Tensorization of quasi-Hilbertian Sobolev spaces

This is a self-archived version of an original article. This version may differ from the original in pagination and typographic details. Author(s): Title: Year: Version: Copyright: Rights: Rights url: Please cite the original version: CC BY 4.0 https://creativecommons.org/licenses/by/4.0/ Tensorization of quasi-Hilbertian Sobolev spaces © 2023 Real Sociedad Matemática Española Published version Eriksson-Bique, Sylvester; Rajala, Tapio; Soultanis, Elefterios Eriksson-Bique, S., Rajala, T., & Soultanis, E. (2024). Tensorization of quasi-Hilbertian Sobolev spaces. Revista Matematica Iberoamericana, 40(2), 565-580. https://doi.org/10.4171/rmi/1433 2024 Rev. Mat. Iberoam. (Online first) DOI 10.4171/RMI/1433 © 2023 Real Sociedad Matemática Española Published by EMS Press Tensorization of quasi-Hilbertian Sobolev spaces Sylvester Eriksson-Bique, Tapio Rajala and Elefterios Soultanis Abstract. The tensorization problem for Sobolev spaces asks for a characterization of how the Sobolev space on a product metric measure space XYcan be determined from its factors. We show that two natural descriptions of the Sobolev space from the literature coincide, W1;2.X Y / DJ1;2.X; Y /, thus settling the tensorization problem for Sobolev spaces in the case pD2, when Xand Yare infinitesimally quasi-Hilbertian, i.e., the Sobolev space W1;2 admits an equivalent renorming by a Dirichlet form. This class includes in particular metric measure spaces X; Y of finite Hausdorff dimension as well as infinitesimally Hilbertian spaces. More generally, for p2.1; 1/we obtain the norm-one inclusion kfkJ1;p .X;Y /  kfkW1;p.XY / and show that the norms agree on the algebraic tensor product W1;p.X/ ˝W1;p.Y / W1;p.X Y /: When pD2and Xand Yare infinitesimally quasi-Hilbertian, standard Dirichlet forms theory yields the density of W1;2.X/ ˝W1;2.Y / in J1;2.X; Y /, thus implying the equality of the spaces. Our approach raises the question of the density of W1;p.X/ ˝W1;p.Y / in J1;p.X; Y / in the general case. 1. Introduction Over the last three decades, Sobolev spaces over metric spaces have become a prominent feature in a plethora of geometric problems ranging from Plateau-type problems [15,23,24] to quasiconformal uniformization questions [25,27] and structural problems of spaces with Ricci curvature bounds [4,5,17,19]. During that time, their theory has been studied intensively and significant developments include the unification of different definitions of Sobolev spaces, the density (in energy) of Lipschitz functions in, and the reflexivity of Sobolev spaces over metric spaces in a very general setting – see, e.g., [1,9,14,18,28]. The tensorization problem for Sobolev spaces, first considered in [5], asks whether Sobolev regularity of a function of two variables can be deduced from the existence and integrability of directional derivatives. More precisely, let XD.X; dX; / and YD 2020 Mathematics Subject Classification: Primary 46E36; Secondary 31C25. Keywords: Sobolev spaces, tensorization, Dirichlet forms, metric measure spaces, analysis on metric spaces, minimal upper gradient. S. Eriksson-Bique, T. Rajala and E. Soultanis 2 .Y;dY;/ be two metric measure spaces, let p2Œ1;1/and let XY;pd2 XCd2 Y;  be their (Euclidean) product. Given p1, the tensorization problem