A new notion of subharmonicity on locally smoothing spaces, and a conjecture by Braverman, Milatovic, Shubin

Given a strongly local Dirichlet space and $\lambda\geq 0$, we introduce a new notion of $\lambda$--subharmonicity for $L^1_\loc$--functions, which we call \emph{local $\lambda$--shift defectivity}, and which turns out to be equivalent to distributional $\lambda$--subharmonicity in the Riemannian case. We study the regularity of these functions on a new class of strongly local Dirichlet, so called locally smoothing spaces, which includes Riemannian manifolds (without any curvature assumptions), finite dimensional RCD spaces, Carnot groups, and Sierpinski gaskets. As a byproduct of this regularity theory, we obtain in this general framework a proof of a conjecture by Braverman, Milatovic, Shubin on the positivity of distributional $L^q$-solutions of $\Delta f\leq f$ for complete Riemannian manifolds.


Introduction
Given q P r1, 8s one says that a connected Riemannian manifold X is L q -positivity preserving, if for every f P L q pXq the following implication holds true: ∆f ď f in the sense of distributions ñ f is nonnegative. ( Here ∆ " ř ij g ij B i B j is the negative-definite Laplace-Beltrami operator.The importance and subtlety of this property is reflected at least by the following observations: ‚ if X is L 2 -positivity preserving, then ´∆ (defined on C 8 c pXq) is essentially self-adjoint (in L 2 pXqq [12], ‚ X is stochastically complete, if and only if X is L 8 -positivity preserving [8], ‚ there exist complete X's which are not L 1 -positivity preserving [8], ‚ a conjecture by Braverman-Milatovic-Shubin (BMS) from 2002 [12] states that if X is complete, then X is L 2 -positivity preserving.The BMS-conjecture was formulated in an L 2 -setting as the authors were interested in essential self-adjointness problems and was based on the classical fact that ´∆ is essentially self-adjoint, if X is complete.More generally, one can ask, whether completeness implies L q -positivity preservation for all q P p1, 8q.Let us point out that the main technical problem in proving such a conjecture is the possible lack of regularity of f above.For if f was in W 1,2  loc pXq, then one can integrate by parts once to see that the inequality in (1.1) is equivalent to ż X p∇f, ∇φq `żX f φ ě 0 for all 0 ď φ P W 1,2 c pXq, (1.2) in other words, f is weakly 1-superharmonic.If f was even in W 1,2 0 pXq, then it would be easy to show that (1.2) implies f ě 0 using first order cut-off functions in the sense of [33], the existence of which is equivalent to the completeness of X.On another note, it is straightforward to see that, if X admits a sequence of Laplacian cut-off functions in the sense of [33], then X is L q -positivity preserving for all q P p1, 8q.However, in general, this procedure (which relies on two integrations by parts [34,33]) requires [7] a lower control of the Ricci curvature in addition to the completeness of X.
1 It is well-known that a convenient abstract setting to formulate an inequality of the form (1.2) is provided by a (regular and symmetric) Dirichlet space pX, m, E , F pXqq [30,49,31,15].Indeed, given a nonnegative real number λ, one has a natural notion of weakly λ-subharmonic and weakly λ-superharmonic functions at hand [62], where the role of W 1,2  loc pXq and W 1,2 c pXq is, respectively, played by F loc pXq and F c pXq, with F pXq the domain of definition of the underlying quadratic form E .Note that in the Riemannian case, one has F pXq " W 1,2 0 pXq, E pf, gq " ż X p∇f, ∇gq, the usual energy form.
In this paper, we will focus on strongly local Dirichlet spaces.The purpose of this paper is to address the following questions: ‚ How can one define λ-subharmonicity (resp.λ-superharmonicity) for L 1 loc -functions on such spaces, in a way that the definition is consistent with the distrubutional one in the Riemannian case?Note that this will lead naturally to the notion of an L q -positivity preserving strongly local Dirichlet space.‚ What is the regularity of λ-subharmonic L 1 loc -functions on strongly local Dirichlet spaces?‚ Which class of strongly local Dirichlet spaces is L q -positivity preserving?Addressing the first question, note first that on a general strongly local Dirichlet space there is no natural substitute for test functions.To deal with this problem, for every open U Ă X let P U t denote the semigroup of pU, m, E , F pUqq with Dirichlet type boundary conditions.We define f P L 1 loc pX, mq to be locally λ-shift defective, if for all open relatively compact sets U Ă X there exists a weakly λ-harmonic function g P C b pUq on U such that e ´tλ P U t pf | U ´gq ě f | U ´g for all t ą 0. This new notion, which can be considered as a localized and unsigned variant of the classical concept of defective functions, turns out on Riemannian manifolds to be equivalent to λ-subharmonicity in the distributional sense (cf.Proposition 4.3).To proceed further with the theory, we introduce a new class of strongly local Dirichlet spaces, which we call locally smoothing spaces.This assumption is entirely local, in the sense that all Riemannian manifolds (without any curvature assumptions) are locally smoothing and all open subsets of locally smoothing spaces again have this property.In addition, we show that all finite dimensional RCD-spaces, all Carnot groups and the unbounded Sierpinski gasket belongs to this category of spaces.In order to establish a regularity theory for locally λ-shift defective functions, we assume throughout that pX, m, E , F pXqq is locally smoothing.A central technical step, more or less intrinsic in the definition, is an approximation result, Lemma 4.7, which reveals that locally λshift defective functions always have upper semicontinuous representatives.Furthermore, Theorem 4.8 states that every locally λ-shift defective function in F loc pXq is weakly λ-subharmonic, and conversely, that every weakly λ-subharmonic which is locally bounded above is locally λshift defective (noting that on many spaces, every weakly λ-subharmonic is automatically locally bounded above; cf.Remark 4.9).Our main regularity result, Theorem 5.4, states that for every locally λ-shift defective function f one has pf ´cq q{2 `P F loc pXq for all c ě 0, q P p1, 8q, which generalizes earlier results from [62].The proof of this result is based on a Kato-Brezis type result, Theorem 5.3, which states that the pointwise maximum of two locally λ-shift defective functions has again this property.The proof of our Kato-Brezis theorem in turn relies on a probabilistic characterization of locally λ-shift defective functions, Proposition 5.1, which is based on a probabilistic characterization of locally bounded weakly λ-subharmonic functions from [17].We also obtain a strong maximum principle, Theorem 6.1, which does not need an F loc pXq or a continuity assumption.We note in passing that these results entail that if f is a weakly λ-harmonic function on an RCD ˚pK, Nq space with λf P L q loc pX, mq for some q ě maxpN, 4q, then the square |∇f | 2 ˚of the minimal relaxed slope of f admits on every open relatively compact subset an upper semicontinuous representative which is subject to the strong maximum principle (cf.Example 6.2).Note that such a result is rather delicate, as on Alexandrov spaces it may happen that |∇f | 2 ˚need not be continuous [53], which follows essentially from the results obtained