Heat Semigroup and Functions of Bounded Variation on Riemannian Manifolds

M. Miranda Jr^{∗} D. Pallara^{∗} F. Paronetto^{∗} M. Preunkert^{†}

Abstract Let M be a Riemannian manifold with Ricci curvature bounded from below and such that the volume of the geodesic balls of centre x and fixed radius r > 0 have a volume bounded away from 0 uniformly with respect to x, and let (T (t))_{t≥0} be the heat semigroup on M. We show that the total variation of the gradient of a function u ∈ L^{1}(M) equals the limit of the L^{1}-norm of ∇T (t)u as t → 0. In particular, this limit is finite if and only if u is a function of bounded variation.

Mathematics subject classification (2000): 53C21, 49Q15, 58G11, 47D06 Keywords: Functions of bounded variation, Riemannian manifolds, Heat semigroup.

Introduction

Functions of bounded variation in R^{n} are by now deeply studied, and the spaces BV are a well-established tool for studying variational problems, often with some geometric flavour. This is due to the possibility of having discontinuities along (n − 1)-dimensional surfaces, which is not the case for functions in Sobolev spaces. In fact, a very important particular case of BV functions are characteristic functions of sets with finite perimeter in the sense of Caccioppoli-De Giorgi, the most natural class of sets where the isoperimetric problem can be formulated and solved. After various attempts to generalise to R^{n} the classical notion of BV functions of only one real variable, the new idea that opened the way for the modern theory was the definition proposed by E. De Giorgi in [8]. It is based upon a regularisation with a Gaussian convolution kernel and can be rephrased using the heat semigroup (T (t))_{t≥0} in R^{n} as follows. Given a function u ∈ L^{1}(R^{n}), define its variation by

|Du|(R^{n}) = lim_{t→0} ∥∇T (t)u∥_{L^{1}(R^{n})}  (1)

and say that u has bounded variation, u ∈ BV (R^{n}), if |Du|(R^{n}) is finite. It has been shown in [8] that u has finite variation if and only if its distributional gradient is a R^{n}- valued measure with finite total variation (in the sense of measures) given by |Du|(R^{n}). As a consequence, we have the equality

|Du|(R^{n}) = sup{∫_{R^{n}} u div g dx : g ∈ [𝒞_{c}^{1}(R^{n})]^{n}, ∥g∥_{∞} ≤ 1}.  (2)

∗Dipartimento di Matematica “Ennio De Giorgi”, Università di Lecce, C.P.193, 73100, Lecce, Italy.

e-mail: michele.miranda@unile.it, diego.pallara@unile.it, fabio.paronetto@unile.it

†Mathematisches Institut, Universität Tübingen, Auf der Morgenstelle 10, D-72076 Tübingen, Germany.

e-mail: mapr@fa.uni-tuebingen.de

This last formula, first used in [17], can then be used directly to define BV functions, and can be easily generalised to Riemannian manifolds (see (1.3) below). By the way, let us just mention that further characterisations of BV functions are available, which can be used in even more general contexts, such as metric spaces endowed with a doubling measure (see [18], [2]). Moreover, notice that further connections between isoperimetric inequalities and the heat semigroup have been pointed out, even in non-euclidean contexts, such as Gaussian spaces (see e.g. [14] and also [19]).

In this paper, we address the question whether equality (1) holds on a Riemannian manifold M, where the left hand side is defined as in (2), see (1.3), and in the right hand side the heat semigroup on M is used. The answer is affirmative, at least under some geometric hypotheses, see (H1), (H2) below. We require that the Ricci curvature of M is bounded below, and that the volume of the (geodesic) balls of a given radius is bounded below by a constant independent of the centre. These assumptions bound the geometry of M in two opposite directions, see Remarks 1.2 and 1.3, and in fact both hold trivially for compact manifolds, and more generally for manifolds of bounded geometry. We deeply use hypotheses (H1) and (H2) and their relevant consequences, such as the Sobolev embedding and the induced regularity of the heat semigroup (see e.g. Theorem 2.1 and Theorem 2.7). We point out that we have used two different approaches for the proof of Theorem 2.7; in the compact case we have used a direct approach as in [3], whereas in the non-compact case we have used Gaussian estimates, that hold true under assumptions (H1) and (H2) (see Subsection 2.1).

It seems to be worth mentioning that the equality

lim_{t→0} ∫_{M} |∇T (t)u(x)|dμ(x) = |Du|(M)

yields an approximation of BV functions in variation through regular functions. An approximation result of this kind can be shown on manifolds using a partition of unity argument and the corresponding result on R^{n} (see Proposition 1.4 below), and in fact we also use this as an intermediate result, but we point out that the approximation given by the heat semigroup is global and intrinsic.

The main steps in our proof are two. First, we show that the limit in (1), with M in place of R^{n}, exists (this is trivial in R^{n} by monotonicity with respect to time), and then, we show that the value of the limit is the variation of u defined in (1.3).

1 Notation and Preliminaries

In this section we recall some basic facts concerning Riemannian manifolds and Sobolev and bounded variation functions.

1.1 Riemannian Manifolds and Geometric Hypotheses

The framework of this paper is given by an n–dimensional manifold M (not necessarily compact) without boundary, i.e., ∂M = ∅, with n ≥ 2, verifying the geometric hypotheses (H1), (H2) stated below in this subsection. Let us start by fixing some notation. We denote by T M and T^{∗}M the tangent and co–tangent bundles of M, respectively. Given a vector bundle E, we denote by Γ(E) the smooth sections of E; in particular, 𝒯_{q}^{p}(M) = Γ((T^{∗}M)^{p} × (T M)^{q}) is the space of tensors of type (p, q) on M, Γ(T M) is the space of vector fields on M and Γ(T^{∗}M) is the space of 1–forms on M.

The Riemannian structure on M is defined by a symmetric metric tensor g ∈ 𝒯_{0}^{2}(M). The metric g defines an isomorphism i : Γ(T M) → Γ(T^{∗}M) in such a way that for any X ∈ Γ(T M),

i(X)(Y) = g(X, Y), ∀ Y ∈ Γ(T M).

