The optimal exponent in the embedding into the Lebesgue spaces for functions with gradient in the Morrey space

Advances in Mathematics 380 (2021) 107592 Contents lists available at ScienceDirect Advances in Mathematics www.elsevier.com/locate/aim The optimal exponent in the embedding into the Lebesgue spaces for functions with gradient in the Morrey space ✩ Xavier Cabré a,b,c, Fernando Charro d,∗ aICREA, Pg. Lluis Companys 23, 08010 Barcelona, Spain bUniversitat Politècnica de Catalunya, Departament de Matemàtiques, Diagonal 647, 08028 Barcelona, Spain cBGSMath, Campus de Bellaterra, Edifici C, 08193 Bellaterra, Spain dDepartment of Mathematics, Wayne State University, 656 W. Kirby, Detroit, MI 48202, USA a r t i c l e i n f o a b s t r a c t Article history: Received 30 July 2019 Received in revised form 21 December 2020 Accepted 7 January 2021 Available online xxxx Communicated by YanYan Li MSC: 42B37 46E35 Keywords: Morrey spaces Optimal embeddings Cantor sets We study a natural question that, apparently, has not been well addressed in the literature. Given functions uwith support in the unit ball B1⊂Rnand with gradient in the Morrey space Mp,λ(B1), where 1 <p <λ <n, what is the largest range of exponents qfor which necessarily u ∈Lq(B1)? While David R. Adams proved in 1975 that this embedding holds for q≤λp/(λ −p), an article from 2011 claimed the embedding in the larger range q<np/(λ −p). Here we disprove this last statement by constructing a function that provides a counterexample for q>λp/(λ −p). The function is basically a negative power of the distance to a set of Hausdorff dimension n −λ. When λ /∈Z, this set is a fractal. We ✩The authors were supported by MICINN grant MTM2017-84214-C2-1-P (Spain). X. Cabré is member of the research group 2017 SGR 1392 (Catalonia). F. Charro was also partially supported by a Juan de la Cierva fellowship and MICINN grants MTM2016-80474-P and PID2019-110712GB-I100 (Spain). *Corresponding author. E-mail addresses: [email protected] (X. Cabré), [email protected] (F. Charro). https://doi.org/10.1016/j.aim.2021.107592 0001-8708/© 2021 Elsevier Inc. All rights reserved. 2X. Cabré, F. Charro / Advances in Mathematics 380 (2021) 107592 also make a detailed study of the radially symmetric case, a situation in which the exponent qcan go up to np/(λ −p). © 2021 Elsevier Inc. All rights reserved. 1. Introduction This article originated from the following natural question: Given functions uwith support in the unit ball B1(0) ⊂Rnand with gradient in the Morrey space Mp,λ(B1(0)), where 1 <p <λ <n, what is the optimal range of exponents qsuch that necessarily u ∈Lq(B1(0))?Apparently, this question has not been well addressed in the literature. In fact, the authors of [3, Theorem 2.5] claimed a range of exponents which, as we will prove in the current paper, turns out to be larger than the correct one. Our motivation came from the recent work [7]of the first author in collaboration with A. Figalli, X. Ros-Oton, and J. Serra, on the regularity of stable solutions to semilinear elliptic equations. Actually, the results of [7]are deduced from a Morrey type bound for the gradient of a stable solution, among other tools (see Remark 1.2 below for more details). The following is the precise statement of the question that we are concerned with. Given real numbers pand λsuch that 1<p<λ<n, we wish to know for which exponents qthe inequality uLq(B1(0)) ≤C∇uMp,λ(B1(0)) (1.1) holds true for functions uwith support in B1(0) ⊂Rnand for a constant Cindependent of u, where ∇up Mp,λ(Ω) := sup r>0,y∈Ωrλ−n Ω∩Br(y) |∇u(x)|pdx is the Morrey norm of ∇uin a domain Ω ⊂Rn. Notice that when λequals the dimension nand Ω =B1(0), (1.1) corresponds to the Sobolev inequality in B1(0) ⊂Rn. In 1975, D.R. Adams [1, Theorem 3.1] proved the following result. Let us denote in the sequel p1:= λp λ−pand p2:= np λ−p. Observe that, clearly, p1<p 2. X. Cabré, F. Charro / Advances in Mathematics 380 (2021) 107592 3 Theorem 1.1 (D.R. Adams [1]). Let p, λ ∈Rsatisfy 1 <p <λ <nand let u :Rn→R be a Lipschitz function with u ≡0in Rn\B1(0). Then, for every q≤p1, inequality (1.1) holds for a constant Cdepending only on n, p, and λ. For the reader’s convenience, in Section 4we will include the proof of the theorem, as given by Adams [1]. The case p =1, which involves weak spaces, is also treated in [1]. In fact, Adams [1, Proposition 3.1 and Theorems 3.1 and 3.2] proved the following stronger embedding: uMp1,λ(B1(0)) ≤C∇uMp,λ(B1(0)).