and 7.8], inspired by the argument of the proof of [6, Theorem 2.5’]), which allows us to pass to arbitrary norms for b or c. To the best of our knowledge, the combination of the symmetrization technique with the fore mentioned argument in order to obtain local estimates has not appeared in the literature before (with the exception of [1, Proposition 7.8], which used estimates on Green’s function), and it is one of the novelties of this article. Since we do not obtain Moser’s estimate (1.2) using test functions and Moser’s iteration, in order to deduce the reverse Moser estimate (1.3) we transform supersolutions to subsolutions via exponentiation (in Section 5). The advantage of this procedure is that, if the exponent is negative and close to 0, we obtain a subsolution to an equation with the coefficients b, d being small, thus we can apply (1.2) to obtain a scale invariant estimate with “good” constants, without any smallness assumptions (when q < ∞). This estimate has negative exponents appearing on the left hand side, and we show (1.3) passing to positive exponents using an estimate for supersolutions and the John-Nirenberg inequality (as in [7]). One drawback of this technique is that we do not obtain the full range α ∈ (0, n−2n ) for the left hand side, as in [2, Theorem 8.18], but this does not affect the proof of the Harnack inequality. Then, the Harnack inequality and continuity of solutions are deduced combining (1.2) and (1.3). Finally, the optimality of our assumptions is shown in Section 6. In particular, the sharpness of our spaces to guarantee some type of estimates (either having “good” constants, or not) is shown, and the failure of scale invariant estimates with “good” constants is exhibited by the construction in Proposition 6.3. Past works. The first fundamental contribution to regularity for equations with rough coefficients was made by De Giorgi [8] and Nash [9] and concerned Hölder continuity of solutions to the operator − div(A∇u) = 0; a different proof, based on the Harnack inequality, was later given by Moser in [7]. The literature concerning this subject is vast, and we refer to the books by Ladyzhenskaya and Ural’tseva [10] and Gilbarg and Trudinger [2], as well as the references therein, for equations that also have lower order coefficients in Lp . However, in these results, the norms of those spaces are not scale invariant under the natural scaling of the equation, so it is not possible to obtain scale invariant estimates without extra assumptions on the coefficients (like smallness, for example). One instance of a scale invariant setting where b, d, f, g ≡ 0 and c ∈ Ln was later treated by Nazarov and Ural’tseva in [6]. G. Sakellaris / J. Math. Pures Appl. 156 (2021) 179–214 183 Another well studied case of coefficients is the class of Kato spaces. The first work on estimates for Schrödinger operators with the Laplacian and potentials in a suitable Kato class was by Aizenman and Simon in [11] using probabilistic techniques, which was later generalized (with nonprobabilistic techniques) by Chiarenza, Fabes and Garofalo in [12], allowing a second order part in divergence form. The case in [11] was also later treated using nonprobabilistic techniques by Simader [13] and Hinz and Kalf [14]. In these works, b, c ≡ 0, while d is assumed to belong to Knloc (Ω), which is comprised of all functions d in Ω such that ηΩ1 ,d (r) → 0 as r → 0, for all Ω1 compactly supported in Ω, where ˆ |d(y)| ηΩ,d (r) = sup dy x∈Rn |x − y|n−2 Ω∩Br (x) (or, in some works, the supremum is considered over x ∈ Ω). Moreover, adding the drift term c∇u, regularity estimates for c in a suitable Kato class were shown by Kurata in [15]. Finally, considering all the lower order terms, Mourgoglou in [16] shows regularity estimates when the coefficients b, d belong to the scale invariant Dini type Kato-Stummel classes (see [16, Section 2.2]), and also constructs Green’s functions. However, the framework we consider in this article for the Moser estimate and Harnack’s inequality, as well as our techniques, are different from the ones in [16]. For example, focusing on the case when c, d ≡ 0, the coefficient b in [16, Theorems 4.4, 4.5 and 4.12] is assumed to be such that |b|2 ∈ −a KDini,2 , which does not cover the case b ∈ Ln,1 , since for any α > 1, the function b(x) = x|x|−2 (− ln |x|) is a member of Ln,1 (B1/e ), while |b|2 ∈ / KDini,2 (B1/e ). We conclude with a brief discussion on symmetrization techniques. Such a technique was used by Wein- berger in [17] in order to show boundedness of solutions with vanishing trace to − div(A∇u) = − div f and p − div(A∇u) = g, where f ∈ Lp and g ∈ L 2 , p > n. Another well known technique consists of a use of test functions that leads to bounds for the derivative of the integral of |∇u|2 over superlevel sets of u, where u is a subsolution to Lu ≤ − div f + g. This bound, combined with Talenti’s inequality [3, estimate (40)], gives an estimate for the derivative of the decreasing rearrangement of u, leading to bounds for u in various spaces and comparison results. This technique has been used by many authors in order to study regularity properties of solutions to second order pdes, some works being [18], [19], [5], [20], [21], [22], [23], [24], [25]. However, as we mentioned above, to the best of our knowledge, no local boundedness results have been deduced using this method so far. We also mention that, in order to treat lower order coefficients, pseudo-rearrangements of functions are also considered the literature, which are derivatives of integrals over suitable sets Ω(s) ⊆ Ω (see, for example, [1, page 11]). On the contrary, in this work we avoid this procedure, and as we mentioned above we rely instead on a slightly different approach, inspired by [4]. Acknowledgments We would like to thank Professors Carlos Kenig and Andrea Cianchi for useful conversations regarding some parts of this article. 2. Preliminaries 2.1. Definitions If Ω ⊆ Rn is a domain, W01,2 (Ω) will be the closure of Cc∞ (Ω) under the W 1,2 norm, where uW 1,2 (Ω) = uL2 (Ω) + ∇uL2 (Ω) . 184 G. Sakellaris / J. Math. Pures Appl. 156 (2021) 179–214 When Ω has infinite measure, the space W 1,2 (Ω) is not well suited to the problems we consider. For this reason, we let Y01,2 (Ω) be the closure of Cc∞ (Ω) under the Y 1,2 norm, where uY 1,2 (Ω) = uL2∗ (Ω) + ∇uL2 (Ω) , (2.1) and 2∗ = 2n n−2 is the Sobolev conjugate to 2. From the Sobolev inequality φL2∗ (Ω) ≤ Cn ∇φL2 (Ω) , for all φ ∈ Cc∞ (Ω), we have that Y01,2 (Ω) = W01,2 (Ω) in the case |Ω| < ∞. We also set Y 1,2 (Ω) to be the ∗ space of weakly differentiable u ∈ L2 (Ω), such that ∇u ∈ L2 (Ω), with the Y 1,2 norm. If u is a measurable function in Ω, we define the distribution function μu (t) = |{x ∈ Ω : |u(x)| > t}| , t > 0. (2.2) If u ∈ Lp (Ω) for some p ≥ 1, then μu (t) < ∞ for any t > 0. Moreover, we define the decreasing rearrangement of u by u∗ (τ ) = inf{t > 0 : μu (t) ≤ τ }, (2.3) as in [26, (1.4.2), page 45]. Then, u∗ is equimeasurable to u: that is, |{x ∈ Ω : |u(x)| > t}| = |{s > 0 : u∗ (s) > t}| for all t > 0. (2.4) Given a function f ∈ Lp (Ω), we consider its maximal function ˆτ 1 Mf (τ ) = f ∗ (σ) dσ, τ > 0. (2.5) τ 0 Let p ∈ (0, ∞) and q ∈ (0, ∞]. If f is a function defined in Ω, we define the Lorentz seminorm ⎧⎛ ⎞ q1 ⎪ ⎪ ˆ ∞ ⎪ ⎪ ⎨⎝ q dτ ⎠ τ p f ∗ (τ ) 1 , q<∞ f Lp,q (Ω) = τ (2.6) ⎪ ⎪ 0 ⎪ ⎪ 1 ∗ ⎩ sup τ f (τ ), p q = ∞, τ >0 as in [26, Definition 1.4.6]. We say that f ∈ Lp,q (Ω) if f Lp,q (Ω) < ∞. Then  · p,q is indeed a seminorm, since f + gp,q ≤ Cp,q f p,q + Cp,q gp,q , (2.7) from [26, (1.4.9), page 50]. In addition, from [26, Proposition 1.4.10], Lorentz spaces increase if we increase the second index, with f Lp,r ≤ Cp,q,r f Lp,q for all 0 < p < ∞, 0 < q < r ≤ ∞. (2.8) Hölder’s inequality for Lorentz functions states that f gLp,q ≤ Cp1 ,q1 ,p2 ,q2 f Lp1 ,q1 gLp2 ,q2 , (2.9) G. Sakellaris / J. Math. Pures Appl. 156 (2021) 179–214 185 whenever 0 < p, p1 , p2 < ∞ and 0 < q, q1 , q2 ≤ ∞ satisfy the relations p1 = p11 + 1 1 p2 , q = 1 q1 + 1 q2 (see [26, Exercise 1.4.19]). If p ∈ (1, ∞] and q ∈ [1, ∞), then [27, Theorem 3.21, page 204] implies that Mf p,q ≤ Cp f p,q , (2.10) where Mf is the maximal function defined in (2.5). For a function u ∈ Y 1,2 , we will say that u ≤ s on ∂Ω if (u − s)+ = max{u − s, 0} ∈ Y01,2 (Ω). Moreover, sup∂Ω u will be