Using the map i, it is possible to define the tensor ḡ ∈ 𝒯_{2}^{0}(M) through

ḡ(ω^{1}, ω^{2}) = g(i^{−1}ω^{1}, i^{−1}ω^{2}).

where i^{−1} : Γ(T^{∗}M) → Γ(T M) is the inverse of i. This tensor defines an inner product on Γ(T^{∗}M) in such a way that the map i becomes an isometry.

For X ∈ T_{x}M and ω ∈ T_{x}^{∗}M, we set

|X| = \sqrt{g(X, X)}, |ω| = \sqrt{ḡ(ω, ω)}.

We denote by ⟨·, ·⟩ and by | · | the inner product and the norm induced by the metric g on any tensor T of type (p, q) in order to satisfy

|T | = ⟨T, T ⟩^{1/2} = sup{T (X_{1}, …, X_{p}, ω^{1}, …, ω^{q}) : |X_{1}|, … |X_{p}|, |ω^{1}|, …, |ω^{q}| ≤ 1},

A vector bundle E such that the fibres are endowed with an inner product is called a Riemannian vector bundle. If M is not compact, given a general Riemannian vector bundle E, we denote by Γ_{c}(E) the smooth sections of E with compact support, and by Γ_{0}(E) the closure of Γ_{c}(E) with respect to the norm

∥T ∥_{∞} = sup{|T (x)| : x ∈ M}, T ∈ Γ(E).

Of course, if M is compact then Γ_{0}(E) = Γ_{c}(E) = Γ(E).

The Riemannian metric g induces a geodesic distance d on M; we always assume that the metric space (M, d) is complete. Moreover, we denote by B_{r}(x) the geodesic open ball centred at x ∈ M and with radius r > 0. In this setting, there is a natural way of defining a measure μ on M, also without assuming M orientable; the measure μ is given in local coordinates by

dμ = \sqrt{detg} dx.

By ∇ : 𝒯_{q}^{p}(M) → 𝒯_{q}^{p+1}(M) we denote the Levi–Civita connection on M, that is the unique connection compatible with the metric g, in the sense that

X⟨T, S⟩ = ⟨∇_{X}T, S⟩ + ⟨T, ∇_{X}S⟩, ∀ T, S ∈ 𝒯_{q}^{p}(M).

Here the notation ∇_{X}T ∈ 𝒯_{q}^{p}(M) means that

(∇_{X}T)(X_{1}, …, X_{p}, ω^{1}, …, ω^{q}) := (∇T)(X, X_{1}, …, X_{p}, ω^{1}, …, ω^{q}).

The Levi–Civita connection is a torsion–free connection which induces the Riemann curvature tensor R ∈ 𝒯_{0}^{4}(M) defined by

R(X, Y, W, Z) = ⟨∇_{X}∇_{Y} W − ∇_{Y} ∇_{X}W − ∇_{[X,Y]}W, Z⟩, ∀X, Y, W, Z ∈ Γ(T M).

We denote by Ric the Ricci tensor of type (2, 0) defined pointwise by

Ric(X, Y) = ∑_{i=1}^{n} R(X, e_{i}, Y, e_{i}) ∀X, Y ∈ T_{x}M,

where {e_{i}} is an orthonormal basis of T_{x}M. The connection ∇ on a function u simply defines the covariant derivative of u, i.e.,

∇u = i(grad u),

where grad is the standard gradient defined using local coordinates.

We also denote by div : 𝒯_{q}^{p}(M) → 𝒯_{q}^{p−1}(M) the operator defined using the formula

∫_{M} ⟨∇T, S⟩dμ = − ∫_{M} ⟨T, div S⟩dμ, ∀ T ∈ 𝒯_{q}^{p−1}(M), ∀S ∈ 𝒯_{q}^{p}(M).

In this way, given a function u ∈ 𝒞^{∞}(M), the Laplace–Beltrami operator applied to u is defined by

Δu = div ∇u;

notice that the operator defined above is negative definite. The Hessian of a function u ∈ 𝒞^{∞}(M) is given by the tensor Hess u = ∇^{2}u ∈ 𝒯_{0}^{2}(M). Finally, let us recall the Bochner–Lichnerowitz–Weitzenböck formula; for any u ∈ 𝒞^{∞}(M),

\frac{1}{2}Δ|∇u|^{2} = |Hess u|^{2} + ⟨∇Δu, ∇u⟩ + Ric(∇u, ∇u).  (1.1)

In the whole paper, we consider Riemannian manifolds satisfying the following hypotheses:

(H1) there exists K ≥ 0 such that Ric ≥ −K, i.e.,

Ric(X, X) ≥ −K|X|^{2} ∀ X ∈ Γ(T M);

(H2) there exists v > 0 such that

inf_{x∈M} μ(B_{1}(x)) ≥ v.

Hypothesis (H1), in particular, implies that the metric measure space (M, d, μ) is locally doubling. Indeed, Cheeger, Gromov and Taylor [5] (see also Hajłasz and Koskela [12, Section 10.1]) prove that for every x ∈ M and for every R > 0

μ(B_{2R}(x)) ≤ 2^{n} exp\{2R\sqrt{(n − 1)K}\}μ(B_{R}(x)).  (1.2)

The local doubling condition implies the following growth bound for the measure of the balls.

Lemma 1.1 Assuming (H1), for every x ∈ M and 0 < r < R the inequality

\frac{μ(B_{R}(x))}{μ(B_{r}(x))} ≤ 2^{n} (\frac{R}{r})^{n} (\frac{e^{4R}}{e^{2r}} )^{\sqrt{(n−1)K}}

holds.

Proof. Choose j ∈ N in such a way that

2^{j−1}r < R ≤ 2^{j}r;

as an immediate consequence, we get that j satisfies the inequality

2^{j} < \frac{2R}{r}.

Iterating the local–doubling condition (1.2), since B_{R}(x) ⊂ B_{2^{j}r}(x) we obtain that

μ(B_{R}(x)) ≤ μ(B_{2^{j}r}(x)) ≤ 2^{n} exp\{2^{j}r\sqrt{(n − 1)K}\}μ(B_{2^{j−1}r}(x))

≤ 2^{jn} exp \{r ∑_{k=1}^{j} 2^{k}\sqrt{(n − 1)K}\} μ(B_{r}(x))

≤ 2^{n} (\frac{R}{r})^{n} exp\{2r\sqrt{(n − 1)K}(2^{j} − 1)\}μ(B_{r}(x))

≤ 2^{n} (\frac{R}{r})^{n} exp\{2r\sqrt{(n − 1)K}(2R/r − 1)\}μ(B_{r}(x))

and then the desired estimate follows.