(1.2) While inequality (1.2)is dimensionless by scaling, note that the dimensionless exponent for inequality (1.1)is q=p2. This suggests that (1.1)could hold with q=p2, or at least for q<p 2. In fact, it is easy to prove that among radially symmetric functions, (1.1) holds for every q<p 2(see [5, Proposition 1.2(i)] and also Theorem 1.5 below). In 2011, the authors of [3, Theorem 2.5] claimed that (1.1)held for every q<p 2and for general functions, not necessarily radial. Some years later, we realized that the proof of [3, Theorem 2.5] was not correct. After that, the claim was withdrawn by the same authors in the Errata papers [4]and [5]. At the same time, we could not find other works addressing the exact question of what is the optimal exponent. In the present paper we show that actually q=p1is the largest possible exponent in (1.1). To show this, for every q>p 1we construct a non-radial function, described in detail below, for which uLq(B1(0)) =∞while ∇uMp,λ(B1(0)) <∞. An important feature of the function is that it depends only (up to a cutoff function) on kvariables (x1, ..., xk), where kis the smallest integer such that λ ≤k. The function is basically a negative power of the distance to a set of Hausdorff dimension n −λ. When λis not an integer this set is a fractal and, therefore, the structure of the function is not so “simple”. In fact, it will be rather delicate to control the Morrey norm of its gradient. We could not find a simpler counterexample for λ /∈Z, although we had several candidates that finally did not work. The possibility of finding simpler examples remains as an open question. In addition, we also consider a related norm, which we call the “triple norm”, given by |∇u|p p,λ;Ω := sup y∈Ω Ω |∇u(x)|p|x−y|λ−ndx, (1.3) where Ω ⊂Rnis a domain. The article [7]on stable solutions to semilinear equations gives rise naturally to such a norm (see Remark 1.2 below).1Note that, clearly, we have ∇uMp,λ(Ω) ≤|∇u|p,λ;Ω (1.4) 1The triple norm has been previously considered in the setting of the hole-filling technique for integral estimates; see [6, Section 1.2.3] among others. It also appears in [11], where it is called the Cordes-Nirenberg norm. 4X. Cabré, F. Charro / Advances in Mathematics 380 (2021) 107592 for every function u. The function that we construct will also satisfy  |∇u |p,λ;B1(0) <∞, and thus q=p1is the largest possible exponent also for the embedding uLq(B1(0)) ≤C|∇u|p,λ;B1(0).(1.5) Instead, among radial functions we show that inequality (1.5)holds for every q≤p2, in contrast to inequality (1.1)for radial functions, which holds only for q<p 2. Remark 1.2. The regularity results from the recent paper [7]on stable solutions to semilinear equations −Δu =f(u)in a domain Ω ⊂Rnare based on bounds for a Morrey norm with p =2, of ∇uor, given a point y, of the radial derivative ∇u(x) ·(x −y)/|x −y|. For this, see [7, Lemma 2.1, step 2 in the proof of Theorem 1.2, and proof of Theorem 7.1], where the arguments also lead to the triple norm (1.3). The boundedness results from [7]up to dimension n ≤9 correspond to p =2and λ =2, while in the Lqresults for n ≥11 one has p =2 <λ <nas in our paper. The results of the current article are used in [7]to determine optimally a range of exponents qfor which stable solutions necessarily belong to Lqin dimensions n ≥11. Summarizing, our main contribution is the following result. It provides a counterexample to the validity of (1.1)and (1.5)for q>p 1, given by a function uwhich is basically a negative power of the distance to a set of Hausdorff dimension n −λ. When λ /∈Z, this set is a fractal. Theorem 1.3. Let p, λ ∈Rsatisfy 1 <p <λ <n. Then, for every q>p 1:= λp/(λ − p)there exists a function u :Rn→Rwith compact support in B1(0), belonging to Mp,λ(B1(0)), and such that uLq(B1(0)) =∞and |∇u|p,λ;B1(0) <∞.(1.6) In particular, we also have ∇uMp,λ(B1(0)) <∞. If λis an integer, such function ucan be taken to be u(x)=|x|−α−2α+ξ(|x|) (1.7) where x =(x, x) ∈Rλ×Rn−λ, the parameter αsatisfies λ q≤α<λ−p p,(1.8) and ξ:R+→[0, 1] is a cutoff function with ξ≡1in [0, 1/2) and ξ≡0in R+\[0, √3/3). X. Cabré, F. Charro / Advances in Mathematics 380 (2021) 107592 5 If k−1 <λ <kfor some integer k∈[2, n], the function ucan be taken to be u(x)=⎧ ⎨ ⎩dist(x, Cn,λ)−α−4α+if k=n dist(x,Ck,λ)−α−4α+ξ(|x|)ifk<n, (1.9) where x =(x, x) ∈Rk×Rn−k, αsatisfies (1.8), ξis a cutoff function as above, and Ck,λ is a set of Hausdorff dimension k−λin Rkgiven by Ck,λ ={0}×Cγ⊂Rk−1×[−1/2,1/2], where Cγ⊂[−1/2, 1/2] is the generalized Cantor set with parameter γ=1 −21−1 k−λ defined in the following remark. We emphasize that the counterexample to the embedding is therefore given by a function that, up to a cutoff, only depends on kvariables, where kis the smallest integer such that λ ≤k. Remark 1.4. The generalized Cantor set Cγ(see [9]) is obtained from the interval [−1/2, 1/2] by removing at iteration j=1, 2, ... the central interval of length γl j−1 from each remaining segment of length lj−1= ((1 −γ)/2)j−1; see Fig. 2in Section 6. The usual Cantor set corresponds to γ=1/3. The reason for our choice of γis that the Hausdorff dimension of Cγis −log 2 log 1−γ 2 =k−λ∈(0,1) (see [9, Theorem 9.3]). In particular, letting λrange from k−1to kyields any fractal dimension between 0 and 1, and (1.9)somehow interpolates between the integer cases λ =k−1and