defined as the infimum of all s ∈ R such that u ≤ s on ∂Ω. We now turn to the definitions of subsolutions, supersolutions and solutions. For this, let Ω ⊆ Rn be a n domain, and let A be bounded in Ω, b, c, f ∈ Ln,∞ (Ω), d, g ∈ L 2 ,∞ (Ω). If Lu = − div(A∇u +bu) +c∇u +du, we say that u ∈ Y 1,2 (Ω) is a solution to the equation Lu = − div f + g in Ω, if ˆ ˆ A∇u∇φ + b∇φ · u + c∇u · φ + duφ = f ∇φ + gφ, ∀φ ∈ Cc∞ (Ω). Ω Ω Moreover, we say that u ∈ Y 1,2 (Ω) is a subsolution to Lu ≤ − div f + g in Ω, if ˆ ˆ A∇u∇φ + b∇φ · u + c∇u · φ + duφ ≤ f ∇φ + gφ, ∀φ ∈ Cc∞ (Ω), φ ≥ 0. (2.11) Ω Ω We also say that u is a supersolution to Lu ≥ − div f + g, if −u is a subsolution to L(−u) ≤ div f − g. 2.2. Main lemmas We now discuss some lemmas that we will use in the sequel. We begin with the following estimate, in which we show that a function in Ln,q for q > 1 fails to be in Ln,1 by a logarithm, with constant as small as we want. This fact will be useful in the proof of Lemma 3.1. Lemma 2.1. Let f ∈ Ln,q (Ω) for some q ∈ (1, ∞). Then, for any 0 < σ1 < σ2 < ∞ and ε > 0, ˆσ2 σ2 τ n −1 f ∗ (τ ) dτ ≤ ε ln 1 + Cf qn,q , σ1 σ1 where C depends on q and ε. Proof. Let p ∈ (1, ∞) be the conjugate exponent to q. Then, from Hölder’s inequality and (2.6), ˆσ2 ˆσ2 n −1 ∗ τ − p τ n − q f ∗ (τ ) dτ 1 1 1 1 τ f (τ ) dτ = σ1 σ1 ⎛ ⎞ p1 ⎛ ⎞ q1 ˆσ2 ˆσ2 ≤⎝ τ −1 dτ ⎠ ⎝ n −1 f ∗ (τ )q dτ ⎠ q τ σ1 σ1 1 (p ε)− p q σ2 p σ2 · (p ε)− p f n,q ≤ ε ln 1 ≤ p ε ln + f qn,q , σ1 σ1 q where we also used Young’s inequality for the last step.  186 G. Sakellaris / J. Math. Pures Appl. 156 (2021) 179–214 The following describes the behavior of the Lorentz seminorm on disjoint sets. Lemma 2.2. Let Ω ⊆ Rn be a set, and let X, Y be nonempty and disjoint subsets of Ω. If f ∈ Lp,q (Ω) for some p, q ∈ [1, ∞), then f rLp,q (Ω) ≥ f rLp,q (X) + f rLp,q (Y ) , r = max{p, q}. Proof. Let μ, μX , μY be the distribution functions of f, f |X and f |Y , respectively. As in [1, Lemma 2.4], we have that μ ≥ μX + μY . Also, if p ≥ q, then pq ≤ 1, hence the reverse Minkowski inequality shows that ⎛ ⎞ pq ⎛∞ ⎞ pq ⎛ ∞ ⎞ pq ˆ∞ ˆ ˆ ⎝ (μX (t) + μY (t)) p sq−1 ds⎠ ≥ ⎝ μX (t) p sq−1 ds⎠ + ⎝ μY (t) p sq−1 ds⎠ . q q q 0 0 0 q q q On the other hand, if q > p, then q p > 1, hence a p + b p ≤ (a + b) p for all a, b > 0. Therefore, ˆ∞ ˆ∞ ˆ∞ q q q (μX (t) + μY (t)) s p q−1 ds ≥ μX (t) s p q−1 ds + μY (t) p sq−1 ds. 0 0 0 Then, the proof follows from the expression for the Lp,q seminorm in [26, Proposition 1.4.9].  The next lemma will be useful in order to reduce to the case d = 0. n Lemma 2.3. Let Ω ⊆ Rn be a domain, and d ∈ L 2 ,1 (Ω). Then there exists a weakly differentiable vector valued function e ∈ Ln,1 (Ω), with div e = d in Ω and eLn,1 (Ω) ≤ Cn dL n2 ,1 (Ω) . Proof. Extend d by 0 outside Ω, and consider the Newtonian potential v of d; that is, we set ˆ d(y) w(x) = Cn dy. |x − y|n−2 Rn From [2, Theorem 9.9] we have that w is twice weakly differentiable in Ω, and Δw = d. Setting e = ∇w, ´ |d(y)| we have that div e = d. Moreover, |e(x)| = |∇w(x)| ≤ Cn Rn |x−y| n−1 dy, and the estimate follows from the first part of [26, Exercise 1.4.19].  The next lemma shows that u∗ is locally absolutely continuous, when u ∈ Y 1,2 . Lemma 2.4. Let Ω be a domain and u ∈ Y01,2 (Ω). Then u∗ is absolutely continuous in (a, b), for any 0 < a < b < ∞. Proof. Extending u by 0 outside Ω, we may assume that u ∈ Y 1,2 (Rn ). Consider the function u∗ defined in [28, (2), page 153] (this u∗ is not the same as the one in (2.3)!), and the function ũ(|x|) = u∗ (x) (as in [28, page 154]). Then, from the argument for the proof of [1, Lemma 2.6], it is enough to show that ũ is locally absolutely continuous in (0, ∞). To show this, note that the proof of [28, Lemma 2.4] shows that u∗ ∈ Y 1,2 (Rn ) whenever u ∈ Y 1,2 (Rn ) (since Y 1,2 (Rn ) is reflexive, bounded sequences have subsequences that converge weakly, and the rest of the argument runs unchanged). Hence, u∗ ∈ Wloc 1,2 (Rn ), and combining with [28, Proposition 2.5], we obtain that ũ is locally absolutely continuous in (0, ∞), as in [28, Corollary 2.6], which completes the proof.  G. Sakellaris / J. Math. Pures Appl. 156 (2021) 179–214 187 We now turn to the following decomposition, which is similar to [1, Lemma 2.8]. This will be useful in a change of variables that we will perform in Lemma 2.6, as well as in the proof of the estimate in Lemma 3.1. Lemma 2.5. Let Ω ⊆ Rn be a domain, and let u ∈ Y01,2 (Ω). Then we can write (0, ∞) = Gu ∪ Du ∪ Nu , where the union is disjoint, such that the following hold. i) If x ∈ Gu , then u∗ is differentiable at x, μu is differentiable at u∗ (x), and (u∗ ) (x) = 0. Moreover, 1 μu (u∗ (x)) = x and μu (u∗ (x)) = , for all x ∈ Gu . (2.12) (u∗ ) (x) ii) If x ∈ Du , then u∗ is differentiable at x, with (u∗ ) (x) = 0. iii) Nu is a null set. Proof. The proof is the same as the proof of [1, Lemma 2.8], where we use continuity of u∗ shown in Lemma 2.4, instead of [1, Lemma 2.6].  We now turn to the following lemma, which is based on [4, Lemma 3.1]. As we mentioned in the intro- duction, the properties of the function Ψ defined below will be crucial in the proof of Lemma 3.1 and, using this lemma, we avoid the construction of pseudo-rearrangements (as in [1, pages 11 and 12]). Lemma 2.6. Let Ω ⊆ Rn be a domain and u ∈ Y01,2 (Ω) with u ≥ 0. For any f ∈ L1 (Ω), the function ˆ Rf,u (τ ) = |f | [u>u∗ (τ )]  is absolutely continuous in (0, ∞), and if Ψf,u = Rf,u ≥ 0 is its derivative, then for any p > 1 and q ≥ 1, Ψf,u Lp,q (0,∞) ≤ Cp,q f Lp,q (Ω) . (2.13) Moreover, for almost every τ > 0,  (−u∗ ) (τ ) ≤ Cn τ n −1 1 Ψ|∇u|2 ,u (τ ). (2.14) Proof. Let u◦ be the function defined in [4, page 660]; that is, we define u◦ (τ ) = sup{t : μu (t ) ≥ τ }, where μu coincides with our definition of the distribution function (2.2), since u ≥ 0. We will show that u∗ = u◦ , so that Rf,u coincides with the function in [4, Lemma 3.1]. Then, the proof of the same lemma (where for absolute continuity of u∗ , we will use Lemma 2.4) will show absolute continuity of Rf,u . Moreover, (2.13) for q ≥ 1 will follow from [4, (3.12), page 661] and (2.10), while the case q < 1 will follow from [26, Theorem 1.4.19] (note that, in the terminology of the aforementioned theorem, the operator Mf in (2.5) is quasilinear, from [4, (3.2), page 660]. Note first that, from the definitions, u∗ (τ ) ≤ u◦ (τ ) for all τ . If now u∗ (τ ) < u◦ (τ ), then we can find t < t with μu (t) ≤ τ and μu (t ) ≥ τ . Since μu is decreasing, this will imply that μu (t ) ≤ μu (t), hence μu is equal 188 G. Sakellaris / J. Math. Pures Appl. 156 (2021) 179–214 to τ in [t, t ], which is a contradiction with continuity of u∗ from Lemma 2.4. This shows that u◦ = u∗ , and completes the proof of the first part. ˆ To show estimate (2.14), set Tu (t) = |∇u|2 , and note that, from [3, estimate (40)], [u>t] ⎛ ⎞ ˆ ⎜ d ⎟ Cn ≤ μu (t) n −2 (−μu (t)) ⎝− |∇u|2 ⎠ = Cn μu (t) n −2 (−μu (t))(−Tu ) (t), 2 2 (2.15) dt [u>t] for every t ∈ F , where F ⊆ (0, supΩ u) has full measure (this estimate is shown for u ∈ W01,2 (Ω), but the same proof as in [3, pages 711-712] gives the result for u ∈ Y01,2 (Ω)). Consider now the splitting (0, ∞) = Gu ∪ Du ∪ Nu in Lemma 2.5. We claim that u∗ (τ ) ∈ F for almost every τ ∈ Gu : if this is not the case, then there exists G ⊆ Gu , with positive measure, such that if τ ∈ G, then u∗ (τ ) ∈ / F . Then, the set u∗ (G) has measure zero and u∗ is differentiable at every point τ ∈ G, hence [29, Theorem 1] shows that (u∗ ) (τ ) = 0 for almost every τ ∈ G. However, u∗ (τ ) = 0 for every τ ∈ Gu from Lemma 2.5, which is a contradiction with the fact that G has positive measure. So, u∗ (τ ) ∈ F for almost every τ ∈ Gu , and for those τ , plugging u∗ (τ ) in (2.15), we obtain that Cn ≤ μu (u∗ (τ )) n −2 (−μu (u∗ (τ ))(−Tu ) (u∗ (τ )), 2 and using (2.12), we obtain that (−u∗ ) (τ ) ≤ Cn τ n −2 (−Tu ) (u∗ (τ )), 2 for almost every τ ∈ Gu . Moreover, R|∇u|2 ,u = Tu ◦ u∗ , and since Tu is differentiable at u∗ (τ ) for almost every τ ∈ Gu , multiplying the last estimate with (−u∗ ) (τ ) implies that ((−u∗ ) (τ )) ≤ Cn τ n −2 (−Tu ) (u∗ (τ )) · (−u∗ ) (τ ) = Cn τ n −2 R|∇u|  2 2 2 2 ,u (τ ), which shows that (2.14) holds for almost every τ ∈ Gu . On the other hand, (u∗ ) (τ ) = 0 when τ ∈ Du , so (2.14) also holds for almost every τ ∈ Du . Since Nu has measure zero, (2.14) holds almost everywhere in (0, ∞), which completes the proof.  