NOTES ON HW 1

TERENCE TAO

1. Comments on Q1.4

Note that for p ≥ 1 and arbitrary f_{n} we have the pointwise inequality

∑_{n} |f_{n}| ≥ (∑_{n} |f_{n}|^{p})^{1/p}

and hence

∥ ∑_{n} |f_{n}|∥_{p} ≥ (∑_{n} ∥f_{n}∥_{p}^{p})^{1/p}

giving the lower bound in both cases. When p ≤ 1, the inequalities are reversed, giving the upper bound in both cases.

Actually in the first case (dyadic heights) one can show (due to convergence of geometric series) that

∑_{n} |f_{n}| ∼ sup_{n} |f_{n}| ∼_{p} (∑_{n} |f_{n}|^{p})^{1/p}

which gives the bound in both cases.

The upper bound in both cases also follows from Theorem 6.6 from Notes 1.

The lower bound in the second case (dyadic widths) can be obtained as follows. We have

∑_{n} |f_{n}| ∼ ∑_{n} H_{n}1_{E_{n}}

where μ(E_{n}) ∼ 2^{n}. Our objective is then to show that

∥ ∑_{n} H_{n}1_{E_{n}}∥_{p} ≳_{p} ∥H_{n}μ(E_{n})^{1/p}∥_{l_{n}^{p}}.

Suppose instead that the E_{n} grew so rapidly that

μ(E_{n}) ≥ 4μ(E_{n−1})

(say). Then

∑_{m<n} μ(E_{m}) ≤ \frac{1}{2}μ(E_{n})

so if we let E′_{n} := E_{n}\ ⋃_{m<n} E_{m} then

μ(E′_{n}) ≥ \frac{1}{2}μ(E_{n}).

On the other hand, E′_{n} are disjoint and

∑_{n} H_{n}1_{E_{n}} ≥ ∑_{n} H_{n}1_{E′_{n}}

and then the claim follows.

Of course, the E_{n} do not grow this rapidly. However we can sparsify the sequence until we do have this convergence. Indeed we see from the above argument that if K is large enough (depending on the implied constants in the definition of a quasistep function) then

∥ ∑_{n=a mod K} H_{n}1_{E_{n}}∥_{p} ≳_{p} ∥H_{n}μ(E_{n})^{1/p}∥_{l_{n}^{p}(n=a mod K)}

for each a. On the other hand, from the triangle inequality and the pigeonhole principle we know that

∥H_{n}μ(E_{n})^{1/p}∥_{l_{n}^{p}(n=a mod K)} ≳_{K} ∥H_{n}μ(E_{n})^{1/p}∥_{l_{n}^{p}}

for at least one a. The claim follows.

The sparsification argument also works to give the lower bound when the absolute value signs are removed as long as the powers of 2 are replaced by the powers of a sufficiently large number. (I claimed 4, but I forgot about the implicit constants in the definition of a quasistep function and in fact the power needed depends on those constants.)

2. Comment on Q2.5

Note that we have the pointwise bound

\frac{1}{t} ∫_{0}^{t} f(x) dx ≤ Mf(t)

(extending f by zero on the negative real axis), where M is the Hardy-Littlewood maximal function. So, as long as you don’t care about constants, Hardy’s inequality is a consequence of the Hardy-Littlewood maximal inequality.

The sharp constant in Hardy’s inequality demonstrates a number of things. Firstly, the operator norm need not be attainable; for instance, while Hardy’s operator has an L^{2}(R^{+}) operator norm of 2, an inspection of the proof of that estimate shows that equality is only possible when f is a multiple of |x|^{−1/2} - but this is not quite in L^{2}. Secondly, it demonstrates that it is not possible to remove the losses of constant in the Marcinkeiwicz interpolation theorem. (It also demonstrates that the Riesz-Thorin theorem does not always give optimal constants either.)

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

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