In this section, the Marcinkiewicz interpolation theorem and Calderon Zygmund
decomposition will be used to establish a remarkable theorem of Mihlin, a generalization
of Plancherel’s theorem to the L^{p} spaces. It is of fundamental importance in the study of
elliptic partial differential equations and can also be used to give proofs for the theory
of singular integrals. Mihlin’s theorem involves a conclusion which is of the
form
|| ||
||F −1ρ∗ ϕ|| ≤ Ap ||ϕ||p (31.3.7)
p
(31.3.7)
for p > 1 and ϕ ∈G. Thus F^{−1}ρ∗ extends to a continuous linear map defined on L^{p}
because of the density of G. It is proved by showing various weak type estimates and then
applying the Marcinkiewicz Interpolation Theorem to get an estimate like the
above.
estimate. The inequality 31.3.7 is established
under the condition, 31.3.8 and then it is shown there exist conditions which are easier to
verify which imply condition 31.3.8. I think the approach used here is due to Hormander
[?] and is found in Berg and Lofstrom [?]. For many more references and generalizations,
you might look in Triebel [?]. A different proof based on singular integrals is in Stein [?].
Functions, ρ which yield an inequality of the sort in 31.3.7 are called L^{p} multipliers.
Lemma 31.3.1Suppose ρ ∈ L^{∞}
n
(ℝ )
∩ L^{2}
n
(ℝ )
and suppose also there exists aconstant C_{1}such that
Then there exists a constant A depending only on C_{1},
||ρ||
_{∞}, and n such that
m ([x :||F −1ρ∗ϕ(x)|| > α]) ≤ A-||ϕ||
α 1
for all ϕ ∈G.
Proof:Let ϕ ∈G and use the Calderon decomposition to write ℝ^{n} = E ∪ Ω where Ω
is a union of cubes,
{Qi}
with disjoint interiors such that
∫
αm (Q ) ≤ |ϕ(x)|dx ≤ 2nαm (Q ), |ϕ (x)| ≤ α a.e. on E. (31.3.9)
i Qi i
(31.3.9)
The proof is accomplished by writing ϕ as the sum of a good function and a bad
function and establishing a similar weak inequality for these two functions separately.
Then this information is used to obtain the desired conclusion.
{
g(x) = ϕ(x1)-if∫ x ∈ E ,g(x)+ b(x) = ϕ (x). (31.3.10)
m(Qi) Qi ϕ(x)dx if x ∈ Qi ⊆ Ω
(31.3.10)
Thus
∫ ∫ ∫ ∫
Q b(x)dx = Q (ϕ (x)− g(x))dx = Q ϕ(x)dx − Q ϕ(x)dx =(301,.3.11)
i i i i
b (x) = 0 if x∈∕Ω. (31.3.12)