Remark 1.2 Condition (H2) together with Lemma 1.1 gives the following uniform lower bound for balls of a given radius 0 < ϱ < 1:

inf_{x∈M} μ(B_{ϱ}(x)) ≥ v (\frac{ϱ}{2})^{n} exp\{(2ϱ − 4)\sqrt{(n − 1)K}\},

where v is the constant in (H2). Moreover, by Lemma 1.1 the manifold M is compact if and only if μ(M) is finite.

Remark 1.3 A counterpart of the previous remark directly follows from hypothesis (H1); in fact, see e.g. [9, Theorem 3.98], for any x ∈ M

μ(B_{ϱ}(x)) = ω_{n}ϱ^{n} (1 − \frac{s(x)}{6(n + 1)}ϱ^{2} + o(ϱ^{2})) ,

where s(x) is the scalar curvature at x ∈ M, i.e., the trace of the Ricci curvature. This, together with (H1), implies that there are R_{0} > 0, c > 0 such that for any x ∈ M and any 0 < ϱ < R_{0}

μ(B_{ϱ}(x)) ≤ cϱ^{n}.

1.2 Sobolev Spaces and Functions of Bounded Variation

In this subsection we recall the definition and the basic properties of Sobolev spaces and functions of bounded variation on a manifold. We refer to [13] for more information on Sobolev spaces on manifolds, and to [1] for a discussion of BV in the Euclidean setting.

For 1 ≤ p < ∞ and k ∈ N, we denote by H^{k,p}(M) the completion of the space

𝒞_{k}^{p}(M) := {u ∈ 𝒞^{∞}(M) : ∥u∥_{k,p} := ∥u∥_{p} + ∑_{j=1}^{k}(∫_{M} |∇^{j}u|^{p}dμ)^{1/p} < +∞}

with respect to the norm ∥ · ∥_{k,p}.

Given a function u ∈ L^{1}(M), define the variation of u by

|Du|(M) = sup {∫_{M} u div ω dμ : ω ∈ Γ_{c}(T^{∗}M), |ω| ≤ 1} .  (1.3)

A function u ∈ L^{1}(M) has bounded variation, u ∈ BV (M), if and only if |Du|(M) < +∞. Notice that H^{1,1}(M) ⊂ BV (M). A function u ∈ BV (M) defines an element Du ∈ (Γ_{0}(T^{∗}M))′, the dual space of Γ_{0}(T^{∗}M); in fact, it is well defined the map

u ↦ (Du, ω) := − ∫_{M} u div ω dμ, ∀ω ∈ Γ_{c}(T^{∗}M),

and, thanks to condition |Du|(M) < +∞, this map can be extended by continuity to the whole space Γ_{0}(T^{∗}M). This dual space, unlike the Euclidean space, cannot be naturally identified with a vector valued measure space; what is possible to say, is that a BV function u defines, as in the Euclidean case, a finite measure |Du| and a |Du|-measurable section σ_{u} : M → T^{∗}M with |σ_{u}| = 1 almost everywhere and such that the distributional derivative Du of u is given by

(Du, ω) = ∫_{M} ⟨σ_{u}, ω⟩d|Du|, ∀ω ∈ Γ_{0}(T^{∗}M).

Given a measurable set E ⊂ M, we denote by χ_{E} its characteristic function, and we say that E has finite perimeter if Dχ_{E} has bounded variation. We denote the perimeter of E in a Borel set A by

P (E, A) = |Dχ_{E}|(A).  (1.4)

Assumptions (H1) and (H2) are crucial in the context of H^{k,p} and BV spaces. In fact, under assumption (H1), condition (H2) is equivalent to the Sobolev embedding H^{1,p}(M) ⊂ L^{q}(M), 1/q − 1/p = 1/n, BV (M) ⊂ L^{n/(n−1)}(M), see [21] and also [13, Theorem 3.3]. As a consequence of the last embedding, the following isoperimetric inequality holds

min\{μ(E), μ(M \ E)\} ≤ c_{I}P (E, M)^{\frac{n}{n−1}} ,  (1.5)

for every E ⊂ M and for some constant c_{I} > 0 depending only upon n, K, v. If we replace (H1) by Ric ≥ 0, the situation is simpler. In particular, by [5] μ is doubling and then by [4] the isoperimetric inequality and the Sobolev embedding hold and (H2) follows (see also [12, Section 10]).

We recall that, since for every ω ∈ Γ_{c}(T^{∗}M) the map

u ↦ ∫_{M} u div ω dμ

is continuous with respect to the L^{1}(M) topology, then the map

u ↦ |Du|(M)

is L^{1}–lower semi–continuous. Obviously, it is impossible to approximate BV function in norm by smooth functions: what can be done is to get an approximation in variation. This is well-known in the Euclidean setting (see e.g. [1, Theorem 3.9]) and can be adapted to manifolds via a partition of unity argument.

Proposition 1.4 For every u ∈ BV (M) there exists a sequence (f_{j})_{j} ⊂ 𝒞_{c}^{∞}(M) such that f_{j} → u in L^{1}(M) and

|Du|(M) = lim_{j→∞} ∫_{M} |∇f_{j}|dμ.  (1.6)

Proof. Let us fix some notation; given an open set V, for τ > 0 we define the following sets:

V^{τ} = {x ∈ M : dist(x, V) < τ}, V_{τ} = {x ∈ M : dist(x, V^{c}) > τ}.

If M is compact, we may take 𝒞^{∞} instead of 𝒞_{c}^{∞} and we don’t need the following statement. If M is not compact, let us prove the following Claim. For any u ∈ BV (M) and for any ε > 0 there exists a function u_{ε} ∈ BV (M) with compact support in M such that

∥u − u_{ε}∥_{L^{1}(M)} < ε, |Du_{ε}|(M) < |Du|(M) + ε.