in [24,51].The regularity of harmonic and subharmonic functions is a central theme in analysis and beyond, ever since Weyl's seminal work [64].We refer to [61] for a discussion of the latter paper and its impact, and to the textbooks [28,38] for a general PDE point of view.It is also a classic fact that harmonicity properties come in semigroup resp.probabilistic flavors, see Kakutani's and Doob's fundamental papers [40], [25] for the Euclidean space.It is no surprise that this aspect is very well captured within the framework of Dirichlet spaces, as those are in a 1-1 correspondence with Hunt processes [30,49,31,15].An important point of the intrigue is that different notions of (sub-)harmonicity require different a-priori regularity assumptions, while the theory from this paper is based on only an L 1 loc -assumption.In this context, we note that recently a distributional notion of λ-subharmonic functions for finite dimensional RCD-spaces has been given in [53] (see in particular also [32] for very interesting equivalent characterizations of this notion under additional local regularity assumptions).The theory from [53,32] uses the fact that a natural space of test functions is available on RCD-spaces.However, when applied to a complete Riemannian manifold with Ricci curvature bounded from below, this class of test functions is larger than smooth compactly supported functions, so that our definition of the distributional inequality ∆f ď λf seems to be slightly more general even in this setting.Finally, based on our notion of locally λ-shift excessive functions, we define L q -positivity preserving strongly local Dirichlet spaces, and prove the L q -variant, q P p1, 8q, of the BMS conjecture for irreducible and intrinsically complete locally smoothing spaces.Our proof combines the L q -Liouville property from [62] with our main regularity result, and in particular, does not rely on Laplacian cut-off functions, which neither make sense nor are available in this generality (note, however, that Laplacian cut-off functions exist in principle on finite dimensional RCDspaces; cf.Lemma 3.1 in [50]).We remark also that recently it has been shown in [54] that in the Riemannian case, the L q -variant of the BMS conjecture remains true, if one removes from a complete Riemannian manifold a possibly singular set with Hausdorff co-dimension strictly larger than 2q{pq ´1q, the threshold value 2q{pq ´1q being sharp.The present paper is organized as follows: in Section 2 standard notions and notation concerning local Dirichlet spaces, their associated operators and heat semigroups and the corresponding diffusion processes is recorded.In Section 3 we introduce locally smoothing spaces and list a number of classes of strongly local Dirichlet spaces that fall into this framework.In Section 4 we first review some important auxiliary results on weakly λ-subharmonic and λ-defective (signed) functions.We then give an equivalent characterization of distributionally λ-subharmonic functions on Riemannian manifolds in terms of the local heat semigroups, which motivates our notion of locally λ-shift defective functions, introduced next.We go on to prove that on locally smoothing spaces, the concepts 'weak λ-subharmonicity' and 'local λ-shift defectivity' are consistent under natural a priori regularity assumptions.Section 5 contains the aforementioned probabilistic characterization of locally λ-shift defective functions, the Kato-Brezis theorem, and the regularity Theorem 5.4.Section 6 is devoted to the maximum principle.Finally, Section 7 contains a formulation and proof of the BMS-conjecture in our general framework.
Acknowledgements: The authors would like to thank Sebastian Boldt, Kazuhiro Kuwae and Marcel Schmidt for very helpful discussions.We are indebted to the referee for his careful reading and suggestions that improved the article.

Conflicts of interest statement:
In this research there is no conflicts of interests.
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Strongly local Dirichlet spaces
In the sequel we understand our function spaces over R. We follow standard notation and refer to [31,49] for textbooks on Dirichlet forms and [23] for a recent article with an in depth treatment of Dirichlet forms.Definition 2.1.A strongly local Dirichlet space is a quadruple pX, m, E , F pXqq where X is a locally compact, separable, metrizable space, equipped with a Radon measure m having full support, and E is a closed, symmetric, nonnegative bilinear form in L 2 pX, mq with dense domain of definition F pXq Ă L 2 pX, mq, such that ‚ (Markovian property) for all contractions T : R Ñ R with T p0q " 0 and all f P F pXq one has T ˝f P F pXq with the scalar product in L 2 pX, mq, a notation we will use, more generally, as long as uv P L 1 pX, mq.‚ (Strong locality) For all u, v P F pXq such that u is constant on the support of v: E pu, vq " 0.
We fix a strongly local Dirichlet space pX, m, E , F pXqq.For open subsets U, V Ă X we use the notation U Ť V to indicate that U is a compact subset of X with U Ă V .The regularity property implies [31] that F pXq X L 8 pX, mq is an algebra, and that for each U 1 Ť U 2 Ť X there exists ψ P F pXq X C c pXq with ψ " 1 on U 1 and supppψq Ă U 2 .From regularity it also follows that there is a Choquet capacity [31] Capp¨q on X such that for all open U Ă X one has CappUq " inftE 1 pf q | f P F pXq, 1 U ď f u.One has m !Cap on Borel sets, and properties that hold away from a set of zero capacity are said to hold quasi everywhere (q.e).A function f : X Ñ R is called quasi-continuous, if for all ǫ ą 0 there exists an open set U ǫ Ă X with CappU ǫ q ď ǫ such that f | XzUǫ is continuous.Every element of F pXq has a quasi-continuous representative and two such representatives agree q.e., [31].Following [62], see [31]  the so called energy measure of pX, m, E , F pXqq, is defined through polarization by ż X φ dΓpf, f q " E pf, φf q ´1 2 E pf 2 , φq for all f P F pXq X L 8 pX, mq, φ P F pXq X C c pXq.
Whenever it makes sense, we are going to use the standard notation E pf q :" E pf, f q and Γpf q :" Γpf, f q.The energy measure has the following properties: ‚ Γ does not charge sets of zero capacity [62,31], ‚ Γ is strongly local, in the sense that for all U Ă X open, all u P F pXq such that u is constant on U, and all v P F pXq, one has 1 U dΓpu, vq " 0, cf.[62], ‚ Γ satisfies the Leibniz rule [62], in the sense that for all dΓpuv, wq " udΓpv, wq `vdΓpu, wq for all u, v P F pXq X L 8 pX, mq, w P F pXq, ‚ Γ satisfies the chain rule [62], in the sense that dΓpη ˝u, vq " pη 1 ˝uqdΓpu, vq for all u, v P F pXq, where η is any function which is C 1 with a bounded derivative, on an interval which contains the m-essential image of u. ‚ Γ satisfies the truncation property [62], in the sense that dΓpu ^v, wq " 1 tuěvu dΓpu, wq `1tuăvu dΓpv, wq for all u, v, w P F pXq ‚ Γ satisfies the Cauchy-Schwarz inequality ż for all v, w P F pXq, f, g P F pXq X L 8 pX, mq.
(2.1) Again, using the locality of Γ, the energy measure extends to a nonnegative definite symmetric bilinear form Γ : F loc pXq ˆFloc pXq ÝÑ tsigned Radon measures on Xu, namely, given f 1 , f 2 P F loc pXq and U Ť X one sets where h 1 , h 2 P F pXq are chosen such that f j " h j on U.All the above listed properties of the energy measure remain to hold, if one replaces F pXq with F loc pXq and L 8 pX, mq with L 8 loc pX, mq accordingly in the statements.
