Q2. What is the simplest explanation of the dominated convergence?
Since further ‖f‖JNr′‖(aij) ∞ j=1‖(r,s) is summable over i by assumption, yet another application of dominated convergence proves thatΛfg = ∑i〈f, gi〉 = lim N→∞∑iˆQ0fNgi = lim N→∞ˆQ0fNg.
Q3. What is the only place where the properties of real numbers were used?
The only place in the above argument where the properties of real numbers beyond their Banach space structure were used was the representation of Λ̃ ∈ (Ls0(Q0)) ∗ by a function f ∈ Ls ′(Q0).
Q4. How does the dyadics calculate the distance between Î and Î?
If the authors take, without loss of generality, The author′ to be the left half of I, this minimal distance is achieved by always choosing the descendants The author′′ of The author′ on the right, so that Î ′′ gets closer and closer to Î.
Q5. What is the simplest way to define the Lp spaces?
In order to have an access to the simple duality of the reflexive Lp spaces (in contrast to the more complicated situation of L∞), the authors first establish the following reduction to finite indices in their candidate predual space:6.4.
Q6. What is the minimum distance of fI from any supp fI?
Since δI > 0 is the minimal distance of supp fI from any supp fI′′ with I′′ ( I, the authors see that each fI is disjointly supported from its descendants fI′′ with I′′ ( I.
Q7. What are the properties of the Chang–Fefferman atoms?
In contrast, the Chang–Fefferman atoms, while beingthe larger structures in their expansion, have properties closely analogous to those of classical atoms, whereas their elementary particles have additional smoothness properties, which are neither present in the classical H1 theory nor in their new spaces.
Q8. what is the easiest term to estimate?
That is, the authors need to estimate∞ ∑k=0‖gk‖(r,∞) ≤ ∞ ∑k=0(∑ℓ,j|Qℓk,j |‖a ℓ kj‖ r ∞)1/r.∞ ∑k=0Ck ( ∑ℓ,j|Qℓk,j |(Qℓ|Aℓ| s)r/s)1/r.∞ ∑k=1Ck [ ∑ℓ∣ ∣ ∣ { MQℓAℓ > C k(Qℓ|Aℓ| s)1/s} ∣∣ ∣ (Qℓ|Aℓ| s)r/s]1/r+ ( ∑ℓ|Qℓ|(Qℓ|Aℓ| s)r/s)1/r,where the authors separated the easier term corresponding to k = 0.