# Uniform continuity and Brézis–Lieb-type splitting for superposition operators in Sobolev space

• Nils Ackermann

## Abstract

Using concentration-compactness arguments, we prove a variant of the Brézis–Lieb-Lemma under weaker assumptions on the nonlinearity than known before. An intermediate result on the uniform continuity of superposition operators in Sobolev space is of independent interest.

MSC 2010: 47H30; 58E40

## 1 Introduction

In their seminal paper [6] Brézis and Lieb prove a result about the decoupling of certain integral expressions, which has been used extensively in the calculus of variations. Using concentration compactness arguments in the spirit of Lions [16, 14, 15], we prove a variant of this lemma under weaker assumptions on the nonlinearity than known before. To describe a special case of the Brézis–Lieb lemma, suppose that Ω is an unbounded domain in N, p>1, f(t):=|t|p for t, and (un) is a bounded sequence in Lp(Ω) that converges pointwise almost everywhere to some function u. If one denotes by :Lp(Ω)L1(Ω) the superposition operator induced by f, i.e., (v)(x):=f(v(x)), then the result in [6] implies that uLp(Ω) and

(1.1)(un)-(un-u)(u)in L1(Ω) as n.

The same conclusion is obtained in that paper for more general functions, imposing conditions that are satisfied for continuous convex f with f(0)=0 and imposing additional conditions on the sequence (un).

A different approach to the decoupling of superposition operators along sequences of functions rests on certain regularity assumptions on f. For example, assume that fC1() satisfies

(1.2)supt|f(t)||t|p-1<.

Then the proof of [19, Lemma 8.1] can easily be extended to obtain (1.1). See also the slightly more general [7, Lemma 1.3], where f is allowed to depend on x explicitly.

Our aim is to give a decoupling result under a different set of hypotheses that applies to a much larger class of functions f than considered above, within a certain range of exponents p. In particular, we do neither impose any convexity type assumptions on f as was done in [6], nor any regularity assumptions as in [19, 7] apart from continuity. The price we pay for relaxing the hypotheses on f is that we need to restrict the range of allowed growth exponents p in comparison with [6], that we need to assume some type of translation invariance for Ω, and that the decoupling result only applies to a smaller set of admissible sequences, namely sequences that converge weakly in H1(Ω). Nevertheless, the numerous applications in the calculus of variations for PDEs where these extra assumptions are satisfied justify the new set of hypotheses.

To keep the presentation simple and highlight the main idea, we only treat the case Ω=N. From here on, function spaces are taken over N unless otherwise noted. It would be possible to consider other domains or superposition operators between other spaces, and we plan to do so in forthcoming work. Nevertheless, we do allow a periodic dependency of f on the space variable.

To explain our results we formalize the notion of decoupling.

## Definition 1.1.

Suppose that X and Y are Banach spaces. Consider a map :XY, a sequence (un)X and uX. We say that BL-splits along (un) with respect to u (BL being an abbreviation for Brézis–Lieb) if

(un)-(un-u)-(u)Y0.

We say that almost BL-splits along (un) with respect to u if, starting with any subsequence of (un), we can pass to a subsequence such that there is a sequence (vn)X such that vn-uX0 and

(un)-(un-vn)-(u)Y0.

If u is a limit of (un) in some unambiguous sense then we frequently omit to mention that (almost) BL-splitting is with respect to u.

By [6], the map f(u)=|u|p induces a map :LpL1 that BL-splits along pointwise a.e. converging bounded sequences in Lp with respect to their pointwise a.e. limits. On the other hand, the technique used to prove [1, Lemma 3.2] (and the related results in [10, 11]) yields the following: if fC() satisfies

(1.3)supt|f(t)||t|p<,

then the induced superposition operator :LpL1 almost BL-splits along any Llocp-converging bounded sequence in Lp with respect to its limit in Llocp, see Theorem 2.1(a) below. This result is basically Lion’s approach, with a simplifying twist. If in addition is uniformly continuous on bounded subsets of Lp, then it is easy to see that it BL-splits along any Llocp-converging bounded sequence in Lp with respect to its limit in Llocp, see [2, Lemma 6.3]. For example, this holds true if (1.2) is satisfied.

We illustrate the distinction between BL-splitting and almost BL-splitting by the following examples.