For future reference, we also note: Lemma 2.2.Let 0 ď w P L 8 loc pX, mq and assume there exists a sequence pw k q kPN in F loc pXq such that w k Ñ w in L 2 loc pX, mq and Then w P F loc pXq.
Proof.It follows easily from the Markovian property that 0 ď v P L 8 loc pX, mq is in F loc pXq, if and only if for all φ P C c pXq X F pXq one has φ ^v P F pXq. Thus, fixing an arbitrary φ P C c pXq X F pXq, we have φ ^wk P F pXq for all k, and it remains to show φ ^w P F pXq.
To this end, we note φ ^wk Ñ φ ^w in L 2 pX, mq.Moreover, by the truncation property, It follows that pφ ^wk q kPN is bounded in the Hilbert space pF pXq, E 1 q.Weak compactness implies that it has a weakly convergent subsequence.The limit must agree with φ ^w, giving φ ^w P F pXq.
Proof.In view of f 1 _ f 2 P F loc pXq, this statement follows from Theorem 6.4 (iii) in [46].
Let H ě 0 denote the nonnegative self-adjoint operator in L 2 pX, mq induced by E , [42,31], and let P t :" e ´tH for t ě 0 (defined by the functional calculus).We call P " pP t ; t ě 0q the associated (self-adjoint) heat semigroup in L 2 pX, mq.The heat semigroup extends to a positivity preserving Markovian contraction semigroup in L q pXq for all q P r1, 8s, which is consistent in q (therefore we use the same symbol for the formally different semigroups on the different spaces), strongly continuous for q ă 8 and weak-˚-continuous for q " 8. Let BpXq and BpX, mq denote, respectively, the space of Borel functions on X and the space of m-equivalences of Borel functions on X.We stretch the notation further by setting P t f pxq :" lim nÑ8 P t pf _ p´nqqpxq P r´8, 0s for every 0 ě f P BpX, mq, and P t f pxq :" lim nÑ8 P t pf ^nqpxq P r0, 8s for every 0 ď f P BpX, mq.
Let 8 X R X be a cemetery point.We equip the space W pXq of continuous paths taking values in the Alexandrov compactification X of X with the σ-algebra and the filtration which is induced by the coordinate process X : r0, 8q ˆW pXq ÝÑ X, with the usual notation X s :" Xps, ‚q.Let denote the shift operator, and, given an open subset V Ă X, let τ V : W pXq ÝÑ r0, 8s, τ V pγq :" infts ě 0 : X s pγq P XzV u.
It follows from a by now classical result of Fukushima [30] that there exists a family pP x q xPX of diffusion measures on W pXq such that for every f P BpXq X L 2 pX, mq one has where f p8 X q :" 0. See also [49] for the quasi-regular case.
Let U Ă X be an open subset.Whenever convenient, we identify functions (and m´equivalence classes) that vanish outside U with their restriction to U. We get the strongly local Dirichlet space pU, m, E , F pUqq by setting The induced operator, semigroup and diffusion measures are denoted by H U , P U , pP x U q xPX , respectively.It is well-known that for all open U Ă V Ă X one has P U t f ď P V t f for all t ą 0, 0 ď f P BpX, mq.Moreover, from monotone convergence theorems for forms, [42,58], it easily follows that , for all t ą 0, 0 ď f P BpX, mq, whenever pU n q nPN is a sequence of open subsets of V with U n Ò V .

Locally smoothing spaces
We collect some classical notions, slightly adjusted to our situation, in: by which we mean that P t pL 1 pX, mqq Ă L 8 pX, mq (which in turn gives continuity of the induced operator by the closed graph theorem), ‚ doubly Feller, if one has the mapping properties and P t : L 8 pX, mq ÝÑ CpXq for all t ą 0, and in addition lim tÑ0`P t f pxq " f pxq for all f P C 0 pXq, x P X. (3.3) Here C 0 pXq denotes the family of continuous functions vanishing at infinity.‚ irreducible, if every f P F loc pXq with E pf q " 0 (which is equivalent to Γpf q " 0) is constant.
We 3. The space pX, m, E , F pXqq is irreducible [62], if and only if for every Borel set Y Ă X with P t p1 Y f q " 1 Y P t f for all t ą 0, f P L 2 pX, mq one has either mpY q " 0 or mpXzY q " 0. This is equivalent to the following positivity improving property: for all t ą 0, 0 ď f P L 2 pX, mqzt0u one has P t f ą 0.
4. If pX, m, E , F pXqq is irreducible, then for all 0 ď f P L 2 pX, mqzt0u, λ ą 0 one has pH `λq ´1f ą 0. This follows from the previous remark and formula (3.4), which converges for f P L 2 pX, mq in the L 2 -Bochner sense.5.By self-adjointness, (3.1) is equivalent to , mq for all t ą 0, in particular, by the spectral theorem and the semigroup property one then has We record that in the doubly Feller case, the underlying diffusion is pointwise determined by the semigroup: Assume that pX, m, E , F pXqq doubly Feller.Then there exists a uniquely determined family of diffusion measures pP x q xPX on W pXq, such that for every bounded function f : X Ñ R, every t ą 0, and every x P X, one has Moreover, one has the absolute continuity condition pX t q ˚Px !m for all t ą 0, x P X, in particular, for all q P r1, 8s, f P L q pX, mq, t ą 0, the function P t f has a pointwise welldefined m´representative, given by the RHS of (3.5).
Proof.This result is standard, except possibly the fact that P x is concentrated on continuous paths for every x P X (noting that the standard theory of Feller processes only produces measures that are concentrated on cadlag paths).This above pointwise concentration follows from observing that the strong locality of E implies this property for m-a.e.x (in fact, q.e. x P X), and then the doubly Feller property together with Theorem 4.5.4(iii) in [31] allows to conclude the pointwise result.
b) The generator of pU, m, E , F pUqq has purely discrete spectrum and a spectral gap.
Proof.(a) This statement is well-known in various forms (cf.[16,18]).We give a detailed proof for the convenience of the reader: by the semigroup property, we only have to show that P U t f is continuous for all f P L 8 pU, mq.To see this, pick U Ă U 1 open with pU 1 , m, E , F pU 1 qq doubly Feller.We first record the standard fact that for all t ą 0, m-a.e.x P U one has qs, and we consider the LHS to be defined pointwise by the RHS.It follows that for all 0 ă s ă t one has, by a standard calculation that relies on the Markov property (cf.p. 18 in [56]), Thus, ˇˇP U t f pxq ´`P U 1 s P U t´s f ˘pxq ˇˇď }f } 8 P x U 1 ps ě τ U q. Since P U 1 s P U t´s f is continuous, it remains to show that P x U 1 ps ě τ U q converges locally uniformly in x to 0 as s Ñ 0. This, however, is a well-known consequence of the Feller property (3.2) of pU 1 , m, E , F pU 1 qq (cf.Lemma 2.5 in [19]).b) Pick U Ă U 1 open with U 1 zU ‰ H such that pU 1 , m, E , F pU 1 qq is irreducible.Since P U t maps to L 8 pU, mq and mpUq ă 8, P U t is Hilbert-Schmidt for every t ą 0, see [22], 11.2, 11.16 and the discussion in [59], p.418; thus H U has purely discrete spectrum.Assume ϕ U P KerpH U q.We are going to show ϕ U " 0. As we have ϕ U P F pUq Ă F pU 1 q with E pϕ U q " 0, it follows from the irreducibility of pU 1 , m, E , F pU 1 qq that ϕ U is constant m-a.e. on U 1 .On the other hand, mpU 1 zUq ą 0 and we have ϕ U " 0 m-a.e. in U 1 zU, thus ϕ U " 0.