Finally, the following is a Grönwall type lemma, which we prove in the setting that will appear in Lemma 3.1. The reason for this is that the function g2 g3 will not necessarily be integrable close to 0, which turns out to be inconsequential. Lemma 2.7. Let M > 0, and suppose that f, g1 , g2 , g3 are functions defined in (0, M ), with g2 , g3 ≥ 0. Assume that g2 g3 is locally integrable in (0, M ), g3 f ∈ L1 (0, M ) and ⎛ ⎞ ⎛τ ⎞ ε ˆτ ˆ0 ˆ exp ⎝− g2 g3 ⎠ g1 (τ )g3 (τ ) ∈ L1 (0, M ), exp ⎝ g2 g3 ⎠ g3 f −−−→ 0, ε→0 τ0 ε 0 ˆτ for some τ0 ∈ (0, M ). If f (τ ) ≤ g1 (τ ) + g2 (τ ) g3 f in (0, M ), then, for every τ ∈ (0, M ), 0 ⎛ τ ⎞ ˆτ ˆ f (τ ) ≤ g1 (τ ) + g2 (τ ) g1 (σ)g3 (σ) exp ⎝ g2 (ρ)g3 (ρ) dρ⎠ dσ. 0 σ G. Sakellaris / J. Math. Pures Appl. 156 (2021) 179–214 189 ´τ ´τ Proof. Define G(τ ) = 0 g3 f and H(τ ) = τ0 g2 g3 , then G is absolutely continuous in [0, M ] and H is locally absolutely continuous in (0, M ). Then, we have that (e−H G) = e−H (G − H  G) ≤ e−H g1 g3 , and since e−H G is absolutely continuous in (ε, τ ) for 0 < ε < τ < M , we integrate to obtain ˆτ −H(τ ) −H(ε) e G(τ ) − e G(ε) ≤ e−H g1 g3 . ε The proof is complete after letting ε → 0 and plugging the last estimate in the original estimate for f .  3. Global estimates 3.1. The main estimate The following lemma is the main estimate that will lead to global boundedness for subsolutions. The test function we use comes from [4, page 663, proof of Theorem 2.1] and it is a slight modification of test functions that have been used in the literature before (see, for example, the references for the decreasing rearrangements technique in the introduction). Lemma 3.1. Let Ω ⊆ Rn be a domain. Let A be uniformly elliptic and bounded in Ω, with ellipticity λ. n Let also b, f ∈ Ln,1 (Ω) and g ∈ L 2 ,1 (Ω). There exists ν = νn,λ such that, if c = c1 + c2 ∈ Ln,∞ (Ω) with c1 ∈ Ln,q (Ω) for some q < ∞ and c2 n,∞ < ν, then for any subsolution u ∈ Y01,2 (Ω) to − div(A∇u + bu) + c∇u ≤ − div f + g in Ω, and any τ ∈ (0, 1),  −v  (τ ) ≤ C1 τ n −1 Ψ|f |2 (τ ) + C1 τ n −1 Mg (τ ) 1 2 ˆτ  n−2 σ n−2 q 1 3 1 1 C 2 c1 + C1 e n,q τ Ψ|f |2 (σ)Ψ|c|2 (σ) dσ 0 ˆτ  τ n−2 σ n − 2 Mg (σ) Ψ|c|2 (σ) dσ q 1 3 2 1 + C1 eC2 c1 n,q 0  ˆτ  n −1 n−2 σ n − 2 v(σ) Ψ|b|2 (σ)Ψ|c|2 (σ)σ n − 2 dσ, (3.1) 1 q 1 3 1 1 1 1 C 2 c1 + C1 v(τ )τ Ψ|b|2 (τ ) + C1 e n,q τ 0 where C1 depends on n, λ, C2 depends on n, λ, q, and where v = (u+ )∗ is the decreasing rearrangement of u+ , Mg is as in (2.5), and Ψ|b|2 = Ψ|b|2 ,u+ , Ψ|c|2 = Ψ|c|2 ,u+ , Ψ|f |2 = Ψ|f |2 ,u+ are defined in Lemma 2.6. Proof. Fix τ > 0 such that v(τ ) > 0. From right continuity of v (see [26, Proposition 1.4.5, (10)], for h > 0 small, we have that v(τ + h) > 0. Consider now the function ⎧ ⎪ ⎨ 0, 0 ≤ u+ ≤ v(τ + h) ψ = u − v(τ + h), v(τ + h) < u+ ≤ v(τ ) ⎪ ⎩ v(τ ) − v(τ + h) u+ > v(τ ). 190 G. Sakellaris / J. Math. Pures Appl. 156 (2021) 179–214 Since ψ ∈ W01,2 (Ω), ψ ≥ 0 and ψ is supported on a set of finite measure, an approximation argument using [1, Lemma 2.2] and Hölder’s inequality (2.9) shows that we can use it as a test function in (2.11). Then, from ellipticity of A, ˆ ˆ ˆ λ |∇u| ≤ v(τ ) 2 |b∇u| + (v(τ ) − v(τ + h)) |c∇u| [v(τ +h)v(τ +h)] ˆ ˆ + |f ∇u| + (v(τ ) − v(τ + h)) |g|. [v(τ +h)v(τ +h)] Letting Ψ(τ ) = Ψ|∇u|2 ,u+ (as in Lemma 2.6), dividing by h, using the Cauchy-Schwartz inequality and letting h → 0, we obtain that   ˆ Ψ(τ ) ≤ Cλ v(τ ) Ψ|b|2 (τ ) Ψ(τ ) + Cλ (−v  )(τ ) |c∇u| [u>v(τ )]   ˆ + Cλ Ψ|f |2 (τ ) Ψ(τ ) + Cλ (−v  (τ )) |g|, (3.2) [u>v(τ )] where we also used continuity of the functions R|c∇u|,u+ and R|g|,u+ , from Lemma 2.6. Moreover, from absolute continuity of R|c∇u|,u+ and the Cauchy-Schwartz inequality, we obtain ˆ ˆτ ˆτ  √ |c∇u| = Ψ|c∇u|,u+ ≤ Ψ|c|2 Ψ. (3.3) [u>v(τ )] 0 0 Let now μ be the distribution function of u+ , and consider the decomposition (0, ∞) = Gu+ ∪ Du+ ∪ Nu+ from Lemma 2.5. Then, for τ ∈ Gu+ , the Hardy-Littlewood inequality (see, for example, [30, page 44, Theorem 2.2]) and (2.12) show that ˆ ˆ )) μ(v(τ ˆτ   ∗  (−v (τ )) |g| ≤ (−v (τ )) g = (−v (τ )) g ∗ = τ (−v  (τ ))Mg (τ ). [u>v(τ )] 0 0 On the other hand, if τ ∈ Nu+ , then −v  (τ ) = 0, and since Nu+ has measure 0, the last estimate holds almost everywhere. Hence, plugging the last estimate and (3.3) in (3.2), we obtain that   ˆτ  √  Ψ(τ ) ≤ Cλ v(τ ) Ψ|b|2 (τ ) Ψ(τ ) + Cλ (−v )(τ ) Ψ|c|2 Ψ 0   + Cλ Ψ|f |2 (τ ) Ψ(τ ) + Cλ τ (−v  (τ ))Mg (τ ).  Let τ such that Ψ(τ ) > 0. Then, dividing the last estimate by Ψ(τ ) and using (2.14),  ˆτ  Cλ (−v  )(τ )  √ Ψ(τ ) ≤ Cλ v(τ ) Ψ|b| (τ ) + 2  Ψ|c|2 Ψ Ψ(τ ) 0  Cλ τ (−v  (τ ))Mg (τ ) + Cλ Ψ|f |2 (τ ) +  Ψ(τ ) G. Sakellaris / J. Math. Pures Appl. 156 (2021) 179–214 191  1  ≤C Ψ|f |2 (τ ) + Cτ n Mg (τ ) + Cv(τ ) Ψ|b|2 (τ ) ˆτ  √ n −1 1 + Cτ Ψ|c|2 Ψ, 0 where C = Cn,λ . On the other hand, the last estimate holds also when Ψ(τ ) = 0, hence it holds for almost every τ > 0. Note now that, from subadditivity of Ψ and Lemma 2.1 (since we can assume that q > 1) for any ε > 0, ˆτ  ˆτ  ˆτ  n −1 −1 Ψ|c1 |2 (ρ) dρ + ρ n −1 Ψ|c2 |2 (ρ) dρ 1 1 1 ρ Ψ|c|2 (ρ) dρ ≤ ρ n σ σ σ  q τ   ≤ ε ln + Cq,ε  Ψ|c1 |2  σ n,q ˆ τ     + ρ n −1  Ψ|c2 |2  ρ− n dρ 1 1 n,∞ σ τ τ ≤ ε ln + Cn,q,ε c1 qn,q + Cn c2 n,∞ ln . σ σ We choose ε = εn,λ and νn,λ such that C εn,λ +CCn νn,λ ≤ 1 2 − n1 ; then, we will have that ⎛ ⎞ ˆτ  2−n 1 1 τ exp ⎝C ρ n −1 Ψ|c|2 (ρ) dρ⎠ ≤ eC2 1 q c1 n,q , (3.4) σ σ ∗ where C2 depends on n, q and λ. Then, using that v ∈ L2 (0, ∞), (3.4) and Lemma 2.6, it is straightforward to check that the hypotheses of Grönwall’s lemma (Lemma 2.7) are satisfied, hence we obtain that   1  Ψ(τ ) ≤ C Ψ|f |2 (τ ) + Cτ n Mg (τ ) + Cv(τ ) Ψ|b|2 (τ ) ⎛ τ ⎞ ˆτ   ˆ  + Cτ n −1 Ψ|f |2 (σ) Ψ|c|2 (σ) exp ⎝C ρ n −1 Ψ|c|2 (ρ) dρ⎠ dσ 1 1 0 σ ⎛ ⎞ ˆτ  ˆτ  n −1 σ Mg (σ) Ψ|c|2 (σ) exp ⎝C n −1 Ψ|c|2 (ρ) dρ⎠ dσ 1 1 1 + Cτ n ρ 0 σ ⎛ τ ⎞ ˆτ   ˆ  n −1 v(σ) Ψ|b|2 (σ) Ψ|c|2 (σ) exp ⎝C ρ n −1 Ψ|c|2 (ρ) dρ⎠ dσ, 1 1 + Cτ 0 σ √ where C = Cn,λ . Finally, using (2.14) to bound Ψ from below, and (3.4), the proof is complete.  3.2. The maximum principle Using Lemma 3.1, we now show global boundedness of subsolutions. Proposition 3.2. Let Ω ⊆ Rn be a domain. Let A be uniformly elliptic and bounded in Ω, with ellipticity λ. n Let also b, f ∈ Ln,1 (Ω), d, g ∈ L 2 ,1 (Ω), and suppose that c = c1 + c2 ∈ Ln,∞ (Ω) with c1 ∈ Ln,q (Ω) for some q < ∞ and c2 n,∞ < ν, where ν = νn,λ appears in Lemma 3.1. 192 G. Sakellaris / J. Math. Pures Appl. 156 (2021) 179–214 There exists τ0 ∈ (0, ∞), depending on b, c1 , c2 and d such that, for any subsolution u ∈ Y 1,2 (Ω) to − div(A∇u + bu) + c∇u + du ≤ − div f + g we have that sup u+ ≤ C sup u+ + Cv(τ0 ) + Cf n,1 + Cg n2 ,1 , (3.5) Ω ∂Ω where C depends on n, q, λ, bn,1 , c1 n,q and d n2 ,1 . In particular, for any p > 0, ⎛ ⎞ p1 ˆ −1/p ⎝ sup u+ ≤ C sup u+ + Cτ0 |u+ |p ⎠ + Cf n,1 + Cg n2 ,1 . (3.6) Ω ∂Ω Ω Proof. If s = sup∂Ω u+ ∈ (0, ∞), then for every s > s, − div(A∇(u − s ) + b(u − s )) + c∇(u − s ) + d(u − s ) ≤ − div(f − s b) + g − s d, and (u − s )+ ∈ Y01,2 (Ω); hence, we can assume that s = 0, so u+ ∈ Y01,2 (Ω). Consider the function e from Lemma 2.3 that solves the equation div e = d in Rn . Then, if we define b = b − e and c = c − e, u is a subsolution to − div(A∇u + b u) + c ∇u ≤ − div f + g. c1 q Set c1 = c1 − e, then c = c1 + c2 . Let C1 , C2 be the constants in Lemma 3.1, and denote C1 eC2 n,q by C0 . Moreover, set  H(τ ) = C1 τ n −1 Ψ|f |2 (τ ) + C1 τ n −1 Mg (τ ) 1 2 ˆτ  + C0 τ n − 2 σn−2 1 3 1 1 Ψ|f |2 (σ)Ψ|c |2 (σ) dσ (3.7) 0 ˆτ  n−2 σ n − 2 Mg (σ) Ψ|c |2 (σ) dσ. 1 3 2 1 + C0 τ 0 From Lemma 2.6, we have that          Ψ|f |2  ≤ Cn f n,1 ,  Ψ|c |2  ≤ Cn c n,∞ . (3.8) n,1 n,∞ Then, since 1 n − 3 2 < −1, changing the order of integration and using (2.9) and (3.8), we have ˆ∞ ˆτ  ˆ∞  n−2 n−2 Ψ|f |2 (σ)Ψ|c |2 (σ) dσ dτ ≤ Cn τ n −1 Ψ|f |2 (σ)Ψ|c |2 (σ) dσ 1 3 1 1 2 τ σ 0 0 0 ≤ Cn c n,∞ f n,1 , and also, using (2.10), we obtain that G. Sakellaris / J. Math. Pures Appl. 156 (2021) 179–214 193 ˆ∞ ˆτ  ˆ∞  n−2 n−2 Mg (σ) Ψ|c |2 (σ) dσ ≤ Cn σ n −1 Mg (σ) Ψ|c |2 (σ) dσ dτ 1 3 2 1 3 τ σ 0 0 0 ≤ Cn c n,∞ g n2 ,1 . The last two estimates and the definition of H in (3.7) imply that ˆ∞   H ≤ C(c n,∞ + 1) f n,1 + g n2 ,1 , (3.9) 0 where C depends on n, q, λ and c1 n,q . Set now   R(τ ) = C1 τ n −1 Ψ|b |2 (τ ), G(σ) = C1 σ n −1 1 2 Ψ|b |2 (σ)Ψ|c |2 (σ). Then, if c1  = c1 n,q , (3.1) shows that ˆτ  C2 c1 n−2 v(σ)σ 2 − n G(σ) dσ, q 1 3 1 1 −v (τ ) ≤ H(τ ) + R(τ )v(τ ) + e τ 0 ´∞ as long as c2 n,∞ < νn,λ . Since also 0 R ≤ Cn,λ b n,1 from Lemma 2.6, we obtain that ´τ  ´τ − e 0 R v =e 0 R (−v  − Rv) ⎛ ⎞ ´τ ˆτ ⎝H(τ ) + eC2 c1 τ n−2 v(σ)σ 2 − n G(σ) dσ ⎠ q 1 3 1 1 ≤e 0 R (3.10) 0 ˆτ Cn,λ b Cn,λ b c1 n−2 v(σ)σ 2 − n G(σ) dσ. q 1 3 1 1 ≤e n,1 H(τ ) + e n,1 +C2 τ 0 Set B = exp (Cn,λ b n,1 ) and C  = exp (Cn,λ b n,1 + C2 c1 q ), and let τ2 > τ1 > 0. Then v is absolutely continuous in (τ1 , τ2 ), from Lemma 2.4; hence, integrating (3.10) in (τ1 , τ2 ), we obtain that ´τ ´τ ˆτ2 ˆτ2 ˆτ  τ n − 2 v(σ)σ 2 − n G(σ) dσdτ 1 3 1 1 v(τ1 ) ≤ e 1 R 2 R e 0 0 v(τ2 ) + B H +C τ1 τ1 0 (3.11) ˆτ2 ˆτ ≤ B (v(τ2 ) + H1 ) + C  τ n − 2 v(σ)σ 2 − n G(σ) dσdτ. 1 3 1 1 τ1 0 Using Fubini’s theorem, the last integral is equal to ˆτ1 ˆτ2 ˆτ2 ˆτ2 n−2 2−n τ n − 2 v(σ)σ 2 − n G(σ) dτ dσ 1 3 1 1 1 3 1 1 τ σ G(σ) dτ dσ + 0 τ1 τ1 σ ˆτ1 ˆτ2 n−2 1 1 2−n 1 1 ≤ Cn τ1 v(σ)σ G(σ) dσ + Cn v(σ)G(σ) dσ, 0 τ1 194 G. Sakellaris / J. Math. Pures Appl. 156 (2021) 179–214 therefore, plugging the last estimate in (3.11), and using that v is decreasing, we obtain ˆτ1 ˆτ2 n−2 1 1  2−n 1 1  v(τ1 ) ≤ B (v(τ2 ) + H1 ) + C Cn τ1 v(σ)σ G(σ) dσ + C Cn v(τ1 ) G(σ) dσ. (3.12) 0 τ1 Consider now τ0 > 0 such that ˆτ0 1 G≤ ; (3.13) 2C  Cn 0 note that such τ0 always exists, since G ∈ L1 (0, ∞) from Lemma 2.6. Then, if 0 < τ1 ≤ τ0 , setting τ2 = τ0 and plugging (3.13) in (3.12) we obtain that ˆτ1 n−2 1 1  v(σ)σ 2 − n G(σ) dσ. 1 1 v(τ1 ) ≤ 2B (v(τ0 ) + H1 ) + 2C Cn τ1 0 Then, for τ1 ∈ (0, τ0 ), the hypotheses of Lemma 2.7 are satisfied, and we obtain that ˆτ1 ´τ n−2 1 1   2B (v(τ0 ) + H1 ) σ 2 − n G(σ)e2C 1 1 v(τ1 ) ≤ 2B (v(τ0 ) + H1 ) + 2C Cn τ1 Cn 1 G σ dσ 0 ˆτ1  2C  Cn G ≤ 2B (v(τ0 ) + H1 ) + 4C Cn B (v(τ0 ) + H1 ) e 1 G(σ) dσ. 0 This estimate holds for every 0 < τ1 ≤ τ0 , as long as (3.13) holds. Then, letting τ1 → 0+ , and using the definition of B and Lemma 2.3, we obtain that lim v(τ1 ) ≤ 2B (v(τ0 ) + H1 ) τ1 →0+    ≤ exp Cn,λ bn,1 + d n2 ,1 (v(τ0 ) + H1 ) , as long as c2 n,∞ < νn,λ and (3.13) hold. Combining with (3.9) then shows (3.5), and (3.6) follows from the fact that v is decreasing and (2.4).  As a corollary, we obtain the following maximum principle, which generalizes [1, Proposition 7.5]. From Remark 6.4, to have such an estimate with constants depending only on the norms of the coefficients for arbitrary b ∈ Ln,1 requires that c should have small norm; hence, we will assume that c belongs to Ln,∞ and has small norm. Proposition 3.3. Let Ω ⊆ Rn be a domain. Let A be uniformly elliptic and bounded in Ω, with ellipticity λ, n and let b, f ∈ Ln,1 (Ω), g ∈ L 2 ,1 (Ω), with bn,1 ≤ M . n There exists β = βn,λ,M > 0 such that, if c ∈ Ln,∞ (Ω) and d ∈ L 2 ,1 (Ω) with cn,∞ < β and d n2 ,1 < β, then for every subsolution u ∈ Y 1,2 (Ω) to − div(A∇u + bu) + c∇u + du ≤ − div f + g in Ω, we have that G. Sakellaris / J. Math. Pures Appl. 156 (2021) 179–214 195 sup u ≤ C sup u+ + Cf n,1 + Cg n2 ,1 , Ω ∂Ω where C depends on n, λ and M . Proof. Assume that cn,∞ < β and d n2 ,1 < β, for β to be chosen later. Consider the νn,λ from Lemma 3.1, and take c1 ≡ 0 and q = 1 in Proposition 3.2. We will take β ≤ νn,λ , so it is enough to show that we can take τ0 = ∞ in (3.5), since limτ →∞ v(τ ) = 0. Hence, from (3.13), and the definitions of C  and e from the proof of Proposition 3.2, it will be enough to have that ˆ∞  σ n −1 2 Ψ|b−e|2 (σ)Ψ|c−e|2 (σ) dσ ≤ C exp (−Cb − en,1 − Cen,1 ) , (3.14) 0 where C depends on n and λ only. We first bound the left hand side from above using Lemmas 2.6 and 2.3, to obtain ˆ∞      σ n −1 2 Ψ|b−e|2 (σ)Ψ|c−e|2 (σ) dσ =  Ψ|b−e|2 Ψ|c−e|2  n 2 ,1 0         ≤ C  Ψ|b−e|2   Ψ|c−e|2  n,1 n,∞ ≤ Cb − en,1 c − en,∞ ≤ C(M + β)β, while −Cb − en,1 − Cen,1 ≥ −Cbn,1 − Cen,1 ≥ −CM − Cβ. From the last two estimates, (3.14) will be satisfied as long as C(M + β)βeCM +Cβ ≤ 1. So, choosing β > 0 depending on n, λ and M , such that the last estimate is satisfied and also β ≤ νn,λ completes the proof.  In addition, we also obtain the following maximum principle, which concerns perturbations of the operator − div(A∇u) + c∇u. Proposition 3.4. Let Ω ⊆ Rn be a domain. Let A be uniformly elliptic and bounded in Ω, with ellipticity λ, and let q < ∞, c = c1 + c2 ∈ Ln,∞ (Ω), with c2 n,∞ < ν and c1 n,q ≤ M , where ν = νn,λ appears in n Lemma 3.1. Assume also that f ∈ Ln,1 (Ω), g ∈ L 2 ,1 (Ω). n There exists γ = γn,q,λ,M > 0 such that, if b ∈ Ln,1 (Ω) and d ∈ L 2 ,1 (Ω) with bn,1 < γ and d n2 ,1 < γ, then for any subsolution u ∈ Y 1,2 (Ω) to − div(A∇u + bu) + c∇u + du ≤ − div f + g in Ω, we have sup u ≤ C sup u+ + Cf n,1 + Cg n2 ,1 , Ω ∂Ω where C depends on n, q, λ and M . 196 G. Sakellaris / J. Math. Pures Appl. 156 (2021) 179–214 Proof. As in the proof of Corollary 3.3, we will take γ ≤ νn,λ , and it will be enough to have that ˆ∞    σ n −1 2 Ψ|b−e|2 (σ)Ψ|c−e|2 (σ) dσ ≤ C exp −Cb − en,1 − Cc1 − eqn,q , 0 whenever bn,1 < γ and d n2 ,1 < γ, and where C = Cn,q,λ . Then, a similar argument as in the proof of Proposition 3.3 completes the proof.  4. Local boundedness 4.1. The first step: all coefficients are small The first step to obtain the Moser estimate is via a coercivity assumption, which we now turn to. The 2n following lemma is standard, and we only give a sketch of its proof. We will set 2∗ = n+2 . Lemma 4.1. Let Ω ⊆ Rn be a domain, and A be uniformly elliptic and bounded in Ω, with ellipticity λ. There n exists θ = θn,λ > 0 such that, if b ∈ Ln,1 (Ω), c ∈ Ln,∞ (Ω) and d ∈ L 2 ,1 (Ω) with bn,1 ≤ θ, cn,∞ ≤ θ and d n2 ,1 ≤ θ, then the operator Lu = − div(A∇u + bu) + c∇u + du is coercive, and every solution v ∈ W01,2 (Ω) to the equation Lu = − div F +G for F ∈ L2 (Ω) and G ∈ L2∗ (Ω) satisfies the estimate ∇vL2 (B2r ) ≤ Cn,λ F L2 (Ω) + Cn,λ GL2∗ (Ω) . (4.1) Also, if Ω = B2r and w ∈ W 1,2 (B2r ) is a subsolution to − div(A∇w + bw) + c∇w + dw ≤ 0, then ˆ ˆ C |∇w|2 ≤ 2 |w+ |2 , (4.2) r Br B2r where C depends on n, λ and A∞ . Moreover, for any subsolution u ∈ W 1,2 (B2r ) to − div(A∇u) + c∇u ≤ 0 in B2r and α ∈ (1, 2), we have that ⎛ ⎞ 12 C ⎝ sup u ≤ |u+ |2 ⎠ , (4.3) Br (α − 1)n/2 Bαr where C depends on n, λ and A∞ . Proof. We first show (4.3), following the lines of the proof of [2, Theorem 8.17]: if φ is a smooth cutoff function, then an approximation argument using [1, Lemma 2.2] and Hölder’s inequality (2.9) shows that we can use u+ φ2 as a test function, and we obtain ˆ ˆ ˆ A∇u ∇u · φ ≤ −2 + + 2 A∇u ∇φ · u φ − + + c∇u+ · u+ φ2 B2r B2r B2r (4.4) ≤ Cφ∇u+ L2 (B2r ) u+ ∇φL2 (B2r )   + cu+ φL2 (B ) φ∇u+ L2 (B2r ) . 2r G. Sakellaris / J. Math. Pures Appl. 156 (2021) 179–214 197 Then, assuming that cn,∞ ≤ θ, for θ to be chosen later, using Hölder’s estimate (2.9) we have  +  cu φ ≤ Cn cn,∞ u+ φL2∗ ,2 (B2r ) ≤ Cn θu+ φL2∗ ,2 (B2r ) , L2 (B2r ) and combining with [1, Lemma 2.2], we have  +  cu φ ≤ Cn θ∇(u+ φ)L2 (B2r ) L2 (B2r ) (4.5) ≤ Cn θφ∇u+ L2 (B2r ) + Cn θu+ ∇φL2 (B2r ) . So, choosing θ such that Cn θ < λ4 , and plugging in (4.4), we obtain that ˆ ˆ |φ∇u+ |2 ≤ C |u+ ∇φ|2 , B2r B2r where C depends on n, λ and A∞ . This estimate corresponds to [2, (8.53), page 196], and following the lines of the argument on [2, pages 196 and 197] we obtain that ⎛ ⎞ 12 sup u ≤ C ⎝ |u+ |2 ⎠ , Br B2r where C depends on n, λ and A∞ . To complete the proof of (4.3) note that, for all x ∈ Br , the last estimate shows that ⎛ ⎞ 12 ⎛ ⎞ 12 ˆ ⎜ ⎟ C ⎝ sup u≤C⎝ |u+ |2 ⎠ ≤ |u+ |2 ⎠ , B α−1 r (x) (α − 1)n/2 rn/2 2 B(α−1)r (x) Bαr since B(α−1)r (x) ⊆ Bαr , and considering the supremum for x ∈ Br shows (4.3). Finally, coercivity of L, (4.1) and (4.2) follow via a combination of the procedure as in (4.4) and (4.5), where for (4.1) we use v as a test function, and for (4.2) we use w+ φ2 as a test function.  We now turn to local boundedness when all the lower order coefficients have small norms. Lemma 4.2. Let A be uniformly elliptic and bounded in B2r , with ellipticity λ. There exists θ = θn,λ >0 (B2r ) and d ∈ L 2 (B2r ) with bn,1 ≤ θ , cn,∞ ≤ θ and d n2 ,1 ≤ θ ,   n such that, if b ∈ L (B2r ), c ∈ L n,1 n,∞ ,1 then for any subsolution u ∈ W (B2r ) to − div(A∇u + bu) + c∇u + du ≤ 0, 1,2 ⎛ ⎞ 12 sup u+ ≤ C ⎝ |u+ |2 ⎠ , Br B2r where C depends on n, λ and A∞ . Proof. Consider the θn,λ that appears in Lemma 4.1. We will take θ ≤ θn,λ , so that the operator is coercive. Then, if u is a subsolution to Lu ≤ 0, the proof of [31, Theorem 3.5] implies that u+ is a subsolution to Lu+ ≤ 0; therefore, we can assume that u ≥ 0. 