In fact, if we fix ε > 0, there exists a relatively compact open set B such that

∫_{M\B} |u|dμ < ε, ∫_{B^{ε}\B} |u|dμ < \frac{ε^{2}}{2} , |Du|(M \ B) < ε;

moreover, we can consider ζ ∈ 𝒞^{∞}(M) with 0 ≤ ζ ≤ 1, ζ = 1 on B, suppζ ⊂ B^{ε} and |∇ζ| ≤ 2/ε, and take u_{ε} = ζu. For such a function we have

∫_{M} |u − u_{ε}|dμ ≤ ∫_{M\B} |u|dμ < ε,

and, if ω ∈ Γ_{c}(T^{∗}M) with |ω| ≤ 1, then

∫_{M} u_{ε} div ωdμ = ∫_{M} u div(ζω)dμ − ∫_{B^{ε}\B} u⟨ω, ∇ζ⟩dμ.

Since ζω ∈ Γ_{c}(T^{∗}M) and |ζω| ≤ 1, we obtain that

∫_{M} u_{ε} div ωdμ ≤ |Du|(M) + ∫_{B^{ε}\B} |u||⟨ω, ∇ζ⟩|dμ ≤ |Du|(M) + ε.

This in particular implies that u_{ε} ∈ BV (M) and

|Du_{ε}|(M) ≤ |Du|(M) + ε

and the claim follows.

Given u ∈ BV (M) with compact support, we consider a finite family of open bounded sets (U_{i})_{i=1,... ,N} with the following properties

1. U_{i} ∩ U_{j} = ∅ for i ≠ j;

2. ∃τ such that for every 0 < η < τ the family U_{i}^{η} covers the support of u and U_{i}^{τ} is contained in a coordinate chart (V_{i}, ψ_{i});

3. |Du|(∂U_{i}) = 0 for all i.

For any fixed ε > 0, we take 0 < η < τ in such a way that

|Du|(M \ U_{η}) < ε, where U_{η} = ⋃_{i=1}^{N} U_{i,η}.

Moreover, for any i = 1, …, N, since u ∘ ψ_{i}^{−1} ∈ BV (ψ_{i}(U_{i}^{η})), by [1, Theorem 3.9] we can find a function f_{i} ∈ 𝒞^{∞}(M) satisfying

∫_{U_{i}^{η}} |u − f_{i}|dμ < \frac{ηε}{2N} , ∫_{U_{i}^{η}} |∇f_{i}|dμ < |Du|(U_{i}^{η}) + \frac{ε}{N} .

If φ_{i} is a partition of unity of suppu subordinated to the open covering U_{i}^{η} with |∇φ_{i}| ≤ C/η, we define

f = ∑_{i=1}^{N} φ_{i}f_{i};

the function f is a smooth function with

f ≡ 0 on M \ ⋃_{i=1}^{N} U_{i}^{η},

that is f ∈ 𝒞_{c}^{∞}(M). Clearly

∫_{M} |u − f|dμ = ∫_{M\suppu} |f|dμ + ∑_{i=1}^{N} ∫_{suppu} φ_{i}|u − f_{i}|dμ ≤ 2 ∑_{i=1}^{N} ∫_{U_{i}^{η}} |u − f_{i}|dμ < ηε.

Moreover, for x ∈ A_{i,η} = U_{i}^{η} \ U_{i,η}, we denote by

I(i) = {j ∈ {1, …, N} : j ≠ i, U_{i}^{η} ∩ U_{j}^{η} ≠ ∅},
I′(x) = {j ∈ {1, …, N} : x ∈ U_{j}^{η}},
I_{i}(x) = I′(x) \ {i} ⊂ I(i).

Using this notation, we have that

∇φ_{i}(x) = − ∑_{j∈I_{i}(x)} ∇φ_{j}(x),

whence

∇f(x) = ∑_{j∈I′(x)} φ_{j}(x)∇f_{j}(x) + ∑_{j∈I_{i}(x)}(f_{j}(x) − f_{i}(x))∇φ_{j}(x)

and then, since suppf \ U_{η} = ⋃_{i=1}^{N} A_{i,η},

∫M|∇f|dμ ≤ ∫_{U_{η}} |∇f|dμ + ∑_{i=1}^{N} ∫_{A_{i,η}} |∇f|dμ

≤ ∑_{i=1}^{N} ∫_{U_{i,η}} |∇f_{i}|dμ + ∑_{i=1}^{N} ∑_{j∈I(i)} ∫_{A_{i,η}} φ_{j}|∇f_{j}|dμ + ∑_{i=1}^{N} ∑_{j∈I(i)} ∫_{A_{i,η}} |f_{j} − f_{i}||∇φ_{j}|dμ

≤ |Du|(U_{η}) + ε + ∑_{i=1}^{N} ∑_{j∈I(i)} ∫_{A_{i,η}∩U_{j}^{η}} |∇f_{j}|dμ + 2\frac{C}{η} ∑_{i=1}^{N} ∑_{j∈I(i)} ∫_{A_{i,η}∩A_{j,η}} |f_{j} − u|dμ

≤ |Du|(U_{η}) + ε + N ∑_{i=1}^{N} (|Du|(A_{i,η}) + \frac{ε}{N} ) + \frac{2CN}{η} ∑_{i=1}^{N} ∫_{A_{i,η}} |u − f_{i}|dμ

≤ |Du|(U_{η}) + 2ε + N^{2}|Du|(M \ U_{η}) + 2CNε.

2 The heat semigroup on M

In this section, after sketching the construction of the heat semigroup in the manifolds we are considering and recalling some of its properties, we derive some pointwise bounds on the heat kernel and its derivatives. We refer to [6] for the missing proofs, unless otherwise stated. Let us consider the Dirichlet form

𝒟(u, v) = ∫_{M} ⟨∇u, ∇v⟩dμ

with domain H^{1,2}(M). It defines the operator Δ_{2}, that on smooth functions coincides with the Laplace–Beltrami operator Δ, defined as follows:

−Δ_{2}u = f ⟺ 𝒟(u, v) = ∫_{M} fvdμ, ∀v ∈ L^{2}(M).

Since Δ_{2} is self–adjoint and nonpositive on L^{2}(M), it generates a strongly continuous, positive, contractive and analytic semigroup (T_{2}(t))_{t≥0} in L^{2}(M), which extrapolates to a positive strongly continuous contractive semigroup (T_{p}(t))_{t≥0} on L^{p}(M), 1 ≤ p < +∞, which is even analytic for p > 1. The generator of (T_{p}(t))_{t≥0} is denoted by Δ_{p}, and the semigroup gives the solution u(t, x) = T (t)f(x) of the Cauchy problem

\{ \begin{matrix}∂_{t}u = Δ_{p}u & t > 0, x ∈ M \\ u(0, x) = f(x) & x ∈ M. \end{matrix}

The case p = 1 is more delicate, and requires (H1). Analyticity was first proved by N. Varopoulos [21] without calculating the angle of sectoriality. With stronger hypotheses on the curvature (see Theorem 2.1 below) it was E.B. Davies [7] who obtained the angle \frac{π}{2} . We state the following theorem.