Let us give some examples of spaces that are locally smoothing: Example 3.6.Arbitrary Riemannian manifolds: Let X be a connected Riemannian manifold with its Levi-Civita connection ∇ and its Riemannian volume measure m.Then with the usual energy form with domain of definition F pXq " W 1,2 0 pXq, the triple pX, m, E , F pXqq is well-known to be a strongly local Dirichlet space.In this case one has F loc pXq " W 1,2 loc pXq, and F pUq " W 1,2 0 pUq for all open U Ă X, so that the restricted strongly local Dirichlet spaces correspond to Dirichlet boundary conditions.Moreover, pX, m, E , F pXqq is locally smoothing: indeed, given U Ť X, one has (3.1) since for the heat kernels one has p U pt, x, yq ď ppt, x, yq, and ppt, x, yq is jointly smooth in px, yq (in fact in pt, x, yq).Moreover, picking U 1 Ť X connected and with smooth boundary such that U Ă U 1 and U 1 zU ‰ H one has that pU 1 , m, E , F pU 1 qq is doubly Feller, because p U 1 pt, x, yq is smooth, and P U 1 t f extends continously to zero on BU 1 because of Dirichlet boundary conditions (the strong continuity of P U 1 t : C 0 pU 1 q Ñ C 0 pU 1 q is automatic in this case).Moreover, pU 1 , m, E , F pU 1 qq is irreducible, because the connectedness of U 1 implies that p U 1 pt, x, yq ą 0.
Example 3.7.RCD ˚-spaces: Consider a complete, locally compact, separable metric measure space pX, dq, and a Radon measure m on X with full support that gives a finite mass to open balls; a function g P L 2 pX, mq is called a relaxed slope of f P L 2 pX, mq, if there exists g P L 2 pX, mq and a sequence of Lipschitz functions f n P L 2 pX, mq such that ‚ f n Ñ f in L 2 pX, mq and Lippf n q Ñ g weakly in L 2 pX, mq ‚ g ď g, where Lipphqpxq :" lim sup yÑx |hpxq ´hpyq| dpx, yq denotes the local Lipschitz constant of a local Lipschitz function h : X Ñ R. Such a g is called the minimal relaxed slope of f , if its L 2 -norm is minimal amongst all relaxed slopes of f , and then one sets |∇f | ˚:" g.We refer the reader to [1] for equivalent definitions of |∇f | ˚.

With
F pXq :" tf P L 2 pX, mq : f has a relaxed slopeu, the Cheeger form E is the densely defined functional on L 2 pX, mq given by and pX, d, mq is called infinitesimally Hilbertian, if E is a quadratic form.In this case, given f, g P F pXq the limit p∇f, ∇gq :" lim ǫÑ0`1 2ǫ `|∇pf `ǫgq| 2 ˚´|∇f | 2 ˚ȇxists in L 1 pX, mq, and pX, m, E , F pXqq becomes a strongly local Dirichlet space, with dΓpf, gq " p∇f, ∇gqdm for all f, g P F pXq.
In the Hilbertian case, given N ě 1, K P R, one calls pX, d, mq an RCD ˚pK, Nq-space (see [27] and the references therein), if ‚ every f P F pXq with |∇f | ˚ď 1 has a 1-Lipschitz m-representative, ‚ one has ż X e ´dpx 0 ,xq 2 dmpxq ă 8 for all x 0 P X, c ą 0, ‚ for all f P DompHq with Hf P F pXq and all 0 ď g P DompHq X L 8 pX, mq with Hg P L 8 pX, mq one has the Bochner inequality Here H denotes the nonnegative self-adjoint operator in L 2 pX, mq associated with pE , F pXqq.
If pX, d, mq is an RCD ˚pK, Nq-space, then this space satisfies the local volume doubling assumption [27,63], and the heat kernel of P t satisfies the two-sided local Li-Yau heat kernel bounds [37].It follows from Proposition A.1 in the appendix that then pX, m, E , F pXqq is locally smoothing.Note that if pX, d, mq is induced from a complete connected Riemannian manifold X, then the Cheeger form is equal to the energy form, and the RCD ˚pK, Nq-assumption is equivalent to dimpXq ď N and Ric ě K.
Example 3.8.Carnot groups: Assume G is a simply connected Lie group with Lie algebra g, such that there exists N ě 1 and a stratification such that rV i , V j s " V i`j (noting that such stratification is essentially unique [47]), where V k :" t0u for k ą N. In this situation, G is nilpotent and called a Carnot group.Let m denote the Haar measure on G, and let V 1 , . . ., V d denote a basis of V 1 , considered as left invariant vector fields on G. Then the closure pE , F pGqq in L 2 pG, mq of the symmetric bilinear form turns pX, m, E , F pXqq into a strongly local Dirichlet space with dΓpf, gq " pV i f qpV j gqdm for all f, g P F pGq.
The Carnot-Caratheodory distance d on G is complete and induces the original topology on G [26], and the metric measure space pG, d, mq becomes doubling and Ahlfors regular, mpBpx, rqq " r Q , where denotes the homogeneous dimension.Furthermore, the heat kernel of P t satisfies the two-sided Gaussian estimate, as shown in [11]: there exist c 1 , c 2 , c 3 , c 4 ą 0 such that for all t ą 0, x, y P G one has t .It follows from Proposition A.1 in the appendix that pG, m, E , F pGqq is locally smoothing.Note that in general pG, d, mq is not an RCD ˚-space.In fact, it has been shown in [39] that for the Heisenberg group H 2m`1 there exists no pair pN, Kq such that pH 2m`1 , d, mq becomes an RCD ˚pK, Nq space.Example 3.9.Unbounded Sierpinski gasket: Let X Ă R 2 be the unbounded Sierpinski gasket, with its usual graph approximation pX n q Ă X.Then with α :" logp3q{ logp2q the Hausdorff dimension of X and m the α-dimensional Hausdorff measure on X, the symmetric bilinear form E pf, gq :" lim nÑ8 ˆ5 3 ˙n ÿ x,yPXn,x"y pf pxq ´f pyqq 2 , F pXq :" tf P L 2 pX, mq : E pf q ă 8u, turns pX, m, E , F pXqq into a strongly local Dirichlet space.This follows from combining the results in [4] with Proposition A.2.