## Example 1.2.

If p>1 and if either f(t):=cos(πt)|t|p or f(t):=cos(π/t)|t|p, then there is a bounded sequence (un) in Lp that converges in Llocp and pointwise a.e. to a function u such that the induced continuous superposition operator :=LpL1 does not BL-split along any subsequence of (un) with respect to u. On the other hand, almost BL-splits along any Llocp-converging bounded sequence in Lp with respect to its limit in Llocp. Hence is not uniformly continuous on bounded subsets of Lp and neither the general conditions used in [6] nor (1.2) are satisfied for f in these examples.

The sequences mentioned in the example are provided in Section 4 below.

Our main interest is to avoid condition (1.2), or any other conditions on f that ensure uniform continuity on bounded subsets of Lp (e.g., a local Hölder condition, together with an appropriate growth bound on the Hölder constants on bounded intervals). Our result below states that it is sufficient to restrict to bounded subsets of H1 instead.

In this context we now formulate our main theorem, in a slightly more general setting than what we considered above. A function f:N× is a Caratheodory function if f is measurable and if f(x,) is continuous for almost every xN. The induced superposition operator on functions u:N is then given by (u)(x):=f(x,u(x)). If A is a real invertible N×N-matrix, then f is said to be A-periodic in its first argument if f(x+Ak,t)=f(x,t) for all xN, kN, and t.

Denote by 2*:=2N/(N-2) if N3 and 2*:= if N=1 or N=2 the critical Sobolev exponent for H1. Recall the continuous and compact embedding of the Sobolev space H1(U) in Lp(U) for p[2,2*) if UN is a bounded domain.

## Theorem 1.3.

Consider μ>0, ν1, and C0>0 such that p:=μν(2,2*). Suppose that f:RN×RR is a Caratheodory function that satisfies

(1.4)|f(x,t)|C0|t|μfor all xN,t,

and which is A-periodic in its first argument, for some invertible matrix ARN×N. Denote by F:LpLν the continuous superposition operator induced by f. Then F is uniformly continuous on bounded subsets of H1 with respect to the Lp-Lν-norms and hence also with respect to the H1-Lν-norms. Moreover, F:H1Lν BL-splits along weakly convergent sequences in H1 with respect to their weak limit.

Our proof of Theorem 1.3 has similarities with the proof of [18, Theorem 3.1] but involves an intermediate cut-off step in the proof of Theorem 2.1. Essentially, we first prove almost BL-splitting of along weakly converging sequences in H1 with respect to their weak limit, using the concentration function and the compactness of the Sobolev embedding H1(U)Lp(U) for p[2,2*) and for a bounded domain U. Then we collect the possible mass loss at infinity along subsequences with the help of Lions’ Vanishing Lemma, employing the assumption p>2.

## Remark 1.4.

Theorem 1.3 applies in particular to the functions considered in Example 1.2 when ν=1 and μ=p(2,2*). On the other hand, for f(t):=cos(π/t)t2 there is a sequence in H1 that converges weakly but possesses no subsequence along which f:H1L1 BL-splits with respect to the weak limit. The same is true for f(t):=cos(πt)t2*. In this sense, Theorem 1.3 is optimal, that is, it cannot be extended in this generality to include the cases p=μν=2 and p=μν=2*. The existence of these counterexamples is proved in Section 4.

Of course, by Sobolev’s embedding theorem, a map LpLν that BL-splits along Llocp-converging bounded sequences in Lp with respect to their limits in Llocp also BL-splits in H1 along weakly convergent sequences with respect to their weak limits. Therefore, Theorem 2.1(a), together with (1.2) (or the weaker Hölder condition with growth bound), yields BL-splitting maps along weakly convergent sequences in H1 with respect to weak limits even for p=2 and p=2*.

## Remark 1.5.

The result also holds true in a slightly restricted sense for functions f that are sums of functions as in Theorem 1.3, i.e., functions that satisfy merely

|f(x,t)|C0(|t|μ1+|t|μ2)for all xN,t,

where μiν(2,2*) for i=1,2. In that case, :H1Lν is uniformly continuous on bounded subsets of H1 with respect to the H1-Lν norms, and BL-splits along weakly convergent sequences in H1 with respect to their weak limits.

