247A FINAL

TERENCE TAO

Instructions: do all questions. You are encouraged to discuss problems with fellow students, and to conduct research in textbooks, in the library, or on the internet, but the final solutions must be written up (or typed) in your own words. You can of course use results proved in the notes, but if you are to use unproved assertions in the notes (e.g. exercises or problems) then a proof to these assertions must also be supplied.

• Q1. (Dual characterisation of L^{p}) Let 0 < p < ∞, 0 < A < ∞, and f ∈ L^{p,∞}(X, dμ). Show that the following are equivalent (up to changes in the implied constant):

– ∥f∥_{L^{p,∞}(X,dμ)} ≲_{p} A.

– For every set E of finite measure, there exists a subset E′ of E with μ(E′) ≥ \frac{1}{2}μ(E) such that

∫_{X} f1_{E′} dμ = O(μ(E)^{1/p′})

(in particular, we assert that the integral on the left-hand side is absolutely integrable).

Hint: It may be instructive to work out the example f(x) = |x|^{−d/p} on R^{d} by hand to get a sense of what is going on; this should suggest how to prove things in general. The proof is slightly simpler in the case when f is non-negative, so you may want to try that case first.

• Q2. (Weighted one-dimensional Hardy-Littlewood-Sobolev) Let 1 ≤ p, q ≤ ∞, 0 < s < 1, and α, β ∈ R obey the conditions

α > −\frac{1}{p′}
β > −\frac{1}{q′}
1 ≤ \frac{1}{p} + \frac{1}{q} ≤ 1 + s

and the scaling condition

α + β − 1 + s = −\frac{1}{p′} − \frac{1}{q′} ,

with at most one of the equalities

p = 1, p = ∞, q = 1, q = ∞, \frac{1}{p} + \frac{1}{q} = 1 + s

holding. Let f ∈ L^{p}(R) and g ∈ L^{q}(R). Show that

∫_{R} ∫_{R} \frac{|x|^{α}|y|^{β}}{|x − y|^{1−s}} |f(x)||g(y)| dxdy ≲_{α,β,p,q} ∥f∥_{L^{p}(R)}∥g∥_{L^{q}(R)}.

• Q3. (Relationship between dyadic and non-dyadic Hardy-Littlewood maximal inequalities) Let f : R^{d} → C be locally integrable. Establish the pointwise bound

|{M_{Δ}f ≥ C_{d}λ}| ≤ |{Mf ≥ λ}| ≲_{d} |{M_{Δ}f ≥ c_{d}λ}|

for some c_{d}, C_{d} > 0 depending only on d.

• Q4 (Sharp function, I) For any locally integrable function f : R^{d} → C, define the function f^{♯} : R^{d} → R^{+} as

f^{♯}(x) := sup_{B∋x} ∫−_{B}|f − ∫−_{B}f|

where the supremum ranges over all balls containing x. In particular we see that

∥f^{♯}∥_{L^{∞}(R^{d})} = ∥f∥_{BMO(R^{d})}.

Show that we have the pointwise bound

|f^{♯}| ≲_{d} Mf.

• Q5 (Sharp function, II) Let f be locally integrable, let ε, λ > 0, and let B be any ball on which Mf(x) ≲ λ for at least one x ∈ B, and such that f^{♯}(y) ≤ ελ for at least one y ∈ B. Use the Hardy-Littlewood maximal inequality to show that for any sufficiently large K ≫ 1, that

|\{x ∈ B : Mf ≥ Kλ\}| ≲_{d} \frac{ε}{K} |B|.

From this and a Vitali covering lemma or Calderón-Zygmund decomposition argument, conclude the good-λ inequality

|\{Mf ≥ Kλ and f^{♯} ≤ ελ\}| ≲_{d} \frac{ε}{K} |\{Mf ≥ λ\}|

and then (by taking K large, and ε very small) show that

∥f∥_{L^{p}(R^{d})} ∼_{p,d} ∥Mf∥_{L^{p}(R^{d})} ≲_{p,d} ∥f^{♯}∥_{L^{p}(R^{d})}

for all 1 < p < ∞ and f ∈ L^{p}(R^{d}). Use this to give an alternate proof of Lemma 3.7 from Notes 4 (at least in the case 1 < p, q < ∞, although with some application of Hölder’s inequality one can get the full range 0 < p, q < ∞).

• Q6. (Zygmund’s inequality) For each positive integer n, let ξ_{n} be an integer with |ξ_{n}| ∼ 2^{n} and all ξ_{n} distinct, and let c_{n} be a complex number with ∑_{n} |c_{n}|^{2} finite. Show that

∥ ∑_{n} c_{n}e^{2πiξ_{n}x}∥_{L^{p}(T)} ∼_{p} (∑_{n} |c_{n}|^{2})^{1/2}

for all 2 ≤ p < ∞. Dually, show that

(∑_{n} |f̂(ξ_{n})|^{2})^{1/2} ≲_{p} ∥f∥_{L^{p}(T)}

for all 1 < p ≤ ∞ and f ∈ L^{p}(T). (Hint: the Khinchine proof is not easily adaptable here, except in the special case when ξ_{n} is exactly 2^{n}. Instead, either establish a BMO estimate or else compute the L^{p} norms for even integer p.)

• Q7 (Boundary values of H^{p} functions). Let 1 < p < ∞.

• (a) Show that a function g : R/Z → C is the boundary value of an H^{p}(D) function f (thus g = f_{1}, in the notation of Notes 3) if and only if g ∈ L^{p}(R/Z) and ĝ(n) = 0 for all n < 0.

• (b) Show that if u ∈ L^{p}(R/Z) is real-valued, then there exists f ∈ H^{p}(D) whose boundary value f_{1} has real part u, Re(f_{1}) = u. Furthermore, f is unique up to addition by imaginary constants.

• Q8 (Wolff interpolation lemma). Let X and Y be two measure spaces, and T : L^{2}(X) → L^{2}(Y) be a continuous linear operator (but possibly with very large operator norm), with adjoint T^{∗} : L^{2}(Y) → L^{2}(X). Suppose that T and T^{∗} are both of weak-type (1, 1) with bounded norm, thus we have

∥T f∥_{L^{1,∞}(Y)} ≲ ∥f∥_{L^{1}(X)} for all f ∈ L^{1}(X) ∩ L^{2}(X)

and

∥T^{∗}g∥_{L^{1,∞}(X)} ≲ ∥g∥_{L^{1}(Y)} for all g ∈ L^{1}(Y) ∩ L^{2}(Y).

Show that T is of strong-type (p, p) for 1 < p < ∞ with bounded norm, or more specifically that

∥T f∥_{L^{p}(Y)} ≲_{p} ∥f∥_{L^{p}(X)} for all f ∈ L^{p}(X) ∩ L^{2}(X).

(In particular, a posteriori, T had a bounded operator norm on L^{2}. But you are not allowed to assume this a priori; the continuity on L^{2} is merely a qualitative assumption.)

Department of Mathematics, UCLA, Los Angeles CA 90095-1555

E-mail address: tao@math.ucla.edu