asks whether the Sobolev space W1;p.X Y / coincides with the Beppo-Levi space J1;p.X; Y / consisting of functions f2Lp.X Y / for which f .x; /2W1;p.Y / for -almost every x2X, f .; y/ 2W1;p.X/ for -almost every y2Y, and (1.1) .x; y/ 7! pjDf .; y/j2.x/ CjDf .x; /j2.y/ 2Lp.X Y /: In addition, tensorization of Sobolev spaces requires that the minimal p-weak upper gradient of any f2J1;p.X; Y / is given by (1.1). For the definition of W1;p.X/ used in this paper, see Section 1.3. While immediate in Euclidean spaces, a positive answer to the tensorization problem is non-trivial in the non-smooth setting, and needed, e.g., in the splitting theorem for RCD-spaces [17]. Further, it is of crucial importance in a variety of settings where partial derivatives can be bounded, and one wishes to obtain a bound on the full derivative, see e.g. [6,12]. Surprisingly, the problem has remained open, even though tensorization of many other properties such as the doubling property, Poincaré inequalities and curvature lower bounds are well known. Previous partial results for pD2include the work of Ambrosio–Gigli–Savaré [4] for RCD-spaces, of Gigli–Han [20] settling the case where one factor is a closed interval in R, and of Ambrosio–Pinamonti–Speight [6] for PI-spaces. Working in the general case p1(with a finite dimensionality assumption on the factors), the authors of the present manuscript proved tensorization of Sobolev spaces assuming that one of the factors is a PI-space [13] (see also the independent work [16], where warped products are considered). The present work strengthens all of these results in the pD2 case, and proves more general results for p > 1. 1.1. Tensorization in infinitesimally quasi-Hilbertian spaces In this paper, we establish tensorization of Sobolev spaces in the important special case pD2when the factors are infinitesimally quasi-Hilbertian. Definition 1.1. A metric measure space Xis infinitesimally quasi-Hilbertian if there exists a closed Dirichlet form Ewith domain W1;2.X/ such that pkuk2 L2.X/ CE.u; u/ is an equivalent norm on W1;2.X/. See Section 3for the definition of Dirichlet forms. We remark that infinitesimally Hilbertian spaces, as well as spaces admitting a 2-weak differentiable structure (in particular spaces with finite Hausdorff dimension), are infinitesimally quasi-Hilbertian, cf. Proposition 3.6. Theorem 1.2. Suppose that Xand Yare infinitesimally quasi-Hilbertian. Then we have that W1;2.X Y / DJ1;2.X; Y / and, for each f2J1;2.X; Y /, we have that jDf j.x; y/2D jDf .; y/j.x/2CjDf .x; /j.y/2 for -almost every .x; y/ 2XY. Remark 1.3. If the space XYis equipped with a product metric k.dX; dY/kinduced by some norm kkon R2, we obtain that jDf jDk.jDf .;y/j.x/;jDf .x;/j.y//k0, where kk0is a form of dual norm, cf. Theorem 3.3. Tensorization of quasi-Hilbertian Sobolev spaces 3 In particular, we have the following corollary. Corollary 1.4. If each of the factors Xand Yis either infinitesimally Hilbertian or has finite Hausdorff dimension, then W1;2.X Y / DJ1;2.X; Y / with equal norms. Theorem 1.2 follows by combining three ingredients: 1) standard theory of Dirichlet forms and the elementary inclusion W1;2.X Y / J1;2.X; Y /, 2) the non-trivial fact that the inclusion W1;2.X Y / J1;2.X; Y / has norm one, and 3) the equality of the norms on the algebraic