245C Take-home final Due date: June 10 You may use any resources available to you, including textbooks, the

internet, or other students, in order to get hints or other inspiration; however, the work you hand in should be your own (i.e. not an identical copy of someone else’s work). If you cite any result from a textbook other than Folland, please state the result in full. Throughout the final, two functions are considered equal if they agree almost everywhere.

• Q1. Let R^{n} be a Euclidean space endowed with Lebesgue measure dm. Let 1 ≤ p ≤ ∞, and let K : R^{n} → R^{+} be a measurable function. Show that the following three statements are equivalent:

• (a) K is absolutely integrable (i.e. K ∈ L^{1}(R^{n})).

• (b) For every f ∈ L^{p}(R^{n}), the convolution integral f∗K(x) := ∫R^{n} f(x− y)K(y) dy is absolutely convergent for almost every x, and f ∗ K lies in L^{p}(R^{n}).

• (c) For every f ∈ L^{p}(R^{n}), the convolution integral f∗K(x) := ∫R^{n} f(x− y)K(y) dy is absolutely convergent for almost every x, and the map from f to f ∗ K is a bounded map from L^{p}(R^{n}) to L^{p}(R^{n}).

• Q2. Let 1 < p < ∞. For any measurable function f : R^{n} → R^{∗} (where R^{∗} is the extended reals), let ∥f∥_{L_{∗}^{p,∞}(R^{n})} be the quantity

∥f∥_{L_{∗}^{p,∞}(R^{n})} := sup{m(E)^{−1/p′} ∫_{E} |f(x)| dx : E ⊂ R^{n} measurable, 0 < m(E) < +∞}.

Here p′ is the dual exponent of p, defined by 1/p + 1/p′ = 1.

• (a) Show that ∥f∥_{L_{∗}^{p,∞}(R^{n})} is finite if and only if f lies in weak L^{p}, and furthermore there exist constants C_{p}, c_{p} > 0 such that

c_{p}∥f∥_{L^{p,∞}(R^{n})} ≤ ∥f∥_{L_{∗}^{p,∞}(R^{n})} ≤ C_{p}∥f∥_{L^{p,∞}(R^{n})}

where ∥f∥_{L^{p,∞}(R^{n})} (which is called [f]_{p} in Folland) is the weak L^{p} “norm” of f:

∥f∥_{L^{p,∞}(R^{n})} := sup{λm({x ∈ R^{n} : |f(x)| > λ})^{1/p} : λ > 0}.

• (b) Show that if we give weak L^{p} the L_{∗}^{p,∞}(R^{n}) norm, and we identify any two functions in weak L^{p}(R^{n}) which agree almost everywhere, then weak L^{p} becomes a Banach space.

• Q3. Let (X, B_{∗}, dμ) be a probability space (thus X is a set, B_{∗} is a σalgebra on X, and dμ is a positive measure on B_{∗} such that μ(X) = 1). Let B be a sub-σ-algebra of B_{∗}, thus (X, B, dμ) is also a probability space. For any 1 ≤ p ≤ ∞, let L^{p}(B) be the space real-valued of Bmeasurable, p^{th}-power integrable functions, thus L^{p}(B) is a subset of L^{p}(B_{∗}). (We of course identify two functions in L^{p} if they agree almost everywhere.)

• (a) Show that L^{2}(B) is a closed subspace of the real Hilbert space L^{2}(B_{∗}). In particular, we have an orthogonal projection map f ↦ E(f|B) from L^{2}(B_{∗}) to L^{2}(B). The expression E(f|B) is known as the conditional expectation of f with respect to the σ-algebra B.

• (b) Show that if f ∈ L^{2}(B_{∗}) is non-negative, then E(f|B) is also (almost everywhere) non-negative. (Hint: E(f|B) is the closest element of L^{2}(B) to f). Conclude that conditional expectation is a contraction on L^{∞}(B_{∗}) (viewed as a metric space).

• (c) Show that for any 1 ≤ p ≤ ∞, there is a unique way to extend the conditional expectation operator to be a map from L^{p}(B_{∗}) to L^{p}(B) which is still a contraction, thus ∥E(f|B)∥_{L^{p}} ≤ ∥f∥_{L^{p}} for all f ∈ L^{p}(B_{∗}). (Hint: do this for p = 1 first, and then use the complex interpolation method.)

• Q4. Let μ_{1}, μ_{2}, … be a sequence of positive Radon measures on R^{n}, such that sup_{j} μ_{j}(R^{n}) < ∞ (thus the μ_{j} are uniformly finite). Show that there exists a subsequence μ_{j_{1}}, μ_{j_{2}}, … of this sequence which is weakly convergent to another positive Radon measure μ, thus lim_{k→∞} ∫ fdμ_{j_{k}} = ∫ fdμ for all f ∈ C_{0}(R^{n}). (Hint: first show that the space C_{0}(R^{n}) is separable, for instance using the Weierstrass approximation theorem. Then use an Arzela-Ascoli diagonalization argument (or some form of the Banach-Alaoglu theorem)).

• Q5. Let λ ∈ (C_{0}^{∞}(R^{n}))^{∗} be a distribution, and suppose that all sufficiently high order derivatives of λ vanish, thus there exist an integer N > 0 such that

(\frac{∂}{∂x_{1}} )^{α_{1}} … (\frac{∂}{∂x_{n}})^{α_{n}}λ = 0

whenever α_{1}, …, α_{n} ≥ 0 are such that α_{1}+…+α_{n} ≥ N. Show that λ is a polynomial, i.e. there exists a polynomial P : R^{n} → R of n variables such that λ(f) = ∫R^{n} f(x)P (x) dx for all f ∈ C_{0}^{∞}(R^{n}). (Hint: as a warm up, you may try the case n = 1 and α_{1} = 1.)

• Q6. Let λ ∈ (C^{∞}(R))^{∗} be a compactly supported distribution, supported in some open interval I. Show that λ agrees with a high order (distributional) derivative of a continuous function on I. More precisely, there exists a continuous function f ∈ C(R) and integers α ≥ 0 such that the distribution

λ − (\frac{d}{dx})^{α}f

is supported outside of I, where of course we use the distributional derivative. (Hint: first show that there exists an integer k ≥ 0 such that λ extends to be a continuous linear functional on C^{k}(R). Use this and the Riesz representation theorem to show that λ is a high order derivative of a Radon measure. Then show that this measure is in turn the distributional derivative of a continuous function.)