λ =k. Let us describe briefly how we found that p1is the optimal exponent. In the case when λis an integer, the hint came from the number p1=λp/(λ −p), which can be thought of as the Sobolev exponent in dimension λ. It was natural then to choose the function (1.7), since it gives a counterexample for the Sobolev inequality in Rλwhen q>λp/(λ −p) =p1and the exponent αis chosen appropriately. When λ ∈Z, (1.7)is basically a negative power of the distance to a subspace of dimension n −λ. Therefore, when λ /∈Z, a negative power of the distance to a set of Hausdorff dimension n −λbecame a natural candidate to counterexample. This is what the function in (1.9) basically is, a power of the distance to Ck,λ ×Rn−k. It may be of interest to recall here the solutions found by R. Schoen and S.-T. Yau in [12, Section 5] for nonlinear equations with critical exponent. They construct weak solutions which are singular on a Cantor set with fractional Hausdorff dimension; see [12, 6X. Cabré, F. Charro / Advances in Mathematics 380 (2021) 107592 Page 70]. Obviously, nonlinear equations with critical exponent are closely related to the Sobolev embedding. Another result on solutions with a singular set of Cantor type is due to Fonseca, Malý, and Mingione [10], a paper that concerns the minimizers of a certain scalar, convex, and regular Lagrangian. The paper [2]by Adams and Lewis was brought to our attention after the completion of the current article. In [2], the authors proved that functions which satisfy an integrability condition of Morrey-Besov type belong also to a certain Lorentz space. In addition, they construct examples of functions to show that their embeddings are the best possible. The Morrey-Besov norm is a fractional Morrey-type condition involving the α-th difference quotients of a function, where 0 <α<1. Thus, this concerns more exotic norms than the basic and standard ones that we treat. The authors of [2]mention the possibility that the ideas in their proof of [2, Theorem 3] could be extended from the case α∈(0, 1) that they treat to the case α=1(and p =qin their paper). This is something that we have not explored. There could be the usual delicate issues taking limits of fractional integral norms as α→1, or even simply the impossibility of taking this limit or adapting the proof for α=1. However, if this could be done, it would show that actually q=p1 is the largest possible exponent in (1.1). On the other hand, it is not clear at all if their example would allow to recover our result on the optimal exponent for the embedding (1.5) concerning the “triple” norm. Among radial functions, the optimal ranges of exponents in inequalities (1.1)and (1.5)are strictly larger than those of Theorems 1.1 and 1.3. This is the content of the following result, where we show that the exponent qcan go up to q2. Interestingly, here the answer is different for the Morrey and the “triple” norms: we prove that (1.1)is false for q=p2while (1.5)holds for this exponent. Here we can include the exponent p =1. Theorem 1.5. Let p, λ ∈Rsatisfy 1 ≤p <λ <n, and let p2:= np/(λ −p). (a) For every 1 ≤q<p 2and all radially symmetric C1functions uvanishing on ∂B1(0), we have uLq(B1(0)) ≤C∇uMp,λ(B1(0)),(1.10) where Cis a constant depending only on n, p, λ, and q. In addition, this embedding is false for q≥p2. (b) For all radially symmetric C1functions uwith compact support in Rn, we have uLp2(Rn)≤C|∇u|p,λ;Rn,(1.11) where Cis a constant depending only on n, p, and λ. In addition, p2is the optimal exponent in this inequality. The paper is organized as follows. In Section 2, we prove a monotonicity result, Lemma 2.1, that we will use several times throughout the paper to optimize the lo- X. Cabré, F. Charro / Advances in Mathematics 380 (2021) 107592 7 cation of the “singularities” yin the Morrey and “triple” norms, both in the radial and non-radial cases. In Section 3we prove the embeddings for radial functions, Theorem 1.5. In Section 4we provide for the reader’s convenience D.R. Adams’ [1] proof of Theorem 1.1 in the general case of non-radial functions. In Sections 5and 6we prove Theorem 1.3 on the optimality of the embeddings. We consider separately the case when λis an integer in Section 5(for its simplicity) and the case when λis a non-integer in Section 6(which is much more involved). Notation. In the sequel B(m) R(x) denotes the open ball in Rmof radius Rcentered at x. For simplicity, whenever mor xare omitted, we will consider m =nand x = 0 respectively. By C, we denote constants that may change from line to line. For points in Rn, we will write x =(x, x) ∈Rk×Rn−kfor ka positive integer specified from the context. Given a function u, u+=max{u, 0}is its positive part. As mentioned before, we will denote p1=λp/(λ −p)and p2=np/(λ −p). For convenience, we will use the following standard notation for intervals: a(−b, b) +h =(h −ab, h +ab). Finally, dist (t, U)=inf z∈U|t −z| as usual. 