A new notion of subharmonic functions
We start this section with the following classical definition, see [31,49]: Definition 4.1.Given λ ě 0, h P L 1 loc pX, mq is called (i) λ-defective, if h ď 0 and e ´λt P t h ě h for all t ą 0, (ii) λ-excessive, if ´h is λ-defective.
The following characterization of subharmonic functions of fixed sign is well-known, if one considers functions in F loc pXq, see [31], Theorem 2.2.1, [49], Proposition III.1.2.The main point of the result below is that we can drop this local regularity assumption.For λ " 0, it has already been recorded by Sturm in [62], Lemma 3, that this is possible.Note, however, that the methods from therein do not generalize directly to the λ ą 0 case, roughly speaking, as they rely on the strong locality of E (which fails for E λ ).Theorem 4.2.Let h P L 1 loc pX, mq, and let λ ě 0. (a) Assume h ď 0 and consider the following properties: (i) h is weakly λ-subharmonic.(ii) For all f P F c pXq one has h _ f P F pXq and E λ ph _ f, f ´h _ f q ě 0. (iii) For all f P F c pXq one has h _ f P F pXq and E λ pf, f ´h _ f q ě 0. (iv) For all f P F c pXq one has h _ f P F pXq and E λ ph _ f q ď E λ pf q. (v) h is λ-defective.Then the following implications hold: (i)ùñ(ii)ùñ[(iii) and (iv)]ðñ(v).(b) Assume h is λ-defective.Then one has h P F loc pXq ðñ Dh 7 P F loc pXq : h ě h 7 ðñ @V Ť X Dh 7 V P F c pXq : h| V ě h 7 V .
In particular, if h P L 8 loc pX, mq, then h P F loc pXq.(c) If h is λ-defective and h satisfies one of the equivalent conditions from (b), then h is weakly λ-subharmonic.
Proof.We follow an approach which is in the spirit of [57], Section 2.4.(a) (i)ùñ(ii): This is 'algebraic' in nature, we follow the proof of Lemma 2.50 in [57].We let f P F c pXq and consider V Ť X so that supppf q Ă V .Note that h P F loc pXq by assumption and therefore, there is h 5 P F c pXq such that h " h 5 on V.
We can and will assume that h 5 ď 0 and note that h _ f " h 5 _ f P F c pV q.
Having seen this, we omit the superscript 5 for the rest of the calculation.Since " E λ ph, ph ´f q `q ď 0, since h is weakly λ-subharmonic and 0 ď ph ´f q `" h _ f ´f P F c pXq.
From this chain of inequalities we get: which in turn is obvious.From (ii) and (iii) we get (iv): The equivalence [(iii) and (iv)]ðñ(v) is a consequence of the characterization of convex sets that are invariant under the semigroup in terms of the corresponding quadratic form given in [52].Here are the details: For given h as in the assumption, the set is closed and convex in L 2 pX, mq and the projection π : L 2 pX, mq Ñ K is given by πf " h _ f .Clearly (v) implies that the semigroup associated with E λ , given by S t " e ´λt P t for t ě 0, leaves K invariant.Denoting the following stronger versions of the properties in the Theorem by (iii') For all f P F pXq one has h _ f P F pXq and E λ pf, f ´h _ f q ě 0.
A standard approximation argument gives (iv) ñ (iv') and so we get the implications: (ii) ñ [(iii) and (iv)] ñ (iv') ô (v) ô (iii'), which settles (a).(b) For the two equivalences, it remains to show that the last property implies the first.The other implications are evident.Let V Ť X and h 7 V as asserted.Then (iii) implies that h _ h 7 V P F pXq and h _ h 7 V " h on V , giving that h P F loc pXq.The 'in particular' follows, since locally bounded functions can be locally minorized by (negative) equilibrium potentials, see below.
(c) This is a standard semigroup argument plus some approximation argument.It suffices to consider bounded h: in fact, E λ ph, f q " lim nÑ8 E λ ph _ p´nq, f q for any f P F c pXq and h _ p´nq is λ-defective, provided h is.By rescaling, we can and will assume that h ě ´1.Let f P F c pXq, f ě 0 and pick V, W open, V Ť W Ť X such that supppf q Ă V .Let e W be the λ-equilibrium potential of the set W , see [31], which is λ-excessive.Therefore, h 5 :" h _ p´e W q is λ-defective, h " h 5 on W and h 5 P F pXq, because ´h5 is λ-excessive and 0 ď ´h5 ď e W P F pXq.It follows that completing the proof.
The following characterization of λ-subharmonic functions in the distributional sense in the Riemannian case served as the blueprint for our new notion of subharmonicity in Definition 4.4 below.Recall that a distribution g defined on an open set U of a Riemannian manifold X with Laplace-Beltrami operator ∆ is called λ-harmonic, if p∆ ´λqg " 0 on U in the sense of distributions.By local elliptic regularity, this is equivalent to g being a smooth classical solution of the latter equation and thus also to the weak λ-harmonicity of g. (i) p∆ ´λqf ě 0 in the sense of distributions, that is, xf, p∆ ´λqφy ě 0 for all 0 ď φ P C 8 c pXq, (ii) For all U Ť X there exists a λ-harmonic function g P C b pUq on U such for all t ą 0 one has e ´tλ P U t pf | U ´gq ě f | U ´g. Proof.Note first that in the above situation, [8] we can pick open and smooth neighbourhoods by dominated convergence and also that f can be chosen upper semicontinuous.By the latter fact we can pick c P R such that and set h :" f | U 1 ´g.We have that h k :" f k | U 1 ´g is λ-subharmonic on U 1 and h k ď 0 on BU 1 .Thus h " lim kÑ8 h k ď 0 on U 1 by the (weak) maximum principle.Moreover, h k Ñ h in L 1 pU 1 q.Let 0 ď φ P C 8 c pU 1 q.For all t ą 0 one has Since ∆e ´λt P U 1 t φ " e ´λt P U 1 t ∆φ is bounded, the latter is

Next we show
To this end, we compute where we have used h k ď 0 and B ν e ´λt P U 1 t φ ď 0 (because e ´λt P U 1 t φ ě 0) in the first inequality, and e ´λt P U 1 t φ " 0 on BU 1 together with p∆ ´λqh k ě 0 and P U 1 t φ ě 0 on U 1 for the last inequality.We have thus shown e ´λt P U 1 t pf | U 1 ´gq ě f | U 1 ´g for all t ą 0. Since P U 1 t Ñ 0 as t Ñ 8 strongly in L 1 pU, mq (cf. the proof of Lemma 4.6(b) below for a detailed argument), we have f | U 1 ´g ď 0. Thus, e ´λt P U t pf | U ´g| U q ě e ´λt P U 1 t pf | U 1 ´gq on V , and the claim follows from setting g :" g| U .(i) ð (ii): Let U Ť X be arbitrary and pick a λ-harmonic function g on U as in the assumption.
We have The previous result motivates: We say that f is ‚ locally λ-shift defective, if for all U Ť X there exists a weakly λ-harmonic function g P C b pUq such that ‚ locally λ-shift excessive, if ´f is locally λ-shift defective.