198 G. Sakellaris / J. Math. Pures Appl. 156 (2021) 179–214 Assume first that b, c, d are bounded in B2r , then [2, Theorem 8.17] shows that sup u < ∞. (4.6) Br Let 4 ≤ 1 η < η  ≤ 12 . From coercivity of the operator L0 u = − div(A∇u) + c∇u, and since div(bu) − du ∈ ∗ −1,2 W (Bη r ) = W01,2 (Bη r ) , the Lax-Milgram theorem shows that there exists v ∈ W01,2 (Bη r ) such that − div(A∇v) + c∇v = div(bu) − du. If β is as in Proposition 3.3, taking θ ≤ βn,λ,θn,λ the same proposition shows that sup v ≤ Cn,λ buLn,1 (Bη r ) + Cn,λ duL n2 ,1 (B ≤ Cn,λ θ sup u, (4.7) η r ) Bη  r Bη  r since u ≥ 0. In addition, from the Sobolev inequality, estimate (4.1) and the Hölder inequality, vL2∗ (Bη r ) ≤ Cn ∇vL2 (Bη r ) (4.8) ≤ Cn,λ buL2 (Bη r ) + Cn,λ duL2∗ (Bη r ) ≤ Cn,λ uL2∗ (Bη r ) . Moreover, the function w = u − v is a subsolution to − div(A∇w) + c∇w ≤ 0, so (4.3) implies that ⎛ ⎞ 12 C ⎜ ⎟ sup w ≤  ⎝ |w+ |2 ⎠ Bηr ( ηη − 1)n/2 Bη  r ⎛ ⎞ 12 ⎛ ⎞ 12 C ⎜ ⎟ C ⎜ ⎟ ≤ ⎝ |u|2 ⎠ + ⎝ |v|2 ⎠ (η  − η)n/2 (η  − η)n/2 Bη  r Bη  r ⎛ ⎞ 21∗ ⎛ ⎞ 21∗ C ⎜ ∗⎟ C ⎜ ∗⎟ ≤ ⎝ |u|2 ⎠ ≤ ⎝ |u|2 ⎠ , (η  − η)n/2 (η  − η)n/2 Bη  r Br/2 where we also used (4.8) for the penultimate estimate, and C depends on n, λ and A∞ . Hence, the definition of w, the last estimate and (4.7) show that ⎛ ⎞ 21∗ C ⎜ ∗⎟ sup u ≤ sup v + sup w ≤ Cn,λ θ sup u + ⎝ |u|2 ⎠ , Bηr Bηr Bηr Bη  r (η  − η)n/2 Br/2 where C depends on n, λ and A∞ . We now set ηN = 12 − 4−N and apply the previous estimate for η = ηN and η  = ηN +1 . Then, ⎛ ⎞ 21∗ ⎜ ∗⎟ sup u ≤ Cn,λ θ sup u + 2nN C ⎝ |u|2 ⎠ . BηN r BηN +1 r Br/2 Inductively, this shows that, for any N ∈ N, G. Sakellaris / J. Math. Pures Appl. 156 (2021) 179–214 199 ⎛ ⎞ 21∗  N ∗ sup u ≤ (Cn,λ θ )N sup u + C (Cn,λ θ )i−1 2ni · ⎝ |u|2 ⎠ . Bη1 r BηN +1 r i=1 Br We will consider θ such that Cn,λ θ ≤ 12 . Then, letting N → ∞ and using (4.6), we obtain that ⎛ ⎞ 21∗ ∞  ⎜ ∗⎟ (2n Cn,λ θ ) i−1 sup u ≤ C ·⎝ |u|2 ⎠ , Br/4 i=1 Br/2 and choosing θ that also satisfies 2n Cn,λ θ ≤ 1 2 shows that ⎛ ⎞ 21∗ ⎛ ⎞ 12 ⎜ ∗ ⎟ sup u ≤ C ⎝ |u|2 ⎠ ≤C⎝ |u|2 ⎠ , (4.9) Br/4 Br/2 B2r where we used (4.2) and the Sobolev inequality for the last estimate, and where C depends on n, λ and A∞ . In the case that b, c, d are not necessarily bounded, let bj be the coordinate functions of b, and define bN having coordinate functions bjN = bj χ[|bj |≤N ] for N ∈ N; define also similar approximations cN and dN for c, d respectively. We then have that bN n,1 ≤ bn,1 , and similarly for cN and dN . Since θ ≤ θn,λ , from coercivity in Lemma 4.1 and the Lax-Milgram theorem there exists vN ∈ W01,2 (Br/2 ) that solves the equation − div(A∇vN + bN vN ) + cN ∇vN + dN vN = − div(A∇u + bN u) + cN ∇u + dN u in Br/2 . Then, from (4.1), ∇vN L2 (Br/2 ) ≤ CA∇u + bN uL2 (Br/2 ) + CcN ∇u + dN uL2∗ (Br/2 ) C (4.10) ≤ uL2 (B2r ) , r where we also used (4.2) and Hölder’s inequality for the last estimate. So, (vN ) is bounded in W01,2 (Br/2 ), hence from Rellich’s theorem there exists a subsequence (vN  ) such that n vN  → v0 weakly in W01,2 (Br/2 ) and strongly in L n−2 (Br/2 ), (4.11) vN  (x) → v0 (x) ∀ x ∈ F, where F ⊆ Br/2 is a set with full measure. Note now that wN = u − vN is a solution to − div(A∇wN + bN wN ) + cN ∇wN + dN wN = 0 in Br/2 , and bN , cN and dN are bounded, so (4.9) (where B2r is replaced by Br/2 ) is applicable to w+ ; therefore, for x ∈ FN , where FN ⊆ Br/16 has full measure, ⎛ ⎞ 12 ⎛ ⎞ 12 ⎛ ⎞ 12 ⎜ + 2⎟ ⎜ ⎟ ⎜ 2 ⎟ + wN (x) ≤ sup wN + ≤C⎝ |wN | ⎠ ≤C⎝ u2 ⎠ + C ⎝ vN ⎠ , Br/16 Br/2 Br/2 Br/2 where C depends on n, λ and A∞ . Therefore, for all x ∈ FN , 200 G. Sakellaris / J. Math. Pures Appl. 156 (2021) 179–214 ⎛ ⎞ 12 ⎛ ⎞ 12 ⎜ ⎟ ⎜ 2 ⎟ u(x) = vN (x) + wN (x) ≤ vN (x) + C ⎝ u2 ⎠ + C ⎝ vN ⎠ Br/2 Br/2 ⎛ ⎞ 12 ≤ vN (x) + C ⎝ u2 ⎠ , B2r where we used the Sobolev inequality and (4.10) for the last estimate. ∞ Let now F0 = F ∩ N =1 FN , then F0 ⊆ Br/16 has full measure, and if x ∈ F0 , then letting N  → ∞ in the previous estimate, (4.11) implies that ⎛ ⎞ 12 ⎛ ⎞ 12 u(x) ≤ lim sup vN  (x) + C ⎝ u2 ⎠ = v0 (x) + C ⎝ u2 ⎠ , (4.12) N  →∞ B2r B2r for all x ∈ F0 . Finally, note that vN  is a subsolution to − div(A∇vN + bN vN ) + cN ∇vN + dN vN ≤ − div((bN − b)u) + (cN − c)∇u + (dN − d)u n in Br/2 , and since bN  → bN and cN  → cN strongly in L2 (Br/2 ), while dN  → dN strongly in L 2 (Br/2 ), using (4.11) and the variational formulation of subsolutions (2.11) we obtain that v0 is a W01,2 (B2r ) subsolution to − div(A∇v0 + bv0 ) + c∇v0 + dv0 ≤ 0. Hence, since θ ≤ βn,λ,θn,λ , Proposition 3.3 implies that v0 ≤ 0 in Br/2 , and plugging in (4.12) and covering Br with balls of radius r/16 completes the proof.  4.2. The second step: b or c have large norms We now turn to scale invariant estimates with “good” constants when d is small, and either b or c are small as well. We first consider the case of small c and assume that the right hand side is identically 0, for simplicity; the terms on the right hand side will be added in Proposition 4.4. Lemma 4.3. Let A be uniformly elliptic and bounded in B2r , with ellipticity λ, and b ∈ Ln,1 (B2r ) with bn,1 ≤ M . n There exists θ = θn,λ,M > 0 such that, if c ∈ Ln,∞ (B2r ) and d ∈ L 2 ,1 (B2r ) with cn,∞ < θ and d n2 ,1 < θ, then for any subsolution u ∈ W 1,2 (B2r ) to − div(A∇u + bu) + c∇u + du ≤ 0, we have ⎛ ⎞ 12 sup u ≤ C ⎝ |u+ |2 ⎠ , (4.13) Br B2r where C depends on n, λ, A∞ and M .  Proof. We will proceed by induction on M . Consider the θn,λ and the constant C0 = Cn,λ, A ∞ ≥ 1 that  appear in Lemma 4.2. In addition, for any integer N ≥ 0, set Cn,λ,N = Cn,λ,2N/n θn,λ  ≥ 1, where the last constant appears in Proposition 3.3. G. Sakellaris / J. Math. Pures Appl. 156 (2021) 179–214 201  We claim that, if bLn,1 (B2r ) ≤ 2N/n θn,λ , then there exists θn,λ,N > 0 such that, if we have that cLn,∞ (B2r ) < θn,λ,N and dL n2 ,1 (B ) < θn,λ,N , then 2r ⎛ ⎞ 12  N sup u ≤ 8 nN 2 C0  Cn,λ,i ⎝ |u+ |2 ⎠ . (4.14) Br i=0 B2r  For N = 0, letting θn,λ,0 = θn,λ , the previous estimate holds from Lemma 4.2. Assume now that this estimate holds for some integer N ≥ 0, for some constant θn,λ,N . From Propo-  sition 3.3 there exists βn,λ,N = βn,λ,2N/n θn,λ  > 0 such that, if Ω ⊆ Rn is a domain, A is elliptic in Ω with ellipticity λ, b Ln,1 (Ω) ≤ 2N/n θn,λ  , c Ln,∞ (Ω) < βn,λ,N  and d L n2 ,1 (Ω) < βn,λ,N  , then for any     subsolution v ∈ Y (Ω) to − div(A ∇v + b v) + c ∇v + d v ≤ 0 in Ω, we have that 1,2  sup v ≤ Cn,λ,N sup v + . Ω ∂Ω  We then set θn,λ,N +1 = min{θn,λ,N , βn,λ,N +1 }, and assume that  bLn,1 (B2r ) ≤ 2(N +1)/n θn,λ , cLn,∞ (B2r ) < θn,λ,N +1 , and dL n2 ,1 (B < θn,λ,N +1 . (4.15) 2r ) We will show that, in this case, (4.14) holds for N + 1. To show this, we distinguish between two cases:   bLn,1 (B3r/2 ) ≤ 2N/n θn,λ , and bLn,1 (B3r/2 ) > 2N/n θn,λ . In the first case, let x ∈ Br . Then, since θn,λ,N +1 ≤ θn,λ,N and Br/2 (x) ⊆ B3r/2 , we have that  bLn,1 (Br/2 (x)) ≤ 2N/n θn,λ , cLn,∞ (Br/2 (x)) < θn,λ,N , and dL n2 ,1 (B < θn,λ,N . r/2 (x)) Therefore, from (4.14) for N (in the ball Br/2 (x) instead of B2r ), we have ⎛ ⎞ 12  N ⎜ ⎟ nN  sup u ≤ 8 2 C0 Cn,λ,i ⎝ |u+ |2 ⎠ Br/4 (x) i=0 Br/2 (x) ⎛ ⎞ 12 ⎛ ⎞ 12  N  N +1 2n ⎝ |u+ |2 ⎠ ≤ 8 ⎝ u2 ⎠ , n(N +1) nN   ≤8 2 C0 Cn,λ,i 2 C0 Cn,λ,i i=0 B2r i=0 B2r  where we used that Cn,λ,N +1 ≥ 1 for the last step. So, (4.14) holds for N + 1 in this case. In the second case, let y ∈ ∂B7r/4 . Then Br/4 (y) ⊆ B2r \ B3r/2 , therefore, from Lemma 2.2, bnLn,1 (Br/4 (y)) ≤ bnLn,1 (B2r ) − bnLn,1 (B3r/2 )    < 2N +1 (θn,λ )n − 2N (θn,λ )n = (2N/n θn,λ )n . Moreover, from (4.15), we have cLn,∞ (Br/4 (y)) < θn,λ,N and dL n2 ,1 (B < θn,λ,N , hence (4.14) for N r/4 (y)) (in the ball Br/4 (y) instead of B2r ) implies that ⎛ ⎞ 12 ⎛ ⎞ 12  N ⎜ ⎟  N ⎝ u2 ⎠ . n(N +1) nN   sup u ≤ 8 2 C0 Cn,λ,i ⎝ |u+ |2 ⎠ ≤ 8 2 C0 Cn,λ,i Br/8 (y) i=0 i=0 Br/4 (y) B2r 202 G. Sakellaris / J. Math. Pures Appl. 156 (2021) 179–214 Then, the last estimate, (4.15) and Proposition 3.3 show that ⎛ ⎞ 12  N ⎝ |u+ |2 ⎠ , n(N +1)    sup u ≤ Cn,λ,N +1 sup u ≤ Cn,λ,N +1 · 8 2 C0 Cn,λ,i Br ∂B7r/4 i=1 B2r which shows that (4.14) for N + 1 in this case as well. Therefore, (4.14) holds for any N ∈ N, which completes the proof.  