2. On the location of the singularity in the “triple” norm In this section we prove a monotonicity result that we will use several times in the sequel to study which locations of the “singularity” ymake larger the integral in the “triple norm” |∇u|p p,λ;Ω =sup y∈Ω Ω |∇u(x)|p|x−y|λ−ndx. Lemma 2.1. Consider a domain Ω ⊂Rn, convex in the e1direction, and symmetric with respect to {z1=0}. Let J:Rn→Rbe given by J(y):= Ω h(z)|z−y|−θdz with θ>0and ha non-negative function in Ω. Then: (a) Jis non-increasing with respect to y1in {y1≥supz∈Ωz1}, and non-decreasing with respect to y1in {y1≤infz∈Ωz1}. (b1) Suppose that the non-negative function hsatisfies that, for some η∈[0, supz∈Ωz1) and every y1∈[η, supz∈Ωz1), h(z∗)≥h(z)for all z∈Ω∩{z1≥y1}, where z∗=(2y1−z1, z) ∈R ×Rn−1is the reflection of zwith respect to the hyperplane {z1=y1}. Then, Jis non-increasing with respect to y1in {η≤y1< supz∈Ωz1}. 8X. Cabré, F. Charro / Advances in Mathematics 380 (2021) 107592 - 6   B B B B s ss s 0z1 z∗z y1 Fig. 1. Monotonicity argument in the proof of Lemma 2.1. (b2) On the other hand, if the non-negative function his such that, for some η∈ (infz∈Ωz1, 0] and every y1∈(infz∈Ωz1, η], h(z∗)≥h(z)for all z∈Ω∩{z1≤y1} with z∗as before, then Jis non-decreasing with respect to y1in {infz∈Ωz1<y 1≤ η}. Proof. Observe first that for every z∈Ω, the quantity |z−y|is increasing with respect to y1in {y1≥supz∈Ωz1}, and decreasing with respect to y1in {y1≤infz∈Ωz1}. Since h ≥0 and θ>0, we deduce that Jis non-increasing with respect to y1in {y1≥supz∈Ωz1}, and non-decreasing with respect to y1in {y1≤infz∈Ωz1}. This proves part (a). Assume now that 0 ≤η≤y1<supz∈Ωz1and compute ∂y1J(y)=θ Ω h(z)(z1−y1)|z−y|−θ−2dz =θ Ω∩{z1≥y1} h(z)(z1−y1)|z−y|−θ−2dz + Ω∩{z1≤y1} h(z)(z1−y1)|z−y|−θ−2dz. X. Cabré, F. Charro / Advances in Mathematics 380 (2021) 107592 9 For every z∈Ω ∩{z1≥y1}, let z∗=(2y1−z1, z)be its reflection with respect to the hyperplane {z1=y1}; see Fig. 1. Then, |z∗−y| =|z−y|, while h(z∗) ≥h(z)by hypothesis. Therefore, for every z∈Ω ∩{z1≥y1}, we have h(z)(z1−y1)|z−y|−θ−2≤−h(z∗)(z∗ 1−y1)|z∗−y|−θ−2 and hence, using that h ≥0,  Ω∩{z1≥y1} h(z)(z1−y1)|z−y|−θ−2dz ≤−  (Ω∩{z1≥y1})∗ h(z)(z1−y1)|z−y|−θ−2dz ≤−  Ω∩{z1≤y1} h(z)(z1−y1)|z−y|−θ−2dz. Therefore, ∂y1J(y) ≤0for all η≤y1<supz∈Ωz1and the conclusion in part (b1) follows. The statement for infz∈Ωz1<y 1≤ηin part (b2) follows from (b1) by reflection.  3. The radial case: proof of Theorem 1.5 In this section we establish Theorem 1.5 on the embeddings for radial functions. In the proof we apply Lemma 2.1 (the monotonicity result proved in Section 2), which will also be used for the non-radial case in Section 6. We point out that the Sobolev inequalities with monomial weights established in [8]by Ros-Oton and the first author will be of great use. Proof of Theorem 1.5.We structure the proof in four parts. In Part 1 we establish estimate (1.10), while in Part 2 we show that q<p 2is the optimal range of exponents for this estimate. In Part 3a we prove (1.11); here we will use the results of [8]. Part 3b provides an alternative proof of (1.11). Finally, we show in Part 4 that p2is the largest exponent for which (1.11)holds. Part 1. We proceed now to show estimate (1.10)for 1 ≤q<p 2. All the constants C will depend only on n, p, λ, and q. On the one hand we have 16 X. Cabré, F. Charro / Advances in Mathematics 380 (2021) 107592 By the well-known Lpestimate for the maximal function M0when p >1, there exists a constant Cdepending only on nand psuch that M0|∇u|Lp(B1(0)) ≤C∇uLp(B1(0)), and therefore uLq(B1(0)) ≤C|B1(0)|1 q−1 p1∇u p λ+p p1 Mp,λ(B1(0)) ≤C∇uMp,λ(B1(0)) with Cdepending only on n, p, and λas desired. Therefore, it remains to prove claims (4.1)and (4.2). Consider first estimate (4.1). To prove it, notice that for σ∈Rnwith |σ| =1 |u(x)|=− ∞  0 d dru(x+rσ)dr≤ ∞  0|∇u(x+rσ)|dr. Then, integrating on σwe get |u(x)|≤C ∞  0 ∂B1(0) |∇u(x+rσ)|r1−nrn−1dσdr =C(I1|∇u|)(x) and (4.1)is proved. Next, consider estimate (4.2). We reproduce the argument in [1]to show that for a given function fwith compact support in Rn, we have I1f(x)≤CMλ/pf(x)p λM0f(x)1−p λ,(4.3) where Cdepends only on n, p, and λ. For f≡ 0, let δ>0to be determined later and set I1f(x)= Rn f(y)|y−x|1−ndy = {y:|x−y|<δ} f(y)|y−x|1−ndy + {y:|x−y|≥δ} f(y)|y−x|1−ndy =I+I. Let ak(x)={y:2 kδ≤|x−y|<2k+1δ}for k∈Z. Then, X. Cabré, F. Charro / Advances in Mathematics 380 (2021) 107592 17 |I|≤ ∞  k=1  a−k(x) |f(y)||x−y|1−ndy ≤ ∞  k=1 2−kδ1−n2−k+1δnM0f(x)=2 nδM0f(x). Similarly, |I|≤ ∞  k=0  ak(x) |f(y)||x−y|1−ndy ≤ ∞  k=0 2kδ1−n2k+1δn−λ pMλ/pf(x)=Cδ1−λ pMλ/pf(x), since p <λ. The choice δ=δ(x)=Mλ/pf(x) M0f(x)p λ finally gives (4.3).  