Document Type

Article

Publication Date

4-13-2018

Abstract

N.C. Phuc, Nonlinear Muckenhoupt–Wheeden type bounds on Reifenberg flat domains, with applications to quasilinear Riccati type equations, Adv. Math. 250 (2014), 387–419. The proof of Theorem 1.6 in the above paper contains a gap. In general, the continuity of the map S stated in Lemma 5.8 may fail. This error was inherited from our earlier work [28, Theorem 3.4], which has now been corrected in the erratum [2]. The purpose of this note is to fix this gap following the main idea of [2]. We shall use the same notation as in the paper. Given a nonnegative locally finite measure ν in [Formula presented], we define its first order Riesz's potentials by [Formula presented] where [Formula presented]. When [Formula presented], we write [Formula presented] instead of [Formula presented] and note that in this case we have [Formula presented] In what follows, given a finite signed measure in Ω we will tacitly extend it by zero to [Formula presented]. Also, recall that the space [Formula presented] is defined in Definition 5.5. Let [Formula presented] be a constant such that [Formula presented] for all [Formula presented], [Formula presented]. Inequality (0.2) follows from an application of Fubini's Theorem and the fact the function [Formula presented] is an [Formula presented] weight. By an [Formula presented] weight we mean a nonnegative function [Formula presented], [Formula presented], such that [Formula presented] for a constant [Formula presented]. The least possible value of C will be denoted by [Formula presented] and is called the [Formula presented] constant of w. It is well-known that [Formula presented]. With [Formula presented], for each measure [Formula presented] we define the set [Formula presented] Here [Formula presented] is to be determined. Under the strong topology of [Formula presented], we have that [Formula presented] is closed and convex since we may assume that [Formula presented]. Recall that existence results in case [Formula presented] had been obtained, e.g., in [31]. By inequality (2.10) of [28] with [Formula presented], [Formula presented], [Formula presented], [Formula presented] and with q replaced by [Formula presented] we find, for a.e. [Formula presented], [Formula presented] On the other hand, for any [Formula presented] by (0.2) we have [Formula presented] for any [Formula presented]. Hence by Fubini's Theorem we find [Formula presented] which yields [Formula presented] Thus by (0.3) for all [Formula presented] and for a.e. [Formula presented] we have [Formula presented] We are now ready to prove Theorem 1.6. Proof of Theorem 1.6 First we assume that [Formula presented] and let [Formula presented] be defined by [Formula presented] where [Formula presented] is the unique renormalized solution of [Formula presented] We claim that we can find [Formula presented] and [Formula presented] such that [Formula presented] provided (1.7) holds with [Formula presented], i.e., [Formula presented] Indeed, for any weight w with [Formula presented] and for any [Formula presented], by Theorem 1.4 we have [Formula presented] Here [Formula presented] depends only on [Formula presented], and [Formula presented]. Thus by (0.4) we find [Formula presented] We now choose [Formula presented] and any [Formula presented] such that [Formula presented] Then it follows that [Formula presented] provided condition (1.7) holds with [Formula presented]. Thus we obtain the claim (0.6) for this choice of [Formula presented] and [Formula presented]. We next show the continuity of S on [Formula presented]. Let [Formula presented] be a sequence in [Formula presented] such that [Formula presented] converges strongly in [Formula presented] to a function [Formula presented]. Then with [Formula presented], we have [Formula presented] in the renormalized sense. As [Formula presented] in [Formula presented], by the stability result of [8, Theorem 3.4], there exists a subsequence [Formula presented] converging a.e. to [Formula presented]. Moreover, the proof of [8, Theorem 3.4] also yields [Formula presented] Note that it follows from (0.4) that [Formula presented] Thus by de la Vallée–Poussin Lemma on equi-integrability, there exists a non-decreasing and convex function [Formula presented] with [Formula presented] and [Formula presented], such that [Formula presented] Moreover, we may assume that G satisfies a [Formula presented] (moderate growth) condition (see, e.g., [1]): there exists [Formula presented] such that [Formula presented] For any nonnegative functions g and w on Ω using the convexity of G, the [Formula presented] condition, and the formula [Formula presented] we have [Formula presented] where [Formula presented] is defined by [Formula presented] With (0.10) and the [Formula presented] property of G, we can calculate as in proof of Theorem 1.4 but now with [Formula presented] on page 412, [Formula presented], [Formula presented], and [Formula presented], to deduce that[Formula presented] Here the constant C is independent of k. Hence by de la Vallée–Poussin Lemma, it follows that the sequence [Formula presented] is equi-integrable in Ω which by (0.9) and Vitali Convergence Theorem yield [Formula presented] As the above limit is independent of the subsequence, we also have that [Formula presented] strongly in [Formula presented], and thus the continuity of S follows. Similarly, we can show that [Formula presented] is precompact under the strong topology of [Formula presented]. Indeed, if [Formula presented] where [Formula presented], then (0.8) holds in the renormalized sense. Thus from the proof of [8, Theorem 3.4] we also have (0.9) for a subsequence [Formula presented] and a function [Formula presented]. This is possible since [Formula presented] is uniformly bounded in [Formula presented]. With the same argument as above, we also have the equi-integrability of [Formula presented] and hence [Formula presented] strongly in [Formula presented]. At this point, we can apply Schauder Fixed Point Theorem to obtain a renormalized solution [Formula presented] to equation (1.8). Moreover, as (0.4) also holds with u in place of v (since [Formula presented]), by Theorem 1.4 we have [Formula presented] for all weights [Formula presented]. Recall that [Formula presented] is the Bessel kernel of first order. Then as in the proof of Theorem 5.6, we obtain [Formula presented] This completes the proof of Theorem 1.6 in the case [Formula presented]. To remove this assumption, let [Formula presented] be defined as in page 417. Then recall that we have [Formula presented] and [Formula presented] provided [Formula presented], where [Formula presented] is the constant in inequality (0.7). Thus from the above result, for each [Formula presented] there exists a renormalized solution [Formula presented] to the equation [Formula presented] such that [Formula presented] and [Formula presented] Now observe that we have [Formula presented] where M is the Hardy–Littlewood maximal function. Thus, as [Formula presented], we see that the sequence [Formula presented] is equi-integrable in Ω. As above, this implies that the sequence [Formula presented] is also equi-integrable. Since [Formula presented], it follows from equation (0.12) and the proof of [8, Theorem 3.4] that there exists a subsequence [Formula presented] converging a.e. to a function [Formula presented] for which (0.9) holds. Then by Vitali Convergence Theorem we have [Formula presented] strongly in [Formula presented]. This allows us to use the stability result of [8, Theorem 3.4] to infer that u is a renormalized solution of (1.8) as desired. This completes the proof of the theorem. □ Other unrelated minor errors: The phrase “.. [Formula presented] is a weak solution of (1.1) …” in Propositions 3.1, 3.2, 3.3, and Theorem 3.5 should read “… u is a renormalized solution of (1.1) …”.

Publication Source (Journal or Book title)

Advances in Mathematics

First Page

1353

Last Page

1359

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