Finally, we show Moser’s estimate allowing right hand sides to the equation, and considering also different Lp norms on the right hand side of the estimate. Proposition 4.4. Let A be uniformly elliptic and bounded in B2r , with ellipticity λ. Let also b ∈ Ln,1 (B2r ) n with bn,1 ≤ M , and p > 0, f ∈ Ln,1 (B2r ), g ∈ L 2 ,1 (B2r ). n There exists ε = εn,λ,M > 0 such that, if c ∈ Ln,∞ (B2r ) and d ∈ L 2 ,1 (B2r ) with cn,∞ < ε and d n2 ,1 < ε, then for any subsolution u ∈ W 1,2 (B2r ) to − div(A∇u + bu) + c∇u + du ≤ − div f + g, we have that ⎛ ⎞ p1 sup u ≤ C ⎝ |u+ |p ⎠ + Cf Ln,1 (B2r ) + CgL n2 ,1 (B , (4.16) 2r ) Br B2r where C depends on n, p, λ, A∞ and M . Proof. Consider the βn,λ,M from Proposition 3.3. If cn,∞ < βn,λ,M and d n2 ,1 < βn,λ,M , any solution u ∈ W01,2 (B2r ) to the equation − div(A∇u + bu) + c∇u + du = 0 in B2r should be identically 0, from Proposition 3.3. Hence, adding a term of the form +Lu to the operator, for some large L > 0 depending only on n, λ, M , the operator becomes coercive, and a combination of the Lax-Milgram theorem and the Fredholm alternative (as in [32, Theorem 4, pages 303-305], for example) shows that there exists a unique v ∈ W01,2 (B2r ) such that − div(A∇v + bv) + c∇v + dv = − div f + g, in B2r . Then, Proposition 3.3 implies that sup |v| ≤ Cf Ln,1 (B2r ) + CgL n2 ,1 (B , (4.17) 2r ) B2r where C depends on n, λ and M . Consider now the θn,λ,M from Lemma 4.3 and set ε = min{βn,λ,M , θn,λ,M }. Then, assuming that cn,∞ < ε and d n2 ,1 < ε, since w = u − v is a subsolution to − div(A∇w + bw) + c∇w + dw ≤ 0, (4.13) implies that ⎛ ⎞ 12 sup w ≤ C ⎝ |w+ |2 ⎠ , (4.18) Br B2r where C depends on n, λ, A∞ and M . Then, (4.16) for p = 2 follows adding (4.17) and (4.18). Finally, in the case p ≥ 2, (4.16) follows from Hölder’s inequality, while in the case p ∈ (0, 2), the proof follows from the argument on [33, pages 80-82].  G. Sakellaris / J. Math. Pures Appl. 156 (2021) 179–214 203 We now turn to the case when c ∈ Ln,q with q < ∞ is allowed to have large norm. Lemma 4.5. Let A be uniformly elliptic and bounded in B2r , with ellipticity λ. Let also q < ∞ and c1 ∈ Ln,q (B2r ) with c1 n,q ≤ M . There exist ξ = ξn,λ > 0 and ζ = ζn,q,λ,M > 0 such that, if b ∈ Ln,1 (B2r ), c2 ∈ Ln,∞ (B2r ) and n d ∈ L 2 ,1 (B2r ) with bn,1 < ζ, c2 n,∞ < ξ and d n2 ,1 < ζ, then for any subsolution u ∈ W 1,2 (B2r ) to − div(A∇u + bu) + (c1 + c2 )∇u + du ≤ 0, we have that ⎛ ⎞ 12 sup u ≤ C ⎝ |u+ |2 ⎠ , Br/4 B2r where C depends on n, q, λ, A∞ and M . Proof. Let Cn ≥ 1 be such that h1 + h2 n,∞ ≤ Cn h1 n,∞ + Cn h2 n,∞ for all h1 , h2 ∈ Ln,∞ (from (2.7)), and Cn,q ≥ 1 be such that hn,∞ ≤ Cn,q hn,q for all h ∈ Ln,q (from (2.8)). Set 1    ξn,λ = min νn,λ , θn,λ > 0, 2Cn   where νn,λ and θn,λ appear in Proposition 3.3 and Lemma 4.2, respectively. For N ≥ 0, set also Cn,q,λ,N = Cn,q,λ,2N/q Cn,q −1 ξn,λ > 1, where the last constant appears in Proposition 3.4, and consider the constant C0 = Cn,λ, A ∞ ≥ 1 that appears in Lemma 4.2. −1 We claim that, for any integer N ≥ 0, if c1 n,q ≤ 2N/q Cn,q ξn,λ , then there exists ζn,q,λ,N such that, if bn,1 < ζn,q,λ,N , c2 n,∞ < ξn,λ and d 2 ,1 < ζn,q,λ,N , then n ⎛ ⎞ 12  N sup u ≤ 8 nN 2 C0  Cn,q,λ,i ⎝ u2 ⎠ . (4.19) Br/4 i=0 B2r For N = 0 we can take ζn,q,λ,0 = ξn,λ , since we then have that  cn,∞ ≤ Cn Cn,q c1 n,q + Cn c2 n,∞ ≤ 2Cn ξn,λ ≤ θn,λ ,   and also bn,1 ≤ θn,λ , d n2 ,1 ≤ θn,λ , therefore (4.19) for N = 0 holds from Lemma 4.2.  Assume now that (4.19) holds for some N ≥ 0, and set ζn,q,λ,N +1 = min{ζn,q,λ,N , γn,q,λ,N +1 }, where  γn,q,λ,N = γn,q,λ,2N/q Cn,q−1 ξn,λ , and the γ appears in Proposition 3.4. We then continue as in the proof of the Lemma 4.3, using Lemma 2.2 for q > n and Proposition 3.4 instead of Proposition 3.3; this shows that −1 (4.19) holds for N + 1 if c1 n,q ≤ 2(N +1)/q Cn,q ξn,λ , as long as bn,1 < ζn,q,λ,N +1 , c2 n,∞ < ξn,λ and d n2 ,1 < ζn,q,λ,N +1 , and this completes the proof.  Finally, we add right hand sides and allow different Lp norms. Proposition 4.6. Let A be uniformly elliptic and bounded in B2r , with ellipticity λ, and q < ∞, c1 ∈ Ln,q (B2r ) n with c1 n,q ≤ M . Let also p > 0 and f ∈ Ln,1 (B2r ), g ∈ L 2 ,1 (B2r ). There exist ξ = ξn,λ > 0 and δ = δn,q,λ,M > 0 such that, if b ∈ Ln,1 (B2r ), c2 ∈ Ln,∞ (B2r ) and n d ∈ L 2 ,1 (B2r ) with bn,1 < δ, c2 n,∞ < ξ and d n2 ,1 < δ, then for any subsolution u ∈ W 1,2 (B2r ) to − div(A∇u + bu) + (c1 + c2 )∇u + du ≤ − div f + g, we have that 204 G. Sakellaris / J. Math. Pures Appl. 156 (2021) 179–214 ⎛ ⎞ p1 sup u ≤ C ⎝ |u+ |p ⎠ + Cf Ln,1 (B2r ) + CgL n2 ,1 (B , 2r ) Br B2r where C depends on n, p, q, λ, A∞ and M . Proof. The proof is similar to the proof of Proposition 4.4, using Proposition 3.4 instead of Proposition 3.3 and Lemma 4.5 instead of Lemma 4.3.  Remark 4.7. Note that the analogue of Propositions 4.4 and 4.6 will hold under no smallness assumptions for b, d or c, d (when c ∈ Ln,q , q < ∞), but then the constants depend on b, d or c, d and not just on their norms. This can be achieved considering r > 0 small enough, so that the norms of b, d or c, d are small enough in all balls of radius 2r that are subsets of B2r , and after covering Br with balls of radius r . 4.3. Estimates on the boundary We now turn to local boundedness close to the boundary. We will follow the same process as in the case of local boundedness in the interior. The following are the analogues of (4.2) and (4.3) close to the boundary; the proof is similar to the one of Lemma 4.1 (as in [2, proof of Theorem 8.25]) and it is omitted. Lemma 4.8. Let Ω ⊆ Rn be a domain and B2r ⊆ Rn be a ball. Let also A be uniformly elliptic and bounded in Ω ∩ B2r , with ellipticity λ. n There exists θ = θn,λ > 0 such that, if b ∈ Ln,1 (Ω ∩ B2r ), c ∈ Ln,∞ (Ω ∩ B2r ) and d ∈ L 2 ,1 (Ω ∩ B2r ) with bn,1 ≤ θ, cn,∞ ≤ θ and d n2 ,1 ≤ θ, then, if w ∈ W 1,2 (Ω ∩ B2r ) is a subsolution to − div(A∇w + bw) + c∇w + dw ≤ 0 with w ≤ 0 on ∂Ω ∩ B2r , we have that ˆ ˆ C |∇w| ≤ 22 |w+ |2 , r Ω∩Br Ω∩B2r where C depends on n, λ and A∞ . Moreover, for any subsolution u ∈ W 1,2 (Ω ∩B2r ) to − div(A∇u) +c∇u ≤ 0 in Ω ∩B2r and any α ∈ (1, 2), we have that ⎛ ⎞ 12 C ⎝ sup u ≤ v2 ⎠ , Ω∩Br (α − 1)n/2 Bαr where v = u+ χΩ∩B2r , and C depends on n, λ and A∞ . To show local boundedness close to the boundary, we will need the following definition from [2, Theorem 8.25]: if u is a function in Ω and ∂Ω ∩ B2r = ∅, we define  sup{u(x), su }, x ∈ B2r ∩ Ω su = sup u+ , ũ(x) = (4.20) ∂Ω∩B2r su , x ∈ B2r \ Ω where the supremum over ∂Ω ∩ B2r is defined as on [2, page 202]. The following proposition concerns the case of large b. G. Sakellaris / J. Math. Pures Appl. 156 (2021) 179–214 205 Proposition 4.9. Let Ω ⊆ Rn be a domain, and B2r be a ball of radius 2r. Let also A be uniformly elliptic and bounded in Ω ∩B2r , with ellipticity λ, b ∈ Ln,1 (Ω ∩B2r ) with bn,1 ≤ M , and p > 0, f ∈ Ln,1 (Ω ∩B2r ), n g ∈ L 2 ,1 (Ω ∩ B2r ). n There exists ε = εn,λ,M > 0 such that, if c ∈ Ln,∞ (Ω ∩ B2r ) and d ∈ L 2 ,1 (Ω ∩ B2r ) with cn,∞ < ε and d n2 ,1 < ε, then for any subsolution u ∈ W 1,2 (Ω ∩ B2r ) to − div(A∇u + bu) + c∇u + du ≤ − div f + g, we have that ⎛ ⎞ p1 sup ũ ≤ C ⎝ |ũ|p ⎠ + Cf Ln,1 (Ω∩B2r ) + CgL n2 ,1 (Ω∩B , 2r ) Ω∩Br B2r where ũ is defined in (4.20), and where C depends on n, p, λ, A∞ and M . Proof. Subtracting a constant from u, and since ũ ≥ su in B2r , we can reduce to the case when u ≤ 0 on ∂Ω ∩ B2r (that is, su = 0). Then, based on Lemma 4.8 and [2, Theorem 8.25] instead of Lemma 4.1 and [2, Theorem 8.17], respectively, we can show the analogue of Lemma 4.2, replacing all the balls by their intersections with Ω, for subsolutions u ∈ W 1,2 (Ω ∩ B2r ) with u ≤ 0 on ∂Ω ∩ B2r . We then continue with a similar argument as in the proofs of Lemma 4.3 and Proposition 4.4, replacing all the balls by their intersections with Ω.  Finally, using a similar argument to the above, and going through the arguments of the proofs of Lemma 4.5 and Proposition 4.6, we obtain the following estimate close to the boundary, in the case that c is large. Proposition 4.10. Let Ω ⊆ Rn be a domain, and B2r be a ball of radius 2r. Let also A be uniformly elliptic and bounded in Ω ∩ B2r , with ellipticity λ, and consider q < ∞ and c1 ∈ Ln,q (Ω ∩ B2r ) with c1 n,q ≤ M . n Let also p > 0, f ∈ Ln,1 (Ω ∩ B2r ), and g ∈ L 2 ,1 (Ω ∩ B2r ). There exist ξ = ξn,λ > 0 and δ = δn,q,λ,M > 0 such that, if b ∈ Ln,1 (Ω ∩ B2r ), c2 ∈ Ln,∞ (Ω ∩ B2r ) and n d ∈ L 2 ,1 (Ω ∩ B2r ) with bn,1 < δ, c2 n,∞ < ξ and d n2 ,1 < δ, then for any subsolution u ∈ W 1,2 (Ω ∩ B2r ) to − div(A∇u + bu) + (c1 + c2 )∇u + du ≤ − div f + g, we have that ⎛ ⎞ p1 sup ũ ≤ C ⎝ |ũ|p ⎠ + Cf Ln,1 (Ω∩B2r ) + CgL n2 ,1 (Ω∩B , 2r ) Ω∩Br B2r where ũ is defined in (4.20), and where C depends on n, p, q, λ, A∞ and M . Remark 4.11. As in Remark 4.7, the analogues of Propositions 4.9 and 4.10 will hold under no smallness assumptions for b, d or c, d (when c ∈ Ln,q , q < ∞), with constants depending on b, d or c, d and not just on their norms. 