tensor product W1;2.X/ ˝W1;2.Y /. We establish the last two results in the more general setting when p > 1 and the product space XYis equipped with a product metric given by a possibly non-Euclidean planar norm. We remark that the earlier work by the authors [13] gave stronger conclusions on tensorization when p¤2, while also employing stronger assumptions. Indeed, the isometric embedding N1;p.X Y / J1;p.X; Y / (a stronger conclusion than Theorems 1.5 and 1.6) is obtained under the additional assumption that the factors admit a p-weak differentiable structure for all p1. By focusing on the Newtonian space, and using the stronger assumptions, the authors were able to analyse the borderline case pD1(see also Remark 1.8). However, in [13], full tensorization (that is, N1;p.X Y / DJ1;p.X; Y / with equal norms) is only obtained under the rather restrictive assumption that one of the factors supports an appropriate Poincaré inequality. In contrast, Theorems 1.5 and 1.6 are true for any metric measure spaces, while full tensorization for pD2(Theorem 1.2) is obtained when the factors admit p-weak differentiable structures. 1.2. Norm inequalities and equalities in the inclusion W1;p.X Y / J1;p.X; Y / Let .X Y; d;  / be the product of two metric measure spaces XD.X; dX; / and YD.Y; dY; /, where the product metric is given by dD k.dX; dY/kfor a given planar norm kk, and let p > 1. For f2J1;p.X; Y /, we denote by jDXfjand jDYfjthe Lp.X Y /-functions .x; y/ 7! jDf .; y/j.x/ and .x; y/ 7! jDf .x; /j.y/, respectively, and replace (1.1) with the comparable quantity k.jDXfj;jDYfj/k02Lp.X Y /:(1.2) Here k.a; b/k0WDsup¹at Cbs Ws; t 0; k.s; t/k  1ºis the partial dual norm of kk. Notice that the Euclidean norm is its own partial dual, and thus (1.1) and (1.2) coincide for dDpd2 XCd2 Y. The first of the two results states that the minimal p-weak upper gradient always dominates (1.2). Theorem 1.5. Let p2.1; 1/. If f2W1;p.X Y /, then f2J1;p.X; Y / and k.jDXfj;jDYfj/k0 jDf j(1.3) -almost everywhere. In particular, for the Euclidean product metric dDpd2 XCd2 Y, Theorem 1.5 yields the inequality pjDXfj2CjDYfj2 jDf j; f 2W1;p.X Y /: S. Eriksson-Bique, T. Rajala and E. Soultanis 4 Although it is straightforward to obtain the estimate k.jDXfj;jDYfj/k0CjDf j for some C > 0 independent of ffrom the definitions, Theorem 1.5 is new and was previously only known for general spaces when pD2and kkis the Euclidean norm, by work of Ambrosio–Gigli–Savaré [5] via different techniques (see also [6]). Our approach uses a density in energy argument [3,11] to reduce the proof of Theorem 1.5 to a simple, yet novel, inequality for Lipschitz functions (see Proposition 2.1 below). Our next result establishes equality in (1.3) in the algebraic tensor product W1;p.X/˝ W1;p.Y / W1;p.X Y / consisting of finite sums of simple tensor products, i.e., functions of the form f .x; y/ D N X j 'j.x/ j.y/; 'j2W1;p.X/; j2W1;p.Y /; j D1; : : : ; N: Theorem 1.6. Let p2.1; 1/. If f2W1;p.X/ ˝W1;p.Y /, then k.jDXfj;jDYfj/k0D jDf j -almost everywhere. We show, using canonical minimal upper gradients introduced in [14], that (1.2) is ap-weak upper gradient of f2W1;p.X/ ˝W1;p.Y /. Together with Theorem 1.5, this suffices to demonstrate Theorem 1.6. Note that having constant one in (1.3) is important for the validity of this argument. The