## Remark 1.6.

The uniform continuity of operators on bounded subsets of H1 has been used, for example, in the proof of [18, Lemma 3.4]. Nevertheless, we are not aware of a published proof of this fact, which is nontrivial in the generality stated in Theorem 1.3. Note that the uniform continuity of :H1(U)Lν(U) on bounded subsets of H1(U) is trivial if U is bounded, by the compact Sobolev embedding H1(U)Lp(U).

We now discuss additional aspects and applications of the results presented above. To this end we return to a simple setting on N. Suppose that fC() satisfies (1.3) with p(2,2*) and consider the functional Φ:H1 given by

Φ(u):=Nf(u).

To prove the existence of a minimizer in typical variational problems involving Φ, Lions [14, 15] introduces the concentration-compactness principle. It is a tool to exclude the possibility of vanishing and of dichotomy along a minimizing sequence (un), in order to obtain compactness of the sequence. Here we are only concerned with dichotomy. In this case, the sequence (un) is approximated by (un1+un2), where dist(supp(un1),supp(un2)). For local functionals like Φ it then follows easily that Φ(un) is approximated by Φ(un1)+Φ(un2), a fact that yields, together with a hypothesis about energy levels, a contradiction. Clearly, the same can be achieved if Φ BL-splits along (un) in a suitable way. Before our Theorem 1.3, Lions’ approach to concentration compactness was more general, in that, besides continuity and appropriate growth bounds, no extra regularity hypotheses need to be placed on f. On the other hand, the arguments are more involved than when using BL-splitting because one has to insert cut-off functions to obtain sequences un1 and un2 with disjoint supports. As a consequence, it is difficult to give a purely functional (abstract) presentation of Lions’ approach.

To explain the advantage of an abstract presentation using BL-splitting, we note that to treat nonlocal functionals of convolution type, e.g.,

Ψ(u):=N(f*h(u))h(u),

the property of disjoint supports is not as effective anymore. In the convolution, the supports get “smeared out” and one has to control the interaction with more involved estimates, see [14, p. 123]. This is aggravated when one also has to consider the decoupling of derivatives of Ψ. We have shown in [2] that using BL-splitting is effective in situations involving nonlocal functionals. Moreover, BL-splitting even survives certain nonlocal operations, like the saddle point reduction; see [2, Theorem 5.1].

For particular cases there are other approaches to avoid conditions on f besides continuity and growth bounds. We reformulate and simplify the following cited results slightly to adapt them to our setting and notation. In [3] we proved, for fC() satisfying (1.4) with μ:=p-1, and setting ν:=p/(p-1), that the map Γ:H1H-1, given by

Γ(u)v:=Nf(u)v,

BL-splits along a weakly convergent sequence if the weak limit is a function tending to 0 as |x|. Another result was given in [13, Lemma 7.2], when fC() satisfies (1.4) with μ:=p-2 and ν:=p/(p-2), and is as follows: The map Λ:H12(H1,) (here 2(H1,) denotes the space of bounded bilinear maps from H1 into ), given by

Λ(u)[v,w]:=Nf(u)vw,

is uniformly continuous on bounded subsets of H1. Together with the almost BL-splitting of Λ given by Theorem 2.1 below this yields BL-splitting for Λ along weakly convergent sequences. Note that the idea of the proof of the latter result does not apply for the maps Φ and Γ defined above (under the respective growth bounds on f). In both cases our result here is stronger, since we show uniform continuity and BL-splitting into the spaces Lν, which are continuously embedded in H-1 and 2(H1,), respectively.

A different application of Theorem 1.3, that is independent of variational methods, is the general study of maps that are uniformly continuous on a subset of an infinite dimensional Hilbert space. These play a role in infinite dimensional potential theory [12, 5] or, more generally, in the theory of stochastic equations in infinite dimensions [17, 9, 8].

The paper is structured as follows: In Section 2 we treat almost BL-splitting of along bounded sequences in Lp that converge in Llocp and along weakly convergent sequences in H1. In Section 3 we prove the uniform continuity of on bounded subsets of H1 and BL-splitting of along weakly convergent sequences. In Section 4 we prove the claims made in Example 1.2.

