Real Analysis. Instructor: Dmitry Ryabogin Assignment VII.

1. Problem 1.

a) Let μ(X) = 1, and suppose f and g are positive measurable functions on X such that fg ≥ 1. Prove that

∫_{X} fdμ ∫_{X} gdμ ≥ 1.

b) Let X and μ be as above, and let h ≥ 0 be measurable. If A = ∫_{X} hdμ, prove that

\sqrt{1 + A^{2}} ≤ ∫_{X} \sqrt{1 + h^{2}}dμ ≤ 1 + A.

2. Problem 2.

a) Suppose μ is a positive measure on X and a positive f satisfies ∫_{X} fdμ = 1. Prove that, for every E ⊂ X with 0 < μ(E) < ∞, that

∫_{E} logfdμ ≤ μ(E)log \frac{1}{μ(E)}.

b) If f is a positive measurable function on [0, 1], which is larger,

∫_{0}^{1} f(x)logf(x)dx,  or  ∫_{0}^{1} f(x)dx ∫_{0}^{1} logf(x)dx?

3. Problem 3. Suppose 1 < p < ∞, f ∈ L^{p} = L^{p}((0, ∞)), relative to the Lebesgue

measure, and

F (x) := \frac{1}{x} ∫_{0}^{x} f(t)dt, x ∈ (0, ∞).

a) Prove the Hardy inequality

∥F ∥_{p} ≤ \frac{p}{p − 1}∥f∥_{p} ,

which is that the mapping f → F carries L^{p} into L^{p}.

Hint. Write

xF (x) = ∫_{0}^{x} f(t)t^{α}t^{−α}dt, 0 < α < 1/q,

use Holder’s inequality to get an upper bound for F^{p}(x), and integrate to obtain

∫_{0}^{∞}F^{p}(x)dx ≤ (1 − αq)^{1−p}(αp)^{−1} ∫_{0}^{∞}f^{p}(t)dt.

b) Show that the best choice of α yields

∫_{0}^{∞}F^{p}(x)dx ≤ (\frac{p}{p − 1})^{p} ∫_{0}^{∞}f^{p}(t)dt.

c) Prove that the constant p/(p − 1) cannot be replaced by a smaller one. Hint. Take f(x) = x^{−1/p} on [1, A], f(x) = 0 elsewhere, for large A.

d) If f > 0, and f ∈ L^{1}, prove that F ∉ L^{1}.

e) Prove that if

∫_{0}^{∞}f(t)dt = 0,  ∫_{0}^{∞}|f(t)log|t||dt < ∞,

then F ∈ L^{1}.

4. Problem 4. Suppose μ is a positive measure on X, μ(X) < ∞, f ∈ L^{∞}(μ, X),

∥f∥_{∞} > 0, and α_{n} := ∫_{X} |f(x)|^{n}dμ, n ∈ N. Prove that lim_{n→∞} α_{n+1}/α_{n} = ∥f∥_{∞}.

Hint.

a) Prove that for any sequence of positive numbers (α_{n})_{n=1}^{∞}, lim_{n→∞} α_{n+1}/α_{n} = lim_{n→∞} α_{n}^{1/n}, provided the first limit exists.

b) Using Holder’s inequality prove that the limit lim_{n→∞} α_{n+1}/α_{n}, α_{n} := ∫_{X} |f(x)|^{n}dμ, exists.

c) Use ∥f∥_{n} → ∥f∥_{∞} as n → ∞ and a), b).