5. Proof of Theorem 1.3 in the case when λis an integer In this section we prove Theorem 1.3 when λis an integer. The argument is very simple. The case λ /∈Zis the core of our paper and will be considered in Section 6. As mentioned in the introduction, the choice of the counterexample when λ ∈Zwas hinted by the number p1=λp/(λ −p), which can be thought of as the Sobolev exponent in dimension λ. Then, it is natural to choose a function that provides a counterexample for the Sobolev inequality in dimension λand then look at this function embedded in the n-dimensional space. Namely, we take u(x)=|x|−α−2α+ξ(|x|),(5.1) where x =(x, x) ∈Rλ×Rn−λand ξ:R+→[0, 1] is a cutoff function with ξ≡1in [0, 1/2) and ξ≡0in R+\[0, √3/3). Note that clearly uhas support in B1(0) ⊂Rn. The rest of the section is devoted to show the following result, which proves Theorem 1.3 when λis an integer. Proposition 5.1. Let λbe an integer such that 1 <p <λ <nand assume that q>p 1:= λp/(λ −p). Then, for ugiven by (5.1), we have that uLq(B1)=∞and  |∇u |p,λ;B1< ∞if λ q≤α<λ−p p. 18 X. Cabré, F. Charro / Advances in Mathematics 380 (2021) 107592 This proves the optimality of the range q≤p1for (1.5)when λ ∈Z, and in turn also for (1.1). Proof. For every y∈B(n) 1we have that  B(n) 1 |∇u(x)|p|x−y|λ−ndx ≤C B(λ) 1 B(n−λ) 1 |x|−αp−p|x−y|λ−ndxdx =C B(λ) 1 |x|−αp−p B(n−λ) 1 1+|x −y| |x−y|2λ−n 2 |x−y|λ−ndxdx, for some constant Cindependent of y. The change of variables z=x−y |x−y|yields  B(n−λ) 1 1+|x −y| |x−y|2λ−n 2 |x−y|λ−ndx ≤ B(n−λ) 2/|x−y|1+|z|2λ−n 2dz =C 2 |x−y|  01+r2λ−n 2rn−λ−1dr ≤C 2 |x−y|  0 max{1,r}λ−nrn−λ−1dr ≤C(1 + |log |x−y||). Therefore, we have  B(n) 1 |∇u(x)|p|x−y|λ−ndx ≤C B(λ) 1 |x|−αp−p(1 + |log |x−y||)dx. We claim that the last integral is bounded uniformly in y∈B(λ) 1. To verify this, since |log |x−y|| ≤ log 2 for x∈B(λ) 1\B1(y)and λ −αp −p >0, it suffices to control the integral over B1(y). But then, calling z:= y−x, the integral becomes  B(λ) 1 h(z)|z−y|−αp−pdz X. Cabré, F. Charro / Advances in Mathematics 380 (2021) 107592 19 with h(z) =1 +|log |z|| =1 −log |z|for z∈B(λ) 1. Now, since his non-negative and radially decreasing in B(λ) 1, we can apply Lemma 2.1 with η=0and conclude that the largest value of the integral corresponds to y=0. But since we have assumed λ −αp −p >0, the integral with y=0is finite. On the other hand, uq Lq(B1)≥ B(λ) 1/2  B(n−λ) 1/2 uqdxdx =C B(λ) 1/2|x|−α−2αqdx=C 1 2  0r−α−2αqrλ−1dr and the last integral is divergent since λ ≤αq by hypothesis.  6. Proof of Theorem 1.3 in the general case In this section we conclude the proof of Theorem 1.3 by considering the case when λ is not an integer. Let us motivate first the case n −1 <λ <n. As we have seen in Section 5, when λis equal to n −1 expression (5.1) provides a counterexample to the embedding (1.5)when q>p 1, and therefore also to (1.1)in view of (1.4). On the other hand, when λis equal to nthe Morrey and triple norms coincide with the Sobolev norm and u(x) =(|x|−α−2α)+ provides a counterexample to embedding (1.5)for q>p 1=p∗(in this case (1.5)is simply the Sobolev embedding). In both cases the function that yields the counterexample is basically a negative power of the distance function, either to the origin in the case λ =n, or to a line when λ =n −1. Therefore, when λis strictly between n −1and n, a negative power of the distance to a fractal set of non-integer dimension n −λis a natural candidate to be a counterexample to inequality (1.5). Let us describe precisely the functions that provide the counterexample. When n −1 < λ <n, we consider uα,n(x)=dist(x, Cn,λ)−α−4α+(6.1) for Cn,λ ={0} ×Cγ⊂Rn−1×[−1/2, 1/2], where Cγis a generalized Cantor set with parameter γ=1−21−1 n−λ∈(0,1).(6.2) The generalized Cantor set Cγ(see [9]) is obtained from the interval [−1/2, 1/2] by removing at iteration j=1, 2, ...the central interval of length γl j−1from each remaining 20 X. Cabré, F. Charro / Advances in Mathematics 380 (2021) 107592 −1 2 1 2 −γ 2 γ 2 Fig. 2. Construction of the generalized Cantor set Cγ. segment of length lj−1=(1 −γ)/2j−1; see Fig. 2. A precise expression for Cγis given later in (6.8), (6.9), and (6.10). The usual Cantor set corresponds to γ=1/3. The reason for our choice of γin (6.2)is that the Hausdorff dimension of Cγis −log 2 log 1−γ 2 =n−λ (see [9, Theorem 9.3]). Thus, letting λvary between n −1and nyields any fractal dimension between 0 and 1. In particular, (6.1)somehow interpolates the integer cases λ =n −1and λ =n. Note that uα,n has support in B(n) 3/4(0). Indeed, if y∈C n,λ then |y| ≤1/2, and thus |x −y| ≥1/4if |x| ≥3/4; in particular dist(x, Cn,λ) ≥1/4and uα,n(x) =0. In the case when k−1 <λ <kfor some integer k∈{2, ..., n −1}we embed into Rn the counterexample in Rkby means of an appropriate cutoff function (as we did in the previous section when λ ∈Z). In this way, we reduce the proof to the case n −1 <λ <n. More precisely, we consider uα(x)=uα,k(x)ξ(|x|)ifk<n, (6.3) where x =(x, x) ∈Rk×Rn−k, uα,k is given by (6.1)with nreplaced by k, i.e., uα,k(x)=dist(x,Ck,λ)−α−4α+,(6.4) and ξ:R+→[0, 1] is a cutoff function with ξ≡1in [0, 1/2) and ξ≡0in R+\[0, √3/3). Note that if uα(x) =0then necessarily |x| ≤3/4and |x| ≤√3/3; thus |x| <1. The rest of the section is devoted to proving the following result, which together with Proposition 5.1 completes the proof of Theorem 1.3. X. Cabré, F. Charro / Advances in Mathematics 380 (2021) 107592 21 Theorem 6.1. Let p, λ, q∈Rbe such that 1 <p <λ <n, k−1 <λ <kfor some integer k∈{2, ..., n}, and q≥1. Consider uα(x)=⎧ ⎨ ⎩dist(x, Cn,λ)−α−4α+if k=n dist(x,Ck,λ)−α−4α+ξ(|x|)ifk∈{2,...,n−1}, with x =(x, x) ∈Rk×Rn−k, ξ:R+→[0, 1] a cutoff function as described after (6.4), and Ck,λ a set of Hausdorff dimension k−λgiven by Ck,λ ={0}×Cγ⊂Rk−1×[−1/2,1/2], where Cγ⊂[−1/2, 1/2] is a generalized Cantor set with parameter γ=1 −21−1 k−λ. Then, uαLq(B(n) 1)=∞and |∇uα|p,λ;B(n) 1<∞ if λ q≤α<λ−p p.