## 2 Almost BL-splitting

In this section we prove a result on the almost BL-splitting of superposition operators in Lp along bounded sequences that converge in Llocp and in H1 along weakly convergent sequences. This is a variation on Lions’ approach in [14]. Note that here the periodicity assumption in x is not needed.

If r[1,], then we denote by ||r the norm of Lr.

## Theorem 2.1.

Consider μ>0, ν1, and C0>0 such that p:=μν1. Suppose that f:RN×RR is a Caratheodory function that satisfies (1.4). Denote by F the superposition operator on real functions induced by f. Then we have the following results:

1. If (un)Lp is bounded and converges in Llocp to a function u, then uLp and :LpLν almost BL-splits along (un) with respect to u.

2. If p[2,2*) and unu in H1, then :H1Lν almost BL-splits along (un) with respect to u.

3. In (b), if in addition (u¯n)H1 converges weakly and |un-u¯n|p0 as n, then u¯nu in H1 and almost BL-splits along (un) and (u¯n) with respect to u, preserving subsequences and the auxiliary sequence (vn) in the following sense: for any subsequence nkthere are a subsequence nk and (v) such that vu in H1 and, writing u:=unk and u¯:=u¯nk, we have

(u)-(u-v)(u)

and

(u¯)-(u¯-v)(u).

For the proof, let BR denote, for R>0, the open ball in N with center 0 and radius R.

## Proof.

We prove the assertions one by one. (a): From (1.4) and from the theory of superposition operators [4] it follows that :Lp(U)Lν(U) is continuous for any open subset U of N. For n define Qn:[0,)[0,) by

Qn(R):=BR|un|p.

The functions Qn are uniformly bounded and nondecreasing. We may assume that (Qn) converges pointwise almost everywhere to a bounded nondecreasing function Q (see [14]). It is easy to build a sequence Rn such that for every ε>0 there is R>0, arbitrarily large, with

lim supn(Qn(Rn)-Qn(R))ε.

Hence, for all ε>0 there exists R>0 such that

(2.1)lim supnBRn\BR|un|pεandN\BR|u|pε.

Consider a smooth cut off function η:[0,)[0,1] such that η1 on [0,1] and η0 on [2,). Set vn(x):=η(2|x|/Rn)u(x). Then

(2.2)limnvn=uin Lp.

From the continuity of on Lp(BR), vn=u on BR, limnun=u in Lp(BR), and f(x,0)=0 for a.e. xN we obtain

limnBR|f(x,un)-f(x,un-vn)-f(x,vn)|νdx=limnBR|f(x,un)-f(x,un-u)-f(x,u)|νdx=0.

Since vn0 in N\BRn, this in turn yields for any ε>0 and R chosen accordingly, as in (2.1),

lim supnN|f(x,un)-f(x,un-vn)-f(x,vn)|νdx=lim supnBRn\BR|f(x,un)-f(x,un-vn)-f(x,vn)|νdx
Clim supnBRn\BR(|un|μ+|un-vn|μ+|vn|μ)ν
Clim supnBRn\BR(|un|p+|u|p)
Cε,

where C is independent of ε. Letting ε tend to 0 and using (2.2), we obtain

limn|(un)-(un-vn)-(u)|ν=0.

(b): The continuous embedding H1Lp implies that (un) is bounded in Lp, and the compact embedding H1(U)Lp(U) for bounded U implies that unu in Llocp. Defining vn as in (a), we therefore obtain that

vnuin H1,

and almost BL-splits along (un) with respect to u by (a).

(c): Since |un-u¯n|p0 and unv in Llocp, it follows that u¯nv in H1. Taking R large enough, (2.1) also holds true if we replace un by u¯n. Therefore, after passing to a subsequence for (un), and using the same subsequence for (u¯n), we obtain

limn|(u¯n)-(u¯n-vn)-(u)|ν=0.

## 3 Uniform continuity

Here we prove uniform continuity on bounded subsets of H1, making use of the periodicity of f in x. As a consequence, we also obtain BL-splitting along weakly convergent sequences in H1.

For simplicity we will only prove the case A=I (the identity transformation). The general case follows in an analogous manner. Denote the respective translation action of the additive group N on functions u:N by

(au)(x):=u(x-a),aN,xN.