crux of Theorem 1.2 is that, for infinitesimally quasi-Hilbertian spaces, the algebraic tensor product is dense in the Beppo-Levi space (with pD2). Indeed, this is a standard result for domains of Dirichlet forms (see Proposition 3.2), and follows easily for Sobolev spaces under the infinitesimal quasi-Hilbertianity assumption. This completes the proof of Theorem 1.2 and also raises the natural question: when is W1;p.X/ ˝W1;p.Y / dense in J1;p.X; Y /? We expect that some separability assumption might be necessary, and formulate the question accordingly below. Question 1.7. Let p2Œ1; 1/. If W1;p.X/ and W1;p.Y / are separable, is W1;p.X/ ˝ W1;p.Y / dense in J1;p.X; Y /? An affirmative answer to Question 1.7 under the stronger assumption that Xand Y admit p-weak differentiable structures would already be interesting, since it covers all spaces with finite Hausdorff dimension. Remark 1.8. The above Theorems 1.5 and 1.6 are stated for exponents p > 1. In the proofs we use the equality of the Newton–Sobolev space N1;p.X/ defined by Shanmugalingam and Cheeger [9,28], and the plan-Sobolev space W1;p.X/ from Ambrosio, Gigli and Savaré [4]. The equality of these spaces is not yet available in the literature in the case pD1. Once proven, such equality would imply Theorems 1.5 and 1.6 also in the case pD1. Tensorization of quasi-Hilbertian Sobolev spaces 5 1.3. Notation and conventions A map uWX!Rfrom a metric space Xis Lipschitz if Lip.u/ WDsup x¤y jf .x/ f .y/j d.x; y/ <1: The space of bounded Lipschitz functions with bounded support, is denoted LIPb.X/. The asymptotic Lipschitz constant of uis defined as Lipau.x/ WDlim sup r!0 Lip.ujB.x;r//; x 2X: Throughout the paper, XD.X; dX; / and YD.Y; dY; / are metric measure spaces, by which we mean complete separable metric spaces equipped with measures that are finite on bounded sets. Given p > 1, we denote by N1;p.X/ and W1;p.X/ the Newton–Sobolev space, and the Sobolev space via test plans, respectively. Both of these spaces are defined using the upper gradient inequality. A function f2Lp.X/ is in N1;p.X/ if there exists a function g2Lp.X/ so that ju.1/u.0/j  Z1 0 g.t/j0 tjdt(1.4) holds for Modp-a.e. curve. On the other hand, f2W1;p.X/ if (1.4) holds for -a.e.  for every q-test plan . Modulus is an outer measure on curve families, and test plans are a family of measures on curve families. See [4] for a definition of test plans. For the properties the modulus of a curve family, Modp, see [21]. For each u2N1;p.X/ and u2W1;p.X/, there exists a minimal jDuj 2 Lp.X/ so that ju.1/u.0/j  Z1 0jDuj.t/j0 tjdt(1.5) holds for “almost all” absolutely continuous curves WŒ0; 1 !X. For u2N1;p.X/, the inequality (1.5) is required to hold for Modp-almost every curve , whereas for u2 W1;p.X/, (1.5) holds for -a.e. for every q-test plan . The minimal objects jDuj associated to each case agree -almost everywhere (in this notation, we suppress its dependence on pand on the metric) and we have that W1;p.X/ DN1;p.X/ for p > 1 with equal norms1[2,3]. Here the Sobolev space W1;p.X/ is equipped with norm kukW1;p.X/ Dkukp Lp.X/ CkjDujkp Lp.X/1=p: For functions uWXY!R, we will define the sliced functions, for x2X; y 2Y, by uxWDu.x; /WY!Rand uyWDu.; y/ WX!R: 1The equality holds up to the subtle issue of choosing appropriate representatives: N1;p.X/ W1;p.X/, but for