Definition (Construction of the dyadic partition of unity)Fix a mapping withwhich we also regard as a map via . We further define by

Note that

Now, from previous time for this time:

Theorem (Mikhlin multiplier theorem, or possibly Marcinkiewicz, or Hörmander)If obeysThen

maps (in a bounded manner) for .

*Proof:* We will just prove the theorem for obeying “symbol estimates” for multi-indices . (For the full case, see [Stein, *Harmonic Analysis*, VI.4.4]) Clearly is bounded on (just requires , i.e., ). To prove it is bounded on we just need to check that obeys the Calderón–Zygmund cancellation condition:

This is implied by

which is what we will show. Decompose , where . Now

where are some combinatorial coefficients. So,

Hence, because F.T,

and so, choosing various ‘s,

Now we sum

Definition (Littlewood–Paley projections)Let be as above and , thenand similarly

These are not honest projections (namely, ).

Proposition

- If , , then
both a.e. and in .

- We have
for all .

- .
- For ,
- for all and for all .
- for all and for all .

Remark

- norm convergence fails because smooth functions are not dense in . A.e. convergence holds for the second decomposition in but not for the first, consider .
- Note that and so . Similarly for . Bernstein’s name is also attached to inequalities of the form for polynomials .

*Proof:* For assertion 2, this is Minkowski’s inequality:

and the reasoning for is similar.

For assertion 3,

To realize the final inequality, observe

i.e., is a positive linear combination of characteristic functions of balls\footnote{} and so

For assertion 1, when , converges to when , to when , both a.e. and in . This extends to via assertions 3 and 2, respectively.

For assertion 4,

For assertion 5, let and so . Thus, by Minkowski’s inequality

For the converse inequality note that

and so

For assertion 6,

Theorem (Littlewood–Paley Square function estimate)Given when , defineThen

Remark

- We can view as a sub-linear operator, or as a vestige of the linear operator and then the square function estimate is about its mapping properties from
- The Littlewood–Paley operators decompose into its constituent frequencies/wave-lengths/length-scales. The square function says that we do not need to fear subtle cancellations between these different components—note the absolute value signs.
- Compare , the triangle inequality, but this is
notreversible.

Lemma (Khinchin)Let be (statistically) independent identically distributed random variables, Bernoulli with equal probability. For complex numbers ,

for each .

*Proof:* We will just treat as real numbers. Note that for all

and optimizing in ,

where the ‘ed inequality follows from noting . We will finish this proof next time.

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