Remark 4.5.1.We stress that no a priori regularity on f is needed in the above definition, other than the somewhat minimal L 1 loc -assumption.2. Note that in the definition of local λ-shift defectiveness we do not require explicitly that f | U ´g ď 0. The latter will however follow automatically in the locally smoothing case (cf. the transfer principle below).In other words, in the locally smoothing case, f P L 1 loc pX, mq is locally λ-shift defective, if and only if for all U Ť X there exists a λ-harmonic function This justifies the name 'locally λ-shift defective'.3. A function f P L 1 loc pX, mq is obviously locally λ-shift defective, if and only if f | U is locally λ-shift defective for every U Ť X, that is, with respect to pU, m, E , F pUqq.Moreover, the class of locally λ-shift defective functions is stable under taking sums, multiplication by positive constants, and shifting by negative constants.
For locally smoothing spaces we get the following result, which will allow us to transfer results for signed functions to functions of varying sign: Proof.(a) Pick an open set U 1 Ă X such that U Ť U 1 and U 1 zU ‰ H, and such that pU 1 , m, E , F pU 1 qq is doubly Feller and irreducible.Pick 0 ď ψ P C c pU 1 qzt0u with supppψq Ă U 1 zU.Assume first λ ą 0. Then we can define Since h is continuous and the closure of U is a compact subset of U 1 , we have inf h| U ą 0, and so for some a ą 0 we have c ď g :" a ¨h| U P C b pUq X F loc pUq, and given φ P F c pUq we have E pg, φq `λ xg, φy " aE ph, φq `λa xh, φy " axH U 1 h, φy `aλ xh, φy " a xψ, φy " 0.
For λ " 0, we can simply take a constant function.(b) For all t ě 1 we have By spectral calculus we have that P U s converges as s Ñ 8 strongly in L 2 pU, mq to the projection onto KerpH U q, the latter space being equal to t0u, as H U has a spectral gap.
We continue with: Lemma 4.7.(Approximation lemma) Assume pX, m, E , F pXqq is locally smoothing and λ ě 0. Let f be locally λ-shift defective.Given U Ť X, pick a weakly λ-harmonic function g P C b pUq on U as in Definition 4.4.Then, for every k P N, is locally λ-shift defective and weakly λ-subharmonic, and one has f U k Ó f | U m-a.e., and so f ‚ U :" inf kPN f U k is an upper semicontinuous (u.s.c.) m-representative of f | U .In particular, the function f has an u.s.c.m-representative.
Proof.Let V Ť U. We have where have used that f | U ´g ď 0 for the first estimate (part (b) of transfer principle), and that f is locally λ-shift defective.Thus f U k is locally λ-shift defective.Since f U k ´g ď 0 is continuous and λ-defective, it is weakly λ-subharmonic by Theorem 4.2 (c).This sequence converges pointwise in a monotonically decreasing way on U, and the limit function is u.s.c. and coincides m-a.e. with f | U .The 'in particular' follows with a partition of unity argument.
Using the transfer principle, we can establish the following results analogous to those of Theorem 4.2, now for functions of varying sign: Theorem 4.8.Let λ ě 0 and let pX, m, E , F pXqq be locally smoothing.(a) If f P L 1 loc pX, mq is locally λ-shift defective and in F loc pXq, then f is weakly λ-subharmonic.In particular, if f P L 8 loc pX, mq is locally λ-shift defective, then f is weakly λ-subharmonic (b) If f P F loc pXq is weakly λ-subharmonic and locally bounded above, then f is locally λ-shift defective.
Proof.(a) Let U Ť X be arbitrary and pick g as in Definition 4.4.Then f | U ´g P F loc pUq is λ-defective on U by the transfer principle; applying Theorem 4.2(c) with h " f | U ´g on U (instead of X) shows that this function is weakly λ-subharmonic on U, and so f | U " f | U ´g `g is weakly λ-subharmonic on U. (b) Let U Ť X be arbitrary and pick c P R with f | U ď c and using the transfer principle pick g P C b pUq weakly λ-harmonic on U with g ě c.Then f | U ´g ď 0 is weakly λ-subharmonic on U, and so applying Theorem 4.2 (a) to h " f | U ´g on U gives that e ´λt P U t pf | U ´gq ě f | U ´g, completing the proof.
Remark 4.9.The local boundedness from above, required in (b) of the previous Theorem, can be typically deduced from elliptic or parabolic subsolution estimates, which are valid under some mild local assumptions on the space.For the most general result to this effect that we are aware of, we can refer to Theorem 4.7 in [48]: under the assumption that the space pX, m, E , F pXqq has the property that on every U Ť X a volume doubling up to a finite scale and a Poincare inequality up to a finite scale are satisfied, a parabolic subsolution estimate is valid, which shows that every weakly λ-subharmonic function is automatically locally bounded above.Indeed, if f is weakly λ-subharmonic, then so if f `by Lemma 2.3, and then applying Theorem 4.7 in [48] to pt, xq Þ Ñ e ´tλ f `pxq, we get that on such a space f `is automatically locally bounded, and so f is locally bounded above.This applies to all examples of locally smoothing spaces above.