(6.5) This proves the optimality of the range q≤p1=λp/(λ −p)for inequality (1.5), and in turn also for (1.1). The proof of Theorem 6.1 is divided into two parts. In the first part we reduce the computations from dimension nto dimension kwith a similar argument to the one in the proof of Proposition 5.1. More precisely we will show, for uα,k given by (6.4), that |∇uα|p,λ;B(n) 1≤C|∇uα,k|p,λ;B(k) 1 and uαLq(B(n) 1)≥C−1uα,kLq(B(k) 1), for some constant C, and hence that it is enough to study the case k=ntaking uα=uα,n. In the second part of the proof we will show that (6.5)leads to uα,nLq(B(n) 1)=∞and |∇uα,n|p,λ;B(n) 1<∞, as desired; this part is the content of the following two propositions. Proposition 6.2. Let λ, q∈Rbe such that λ >n −1and q≥1. Consider uα,n given by (6.1)with αsuch that λ ≤αq. Then, uα,nLq(B(n) 1)=∞. 22 X. Cabré, F. Charro / Advances in Mathematics 380 (2021) 107592 Proposition 6.3. Let λ, p ∈Rbe such that n −1 <λ <n, p >1and consider uα,n given by (6.1)with α>0satisfying α<λ−p p.(6.6) Then, |∇uα,n|p p,λ;B(n) 1 := sup y∈B(n) 1 B(n) 1 |∇uα,n(x)|p|x−y|λ−ndx < ∞. Let us now prove Theorem 6.1 assuming Propositions 6.2 and 6.3, which will be established afterwards. Proof of Theorem 6.1.Since Propositions 6.2 and 6.3 yield the result when k=n, let us assume k<n. Let y∈B(n) 1. By (6.3)and (6.4)we have |∇uα(x)|≤C|∇uα,k(x)|+uα,k(x)≤C|∇uα,k(x)| for almost every x ∈B(n) 1, where we have used that the modulus of the gradient of a distance function is equal to 1a.e. Therefore,  B(n) 1 |∇uα(x)|p|x−y|λ−ndx ≤C B(k) 1 B(n−k) 1 |∇uα,k(x)|p|x−y|λ−ndxdx =C B(k) 1 |∇uα,k(x)|p|x−y|λ−k B(n−k) 1 1+|x −y| |x−y|2λ−n 2 |x−y|k−ndxdx. The change of variables z=x−y |x−y|yields  B(n−k) 1 1+|x −y| |x−y|2λ−n 2 |x−y|k−ndx ≤ B(n−k) 2/|x−y|1+|z|2λ−n 2dz =C 2 |x−y|  01+r2λ−n 2rn−k−1dr ≤C 2 |x−y|  0 max{1,r}λ−nrn−k−1dr =C⎛ ⎜ ⎝ 1  0 rn−k−1dr + 2 |x−y|  1 rλ−k−1dr⎞ ⎟ ⎠≤C X. Cabré, F. Charro / Advances in Mathematics 380 (2021) 107592 23 independently of y. Therefore,  B(n) 1 |∇uα(x)|p|x−y|λ−ndx ≤Csup y∈B(k) 1 B(k) 1 |∇uα,k(x)|p|x−y|λ−kdx=C|∇uα,k|p,λ;B(k) 1, and Proposition 6.3 applied in Rk, i.e., with nreplaced by k, yields  |∇uα |p,λ;B(n) 1<∞. On the other hand, uαq Lq(B(n) 1)= B(k) 3/4 uq α,k(x) B(n−k) √3/3 ξq(|x|)dxdx=Cuα,kq Lq(B(k) 3/4), which is infinite by Proposition 6.2 applied with n =k, since (as we pointed out) uα,k given by (6.4)has support in B(k) 3/4. We devote the rest of the section to the proofs of Propositions 6.2 and 6.3. In the sequel we assume n−1<λ<n. Recall that uα,n(x)=dist(x, Cn,λ)−α−4α+(6.7) for Cn,λ ={0} ×Cγ⊂Rn−1×[−1/2, 1/2], where Cγis the generalized Cantor set with parameter γ=1 −21−1 n−λdefined in the beginning of this section. We have −1 2,1 2\Cγ:= ∞  l=1 2l−1  m=1 Gl,m,(6.8) where the union is disjoint and Gl,m are the 2l−1gap-intervals introduced in generation l, namely2 Gl,m =1−γ 2l−1−γ 2,γ 2+hl,m =hl,m −γ 21−γ 2l−1 ,h l,m +γ 21−γ 2l−1, (6.9) 2We will not need the following precise expression for the gaps, but only to understand their size and self-similar structure. 24 X. Cabré, F. Charro / Advances in Mathematics 380 (2021) 107592 where hl,m =−1 2+1 21−γ 2l−1 +1+γ 2 l−1  j=1 cj1−γ 2j−1 .(6.10) Here cj∈{0, 1}for all j=1, ..., l−1, and the index m ∈{1, 2, ..., 2l−1}runs through all possible choices of the coefficients c1c2...c l−1; see Fig. 2above. As a consequence of (6.8), we have that Cγis a compact set of Lebesgue measure 0. In addition, the set Cγis self-similar, that is, Cγ=S1(Cγ) ∪S2(Cγ) where S1(t) =(1 −γ)t/2 −(1 +γ)/4and S2(t) =(1 −γ)t/2 +(1 +γ)/4. Finally, the Hausdorff dimension of Cγis − log 2/ log((1 −γ)/2)), see [9, Theorem 9.3]. Notice that we have chosen γ=1 −21−1 n−λsuch that the Hausdorff dimension of Cγis n −λ. 6.1. Proof of Proposition 6.2: computation of the Lqnorm In this subsection we provide the proof of Proposition 6.2. Proof of Proposition 6.2.Recall that Cn,λ ={0} ×Cγ⊂Rn−1×[−1/2, 1/2]. We will denote x =(x, x) ∈Rn−1×R. Since Cγhas zero Lebesgue measure, (6.8)leads to uα,nq Lq(B(n) 1)≥ 1 2  −1 2 B(n−1) 1/4dist(x, Cn,λ)−α−4αq +dxdx = ∞  l=1 2l−1  m=1  Gl,m  B(n−1) 1/4dist(x, Cn,λ)−α−4αq +dxdx, where Gl,m are given by (6.9)and (6.10). An affine change of variables x=1−γ 2l−1t, x =1−γ 