Theorem 2.1 If M be a complete Riemannian manifold satisfying (H1), then the heat semigroup (T (t))_{t≥0} is analytic on L^{1}(M). In addition, if the curvature of M is positive outside some compact subset of M, then it even holds that (T (t))_{t≥0} is bounded analytic on L^{1}(M) with angle \frac{π}{2} .

It is also important to point out that the heat semigroup has the following integral representation; for any u ∈ L^{1}(M),

T (t)u(x) = ∫_{M} p(x, y, t)u(y)dμ(y),

where the heat kernel 0 < p ∈ 𝒞^{∞}(M × M × (0, +∞)) verifies in our setting

∫_{M} p(x, y, t)dμ(y) = 1

for any x ∈ M, t > 0.

2.1 Gaussian estimates for the heat kernel in the non-compact case

Let us discuss some further properties of the heat kernel in the case where M is a noncompact manifold. We start with the following definition (see Grigor’yan, [10, Definition 1.1]).

Definition 2.2 A Λ–isoperimetric inequality is valid for a region Ω ⊂ M if for every subregion D ⊂ Ω the inequality

λ_{1}(D) ≥ Λ(μ(D))  (2.1)

holds, where λ_{1}(D) is the first Dirichlet eigenvalue of D and Λ is a positive decreasing function.

The validity of isoperimetric inequalities on Riemannian manifolds is closely related to the validity of Gaussian estimates for the heat kernel. More precisely, Grigor’yan proves in [10, Theorem 5.2] the following estimate.

Theorem 2.3 Suppose that in any ball B_{R}(x) of a fixed radius R > 0 the Λ–isoperimetric inequality holds with the function Λ = Λ_{x,R} defined as follows

Λ_{x,R}(v) = a(x, R)v^{−ν},

where a(x, R) > 0, ν > 0; then for all x, y ∈ M, for any t > t_{0} > 0

p(x, y, t) ≤ \frac{c_{ν} exp\{λ(t_{0} − t)\}}{min\{t_{0}, R^{2}\}^{1/ν}(a(x, R)a(y, R))^{1/2ν}} ·

·(1 + \frac{d(x, y)^{2}}{t} )^{1+1/ν} exp \{−\frac{d(x, y)^{2}}{4t} \},

(2.2)

where λ = λ_{1}(M) is the spectral radius of the manifold M, i.e.,

λ_{1}(M) := inf_{Ω⊂⊂M} λ_{1}(Ω).

In the same paper, Grigor’yan proves also that an isoperimetric inequality implies the Λ–isoperimetric inequality. More precisely, in [10, Proposition 2.4] it is proved that if for any D ⊂ Ω the inequality

Area(∂D) ≥ g(μ(D))  (2.3)

holds, for a function g : (0, +∞) → (0, +∞) such that g(ξ)/ξ is decreasing, then the Λ–isoperimetric inequality holds for the region Ω with

Λ(ξ) = \frac{1}{4} (\frac{g(ξ)}{ξ})^{2} .

Starting from these considerations and since the isoperimetric inequality (1.5) holds, we can state the following proposition.

Proposition 2.4 Let M be a non-compact Riemannian manifold satisfying (H1) and (H2); then for any 0 < t ≤ 1 and x, y ∈ M, the following Gaussian estimate holds

p(x, y, t) ≤ \frac{c_{G}}{t^{n/2}} (1 + \frac{d(x, y)^{2}}{t} )^{1+n/2} exp \{−\frac{d(x, y)^{2}}{4t} \} ,  (2.4)

where c_{G} = c(n, K, c_{I}) is a constant depending only on the dimension n of the manifold M, the bound K of the Ricci curvature and the isoperimetric constant c_{I}.

Proof. Since μ(M) = +∞, we can apply (2.2) with an arbitrary R > 1 and then the isoperimetric inequality (1.5) reduces, for bounded sets E ⊂ M, to

μ(E) ≤ c_{I}P (E, M)^{n/n−1}.

This last inequality is exactly (2.3) with

g(ξ) = (\frac{ξ}{c_{I}} )^{1−1/n} .

Then B_{R}(x) admits a Λ–isoperimetric inequality with

Λ(ξ) = \frac{ξ^{−2/n}}{4c_{I}^{2−2/n}};

hence, (2.4) follows from (2.2) with t ≤ 1, t_{0} = t/2, ν = 2/n, a(x, R) = 1/(4c_{I}^{2−2/n}) =: a and

c_{G} = \frac{2^{n/2}c_{n} exp \{\frac{λ}{2}\}}{a^{n/2}}.

We also need estimates on the time and space derivatives of the heat kernel. Starting from time derivatives, let us quote the following result of Grigor’yan’s. In [11, Theorem 3.1] it is proved that on a complete manifold M, if R > 0 is a fixed radius, for the k-th time derivative of the heat kernel there holds

|\frac{∂^{k}}{∂t^{k}} p(x, y, t)| ≤ \frac{c_{ν,k}}{(a(x, R)a(y, R))^{1/2ν}} (1 + \frac{d(x, y)^{2}}{t} )^{k+1+1/ν} \frac{exp\{−d(x, y)^{2}/4t\}}{t^{k} min(t, R^{2})^{1/ν}}. (2.5)

Arguing in the same way as in Proposition 2.4, it is possible to deduce the following result.

Proposition 2.5 Let M be a non-compact Riemannian manifold satisfying (H1) and (H2); then for any 0 < t ≤ 1 and x, y ∈ M, the following Gaussian estimate for the time derivative of the heat kernel holds

|\frac{∂}{∂t}p(x, y, t)| ≤ \frac{c_{G}}{t^{1+n/2}} (1 + \frac{d(x, y)^{2}}{t} )^{2+n/2} exp \{−\frac{d(x, y)^{2}}{4t} \} ,  (2.6)

where c_{G} = c(n, K, c_{I}) is a constant depending on the dimension n of the manifold M, the bound K of the Ricci curvature and on the isoperimetric constant c_{I}.

