The Subject is on Semigroup, the Ornstein Uhlenbeck Operators.

Part -1:

Exercise 1. Prove that for every k ∈ N, k ≥ 3, C_b^k(R^d) is dense in L^P(R^d,γd) and that T(t) ∈ ζ(C_b^kt(R^d)) for every t > 0.

Exercise 2. Let {h_j : j ∈ N} be any orthonormal basis of H contained in R_γ(X*). Prove that the set Σ of the cylindrical functions of the type f(x) = φ(h^{^}₁(x).......h^{^}_d(x)) with φ ∈ C_b²(R^d) with some d ∈ N, is dense in LP(X, γ) and in W^2,P(X, y) for every p ∈ [1, +∞).

Exercise 3.

(i) With the help of Proposition 10.1.2, show that if f ∈ W^1,P(X, y) with p ∈ [1, +∞) is such that ∇_Hf = 0 a.e., then f is a.e. constant.

(ii) Use point (i) to show that for every p ∈ [1, +∞) the kernel of L_P consists of the constant functions.

(HINT: First of all, prove that T(t)f = f for all f ∈ D(L_P) such that L_pf = 0 and then pass to the limit as t →+∞ in (12.1.3))

Part -2:

Exercise 1. Show that for every p ≥ W^1,p (R,y₁) is not contained in L^P+ε(R,y₁) for any ε > O.

Exercise 2. Prove that for every f ∈ W^1,P(X,y) the sequence converges to |f| in W^1,P(X, y).

Exercise 3. Prove that for every p > 1 and f ∈ D(L_p), holds.

Hint: for every f ∈ Σ and ε > 0, apply formula (13.2.5) with g = f(f² + ε)^1-p/2 and then let ε → 0.

So in total there are six exercises & you have all the material,

Do with semigroups and is about Gaussian measures, Sobolov spaces etc etc. But still i will give you the website where you can find all the lectures.

http://dmi.unife.it/it/ricerca-dmi/seminari/isem19/lectures/lecture-14

I hope for a good explanation, because i want to understand it.

Understanding is the main point. So write the names of the theorems you are using to prove everything