2l−1t +hl,m and the self-similarity of Cn,λ yield  Gl,m  B(n−1) 1/4dist(x, Cn,λ)−α−4αq +dxdx =1−γ 2(n−αq)(l−1) γ 2  −γ 2 |t|≤ 1 42 1−γl−1dist(t, Cn,λ)−α−1−γ 2α(l−1) 4αq + dtdt ≥1−γ 2(n−αq)(l−1) γ 2  −γ 2 B(n−1) 1/4dist(t, Cn,λ)−α−4αq +dtdt X. Cabré, F. Charro / Advances in Mathematics 380 (2021) 107592 25 for each m ∈{1, ..., 2l−1}. Therefore, adding these 2l−1integrals of generation l, and then summing in l, we have uα,nq Lq(B(n) 1)≥ ∞  l=1 21−γ 2n−αql−1γ 2  −γ 2 B(n−1) 1/4dist(t, Cn,λ)−α−4αq +dtdt. Since the integral on the right-hand side is positive, it is enough to show that the series diverges. This happens whenever 21−γ 2n−αq ≥1, which by our choice of γis equivalent to n−αq ≤−log 2 log 1−γ 2 =n−λ. This inequality holds by hypothesis.  6.2. Proof of Proposition 6.3: bound for the “triple norm” The proof of Proposition 6.3 has two parts. The first one (Lemma 6.6 below) shows that in order to bound the triple norm of ∇uα,n, it suffices to only consider points y∈{0} ×[−1/2, 1/2], instead of the full B1 (n). More precisely, we prove that |∇uα,n|p p,λ;B(n) 1≤Csup y=0,|y|≤ 1 2 1 2  −1 2 B(n−1) 1/4 dist(x, Cn,λ)−αp−p|x−y|λ−ndxdx. (6.11) Since Cγhas zero Lebesgue measure, by (6.8)we can write the outer integral on the right-hand side of (6.11)as an infinite sum of integrals over the disjoint gap-intervals Gl,m of decreasing size. The second part of the proof of Proposition 6.3, and crucial point in the argument, is how to estimate these integrals in terms of the size of the gaps in such a way that the series converges. Here, there are two cases to be considered according to the position of the singularity yrelative to a given gap: the case when ylies on the closure of a gap (and hence the function x →|x −y|λ−nis singular), and the case when the gap is uniformly away from y(and hence |x −y|λ−ncan be bounded above and factored out from the integral). We deal with these two cases in Lemmas 6.4 and 6.5 respectively. 32 X. Cabré, F. Charro / Advances in Mathematics 380 (2021) 107592 (4) We continue the iteration until j=l−2, which starts with only two gaps of generation lleft. The farthest from y, denoted by Gl−2 l,1, satisfies (6.21)with j=l−2. On the other hand, we denote the gap closest to y by Gl,1. Summarizing, among the 2l−1gaps of generation lwe have selected one, called Gl,1, in step 4. The gap Gl,1is the closest to y among those in generation l. The remaining 2l−1− 1 gaps have been clustered into l−1 families {G0 l,m}m, ..., {Gj l,m}m, ..., {Gl−2 l,m }m, where the j-th family contains 2l−2−jgaps of generation lwhich, in addition, satisfy (6.21). With this classification, we have ∞  l=1 2l−1  m=1  Gl,m  B(n−1) 1/4 dist(x, Cn,λ)−αp−p|x−y|λ−ndxdx = ∞  l=1  Gl,1 B(n−1) 1/4 dist(x, Cn,λ)−αp−p|x−y|λ−ndxdx + ∞  l=2 l−2  j=0 2l−2−j  m=1  Gj l,m  B(n−1) 1/4 dist(x, Cn,λ)−αp−p|x−y|λ−ndxdx. (6.22) There are two cases to study in the sequel since integrals over Gl,1and Gj l,m are qualitatively different. The key difference is that (6.21) allows us to control |x −y| from below and the integrals over Gj l,m become independent of y. Thus, we can apply Lemma 6.5 to them. This is not possible for integrals over the gaps Gl,1. For the integrals over a gap Gl,1, if y ∈Gl,1then we can apply Lemma 6.4 directly. Instead, if y /∈Gl,1, we move y to the closest boundary point of Gl,1from y, and with this procedure the integral becomes larger (since all the distances from points in B(n−1) 1/4×Gl,1decrease). With this new point y the integral can be bounded using Lemma 6.4. In fact, when we assume y ∈Gl,1Lemma 6.4 yields ∞  l=1  Gl,1 B(n−1) 1/4 dist(x, Cn,λ)−αp−p|x−y|λ−ndxdx ≤C ∞  l=1 1−γ 2(λ−αp−p)(l−1) +l1−γ 2l−1≤C, (6.23) uniformly in y, since (1 −γ)/2 <1and λ −αp −p >0by hypothesis (6.6). X. Cabré, F. Charro / Advances in Mathematics 380 (2021) 107592 33 On the other hand, given a gap Gj l,m, by (6.21)we have  Gj l,m  B(n−1) 1/4 dist(x, Cn,λ)−αp−p|x−y|λ−ndxdx ≤C1−γ 2(λ−n)j Gj l,m  B(n−1) 1/4 dist(x, Cn,λ)−αp−pdxdx. Then, Lemma 6.5 leads to  Gj l,m  B(n−1) 1/4 dist(x, Cn,λ)−αp−p|x−y|λ−ndxdx ≤C1−γ 2(λ−n)j+(n−αp−p)(l−1) +l1−γ 2(λ−n)j+l−1 uniformly in m. Observe that by our choice of γ, we have 1−γ 2λ−n=2and thus ∞  l=2 l−2  j=0 2l−2−j  m=1  Gj l,m  B(n−1) 1/4 dist(x, Cn,λ)−αp−p|x−y|λ−ndxdx ≤C ∞  l=2 l−2  j=0 2l−2−j1−γ 2(λ−n)j+(n−αp−p)(l−1) +l1−γ 2(λ−n)j+l−1 =C 2 ∞  l=2 l−2  j=0 1−γ 2(λ−αp−p)(l−1) +l1−γ 2(λ−n+1) (l−1) =C 2 ∞  l=2 (l−1) 1−γ 2(λ−αp−p)(l−1) +l(l−1) 1−γ 2(λ−n+1) (l−1)≤C, (6.24) uniformly in y, since (1 −γ)/2 <1, λ −αp −p >0, and λ >n −1by hypothesis. Then, (6.20), (6.22), (6.23), and (6.24)give 1 2  −1 2 B(n−1) 1/4 dist(x, Cn,λ)−αp−p|x−y|λ−ndxdx ≤C uniformly in y, and the proof is complete.  We conclude the article with the proof of Lemma 6.6. 