every f2N1;p.X/ there exists a function Q f2W1;p.X/ with Q fDfalmost everywhere. A further difference is that functions in N1;p.X/ are defined up to capacity-a.e. equivalence, whereas functions in W1;p.X/ are defined up to an almost everywhere equivalence. The proof is contained in Theorem 10.7 of [2]. S. Eriksson-Bique, T. Rajala and E. Soultanis 6 When u2J1;p.X; Y /, we denote by jDXuj;jDYuj2Lp.X Y / the functions such that jDXuj.; y/ D jDuyjfor -a.e. y2Y,jDYuj.x; / D jDuxjfor -a.e. x2X. We equip J1;p.X; Y / with the norm kfkJ1;p.X;Y / DZXYjfjpC.k.jDXfj;jDYfj/k0/pd. /1=p : Remark 1.9. It is straightforward to check that jDXujis given as the minimal p-weak upper gradient of uwhen XYis equipped with the metric dDdXCpdY, and similarly for jDYuj. In particular, jDXujand jDYujcan be chosen Borel measurable. 2. The inclusion W1;p.X Y / J1;p.X; Y / In this section, we prove Theorems 1.5 and 1.6 for a general p > 1. In the proof of Proposition 2.1, below we will shorten the notation by using the evaluation map eXWC.Œ0; 1IX/ Œ0; 1 !XW.; t/ 7! t: We start with a new and seemingly elementary inequality, which however has hitherto not appeared. The authors in [6] and [5] used a substantially different approach employing Hopf-lax equations, heat flows and further results. The following result is the key to our proof of the isometric inclusion, and perhaps gives a more transparent and geometric argument. Proposition 2.1. Let p2.1; 1/and let f2LIPb.X Y /. Then k.jDXfj;jDYfj/k0Lipaf -almost everywhere. Proof of Proposition 2.1.The argument will proceed by finding, for a.e. point .x; y/, a curve in the Xand Y-directions along which the function fhas maximal derivative given by the minimal p-weak upper gradients. We do this by employing a result from [14], but we also outline in Remark 2.2 another argument inspired by one from Cheeger and Kleiner [10] after the proof, which some readers may find helpful. Since f2LIPb.X Y /, we have that f2W1;p.X Y / when XYis equipped with the distance dXCpdY. By Remark 1.9 and Theorem 1.1 in [14], there exists a test plan so that the disintegration ¹.x;y/ºof the measure dWD j0 tjdtdwith respect to the evaluation map eXYWC.Œ0; 1IXY / Œ0; 1 !XYsatisfies (2.1) jDXfj.x; y/ D jDf yj.x/ D   .f yı/0 t j0 tj  L1..x;y// for -almost every .x;y/2¹jDXfj> 0º. (Notice that every rectifiable curve in .X Y; dXCpdY/is of the form .˛; y/, where y2Yis a constant curve and ˛is a rectifiable Tensorization of quasi-Hilbertian Sobolev spaces 7 curve in X. One could obtain (2.1) for p > 1 alternatively via the existence of the master test plans introduced in [26] and by using Fubini’s theorem.) By applying the same argument with metric pdXCdY, we similarly obtain measures ¹z .x;y/ºfor almost every .x; y/ 2 ¹jDYfj> 0ºso that (2.2) jDYfj.x; y/ D jDfxj.y/ D   .fxı/0 t j0 tj  L1.z .x;y//: Let us fix .x; y/ 2XYwhere both (2.1) and (2.2) hold. For any " > 0, there exist .˛; t0/2e1 X.x/ and .