Kato-Brezis theorem and regularity for λ-shift defective functions
The starting point of this section is the following probabilistic characterization of locally λ-shift defective functions: Proposition 5.1.Assume that pX, m, E , F pXqq is locally smoothing and that λ ě 0. A function f P L 1 loc pX, mq is locally λ-shift defective, if and only if for all U Ť X there exists a u.s.c m-representative f ‚ U of f | U such that for all V Ť U with UzV ‰ H one has U pxq for q.e.x P V. (5.1) The proof is based on the following technical result by Chen and Kuwae, which holds on any strongly local Dirichlet space and which follows from applying Theorem 2.9 in [17] (see also [14]) to the Dirichlet form which is shifted by λ, noting that the result therein holds for nonlocal Dirichlet forms: Lemma 5.2.A function h P L 8 loc pXq is weakly λ-subharmonic, if and only if for all V Ť X with XzV ‰ H one has e ´λτ V hpX τ V q ‰ ě hpxq for q.e.x P V. (5.2) Note that, if h P L 8 loc pXq is weakly λ-harmonic, one gets an equality in (5.2).Proof of Proposition 5.1.ñ: Let U Ť X.Consider the functions g, f U k , f ‚ U from the approximation Lemma 4.7.For every k P N the function f U k ´g ď 0 is continuous and λ-defective on U, in particular, weakly λ-subharmonic on U by Theorem 4.2(c).It follows from using the previous lemma with h " ‰ ě pf U k ´gqpxq for q.e.x P V.For k Ñ 8 we arrive at U ´gqpxq for q.e.x P V, which using the previous lemma with h " g gives U " e ´λτ V gpX τ V q ‰ " gpxq for q.e.x P V, (5.3) proving (5.1).ð: The function f ‚ | U is bounded above for given U Ť X.As above, we can pick a weakly λ-harmonic function g P C b pUq such that f ‚ | U ´g ď 0 on U. Given V Ť U with UzV ‰ H, the assumption on f together with (5.3) implies ‰ ě f ‚ U pxq ´gpxq, for q.e.x P V.For arbitrary n P N, this implies U pxq ´gpxqq _ p´nq, for q.e.x P V. and so the locally bounded function pf ‚ U ´gq _ p´nq is weakly λ-subharmonic on U by the previous lemma.From Theorem 4.2(a) we get that pf ‚ U ´gq _ p´nq is λ-defective on U. Since this is true for arbitrary n P N, we get that f ‚ U ´g is λ-defective on U.As noted above, Lemma 5.2 is a consequence of Theorem 2.9 in [17].The latter gives a probabilistic characterization of weak λ-subharmonicity under the a priori assumption that the function is in L 8 loc .This assumption is partially used in [17] to verify necessary measurability proporties and to make the expectation on the LHS of (5.2) well-defined, noting that the proof goes through an intermediate step, giving a characterization in terms of a submartingale property.Note that we do not make an L 8 loc -assumption in Proposition 5.1, replacing the weak λsubharmonicity assumption with a local λ-shift defectiveness assumption in one direction of the equivalence of Lemma 5.2.The main reason for this is that we get a u.s.c.m-respresentative from our local smoothing assumptions on the space.The main strength of Theorem 2.9 in [17], however, clearly lies in the very remarkable fact that this result applies to nonlocal forms (with some mild additional regularity assumptions).As an application of Proposition 5.1 we immediately get the following generalization of a consequence of Kato's celebrated inequality, Lemma A from [41], which states that for all f P L 1 loc pR n q such that ∆f P L 1 loc pR n q, one has ∆|f | ě signpf q∆f (5.4) in the sense of distributions.This readily gives ∆f `ě 1 tf ě0u ∆f. (5.5) This was later sharpened and generalized (see Ancona, [3] for a thorough discussion) to the case that ∆f is a Radon measure in the article [13] by Brezis and Ponce.It is maybe superfluous to add that some regularity assumption on ∆f is needed to ensure that the RHS of (5.4) and (5.5) are well-defined distributions.An immediate consequence of the latter estimate is that ∆f `is a positive distribution, in other words, f `is subharmonic in the distributional sense.Remarkably, using the results we have obtained so far, we get a considerably strengthened variant of the statement f P L 1 loc is subharmonic in the distributional sense ñ f `is subharmonic in the distributional sense in our general setting, where neither test functions nor distributions are available.Indeed, the the following result follows immediately from Proposition 5.1: Theorem 5.3.(Kato-Brezis) Assume that pX, m, E , F pXqq is locally smoothing and that λ ě 0. If f, g P L 1 loc pX, mq are locally λ-shift defective, then so is f _ g.If 0 ď f P L 1 loc pX, mq is locally λ-shift defective, then f is locally shift defective.
The reader may find similar statements in [54] for Riemannian manifolds and in [20] for Carnot groups.
As a first application of the above theorem, we are going to prove: Theorem 5.4.(Regularity theorem) Let pX, m, E , F pXqq be locally smoothing, let λ ě 0 and let f P L 1 loc pX, mq be locally λ-shift defective.Then for every c ě 0, q ą 1 one has pf ´cq q{2 `P F loc pXq, and for q ě 2 this function is weakly subharmonic.
Proof.The function f ´c is locally λ-shift defective and so by Kato-Brezis pf ´cq `is locally λ-shift defective.Thus we can assume c " 0 and f ě 0.Then, again by Kato-Brezis, f is locally shift defective.As f has a u.s.c.m-representative, f is locally bounded above (and by f ě 0 then locally bounded) and thus weakly subharmonic by Theorem 4.8 (a).It then follows immediately from the chain rule that for q ě 2 the function f q{2 is weakly subharmonic as well (cf.the calculation in the proof of Lemma 2 a) in [62]).For arbitrary q ą 1 we proceed as follows to show that f q{2 P F loc pXq: let V Ť X be arbitrary and pick U with V Ť U Ť X.We also pick 0 ď φ P C c pUq X F pUq such that 1 V ď φ ď 1 U , and a sequence `f U k ˘kPN as in Lemma 4.7 and set 0 ď f | U ď f k :" f U k P F loc pUq X C b pUq for notational convenience.Replacing f k with f k `1{k if necessary, we can assume f k ą 0 and still get f k Ó f | U .We have f q´1 k P F loc pUq by the chain rule, so where have used the chain rule.Using the Cauchy-Schwarz inequality (2.1) with k , on the second term we get, for ǫ ą 0: For ǫ ą 0 small enough (depending only on q), we deduce that Using the chain rule we end up with the second estimate in ż 1 dΓpφq, the latter expression being finite, as f q 1 P L 8 pU, mq.Finally, applying Lemma 2.2 with w k :" loc pV, mq (in fact, in L 2 pV, mq).

Maximum principle
The main result of this section is: Theorem 6.1 (Maximum principle).Let pX, m, E , F pXqq be locally smoothing and let f P L 1 loc pX, mq.a) If f is locally shift defective, then for every V Ť X there exists a u.s b) If λ ě 0 and if f is be locally λ-shift defective, then for every V Ť X there exists a u.s.c.
Note that part b) follows immediately from part a), as by the Kato-Brezis theorem, f `is locally shift defective.Note also that it follows immediately from Proposition 5.1 that In this sense, the main point of Theorem 6.1 is to show that one can guarantee a pointwise inequality of this form.
Proof of Theorem 6.1 a).From the definition of local shift defectiveness, it is easy to see that it is enough to prove that for every defective function h, which is defined in an open neighbourhood U Ť X of V , there exists an m-representative h ‚ of h such that As P U t hpxq is continuous in x and increasing in t, we define h ‚ :" lim nÑ8 P U 1{n h.
Assume, by contradiction, that there exists x P V such that h ‚ pxq ą sup BV h ‚ .We choose δ 1 , δ 2 P R such that h ‚ pxq ą δ 1 ą δ 2 ą sup Since h ‚ is u.s.c., we have that h ‚ ă δ 2 in some open neighborhood N BV of BV in V and, up to choosing a smaller set, we can always assume that We claim that, by a variation of the proof of Dini's theorem, there exists t 0 ą 0 small enough such that, for all 0 ă t ă t 0 , P U t h ă δ 1 in Ě N BV .Indeed, let E t :" tx P V : P U t h ă δ 1 u.Since h is defective, E t Ď E s for t ą s, and by pointwise convergence we get Hence E t 0 Ą Ě N BV for some t 0 small enough, and the claimed property follows.On the other hand P U t hpxq ě h ‚ pxq ą δ 1 .Now, P U t h is defective on U and in F pUq X C b pUq.Thus, Theorem 4.2 (c) implies that P U t h, and hence u :" P U t h ´δ1 , is weakly subharmonic.Since u ď 0 in V zY where Y :" V zN BV , the weak maximum principle of [62,Lemma 4] gives that u ď 0 on V .In particular, P U t hpxq ď δ 1 .The contradiction completes the proof.Example 6.2.Assume that pX, m, E , F pXqq stems from an RCD ˚p0, Nq space for some N ě 1, see Example 3.7 above.Since |∇ ¨|˚i s local, one can define |∇f | ˚also for f P F loc pXq.Let f be weakly λ-harmonic for some λ ě 0, with λf P L q loc pX, mq for some q ě maxpN, 4q.It then follows from the localized Bochner inequality (cf.[36]) that |∇f | 2 ˚is weakly subharmonic.By Remark 4.9 this function is locally bounded (above), and so Theorem 4.8 (b) shows that this function is locally shift defective.We get that for every V Ť X there exists a u.s.c.