Let , denote the standard scalar product in H1, defined by

u,v:=N(uv+uv),

and let denote the associated norm. Also denote by w-lim the weak limit of a weakly convergent sequence.

We first recall a functional consequence of Lions’ Vanishing Lemma [15, Lemma I.1.].

## Lemma 3.1.

Suppose for a sequence (un)H1 that anun0 in H1 for every sequence (an)ZN. Then un0 in Lp for all p(2,2*).

## Proof.

Note first that (un) is bounded in H1 since un0 in H1. We claim that

(3.1)supyNy+B1|un|20as n.

If the claim were not true there would exist ε>0 and a sequence (yn)N such that, after passing to a subsequence of (un), we would have

yn+B1|un|2ε.

Pick (an)N such that |an+yn|<1 for all n. With R:=N+1 it follows that an+yn+B1BR and hence

BR|anun|2ε

for all n. We reach a contradiction since anun0 in H1 and hence anun0 in L2(BR) by the theorem of Rellich and Kondrakov. Therefore, (3.1) holds true.

The claim of the theorem now follows from [15, Lemma I.1.] with p=q=2. Compare also with [18, Lemma 3.3]. ∎

## Proof of Theorem 1.3.

We start by proving the uniform continuity. Let (ui,n0)n0 be bounded sequences in H1 for i=1,2 and set C1:=maxi=1,2lim supnui,n0. Suppose for a contradiction that

(3.2)|u1,n0-u2,n0|p0as n

and that there is C2>0 such that

(3.3)|(u1,n0)-(u2,n0)|νC2for all n.

Successively we will define infinitely many sequences (ank)nN and (ui,nk)nH1, i=1,2, indexed by k0 and strictly increasing functions φk: with the following properties:

(3.4)maxi=1,2lim supnui,nkC1,
(3.5)limn|u1,nk-u2,nk|p=0,
(3.6)lim infn|(u1,nk)-(u2,nk)|νC2,
(3.7)w-limn(-aψk-1(n))ui,nk=0in H1, if 0<k, for i=1,2,

and

(3.8)limn|aψm(n)m-an|=if 0m<<k.

Here,

ψk:=φ+1φ+2φkif =-1,0,1,,k-1
ψkk:=id.

We need to say something about the extraction of subsequences. In order to obtain φk, (ank)n, and (ui,nk+1)n from (ui,nk), we first pass to a subsequence (ui,φk(n)k)n of (ui,nk)n and then use its terms in the construction. Once the new sequences (ank)n and (ui,nk+1)n are built we may remove a finite number of terms at their start, modifying φk accordingly, with the goal of obtaining additional properties. Beginning with the following iteration there are no more retrospective changes to the sequences already built. This is to assure a well defined infinite sequence of sequences, from which eventually we take the diagonal sequence. In this setting it seems clearer to make the selection of subsequences explicit, contrary to what is usually done when using concentration compactness methods [16, 14, 15] or when proving a variational splitting lemma.

For k=0 properties (3.4)–(3.8) are fulfilled by the definition of C1 and by (3.2) and (3.3). Assume now that (3.4)–(3.8) hold for some k0. Denote by Wk the set of vH1 such that there are a sequence (an)N and a subsequence of (u1,nk) with

w-limnanu1,nk=vin H1.

If w-limnanu1,nk=0 in H1 were true for all sequences (an)N, by Lemma 3.1 it would follow that limnu1,nk=0 in Lp. Equation (3.5) and the continuity of on Lp would lead to a contradiction with (3.6). Therefore,

qk:=supvWkv(0,C1].

Pick vkWk such that

(3.9)vkqk2>0.

There are (ank)nN and a strictly increasing function φk: such that

w-limn(-ank)u1,φk(n)k=vkin H1.

By (3.5) and by Theorem 2.1(b) and (c) there exists a sequence (vnk)nH1 such that

(3.10)limnvnk=vkin H1,
(3.11)w-limn(-ank)ui,φk(n)k=vkin H1, for i=1,2,

and

limn|((-ank)ui,φk(n)k)-((-ank)ui,φk(n)k-vnk)-(vk)|ν=0,i=1,2.

Set

ui,nk+1:=ui,φk(n)k-ankvnk.