ˇ; s0/2e1 Y.y/ such that .1 "/jDXfj.x; y/ .f yı˛/0 t0 j˛0 t0jand .1 "/jDYfj.x; y/ .fxıˇ/0 s0 jˇ0 s0j;(2.3) and the limits in all the relevant quantities exist. Let a; b 0and define the curves Q˛.t/ D˛t0Ca j˛0 t0jtand Q ˇ.s/ Dˇs0b jˇ0 s0js in a small neighbourhood of the origin. Then a.f yı˛/0 t0 j˛0 t0jCb.fxıˇ/0 s0 jˇ0 s0jD.f yı Q˛/0 0.fxıQ ˇ/0 0 Dlim h!0C Œf . Q˛.h/; y/ f .x; y/ Œf .x; Q ˇ.h// f .x; y/ h Dlim h!0C f . Q˛.h/; y/ f .x; Q ˇ.h// h Lipaf .x; y/ lim sup h!0C d.. Q˛.h/; y/; .x; Q ˇ.h/// h Note however that d.. Q˛.h/; y/; .x; Q ˇ.h/// hD  dX.Q˛.h/; x/ h;dY.Q ˇ.h/; y/ h   h!0C ! k.a; b/k: Using (2.3), we arrive at (2.4) .1 "/ ŒajDXfj.x; y/ CbjDYfj.x; y/  k.a; b/kLipaf .x; y/: Taking supremum over all a; b 0with k.a; b/k D 1in (2.4) yields .1 "/k.jDXfj.x; y/; jDYfj.x; y//k0Lipaf .x; y/  -a.e. .x; y/ 2XY: Since ">0is arbitrary, the claim now follows. Remark 2.2. In the previous proof, the use of [14] is convenient, but the existence of curves ˛and ˇas in (2.3) with nearly maximal derivative is actually a much weaker conclusion. In fact, in the category of doubling spaces satisfying a Poincaré inequality, S. Eriksson-Bique, T. Rajala and E. Soultanis 8 their existence follows from the work of Cheeger and Kleiner, see Theorem 4.2 in [10]. They gave a characterization of the minimal p-weak upper gradient of a Lipschitz function as a maximal directional derivative. In fact, the first part of the proof, which does not use the doubling or Poincaré assumptions, shows that a function Ogdefined using the maximal directional derivatives is an upper gradient. A minimal p-weak upper gradient is a.e. less than this upper gradient, and from this the existence of ˛and ˇcan be deduced. This idea played a central role in later developments, such as the seminal work of Bate [7] characterizing Lipschitz differentiability spaces. Proposition 2.1 now implies Theorem 1.5. Proof of Theorem 1.5.Let f2W1;p.X/. By the density in energy (cf. [3] or [11] for an alternate proof), there exists a sequence .fj/LIPb.X/ such that fj!fand Lipafj! jDf jin Lp.X/ as j! 1. We also have, for a; b 0and any non-negative '2Cb.X/, that ZXY '.ajDXfjCbjDYfj/d./ lim inf j!1 ZXY '.ajDXfjjCbjDYfjj/d./; (cf. Remark 1.9 and the lower semicontinuity of the Cheeger energy [3,9]). By Proposition 2.1 (and the definition of the partial dual norm kk0), this implies that ZXY '.ajDXfjCbjDYfj/d. /  k.a; b/klim inf j!1 ZXY 'Lipafjd. / D k.a; b/kZXY 'jDf jd. / for arbitrary a; b and '. Thus ajDXfj C bjDYfj  k.a; b/kjDf j-a.e. for every a; b 0. Then, by choosing a countable dense set of real numbers a; b 0, we obtain the pointwise inequality k.jDXfj;jDYfj/k0Dsup .a;b/ ajDXfjCbjDYfj k.a; b/k jDf j-a.e. Next we prove Theorem 1.6. In the proof, we identify RNwith .RN/in the standard way by identifying Na2RNwith the functional x7! Nax2.RN/. Proof of Theorem 1.6.Let h.x; y/ D N X iD1 fi.x/ gi.y/ 2W1;p.X/ ˝W1;p.Y /; where f1;: : :;fN2W1;p.X/ and g1;: : :;gN2W1;p.Y /. Since each function in W1;p.X/ is a.e. equal to a Newton–Sobolev function (Theorem 10.7 in [2]), we can choose Newton– Sobolev representatives for each fjand gjand consider the maps 'D.f1; : : : ; fN/2 N1;p.XIRN/and D.g1; : : : ; gN/2N1;p.Y IRN/. 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Box 3000, 90014 Oulu; (Current address): Department of Mathematics and Statistics, University of Jyvaskyla, 40014 University of Jyvaskyla, Finland; sylvester.d.eriksson-[email protected] Tapio Rajala Department of Mathematics and Statistics, University of Jyvaskyla, 40014 University of Jyvaskyla, Finland; [email protected] Elefterios Soultanis Department of Mathematics, Radboud University, P.O. Box 9010, Postvak 59, 6500 GL Nijmegen, Netherlands; [email protected]