Let us come to Gaussian estimates on the space derivative of the heat kernel; P. Li and S.T. Yau [16, Theorem 1.4] (see also the Introduction of J. Li [15]) prove that on a manifold M satisfying (H1), the following estimate holds

\frac{|∇p(x, y, t)|^{2}}{p(x, y, t)^{2}} − γ \frac{1}{p(x, y, t)} \frac{∂}{∂t}p(x, y, t) ≤ \frac{nKγ^{2}}{2(γ − 1)} + \frac{nγ^{2}}{t} , ∀γ > 1.  (2.7)

Starting from this estimate, we can state the following proposition.

Proposition 2.6 Let M be a non-compact Riemannian manifold satisfying (H1) and (H2); then for any 0 < t ≤ 1 and x, y ∈ M, the following Gaussian estimate for the space derivative of the heat kernel holds:

\begin{aligned}|∇p(x, y, t)| ≤ & \frac{c_{G}}{t^{n/2}} (1 + \frac{d(x, y)^{2}}{t} )^{1+n/2}· \\ & ·exp \{−\frac{d(x, y)^{2}}{4t} \} \sqrt{\frac{1}{t} (4n + 1 + \frac{d(x, y)^{2}}{t} ) + 2nK}, \end{aligned}  (2.8)

where c_{G} = c(n, K, c_{I}) is a constant depending on the dimension n of the manifold M, the bound K of the Ricci curvature and on the isoperimetric contant c_{I}. Moreover, there exists c > 0 such that for any y ∈ M

∫_{M} |∇p(x, y, t)|dμ(x) ≤ \frac{c}{\sqrt{t}}.  (2.9)

Proof. The proof of (2.8) simply follows from (2.4), (2.6), and (2.7) with γ = 2. Concerning (2.9), we notice that the function

f_{t}(s) = (1 + s^{2})^{1+n/2} exp \{−\frac{s^{2}}{8} \} \sqrt{4n + 1 + s^{2} + 2nKt}

is bounded on (0, +∞) for t ≤ 1, say

f_{t}(s) ≤ c, ∀s > 0, t ≤ 1.

Therefore, it suffices to prove that there exists c > 0 such that for any y ∈ M

∫_{M} exp \{−\frac{d_{t}(x, y)^{2}}{8} \} dμ_{t}(x) ≤ c, ∀y ∈ M,

where we have introduced the metric tensor g_{t} = t^{−1}g, with associated distance d_{t} = t^{−1/2}d and measure dμ_{t} = t^{−n/2}dμ. For the measure μ_{t}, since B_{R}^{t}(y) = {x : d_{t}(x, y) < R} = B_{R\sqrt{t}}(y), by Lemma 1.1 the inequality

\frac{μ_{t}(B_{R}^{t}(x))}{μ_{t}(B_{r}^{t}(x))} ≤ 2^{n} (\frac{R}{r})^{n} (\frac{e^{4R\sqrt{t}}}{e^{2r\sqrt{t}}} )^{\sqrt{(n−1)K}}  (2.10)

holds for any y ∈ M and 0 < r < R. Then, setting B = B_{R_{0}}^{t}(y), with R_{0} the radius coming from Remark 1.3, we can write

M = B ∪ ⋃_{j=0}^{∞} A_{j}

where

A_{j} = B_{2^{j+1}R_{0}}^{t}(y) \ B_{2^{j}R_{0}}^{t}(y).

For x ∈ A_{j} we have that 2^{j}R_{0} ≤ d_{t}(x, y) < 2^{j+1}R_{0}, and then

exp \{−\frac{d_{t}(x, y)^{2}}{8} \} ≤ exp \{−\frac{2^{2j}R_{0}^{2}}{8} \};

as a consequence, using (2.10) we obtain that

∫A_{j} exp \{−\frac{d_{t}(x, y)^{2}}{8} \} dμ_{t}(x)

≤ μ_{t}(A_{j}) exp \{−\frac{2^{2j}R_{0}^{2}}{8} \}

≤ μ_{t}(B_{2^{j+1}R_{0}}^{t}(y)) exp \{−\frac{2^{2j}R_{0}^{2}}{8} \}

≤ μ_{t}(B) exp \{−\frac{2^{2j}R_{0}^{2}}{8} \} 2^{n(j+2)} exp \{2^{j+3}R_{0}\sqrt{(n − 1)Kt}\}

= μ_{t}(B) exp \{−\frac{4^{j}R_{0}^{2}}{8} + 2^{j+3}R_{0}\sqrt{(n − 1)Kt} + n(j + 2) ln 2\}

= μ_{t}(B)a_{j},

where a_{j} = a_{j}(n, K) are numbers depending only upon the indicated parameters. A direct calculation gives that

∑_{j=0}^{∞} a_{j} < ∞,

and then we have

∫M exp \{−\frac{d_{t}(x, y)^{2}}{8} \} dμ_{t}(x)

= ∫_{B} exp \{−\frac{d_{t}(x, y)^{2}}{8} \} dμ_{t}(x) + ∑_{j=0}^{∞} ∫_{A_{j}} exp \{−\frac{d_{t}(x, y)^{2}}{8} \} dμ(x)

≤ μ_{t}(B) (1 + ∑_{j=0}^{∞} a_{j}) ,

and then the assertion then follows since, for t ≤ 1 and Remark 1.3,

μ_{t}(B) = t^{−n/2}μ(B_{R_{0}\sqrt{t}}(y)) ≤ cR_{0}^{n}.

2.2 Regularity of heat semigroup

In this Subsection we investigate some regularity properties of the heat semigroup.