By the equivariance of and the invariance of the involved norms under the N-action,

(3.12)limn|(ui,φk(n)k)-(ui,nk+1)-(ankvk)|ν=0for i=1,2,

and, since by (3.11) the map 2 BL-splits along (-ank)ui,φk(n)k with respect to vk, we have

(3.13)limn|ui,φk(n)k2-ui,nk+12-vk2|=0for i=1,2.

Equations (3.13) and (3.4) (for k) imply that

maxi=1,2lim supnui,nk+1C1,

hence (3.4) for k+1. The definition of the sequences ui,nk+1 and (3.5) (for k) imply that

(3.14)limn|u1,nk+1-u2,nk+1|p=limn|u1,φk(n)k-u2,φk(n)k|p=0,

hence (3.5) for k+1. It follows from (3.12) and (3.6) (for k) that

(3.15)lim infn|(u1,nk+1)-(u2,nk+1)|ν=lim infn|(u1,φk(n)k)-(u2,φk(n)k)|νC2,

hence (3.6) for k+1. Last but not least, from (3.7) (for k), (3.9), and (3.11) it follows that

(3.16)limn|aψmk(n)m-ank|=if m<k.

Since (3.8) is true for k, together with (3.16) we obtain (3.8) for k+1. Moreover, (3.16), (3.7) (for k) and (3.10) yield

w-limn(-aψk(n))ui,nk+1=w-limn((-aψk-1(φk(n)))ui,φk(n)k-(ank-aψk(n))vnk)=0in H1 if <k.

By the definition of ank, we have

w-limn(-ank)ui,nk+1=w-limn((-ank)ui,φk(n)k-vnk)=0in H1.

This proves (3.7) for k+1.

We now skip a finite number of elements of the sequences constructed in this induction step and adapt φk accordingly. Choosing m large enough, by (3.14) and (3.15) we obtain

|u1,m+nk+1-u2,m+nk+1|p1k+1

and

|(u1,m+nk+1)-(u2,m+nk+1)|νC2-1k+1

for all n. Property (3.8) (for k+1) implies that

limn|aψmk(n)m-aψk(n)|=limn|aψm(ψk(n))m-aψk(n)|=if m<k.

Since 2 BL-splits along weakly convergent sequences, this yields, together with (3.10), that

limnj=kaψjk(n)jvψjk(n)j2=j=kvj2

for all k. For large enough m this implies

j=kaψjk-1(φk(m+n))jvψjk-1(φk(m+n))j22j=kvj2for all n and k.

We fix m with these properties, we write ui,nk+1, ank, and vnk instead of ui,m+nk+1, am+nk, and vm+nk, respectively, and we write φk(n) instead of φk(m+n), thus all properties proved above remain valid, and, in addition, the following hold true:

(3.17)|u1,nk+1-u2,nk+1|p1k+1

and

(3.18)|(u1,nk+1)-(u2,nk+1)|νC2-1k+1

for all n and

(3.19)j=kaψjk(n)jvψjk(n)j22j=kvj2for all n and k.

Now we consider the process of constructing sequences as finished and proceed to prove properties of the whole set. By induction, (3.13) leads to

u1,nk+12=u1,ψ-1k(n)02-j=0kvj2+o(1)as n,

and hence j=0vj2C1 by (3.4). In view of (3.9) this yields

(3.20)qk0as k.

We claim that the diagonal sequence (u1,nn) satisfies

(3.21)bnu1,nn0in H1 as n, for every sequence (bn).

Note that by construction, for all k we have

u1,nk=u1,ψ-1k-1(n)-j=k-1aψjk-1(n)jvψjk-1(n)j.

Hence we have the representation

(3.22)u1,nn=u1,ψk-1n-1(n)k-j=kn-1aψjn-1(n)jvψjn-1(n)jif nk.

First we show that

(3.23)w-limn(-aψkn-1(n)k)u1,nn=0in H1, for all k0.

Fix k0. For every wH1 and ε>0 there is 0k+1 such that

w2j=0vj2ε2/2.