5. The reverse Moser estimate and the Harnack inequality 5.1. The lower bound In order to deduce the Harnack inequality, we will consider negative powers of positive supersolutions to transform them to subsolutions of suitable operators, where the coefficients b, d will be small. This is the context of the following lemma. 206 G. Sakellaris / J. Math. Pures Appl. 156 (2021) 179–214 n Lemma 5.1. Let Ω ⊆ Rn be a domain, b, c, f ∈ Ln,∞ (Ω) and d, g ∈ L 2 ,∞ (Ω). Let also u ∈ W 1,2 (Ω) be a supersolution to − div(A∇u + bu) + c∇u + du ≥ − div f + g with inf Ω u > 0, and consider the function v = u + f Ln,1 (Ω) + gL n2 ,1 (Ω) . Then, for any k < 0, v k is a W 1,2 (Ω) subsolution to k(bu − f ) k (k − 1)(bu − f ) k(du − g) k − div A∇(v k ) + v + + c ∇(v k ) + v ≤ 0. (5.1) v v v Proof. We compute − div(A∇(v k )) = −k div(A∇v)v k−1 − k(k − 1)A∇v∇v · v k−2 . From ellipticity of A we have that A∇v∇v ≥ 0. Since also k < 0, the last identity shows that − div(A∇(v k )) ≤ − div(A∇u) · kv k−1 . Since k < 0, v k−1 > 0 and u is a supersolution, we have − div(A∇(v k )) ≤ (div(bu) − c∇u − du − div f + g)kv k−1 , and the proof is complete after a straightforward computation.  The next lemma bridges the gap between Lp averages for positive and negative p. Lemma 5.2. Let A be uniformly elliptic and bounded in B2r , with ellipticity λ, and b, c ∈ Ln,∞ (B2r ), n d ∈ L 2 ,∞ (B2r ). Let also u ∈ W 1,2 (B2r ) be a supersolution to − div(A∇u + bu) + c∇u + du ≥ 0 in B2r , with inf B2r u > 0. Then there exists a constant a = an such that ua u−a ≤ C, Br Br where C depends on n, λ, A∞ , bn,∞ , cn,∞ and d n2 ,∞ . Proof. We use the test function from [7, page 586] (see also [2, page 195]): let B2s be a ball of radius 2s, contained in B2r . If φ ≥ 0 be a smooth cutoff supported in B2s , with φ ≡ 1 in Bs and |∇φ| ≤ Cs , then the function φ2 u−1 is nonnegative and belongs to W01,2 (B2s ). Hence, using it as a test function, we obtain that ˆ ˆ 2φ∇φ φ2 ∇u 2φ∇φ φ2 ∇u φ2 φ2 A∇u − A∇u 2 + b u−b 2 u + c∇u + du ≥ 0, u u u u u u B2s B2s hence ˆ ˆ ∇u 2 2φ∇φ ∇u 2 φ2 A∇u φ ≤ A∇u + 2b∇φ · φ − b φ + c∇u + dφ2 . u2 u u u B2s B2r Using ellipticity of A, the Cauchy-Schwartz inequality, and Cauchy’s inequality with ε, we obtain ˆ ˆ |∇u|2 2   φ ≤C |∇φ|2 + |b∇φ|φ + (|b|2 + |c|2 + |d|)φ2 u2 B2s B2s ≤ Csn−2 + Cs−1 bn,∞ 1L n−1 n ,1 (B2s )  2  + C |b| + |c| + |d| n 1 n 2 ,∞ L n−2 ,1 (B2s ) 2 ≤ Csn−2 , G. Sakellaris / J. Math. Pures Appl. 156 (2021) 179–214 207 where C depends on n, λ, A∞ , bn,∞ , cn,∞ and d n2 ,∞ , and where we used (2.9) for the second estimate. The proof is complete using the Poincaré inequality and the John-Nirenberg inequality, as on [7, page 586].  The next bound is a reverse Moser estimate for supersolutions. Surprisingly, if we assume that the coefficient c belongs to Ln,q for some q < ∞, then we obtain a scale invariant estimate with “good” constants under no smallness assumption on the coefficients. As mentioned before, for the Moser estimate in Propositions 4.4 and 4.6, such a bound cannot hold with “good” constants under these assumptions. Proposition 5.3. Let A be uniformly elliptic and bounded in B2r , with ellipticity λ. Let also b, f ∈ Ln,1 (B2r ), n c1 ∈ Ln,q (B2r ) for some q < ∞, and d, g ∈ L 2 ,1 (B2r ), with bn,1 ≤ Mb , c1 n,q ≤ Mc and d n2 ,1 ≤ Md . There exist a = an > 0 and ξ = ξn,λ > 0 such that, if c2 ∈ Ln,∞ (B2r ) with c2 n,∞ < ξ, then for any nonnegative supersolution u ∈ W 1,2 (B2r ) to − div(A∇u + bu) + (c1 + c2 )∇u + du ≥ − div f + g, we have that ⎛ ⎞ a1 ⎝ ua ⎠ ≤ C inf u + Cf Ln,1 (B2r ) + CgL n2 ,1 (B , Br/2 2r ) Br where C depends on n, q, λ, A∞ , Mb , Mc and Md . Proof. Adding a constant δ > 0 to u, we may assume that inf B2r u > 0; the general case will follow by letting δ → 0. Set v = u + f Ln,1 (B2r ) + gL n2 ,1 (B ) , then v is a supersolution to 2r bu − f du − g − div A∇v + v + c∇v + v ≥ 0, v v with      bu − f   du − g       v  ≤ Cn bn,1 + Cn ,  v  n ≤ Cn d n2 ,1 + Cn . n,1 2 ,1 Then, since inf B2r v > 0, Lemma 5.2 implies that there exists a = an such that va v −a ≤ C, (5.2) Br Br where C depends on n, q, λ, A∞ , Mb , Mc and Md . For k ∈ (−1, 0) to be chosen later, v k is a W 1,2 (B2r ) subsolution to (5.1) for c = c1 + c2 , and      (k − 1)(bu − f )   bu    + c1  ≤ Cn,q (1 − k)    v v n,q n,q   f  + Cn,q (1 − k)    v  + Cn,q c1 n,q n,q ≤ M, where M depends on n, q, Mb and Mc . Then, for the ξn,λ and the δn,q,λ,M > 0 from Proposition 4.6 and (5.1), if 208 G. Sakellaris / J. Math. Pures Appl. 156 (2021) 179–214      k(bu − f )   k(du − g)    < δn,q,λ,M , c2 Ln,∞ (Br ) < ξn,λ ,   < δn,q,λ,M , (5.3)  v  n,1  v  n ,1 L (Br ) L 2 (Br ) then v k satisfies the estimate sup v k ≤ C vk , Br/2 Br where C depends on n, q, λ, A∞ and M . It is true that (5.3) holds for some k ∈ (−a, 0), depending on n, q, λ, Mb , Mc and Md ; hence, for this k, ⎛ ⎞ k1 ⎝ v k ⎠ ≤ C( sup v k ) k = C inf v, 1 (5.4) Br/2 Br/2 Br where C depends on n, q, λ, A∞ , Mb , Mc and Md . Since − ka > 1, Hölder’s inequality implies that ⎛ ⎞− ka ⎛ ⎞ k1 ⎛ ⎞− a1 ⎛ ⎞ a1 vk ≤ ⎝ v −a ⎠ ⇒⎝ vk ⎠ ≥ ⎝ v −a ⎠ ≥C⎝ va ⎠ , Br Br Br Br Br where we used (5.2) for the last step. Then, plugging the last estimate in (5.4), and using the definition of v, the proof is complete.  5.2. Estimates on the boundary We now consider the analogue of Proposition 5.3 close to the boundary. We will need the analogue of the definition of ũ in (4.20), from [2, Theorem 8.26]: if u ≥ 0 is a function in Ω and ∂Ω ∩ B2r = ∅, we define  inf{u(x), mu }, x ∈ B2r ∩ Ω mu = inf u, ū(x) = . (5.5) ∂Ω∩B2r mu , x ∈ B2r \ Ω The following is the analogue of Lemma 5.2 close to the boundary. Lemma 5.4. Let A be uniformly elliptic and bounded in Ω ∩ B2r , with ellipticity λ, and b, c ∈ Ln,∞ (Ω ∩ B2r ), n d ∈ L 2 ,∞ (Ω∩B2r ). Let also u ∈ W 1,2 (B2r ) be a nonnegative supersolution to − div(A∇u+bu)+c∇u+du ≥ 0 in B2r , and consider the function ū from (5.5). If inf Ω∩B2r u > 0 and mu > 0, then there exists a constant a = an such that ūa ū−a ≤ C, Br Br where C depends on n, λ, A∞ , bn,∞ , cn,∞ and d n2 ,∞ . Proof. As in the proof of [2, Theorem 8.26], set v = ū−1 − m−1 u ∈ W 1,2 (Ω ∩ B2r ), which is nonnegative in Ω ∩ B2r and vanishes on ∂Ω ∩ B2r . Then, considering the test function vφ2 , where φ is a suitable cutoff function, and using that v > 0 if and only if ū = u, the proof follows by an argument as in the proof of Lemma 5.2.  Using the previous lemma, we can show the following estimate. G. Sakellaris / J. Math. Pures Appl. 156 (2021) 179–214 209 Proposition 5.5. Let Ω ⊆ Rn be a domain, B2r be a ball of radius 2r, and let A be uniformly elliptic and bounded in Ω ∩ B2r , with ellipticity λ. Let also b, f ∈ Ln,1 (Ω ∩ B2r ), c1 ∈ Ln,q (Ω ∩ B2r ) for some q < ∞, n and d, g ∈ L 2 ,1 (Ω ∩ B2r ), with bn,1 ≤ Mb , c1 n,q ≤ Mc and d n2 ,1 ≤ Md . There exist a = an > 0 and ξ = ξn,λ > 0 such that, if c2 ∈ Ln,∞ (Ω ∩ B2r ) with c2 n,∞ < ξ, then for any nonnegative supersolution u ∈ W 1,2 (Ω ∩ B2r ) to − div(A∇u + bu) + (c1 + c2 )∇u + du ≥ − div f + g, we have that ⎛ ⎞ a1 ⎝ ūa ⎠ ≤ C inf ū + Cf Ln,1 (Ω∩B2r ) + CgL n2 ,1 (Ω∩B , Br/2 2r ) Br where ū is defined in (5.5), and where C depends on n, q, λ, A∞ , Mb , Mc and Md . Proof. As in the proof of Proposition 5.3, we can assume that inf Ω∩B2r u > 0, mu > 0, and f, g ≡ 0. Let a = an be as in Lemma 5.4. Then, Lemma 5.1 and Proposition 4.10 show that, for suitable k ∈ (−a, 0), if wk = uk and w˜k is as in (4.20), we have that ⎛ ⎞− ka sup w˜k ≤ C w˜k ≤ C ⎝ w˜k −a k ⎠ . Ω∩Br/2 Br Br Since w˜k = ūk , the proof is complete using also Lemma 5.4.  5.3. The Harnack inequality, and local continuity We now show the Harnack inequality in the cases when b, d are small, or when c, d are small. Theorem 5.6. Let A be uniformly elliptic and bounded in B2r , with ellipticity λ. Let also b, f ∈ Ln,1 (B2r ) n with bn,1 ≤ M , and g ∈ L 2 ,1 (B2r ). n There exists εn,λ,M > 0 such that, if c ∈ Ln,∞ (B2r ) and d ∈ L 2 ,1 (B2r ) with cn,∞ < ε and d n2 ,1 < ε, then for any nonnegative solution u ∈ W 1,2 (B2r ) to − div(A∇u + bu) + c∇u + du = − div f + g, we have that sup u ≤ C inf u + Cf Ln,1 (B2r ) + CgL n2 ,1 (B , Br 2r ) Br where C depends on n, λ, A∞ and M . Proof. The proof is a combination of Proposition 4.4 (choosing p = an in (4.16), as in Proposition 5.3), and Proposition 5.3, (considering q = n and c1 ≡ 0), after also covering Br with balls of radius r/4.  