34 X. Cabré, F. Charro / Advances in Mathematics 380 (2021) 107592 Proof of Lemma 6.6.Note first that the support of uα,n, given by (6.7), is included in B(n−1) 1/4×[−1, 1]. Hence, for any given y∈B(n) 1we have  B(n) 1 |∇uα,n(x)|p|x−y|λ−ndx ≤C 1  −1 B(n−1) 1/4 dist(x, Cn,λ)−αp−p|x−y|λ−ndxdx, where we have used that the modulus of the gradient of a distance function is equal to 1a.e. Then, our goal is to show that sup y∈B(n) 1 1  −1 B(n−1) 1/4 dist(x, Cn,λ)−αp−p|x−y|λ−ndxdx ≤Csup y=0,|y|≤ 1 2 1 2  −1 2 B(n−1) 1/4 dist(x, Cn,λ)−αp−p|x−y|λ−ndxdx, from which (6.19) follows. First we will prove that the supremum over y∈B(n) 1is bounded by the supremum over the axis, i.e., y∈{0} ×[−1, 1]. Then, that it actually suffices that |y| ≤1/2instead of |y| ≤1, and finally that it is enough to integrate x over [−1/2, 1/2] instead of the whole [−1, 1]. In doing these we will use twice the monotonicity result in Lemma 2.1. Therefore, consider y∈B(n) 1, and let us show that J1(y):= 1  −1 B(n−1) 1/4 dist(x, Cn,λ)−αp−p|x−y|λ−ndxdx ≤J1(0,y), where y=(y, y) ∈B(n) 1, and thus y ∈[−1, 1]. In fact, upon a rotation in the x variables, we can assume that y=(y1, 0) ∈R ×Rn−2with y1≥0. Hence, we are under the hypotheses of Lemma 2.1 with Ω =B(n−1) 1/4×[−1, 1], h(z) =dist(z, Cn,λ)−αp−p (which is non-increasing with respect to z1in {z1≥0}), θ=n −λ, and η=0. Therefore, Lemma 2.1 gives that J1is non-increasing with respect to y1in [0, ∞). Hence we conclude that J1(y) ≤J1(0, y), as desired. A similar argument shows that we just need to consider the case |y| ≤1/2instead of |y| ≤1. In fact, define J2(y):= 1  −1 B(n−1) 1/4 dist(x, Cn,λ)−αp−p|x|2+(x −y)2λ−n 2dxdx X. Cabré, F. Charro / Advances in Mathematics 380 (2021) 107592 35 - 6  PPPs s s x |x| x x∗ y 1 1 2 −1 2 −1 Fig. 4. Monotonicity argument for J2in the proof of Lemma 6.6. and assume |y| ≥1/2. By symmetry, we can assume y ≥1/2. In order to apply Lemma 2.1, we take x as the direction e1which is “privileged” in the lemma, while the rest of the hypotheses are fulfilled for (z, z) ∈Ω =[−1, 1] ×B(n−1) 1/4, h(z, z) = dist((z, z), Cn,λ)−αp−p(which is non-increasing with respect to z in {z ≥1/2}), θ=n −λ, and η=1/2, since the set Cn,λ is contained in {0} ×[−1/2, 1/2] (see Fig. 4). Then, Lemma 2.1 gives that J2(y)is non-increasing with respect to y in [1/2, ∞), and therefore it is enough to study the case y ≤1/2. The case y ≤−1/2 follows similarly by taking η=−1/2in the lemma. Finally, let y=(0, y)with |y| ≤1/2. Notice that for every x ∈B(n−1) 1/4×[1/2, 1] and its reflected x∗with respect to {x =1/2}, we have |x∗−y| ≤|x −y|and dist(x∗, Cn,λ) ≤ dist(x, Cn,λ), which give dist(x, Cn,λ)−αp−p|x−y|λ−n≤dist(x∗,Cn,λ)−αp−p|x∗−y|λ−n. This leads to 1 1 2 B(n−1) 1/4 dist(x, Cn,λ)−αp−p|x−y|λ−ndxdx ≤ 1 2  0 B(n−1) 1/4 dist(x, Cn,λ)−αp−p|x−y|λ−ndxdx, 36 X. Cabré, F. Charro / Advances in Mathematics 380 (2021) 107592 and similarly for the integral over B(n−1) 1/4×[−1, −1 2]. Therefore, 1  −1 B(n−1) 1/4 dist(x, Cn,λ)−αp−p|x|2+(x −y)2λ−n 2dxdx ≤2 1 2  −1 2 B(n−1) 1/4 dist(x, Cn,λ)−αp−p|x|2+(x −y)2λ−n 2dxdx, which completes the proof of the lemma.  Acknowledgments The first author would like to thank Joan Orobitg and Joan Verdera for a stimulating discussion on the topic of this paper. The authors also thank Giuseppe Mingione for interesting comments and for bringing [2,6,10]to their attention after the completion of this manuscript, as well as the referee for some appropriate remarks and references [11,13]. References [1] D.R. Adams, A note on Riesz potentials, Duke Math. J. 42 (1975) 765–778. [2] D.R. Adams, J.L. Lewis, On Morrey-Besov inequalities, Stud. Math. 74 (1982) 169–182. [3] D.R. Adams, J. Xiao, Morrey potentials and harmonic maps, Commun. Math. Phys. 308 (2011) 439–456. [4] D.R. Adams, J. Xiao, Erratum to: Morrey potentials and harmonic maps, Commun. Math. Phys. 339 (2015) 769–771. [5] D.R. Adams, J. Xiao, Restrictions of Riesz-Morrey potentials, Ark. Mat. 54 (2016) 201–231. [6] A. Bensoussan, J. Frehse, Regularity Results for Nonlinear Elliptic Systems and Applications, Applied Mathematical Sciences, vol. 151, Springer, Berlin, 2002. [7] X. Cabré, A. Figalli, X. Ros-Oton, J. Serra, Stable solutions to semilinear elliptic equations are smooth up to dimension 9, Acta Math. 224 (2020) 187–252. [8] X. Cabré, X. Ros-Oton, Sobolev and isoperimetric inequalities with monomial weights, J. Differ. Equ. 255 (2013) 4312–4336. [9] K. Falconer, Fractal Geometry: Mathematical Foundations and Applications, John Wiley & Sons, 2004. [10] I. Fonseca, J. Malý, G. Mingione, Scalar minimizers with fractal singular sets, Arch. Ration. Mech. Anal. 172 (2004) 295–307. [11] S. Hou, J. Xiao, Cordes-Nirenberg’s imbedding and restricting with application to an elliptic equation, Commun. Contemp. Math. 20 (7) (2018) 1750080. [12] R. Schoen, S.-T. Yau, Conformally flat manifolds, Kleinian groups and scalar curvature, Invent. Math. 92 (1988) 47–71. [13] G. Talenti, Best constant in Sobolev inequality, Ann. Mat. Pura Appl. 110 (1976) 353–372. [14] G. Talenti, A weighted version of a rearrangement inequality, Ann. Univ. Ferrara 43 (1997) 121–133.