Then (3.19), (3.22), and the translation invariance of the norm yield for n0 the following:

|(-aψkn-1(n)k)u1,nn,w||(-aψkn-1(n)k)u1,ψkn-1(n)k+1,w|+|j=k+10-1(aψjn-1(n)j-aψkn-1(n)k)vψjn-1(n)j,w|
+wj=0n-1aψjn-1(n)jvψjn-1(n)j
|(-aψkn-1(n)k)u1,ψkn-1(n)k+1,w|+|j=k+10-1(aψjn-1(n)j-aψkn-1(n)k)vψjn-1(n)j,w|+ε.

It is easy to see that the sequence (ψkn-1(n))n is strictly increasing. Hence the first term in the last expression tends to 0 as n by (3.7), and the second term tends to 0 by (3.10) and (3.16). Since ε>0 and wH1 were arbitrary, this proves (3.23).

To finish the proof of (3.21), suppose for a contradiction that w-limnbnu1,nn=v0 in H1, for a subsequence. Equation (3.23) implies that

limn|bn+aψkn-1(n)k|=

for every k0. Pick k0 such that qk<v. This is possible by (3.20). Then, for every wH1, it follows from (3.19) and (3.22) that

|bnu1,ψk-1n-1(n)k-v,w||bnu1,nn-v,w|+|j=kn-1(bn+aψjn-1(n)j)vψjn-1(n)j,w|0

as n, as above. Hence,

w-limn(bnu1,ψk-1n-1(n)k)=v

with v>qk. Since

(u1,ψk-1n-1(n)k)n

is a subsequence of (u1,nk)n, this contradicts the definition of qk and proves (3.21).

We are now in the position to finish the proof of uniform continuity of . Equations (3.17) and (3.18) imply that

(3.24)limn|u1,nn-u2,nn|p=0

and

(3.25)lim infn|(u1,nn)-(u2,nn)|νC2.

By Lemma 3.1 and (3.21), we have u1,nn0 in Lp. Together with (3.24) and (3.25) this contradicts the continuity of on Lp and therefore proves the assertion about uniform continuity.

It only remains to prove BL-splitting for along weakly convergent sequences in H1 with respect to their weak limits. Suppose that unv in H1. By Theorem 2.1(b) there is a sequence (vn)H1 such that vnv in H1 and, after passing to a subsequence of (un), we have

(3.26)(un)-(un-vn)(v)in Lν

as n. Since (un) and (vn) are bounded in H1, and by the uniform continuity of on bounded subsets of H1 with respect to the Lp-norm (and hence also with respect to the H1-norm), it follows that we may replace vn by v in (3.26). Using this, a standard reasoning by contradiction yields the claim. ∎

## Proof of Example 1.2.

We first treat the case f(t):=cos(πt)|t|p. Set Rn:=n-p/N and fix a sequence (xn)N such that |xn| and

BRm(xm)BRn(xn)=

for mn. Define real functions u and un on N by setting

u:=k=1χBRk(xk),wn:=2nχBRn(xn),andun:=u+wn

for each n. It is straightforward to show that uLp, that (un) is a bounded sequence in Lp, and that unu pointwise and in Llocp. On the other hand, denoting by ωN the volume of the unit ball in N, we obtain

|Nf(un)-f(un-u)-f(u)|=|BRn(xn)f(u+wn)-f(wn)-f(u)|
=|BRn(xn)cos((2n+1)π)(2n+1)p-cos(2nπ)(2n)p-cosπ|
=|BRn(xn)(-(2n+1)p-(2n)p+1)|
(4.1)=ωN((2+1n)p+2p-(1n)p)2p+1ωN

as n. Since 2p+1ωN>0, this implies the claim.

For the other example, f(t):=cos(π/t)|t|p, we set Rn:=np/N and fix a sequence (xn)N such that |xn|/Rn and BRm(xm)BRn(xn)= for mn. We define

u:=k=112n(2n-1)χBRk(xk),wn:=12nχBRn(xn),andun:=u+wn

for each n. Then again, uLp, (un) is a bounded sequence in Lp, and unu pointwise and in Llocp. For xBRn(xn) we obtain

