The Lp theory of heat semigroups

Our goal, in this section, is to define, for 1 ≤ p ≤ +∞, P_t on L^p := L^p(X,μ). This may be done in a natural way by using the Riesz-Thorin interpolation theorem that we recall below.

Theorem (Riesz-Thorin interpolation theorem)

Let 1 ≤ p₀, p₁, q₀, q₁ ≤ ∞, and θ ∈ (0,1). Define 1 ≤ p,q ≤ ∞ by

1/p = (1-θ)/p₀ + θ/p₁,     1/q = (1-θ)/q₀ + θ/q₁.

If T is a linear map such that

T : L^p₀ → L^q₀, ||T||_L^p0→L^q0 = M₀

T : L^p₁ → L^q₁, ||T||_L^p1→L^q1 = M₁,

then, for every f ∈ L^p₀ ∩ L^p₁,

||Tf||_q ≤ M₀^1-θM₁^θ ||f||_p.

Hence T extends uniquely as a bounded map from L^p to L^q with

||T||_L^p→L^q ≤ M₀^1-θM₁^θ.

Remark The statement that T is a linear map such that

T : L^p₀ → L^q₀, ||T||_L^p0→L^q0 = M₀

T : L^p₁ → L^q₁, ||T||_L^p1→L^q1 = M₁

means that there exists a map T : L^p₀ ∩ L^p₁ → L^q₀ ∩ L^q₁ with

sup_{f ∈ L^p0 ∩ L^p1 , ||f||p0 ≤ 1} ||Tf||_q0 = M₀

and

sup_{f ∈ L^p0 ∩ L^p1 , ||f||p1 ≤ 1} ||Tf||_q1 = M₁.

In such a case, T can be uniquely extended to bounded linear maps T₀ : L^p₀ → L^q₀ , T₁ : L^p₁ → L^q₁. With a slight abuse of notation, these two maps are both denoted by T in the theorem.

Remark If f ∈ L^p₀ ∩ L^p₁ and p is defined by

1/p = (1-θ)/p₀ + θ/p₁,

then by Hölder’s inequality, f ∈ L^p and

||f||_p ≤ ||f||_p0^1-θ ||f||_p1^θ.

We now are in position to state the following theorem:

Theorem Let (P_t)_t≥0 be a strongly continuous self-adjoint contraction Markovian semigroup on L²(X,μ). The space L¹ ∩ L^∞ is invariant under P_t and P_t may be extended from L¹ ∩ L^∞ to a contraction semigroup (P_t^(p))_t≥0 on L^p for all 1 ≤ p ≤ ∞: For f ∈ L^p,

||P_tf||_L^p ≤ ||f||_L^p.

These semigroups are consistent in the sense that for f ∈ L^p ∩ L^q,

P_t^(p)f = P_t^(q)f.

Exercise Show that if f ∈ L^p and g ∈ L^q with 1/p + 1/q = 1 then,

∫_ℝⁿ f P_t^(q) g dμ = ∫_ℝⁿ g P_t^(p) f dμ.

Exercise 

1. Show that for each f ∈ L¹, the L¹-valued map t → P_t⁽¹⁾f is continuous.
2. Show that for each f ∈ L^p, 1 < p < 2, the L^p-valued map t → P_t^(p)f is continuous.
3. Finally, by using the reflexivity of L^p, show that for each f ∈ L^p and every p ≥ 1, the L^p-valued map t → P_t^(p)f is continuous.

We mention, that in general, the L^∞ valued map t → P_t^(∞)f is not continuous.

Diffusion operators as generators of Dirichlet forms

Consider a diffusion operator

L = ∑_i,j=1ⁿ σ_ij(x) ∂²/∂x_i∂x_j + ∑_i=1ⁿ b_i(x) ∂/∂x_i,

where b_i and σ_ij are continuous functions on ℝⁿ and for every x ∈ ℝⁿ, the matrix (σ_ij(x))_{1 ≤ i,j ≤ n} is a symmetric and non-negative matrix.

Assume that there is Borel measure μ on ℝⁿ which is equivalent to the Lebesgue measure and that symmetrizes L in the sense that for every smooth and compactly supported functions f, g : ℝⁿ → ℝ,

∫_ℝⁿ gLf dμ = ∫_ℝⁿ fLg dμ.

For instance, if one can write

Lf = -div(a ∇f),

where a is a smooth field of positive and symmetric matrices, then the Lebesgue measure symetrizes L. From a previous lemma  the quadratic form

ℰ(f,g) = -∫_ℝⁿ g Lf dμ, f,g ∈ C_c^∞(ℝⁿ)

is closable. Let ℰ̄ denotes its closure in L²(ℝⁿ,μ).

Proposition The quadratic form ℰ̄ is a Dirichlet form.