A proof of the BMS-conjecture on locally smoothing spaces
We still assume that pX, m, E , F pXqq is a strongly local Dirichlet space.Then there exists a possibly degenerate pseudo metric d E on X (that is, it may happen that d E px, yq " 8 for some x, y P X, or that d E px, yq " 0 for some x, y P X with x ‰ y), called the intrinsic metric, given by d E px, yq :" suptf pxq ´f pyq : f P F loc pXq X CpXq, dΓpf q ď dmu, where the condition dΓpf q ď dm means that Γpf q is absolutely continuous with respect to m, with a density ď 1.
Definition 7.1.We say that pX, m, E , F pXqq is ‚ strictly local, if d E induces the original topology on X, ‚ intrinsically complete, if this space is strictly local and complete.
If pX, m, E , F pXqq is strictly local, then intrinsic completeness is equivalent to properness.This follows from the Hopf-Rinow theorem for locally compact length spaces, in combination with the fact that strictly local Dirichlet spaces are lenght spaces [60].
In spirit of [62], we say that pX, m, E , F pXqq has the L q -Liouville property, for given q P r1, 8s, if every weakly subharmonic function 0 ď f P L q pX, mq is constant.One of the main results from Sturm's seminal paper [62] states: Theorem 7.2.If pX, m, E , F pXqq is intrinsically complete and irreducible, then it has the L q -Liouville property for every q P p1, 8q.
The following definition is adapted from [33], where the setting of Riemannian manifolds has been considered: Definition 7.3.Given q P r1, 8s, one says that pX, m, E , F pXqq is L q -positivity preserving, if for every locally 1-shift-excessive function f P L q pX, mq one has f ě 0.
With this definition, we easily get: Proposition 7.4.If pX, m, E , F pXqq is locally smoothing with the L q -Liouville property for some q P p1, 8q, then it is L q -positivity preserving.
Proof.If f P L q pX, mq is locally 1-shift excessive, then ´f is locally 1-shift defective, so 0 ď p´f q `P L q pX, mq is weakly subharmonic by the regularity Theorem 5.4; thus this function must be constant by the L q -Liouville property, and thus " 0.
Finally, as stated in the introduction, Theorem 7.2 in combination with Proposition 7.4 immediately leads to a proof of a conjecture by Braverman, Milatovic and Shubin extended to our general setting (see Conjecture P in Appendix B, p. 679, and also the discussion in [35]): Theorem 7.5.(BMS-conjecture) If pX, m, E , F pXqq is locally smoothing, intrinsically complete and irreducible, then it is L q -positivity preserving for all q P p1, 8q.
Note that every connected and complete Riemannian manifold, every RCD ˚-space, as well as every Carnot group satisfies the assumptions of this theorem.Indeed, in the Riemannian case the intrinsic completeness is well-known, in the RCD ˚-case it is shown in [2], and for the Carnot group in [26].In all cases, the irreducibility follows from the strict positivity of the heat kernel.
Then for all U Ť X the space pU, m, E , F pUqq is ultracontractive, and pX, m, E , F pXqq is doubly Feller and irreducible.In particular, pX, m, E , F pXqq is locally smoothing.

Proposition 4 . 3 .
Assume X is a connected Riemannian m-manifold (considered as a smoothing strongly local Dirichlet space in the sense of Example 3.6) and let f P L 1 loc pX, mq, λ ě 0. The following two properties are equivalent:

Lemma 4 . 6 .
(Transfer principle) Let pX, m, E , F pXqq be locally smoothing and λ ě 0. (a) For every c P R, U Ť X there exists g P C b pUq weakly λ-harmonic on U, such that g ě c.(b) If f P L 1 loc pX, mq is locally λ-shift defective, then with U, g as in Definition 4.4 one has f | U ´g ď 0.
Regularity) The space F pXq X C c pXq is dense in C c pXq with respect to }‚} 8 and in F pXq with respect to the scalar product E 1 pu, vq :" E pu, vq `xu, vy , [23]29], let F loc pXq :" f P L 2 with the above equality easily checked by regularity.By what we mentioned earlier, one has 1 P F loc pXq.Note that there are different ways of introducing F loc pXq, see[45,29]and the discussion in[23].We also note that thanks to locality, E has a canonical extension to a bilinear map F loc pXq ˆFc pXq Ñ R, where F c pXq denotes the elements of F with compact m-support, given by E pf, φq :" E ph, φq, where h P F pXq is such that h " f in an open neighbourhood of the m-support of φ.
loc pX, mq ˇˇfor all U Ť X there exists h P F pXq with f " h on U ( " f P L 2 loc pX, mq ˇˇfor all φ P F pXq X C c pXq one has φf P F pXq ( , say that [the generator of] pX, m, E , F pXqq ‚ has a spectral gap, if H ą 0, i.e., infpσpHqq ą 0. ‚ has purely discrete spectrum, if the spectrum σpHq of H is purely discrete, that is, consists of isolated eigenvalues having a finite multiplicity., one gets that P t : C 0 pXq Ñ C 0 pXq, t ą 0, is strongly continuous.In particular, then one also has the mapping property pH `λq ´1 : C 0 pXq ÝÑ C 0 pXq, for all λ ą 0, where the improper Riemannian integral converges in the uniform norm.2. If pX, m, E , F pXqq has purely discrete spectrum, then pX, m, E , F pXqq has a spectral gap, if and only if KerpHq " t0u.
Definition 3.4.pX, m, E , F pXqq is called locally smoothing, if for all U Ť X it holds that ‚ pU, m, E , F pUqq is ultracontractive, ‚ there exists U 1 Ă X open with U Ă U 1 and U 1 zU ‰ H such that pU 1 , m, E , F pU 1 qq is doubly Feller and irreducible.Note that if pX, m, E , F pXqq is locally smoothing, then so is pU, m, E , F pUqq for every open U Ă X. Locally smoothing spaces have the following self-improvement properties: Proposition 3.5.Assume that pX, m, E , F pXqq is locally smoothing, and let U Ť X.
a) For all t ą 0 one has , mq, and so e ´λt P U t pf | U ´gq Ñ f | U ´g as t Ñ 0à s distributions.Since p∆ ´λqe ´λt P U t pf | U ´gq " B t e ´λt P U t pf | U ´gq ě 0, as e ´λt P U t pf | U ´gq is increasing in t, we get for all 0 ď φ P C 8 p∆ ´λqe ´λt P U t pf | U ´gq, φ D ě 0,and the proof is complete.