Theorem 2.7 Under hypotheses (H1), (H2)

D(Δ_{1}) ⊂ H^{1,1}(M),  (2.11)

ΔT (t)u ∈ H^{1,1}(M), ∀t > 0, u ∈ L^{1}(M).  (2.12)

Proof. If M is a compact manifold the proof is achieved either by duality relying on [20, Théorème 4.2], or by the same direct argument as in [3], which we sketch below. Assume n ≥ 3 at first, let T_{k}(y) = (−k) ∨ (y ∧ k) be the truncation function, and let us denote by {u < k} the set {x ∈ M : u(x) < k}. Multiplying the equation u − Δu = f by T_{k}(u) and using the divergence theorem, for f ∈ L^{1}(M) we get

∫_{{|u|<k}} |∇T_{k}(u)|^{2}dμ ≤ k∥f∥_{L^{1}(M)}.  (2.13)

By the Sobolev embedding we know that

(∫_{M} |T_{k}(u)|^{2^{∗}}dμ)^{2/2^{∗}} = (∫_{{|u|<k}} |u|^{2^{∗}}dμ + k^{2^{∗}}μ({|u| ≥ k}))^{2/2^{∗}}
≤ ∫_{{|u|<k}} |∇T_{k}(u)|^{2}dμ,

(where 2^{∗} = 2n/(n − 2)) whence, using (2.13), we deduce

μ(\{|u| ≥ k\}) ≤ \frac{∥f∥_{1}}{k^{n/(n−2)}} ,  (2.14)

which is true for every k. Let us come to the estimate of the gradient. Using (2.14) and (2.13) we get

μ(\{|∇u| ≥ λ\}) ≤ μ(\{|u| ≥ k\}) + \frac{1}{λ^{2}} ∫_{\{|u|<k\}∩\{|∇u|≥λ\}} |∇u|^{2}dμ

≤ \frac{∥f∥_{1}}{k^{n/(n−2)}} + \frac{k}{λ}∥f∥_{1},

which is true for every k > 0, λ > 0. Minimising over k we find a constant C such that μ({|∇u| ≥ λ}) ≤ Cλ^{n/(n−1)} for every λ > 0, and therefore |∇u| ∈ L^{p}(M) for all p < n/(n − 1). If n = 2, the same argument can be used, with an arbitrary exponent 1 < q < ∞ in place of 2^{∗}.

If M is non-compact, we show that there is C > 0 such that if u − Δu = f ∈ L^{1}(M), then ∥∇u∥_{1} ≤ C∥f∥_{1}. From the representation of the resolvent operator

R(λ, Δ_{1})f = ∫_{0}^{∞} e^{−λt}T (t)fdt

we see that it is sufficient to prove that

∫_{0}^{∞} e^{−λt}∥∇T (t)f∥_{1}dt ≤ C∥f∥_{1}.  (2.15)

From (2.9) we deduce for t ≤ 1

∥∇T (t)f∥_{1} = ∫_{M} | ∫_{M} ∇_{x}p(x, y, t)f(y)dμ(y)|dμ(x)

≤ ∫_{M} ∫_{M} |∇_{x}p(x, y, t)| |f(y)|dμ(y)dμ(x)

= ∫_{M} |f(y)| ∫_{M} |∇_{x}p(x, y, t)|dμ(x)dμ(y) ≤ \frac{c}{\sqrt{t}}∥f∥_{1}

whereas for t > 1 we have

∥∇T (t)f∥_{1} ≤ ∥∇T (1)T (t − 1)f∥_{1} ≤ c∥T (t − 1)f∥_{1} ≤ c∥f∥_{1}

by contractivity. Summing up,

∥ ∫_{0}^{∞} e^{−λt}∇T (t)fdt∥_{1} ≤ c ∫_{0}^{1} \frac{e^{−λt}}{\sqrt{t}} ∥f∥_{1}dt + c ∫_{1}^{∞} e^{−λt}dt ≤ C∥f∥_{1},

where the last constant C depends only upon the constants in the preceding inequalities.

Let us come to (2.12). Since, as a consequence of analyticity, we know that T (t)u ∈ D(Δ_{1}) for every t > 0, we may write

ΔT (t)u = ΔT (t/2)T (t/2)u = T (t/2)ΔT (t/2)u.

As noticed before, T (t/2)u ∈ D(Δ_{1}), so that ΔT (t/2)u ∈ L^{1}(M) and T (t/2)ΔT (t/2)u ∈ D(Δ_{1}) ⊂ H^{1,1}(M).

Remark 2.8 It will be important in the sequel to notice that Proposition 1.4 can be stated by saying that BV functions can be approximated in variation by functions in the domain of the Laplace–Beltrami operator, i.e., for every u ∈ BV (M) there exists a sequence (f_{j}) contained in D(Δ_{1}) such that f_{j} → u in L^{1}(M) and (1.6) holds.

3 A characterisation of total variation via heat semigroup

In this section we prove that, given a function u ∈ L^{1}(M), the following limit exists

lim_{t→0} ∫_{M} |∇T (t)u|dμ.  (3.1)

and it is equal to the total variation of u defined by (1.3). We need an elementary calculus lemma.

Lemma 3.1 Let f ∈ 𝒞^{1}(0, +∞) be a positive function satisfying the condition

f′(t) ≤ Kf(t), ∀t > 0,

for a positive constant K > 0. Then, the function t ↦ e^{−Kt}f(t) is non–increasing; in particular, the limit

lim_{t→0} f(t)

exists, either finite or infinite.

Proof. From the condition

f′(t) ≤ Kf(t)

and the fact that f(t) > 0, we can deduce that

\frac{f′(t)}{f(t)} ≤ K.

Then, fixed 0 < s < t and integrating the previous condition from s to t, we get

ln \frac{f(t)}{f(s)} ≤ K(t − s),

and then

e^{−Kt}f(t) ≤ f(s)e^{−Ks},

whence the monotonicity, and the thesis follows.

We are now in a position to prove the following result.

Theorem 3.2 Let M be a Riemannian manifold satisfying (H1), (H2). Then, for every u ∈ L^{1}(M), the function

t ↦ e^{−Kt} ∫_{M} |∇T (t)u|dμ,  (3.2)

where K is the constant in (H1), is non–increasing; in particular, the limit

lim_{t→0} ∫_{M} |∇T (t)u(x)|dμ,

exists, either finite or infinite.