(4.2)u(x)+wn(x)=12n(2n-1)+12n=12n-1

and hence

|Nf(un)-f(un-u)-f(u)|=|BRn(xn)f(u+wn)-f(wn)-f(u)|
|BRn(xn)cos((2n-1)π)(12n-1)p-cos(2nπ)(12n)p|-BRn(xn)(12n(2n-1))p
=|BRn(xn)-(12n-1)p-(12n)p|-BRn(xn)(12n(2n-1))p
(4.3)=ωN((12-1n)p+(12)p-(12(2n-1))p)ωN2p-1

as n. This yields the claim. ∎

## Proof of Remark 1.4.

The construction of these counterexamples is closely related to Example 1.2. First consider the function f(t):=cos(π/t)t2. We define the Lipschitz-continuous cut-off function η: by

η(t):={1,t0,1-t,0<t<1,0,1t,

introduce Rn:=n2/N, pick a sequence (xn)N such that |xn|/Rn and BRm+1(xm)BRn+1(xn)= for mn, and define

u(x):=k=112n(2n-1)η(|x-xk|-Rk),wn(x):=12nη(|x-xn|-Rn),

and un:=u+wn for each n and xN. It is straightforward to check that u,wnH1 and that (wn) is bounded in H1. Since wn0 a.e., wn0 in H1. Using (4.2), we estimate

BRn+1(xn)\BRn(xn)|f(u+wn)-f(wn)-f(u)|BRn+1(xn)\BRn(xn)((12n-1)2+(12n)2+(12n(2n-1))2)
3ωNn2((Rn+1)N-RnN)=3ωN((1+n-2/N)N-1)0

as n. Hence by the calculation in (4.3), we have

|Nf(un)-f(un-u)-f(u)|=|BRn+1(xn)f(u+wn)-f(wn)-f(u)|
|BRn(xn)f(u+wn)-f(wn)-f(u)|-BRn+1(xn)\BRn(xn)|f(u+wn)-f(wn)-f(u)|
ωN2p-1

and the claim follows. Note that the example above has no simple analogue in the case f(t):=cos(π/t)|t|p for p>2, using Rn:=np/N as in the proof of the second case of Example 1.2. The reason is that the analogously defined sequence (wn) is not bounded in L2 in that case.

Now we treat the function f(t):=cos(πt)|t|2*. To this end put Rn:=n-2*/N, fix a sequence (xn)N such that |xn| and B2Rm(xm)B2Rn(xn)= for mn, and choose γ(0,1) small enough such that

(4.4)(32*+22*+1)((1+γ)N-1)22*+12.

Define

u(x):=k=1η(|x-xk|-RkγRk),wn(x):=2nη(|x-xk|-RnγRn),

and un:=u+wn for each n and xN. It follows that supp(wn)=B¯(1+γ)Rn(xn) for each n, where B¯r(z) denotes the closed ball in N with radius r and center z. Again, it is straightforward to check that u,wnH1, that (wn) is bounded in H1, and that wn0 in H1. Using (4.4), we estimate

B(1+γ)Rn(xn)\BRn(xn)|f(u+wn)-f(wn)-f(u)|B(1+γ)Rn(xn)\BRn(xn)((1+2n)2*+(2n)2*+1)
ωN((3n)2*+(2n)2*+n2*)(((1+γ)Rn)N-RnN)
=ωN(32*+22*+1)((1+γ)N-1)
22*+1ωN2

for all n. Hence by the calculation in (4.1), we have

|Nf(un)-f(un-u)-f(u)|=|B(1+γ)Rn(xn)f(u+wn)-f(wn)-f(u)|
|BRn(xn)f(u+wn)-f(wn)-f(u)|-B(1+γ)Rn(xn)\BRn(xn)|f(u+wn)-f(wn)-f(u)|
|BRn(xn)f(u+wn)-f(wn)-f(u)|-22*+1ωN2
22*+1ωN2

and the claim follows. Note that this example has no simple analogue in the case f(t):=cos(πt)|t|p for p<2*, using Rn:=n-p/N as in the proof of the first case of Example 1.2. Here the reason is that for the analogously defined sequence (wn), the sequence (wn) is not bounded in L2. ∎

Award Identifier / Grant number: 237661

Award Identifier / Grant number: IN104315

Funding statement: This research was partially supported by CONACYT grant 237661 and UNAM-DGAPA-PAPIIT grant IN104315 (Mexico).

## Acknowledgements

I would like to thank Kyril Tintarev for an informative exchange on this subject. Moreover, I thank the referee for suggesting to analyze the limiting cases in Theorem 1.3.

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