33.3. MIHLIN’S THEOREM 1123

Let x be a point of Ω and let x be in a cube of Tm such that m is the first index for whichthis happens. Let Q be the cube in Sm−1 containing x and let Q∗ be the cube in the bisectionof Q which contains x. Therefore 33.2.6 does not hold for Q∗. Thus

α <1

m(Q∗)

∫Q∗

f dx≤ m(Q)

m(Q∗)

≤α︷ ︸︸ ︷1

m(Q)

∫Q

f dx≤ 2nα

which shows Ω is the union of cubes having disjoint interiors for which 33.2.5 holds.Now a.e. point of F is a Lebesgue point of f . Let x be such a point of F and suppose x

∈ Qk for Qk ∈ Sk. Let dk ≡ diameter of Qk. Thus dk→ 0.

1m(Qk)

∫Qk

| f (y)− f (x)|dy≤ 1m(Qk)

∫B(x,dk)

| f (y)− f (x)|dy

=m(B(x,dk))

m(Qk)

1m(B(x,dk))

∫B(x,dk)

| f (x)− f (y)|dy

≤ Kn1

m(B(x,dk))

∫B(x,dk)

| f (x)− f (y)|dy

where Kn is a constant which depends on n and measures the ratio of the volume of a ballwith diamiter 2d and a cube with diameter d. The last expression converges to 0 because xis a Lebesgue point. Hence

f (x) = limk→∞

1m(Qk)

∫Qk

f (y)dy≤ α

and this shows f (x)≤ α a.e. on F . This proves the theorem.

33.3 Mihlin’s TheoremIn this section, the Marcinkiewicz interpolation theorem and Calderon Zygmund decom-position will be used to establish a remarkable theorem of Mihlin, a generalization ofPlancherel’s theorem to the Lp spaces. It is of fundamental importance in the study ofelliptic partial differential equations and can also be used to give proofs for the theory ofsingular integrals. Mihlin’s theorem involves a conclusion which is of the form∣∣∣∣F−1

ρ ∗φ∣∣∣∣

p ≤ Ap ||φ ||p (33.3.7)

for p> 1 and φ ∈G . Thus F−1ρ∗ extends to a continuous linear map defined on Lp becauseof the density of G . It is proved by showing various weak type estimates and then applyingthe Marcinkiewicz Interpolation Theorem to get an estimate like the above.

Recall that by Corollary 32.3.19, if f ∈ L2 (Rn) and if φ ∈ G , then f ∗φ ∈ L2 (Rn) and

F ( f ∗φ)(x) = (2π)n/2 Fφ (x)F f (x).

The next lemma is essentially a weak (1,1) estimate. The inequality 33.3.7 is establishedunder the condition, 33.3.8 and then it is shown there exist conditions which are easier to