Proof. Let u ∈ L^{1}(M) be a given function; we are going to prove that the function f : (0, +∞) → R defined by

f(t) = ∥∇T (t)u∥_{L^{1}}

is differentiable and satisfies f′ ≤ Kf for every t > 0. We notice that for every t > 0 the function T (t)u is analytic, and then the equality

∂_{t}|∇T (t)u| = \frac{1}{|∇T (t)u|}⟨∇ΔT (t)u, ∇T (t)u⟩

holds μ–a.e. in M. Since

|\frac{1}{|∇T (t)u|}⟨∇ΔT (t)u, ∇T (t)u⟩| ≤ |∇ΔT (t)u|,

by (2.12), we can conclude that f is differentiable and we can differentiate under the integral sign. Using the Bochner–Lichnerowitz–Weitzenböck formula (1.1), we obtain

f′(t) = \frac{d}{dt} ∫_{M} |∇T (t)u|dμ

= \frac{d}{dt} ∫_{M} ⟨∇T (t)u, ∇T (t)u⟩^{1/2}dμ

= \frac{1}{2} ∫_{M} \frac{1}{|∇T (t)u|}∂_{t}⟨∇T (t)u, ∇T (t)u⟩dμ

= ∫_{M} \frac{1}{|∇T (t)u|}⟨∂_{t}∇T (t), ∇T (t)u⟩dμ

= ∫_{M} \frac{1}{|∇T (t)u|}⟨∇ΔT (t)u, ∇T (t)u⟩dμ

= \frac{1}{2} ∫_{M} \frac{Δ|∇T (t)u|^{2}}{|∇T (t)u|} dμ − ∫_{M} \frac{|Hess T (t)u|^{2}}{|∇T (t)u|} dμ − ∫_{M} \frac{Ric(∇T (t)u, ∇T (t)u)}{|∇T (t)u|}dμ.

Let us now consider the following equality:

∫_{M} \frac{Δ|∇T (t)u|^{2}}{|∇T (t)u|} dμ = ∫_{M} \frac{1}{|∇T (t)u|}Δ|∇T (t)u|^{2}dμ

= − ∫_{M} ⟨∇\frac{1}{|∇T (t)u|}, ∇|∇T (t)u|^{2}⟩dμ

= \frac{1}{2} ∫_{M} \frac{1}{|∇T (t)u|^{3}} ⟨∇|∇T (t)u|^{2}, ∇|∇T (t)u|^{2}⟩dμ

= \frac{1}{2} ∫_{M} \frac{1}{|∇T (t)u|^{3}} |∇|∇T (t)u|^{2}|^{2} dμ.

Taking into account that

|∇|∇T (t)u|^{2}| = sup_{|X|≤1} |∇_{X}|∇T (t)u|^{2}|

= 2 sup_{|X|≤1} ⟨∇_{X}∇T (t)u, ∇T (t)u⟩

≤ 2 sup_{|X|≤1} |∇_{X}∇T (t)u| · |∇T (t)u|

≤ 2|Hess T (t)u| · |∇T (t)u|,

we obtain

∫_{M} \frac{Δ|∇T (t)u|^{2}}{|∇T (t)u|} dμ = −\frac{1}{2} ∫_{M} \frac{1}{|∇T (t)u|^{3}} ⟨∇|∇T (t)u|^{2}, ∇|∇T (t)u|^{2}⟩dμ

= −\frac{1}{2} ∫_{M} \frac{1}{|∇T (t)u|^{3}} |∇|∇T (t)u|^{2}|^{2} dμ

≤ 2 ∫_{M} \frac{1}{|∇T (t)u|}|Hess T (t)u|^{2}dμ.

In conclusion, taking into account that Ric(∇T (t)u, ∇T (t)u) ≥ −K|∇T (t)u|^{2}, we have obtained that

f′(t) ≤ K ∫_{M} |∇T (t)u|dμ = Kf(t);

applying Lemma 3.1, the conclusion follows.

Remark 3.3 Notice that, since T (t)u → u in L^{1} as t → 0, by lower semicontinuity we have that

|Du|(M) ≤ lim inf_{t→0} ∫_{M} |∇T (t)u|dμ;

then, if u ∈ L^{1}(M) but not in BV (M), the equality

|Du|(M) = lim_{t→0} ∫_{M} |∇T (t)u|dμ  (3.3)

holds trivially, being both sides equal to +∞. Formula (3.3) holds true also for any u ∈ BV, but the proof of it requires some additional results that we are going to prove.

We can now prove that the limit (3.1) coincides with the total variation of u ∈ BV (M).

Theorem 3.4 Let M be a Riemannian manifold satisfying (H1), (H2). Then, for every u ∈ L^{1}(M), the following formula holds

lim_{t→0} ∫_{M} |∇T (t)u(x)|dμ(x) = |Du|(M).  (3.4)

Proof. Thanks to Remark 3.3, we can take u ∈ BV (M). By Remark 2.8, there exists a sequence f_{j} ∈ D(Δ_{1}) converging to u in L^{1}(M) such that

lim_{j→+∞} ∫_{M} |∇f_{j}|dμ = |Du|(M).

Since (f_{j}) ⊂ D(Δ_{1}), we have T (t)f_{j} → f_{j} as t → 0 in the graph norm, and then, by the inclusion D(Δ_{1}) ⊂ H^{1,1}(M), also in the H^{1,1}(M) norm. Therefore, for every ε > 0 we can find a sequence (t_{j}) converging to 0 such that ∥∇T (t_{j})f_{j} − ∇f_{j}∥_{1} < ε, whence

∫_{M} |∇T (t_{j})f_{j}|dμ < ∫_{M} |∇f_{j}|dμ + ε.

Moreover, for any f ∈ D(Δ_{1}) and t > 0, from Theorem 3.2 we get

e^{−Kt} ∫_{M} |∇T (t)f|dμ ≤ ∫_{M} |∇f|dμ;  (3.5)

then, defining g_{j} = T (t_{j})f_{j}, the sequence (g_{j}) converges to u in L^{1}(M), and also T (t)g_{j} → T (t)u in L^{1}(M) as j → +∞ for every t > 0. From the lower semicontinuity of the total variation and (3.5) it follows that

|Du|(M) ≤ lim inf_{t→0} e^{−Kt} ∫_{M} |∇T (t)u|dμ

≤ lim sup_{t→0} (lim inf_{j→+∞} e^{−Kt} ∫_{M} |∇T (t)g_{j}|dμ)

≤ lim sup_{j→+∞} ∫_{M} |∇g_{j}|dμ = lim sup_{j→+∞} ∫_{M} |∇T (t_{j})f_{j}|dμ

≤ lim sup_{j→+∞} ∫_{M} |∇f_{j}|dμ + ε = |Du|(M) + ε.

Since ε was arbitrary, the proof is complete.

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