NOTES ON HW 4

TERENCE TAO

1. Comments on Q3.4

Almost everybody got this one right. There is just one point that needs to be observed: if a lower dimensional operator, such as M_{□}^{d−1}, is bounded on L^{p}(R^{d−1}), then it is automatically bounded also on L^{p}(R^{d}) when applied “horizontally” or “vertically” to each “slice” of R^{d} separately. This follows from two applications of the Fubini-Tonelli theorem.

Incidentally, if the rectangles are not enforced to be parallel to the axes, but can point in arbitrary directions, then the maximal operator becomes unbounded on any L^{p}(R^{d}) with p < ∞ and d ≥ 2. This is due to the existence of the infamous Besicovitch sets, and causes a great deal of trouble in many advanced areas of harmonic analysis.

2. Comments on Q3.5

Note that there are three symmetries available for this problem: translation f(x) ↦ f(x − x_{0}), homogeneity f(x) ↦ cf(x), and scaling f(x) ↦ f(x/λ) (one can also use f(x) ↦ λ^{−d/p}f(x/λ), but this is equivalent to the other scaling modulo homogeneity). Using all three one can normalise x, ∥f∥_{L^{p}(R^{d})}, and Mf(x) to be 0, 1, and 1 respectively.

3. Comments on Q3.11

Several of you reduced to the case when f_{1} and f are non-negative. Strictly speaking, the question as stated does not permit you to do this, because the real or positive part of an H^{p}(D) function is not in H^{p}(D) (real-valued functions tend to be non-holomorphic). However, one can get around this by generalising the claim, namely to prove that for any f_{1} ∈ L^{p}(R/Z) (not necessarily the boundary values of an H^{p} function), the functions f_{1} ∗ K_{t} converge non-tangentially a.e. to f_{1} as t → 1. In this more general context there is no difficulty reducing to the real or non-negative case.

As observed by several people, it is not only true that we have non-tangential convergence a.e.; we in fact have non-tangential convergence at every Lebesgue point. Proving this stronger fact obviates the need to treat each non-tangential approach region (e.g. those in which the angle is less than π/2 − 1/n) separately, and then sum up (using the fact that if a countable number of properties each hold almost everywhere seperately, then they also hold jointly almost everywhere).

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

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