Theorem 5.7. Let A be uniformly elliptic and bounded in B2r , with ellipticity λ, and q < ∞, c1 ∈ Ln,q (B2r ) n with c1 n,q ≤ M . Let also f ∈ Ln,1 (B2r ), g ∈ L 2 ,1 (B2r ). There exist ξ = ξn,λ > 0 and δ = δn,q,λ,M > 0 such that, if b ∈ Ln,1 (B2r ), c2 ∈ Ln,∞ (B2r ) and d ∈ n L 2 ,1 (B2r ) with bn,1 < δ, c2 n,∞ < ξ and d n2 ,1 < δ, then for any nonnegative solution u ∈ W 1,2 (B2r ) to − div(A∇u + bu) + c∇u + du = − div f + g, we have that sup u ≤ C inf u + Cf Ln,1 (B2r ) + CgL n2 ,1 (B , Br 2r ) Br where C depends on n, q, λ, A∞ and M . 210 G. Sakellaris / J. Math. Pures Appl. 156 (2021) 179–214 Proof. The proof follows by a combination of Propositions 4.6 and 5.3.  We now turn to local continuity of solutions. For the following theorem, for ρ ≤ 2r, we set   Qb,d (ρ) = sup bLn,1 (Bρ ) + dL n2 ,1 (B  ) : Bρ ⊆ B2r , (5.6) ρ where Bρ runs over all the balls of radius ρ that are subsets of B2r . Also, we will follow the argument on [2, pages 200-202]. Theorem 5.8. Let A be uniformly elliptic and bounded in B2r , with ellipticity λ. Let also b, f ∈ Ln,1 (B2r ) n with bn,1 ≤ M , g ∈ L 2 ,1 (B2r ), and μ ∈ (0, 1). For every μ ∈ (0, 1), there exists ε = εn,λ,M > 0 and α = αn,λ, A ∞ ,M,μ ∈ (0, 1) such that, if c ∈ n Ln,∞ (B2r ) and d ∈ L 2 ,1 (B2r ) with cn,∞ < ε and d n2 ,1 < ε, then for any solution u ∈ W 1,2 (B2r ) to − div(A∇u + bu) + c∇u + du = − div f + g, we have that ⎛ ⎞ |x − y|α |u(x) − u(y)| ≤ C + Qb,d (|x − y|μ r1−μ ) ⎝ |u| + Qf,g (2r)⎠ + CQf,g (|x − y|μ r1−μ ), rα B2r for any x, y ∈ Br , where Q is defined in (5.6) and C depends on n, λ, A∞ and M . Proof. Let ρ ∈ (0, r], and set M (ρ) = supBρ u, m(ρ) = inf Bρ u. Then v1 = M (ρ) − u is nonnegative in Bρ , and solves the equation − div(A∇v1 + bv1 ) + c∇v1 + dv1 = − div(M (ρ)b − f ) + (M (ρ)d − g) in Bρ . Hence, from Theorem 5.6, (2.7) and (5.6), we obtain that ρ M (ρ) − m = sup v1 2 Bρ/2 ≤ C inf v1 + CM (ρ)b − f Ln,1 (Bρ ) + CM (ρ)d − gL n2 ,1 (B (5.7) Bρ/2 ρ) ρ = C M (ρ) − M + C sup |u| · Qb,d (ρ) + CQf,g (ρ), 2 Br where C depends on n, λ, A∞ and M . Moreover, v2 = u − m(ρ) is nonnegative in Bρ , and solves the equation − div(A∇v2 + bv2 ) + c∇v2 + dv2 = − div(f − m(ρ)b) + (g − m(ρ)d) in Bρ . Hence, from Theorem 5.6, as in (5.7), ρ ρ M − m(ρ) ≤ C m − m(ρ) + C sup |u| · Qb,d (ρ) + CQf,g (ρ). (5.8) 2 2 Br Adding (5.7) and (5.8) and defining ω(ρ) = M (ρ) − m(ρ), we obtain that ρ ω ≤ θ0 ω(ρ) + C sup |u| · Qb,d (ρ) + CQf,g (ρ), 2 Br where θ0 = C−1 C+1 ∈ (0, 1). Then, [2, Lemma 8.23] shows that, for ρ ≤ r, G. Sakellaris / J. Math. Pures Appl. 156 (2021) 179–214 211 ρα ω(ρ) ≤ C ω(r) + C sup |u| · Qb,d (ρμ r1−μ ) + CQf,g (ρμ r1−μ ), rα Br where C depends on n, λ, A∞ , M , and α = αn,λ, A ∞ ,M,μ . We then bound supBr |u| using Proposition 4.4 (applied to u and −u, for p = 1), which completes the proof.  Finally, based on Proposition 4.6 and Theorem 5.7, we obtain the following theorem when b, d are small. Theorem 5.9. Let A be uniformly elliptic and bounded in B2r , with ellipticity λ, and q < ∞, c1 ∈ Ln,q (B2r ) n with c1 n,q ≤ M . Let also f ∈ Ln,1 (B2r ), g ∈ L 2 ,1 (B2r ). For every μ ∈ (0, 1), there exist ξ = ξn,λ > 0, δ = δn,q,λ,M > 0 and α = αn,λ, A ∞ ,M,μ such that, if n b ∈ Ln,1 (B2r ), c2 ∈ Ln,∞ (B2r ) and d ∈ L 2 ,1 (B2r ) with bn,1 < δ, c2 n,∞ < ξ and d n2 ,1 < δ, then for any solution u ∈ W 1,2 (B2r ) to − div(A∇u + bu) + c∇u + du = − div f + g, we have that ⎛ ⎞ |x − y|α |u(x) − u(y)| ≤ C + Qb,d (|x − y|μ r1−μ ) · ⎝ |u| + Qf,g (2r)⎠ + CQf,g (|x − y|μ r1−μ ), rα B2r for any x, y ∈ Br , where Q is defined in (5.6) and C depends on n, q, λ, A∞ and M . Remark 5.10. As in Remarks 4.7 and 4.11, the analogues of Theorems 5.6 - 5.9 will hold under no smallness assumptions for b, d and c, d (when c ∈ Ln,q , q < ∞), but then the constants depend on b, d or c, d and not just on their norms. 6. Optimality of the assumptions We now turn to showing that our assumptions are optimal in order to deduce the estimates we have shown so far, in the setting of Lorentz spaces. We first show optimality for b and d. Remark 6.1. Considering the operators L1 u = −Δu − div(bu) and L2 u = −Δu + du, an assumption of the n form b ∈ Ln,q , d ∈ L 2 ,q for some q > 1, with bn,q , d n2 ,1 being as small as we want, is not enough to guarantee the pointwise bounds in the maximum principle and Moser’s estimate. Indeed, as in Lemma [34, δ Lemma 7.4], set uδ (x) = (− ln |x|) and bδ (x) = − |x|2δx ln |x| . Then, for δ ∈ (−1, 1), b ∈ L n,q (B1/e ) for all q > 1, uδ ∈ W (B1/e ), and uδ solves the equation 1,2 −Δu − div(bδ uδ ) = 0 in B1/e . However, vδ ≡ 1 on ∂B1/e , and vδ → ∞ as |x| → 0 for δ > 0, so the assumption b ∈ Ln,1 is optimal for the maximum principle and the Moser estimate. Note that uδ also solves the equation δ(δ − 1) δ(n − 2) −Δuδ + dδ uδ = 0, dδ (x) = + 2 , |x| ln |x| 2 2 |x| ln |x| n n and dδ ∈ L 2 ,q (B1/e ) for every q > 1; hence, the assumption d ∈ L 2 ,1 is again optimal. The same functions bδ and dδ serve as counterexamples to show optimality for the spaces of b, d in the reverse Moser estimate. In particular, considering δ < 0, we have that uδ (0) = 0, while uδ does not identically vanish close to 0, therefore the reverse Moser estimate cannot hold. We now turn to optimality for smallness of c, when c ∈ Ln,∞ . 212 G. Sakellaris / J. Math. Pures Appl. 156 (2021) 179–214 Remark 6.2. In the case of the operator L0 u = −Δu + c∇u with c ∈ Ln,∞ , smallness in norm is a necessary condition, in order to obtain all the estimates we have considered. Indeed, if u(x) = − ln |x| − 1, then u ∈ W01,2 (B1/e ), and u solves the equation (2 − n)x −Δu + c∇u = 0, c= ∈ Ln,∞ (B1/e ). |x|2 However, u is not bounded in B1/e , so the maximum principle, as well as Moser’s and Harnack’s estimates fail. On the other hand, the function v(x) = (− ln |x|)−1 ∈ W01,2 (B1/e ) solves the equation (n − 2)x 2x −Δv + c ∇v = 0, c = − 2 ∈ Ln,∞ (B1/e ), |x|2 |x| ln |x| with v(0) = 0 and v not identically vanishing close to 0, therefore smallness for c ∈ Ln,∞ in the reverse Harnack estimate is necessary. Finally, we show the optimality of the assumption that either b, d should be small, or c, d should be small, so that in the maximum principle, as well as Moser’s and Harnack’s estimates, the constants depend only on the norms of the coefficients. The fact that d should be small is based on the following construction. n Proposition 6.3. There exists a bounded sequence (dN ) in L 2 ,1 (B1 ) and a sequence (uN ) of nonnegative W01,2 (B1 ) ∩ C(B 1 ) functions such that, for all N ∈ N, uN is a solution to the equation −ΔuN + dN uN = 0 in B1 , and uN W01,2 (B1 ) ≤ C, while uN (0) −−−−→ ∞. N →∞ Proof. We define    n 2 + 1 − n2 r2 , 0 1. Set u(x) = v(|x|), then it is straightforward to check that u is radially decreasing, u ≥ 1 in B1 , u ≤ n2 in Rn , and u ∈ Y 1,2 (Rn ) ∩ C 1 (Rn ). Then, the function d = n(2 − n)u−1 χB1 is bounded and supported in B1 , and u is a solution to the equation −Δu + du = 0 in Rn . We now let N ∈ N with N ≥ 2, and set BN to be the ball of radius N , centered at 0. We will modify u to be a W01,2 (BN ) solution to a slightly different equation: for this, set wN = u − v(N ), and also du dN = . u − v(N ) Since d is supported in B1 , dN is well defined. Note also that wN ∈ W01,2 (BN ), and wN is a solution to the equation −ΔwN + dN wN = 0 in BN . Moreover, since d is supported in B1 , u ≥ 1 in B1 and v is decreasing, we have that dL∞ (B1 ) uL∞ (B1 ) dN L n2 ,1 (B ≤ Cn dN L∞ (B1 ) ≤ Cn ≤ Cn . N) 1 − v(N ) Let now d˜N (x) = N 2 dN (N x) and w̃N (x) = wN (N x), for x ∈ B1 . Then w̃N ∈ W01,2 (B1 ), (d˜N ) is bounded in n L 2 ,1 (B1 ), and w̃N is a solution to the equation −Δw̃N + d˜N w̃N = 0 in B1 . Moreover, w̃N (0) ≥ Cn , while G. Sakellaris / J. Math. Pures Appl. 156 (2021) 179–214 213 ˆ ˆ ˆ |∇w̃N | = N 2 2−n |∇wN | = N 2 2−n |∇u|2 −−−−→ 0, N →∞ B1 BN BN since ∇u ∈ L2 (Rn ). Hence, considering the function w̃N ∇w̃N L2 (B completes the proof.  1) Remark 6.4. 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