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The Hilbert and Riesz transforms map L-infinity to BMO

Statement

Assume Countable Choice. The Hilbert transform $H$ on $R$ and the Riesz transforms $R_{1}, \dots, R_{n}$ on $R^{n}$ extend to bounded maps $L^{\infty} (R^{d}) \to B M O (R^{d}) / C$ , where $d = 1$ for $T = H$ and $d = n$ for $T = R_{j}$ : for every $b \in L^{\infty} (R^{d})$ the class of $T b$ is well defined modulo constants, agrees with the $L^{2}$ action of $T$ when $b \in L^{2}$ , and satisfies $∥ T b ∥_{B M O} \leq C_{d} ∥ b ∥_{L^{\infty}}$ with a dimensional constant.

Facts & Assumptions

Given: Countable Choice, the Hilbert transform $H$ and the Riesz transforms $R_{1}, \dots, R_{n}$ , and a bounded function $b \in L^{\infty} (R^{d})$ , with $d = 1$ for $H$ and $d = n$ for $R_{j}$ .

[F1]

The Hilbert transform and each Riesz transform are Calderon-Zygmund operators whose kernels are standard $1$ -Holder, with $L^{2}$ operator norm at most $1$ ; the Hilbert transform is the case $n = 1$ with kernel $1 / (π x)$ and the Riesz transforms have kernels $c_{n} x_{j} / ∣ x ∣^{n + 1}$ (The Hilbert and Riesz transforms are Calderon-Zygmund operators).

[F2]

Every Calderon-Zygmund operator with a standard $δ$ -Holder kernel and $L^{2}$ norm at most $1$ maps $L^{\infty}$ into $B M O (R^{n}) / C$ : the class of $T b$ is well defined modulo constants, agrees with the $L^{2}$ action when $b \in L^{2}$ , and has $∥ T b ∥_{B M O} \leq C_{n, δ} (A_{2}^{'} + 1) ∥ b ∥_{L^{\infty}}$ (Calderon-Zygmund operators map L-infinity to BMO).

Proof

technique · direct

1.1F1

The hypotheses of [F2] are satisfied with $δ = 1$ : [F1] supplies the Calderon-Zygmund operator, the standard $1$ -Holder kernel with its constant, and the $L^{2}$ bound by $1$ ; for the Hilbert transform this is the case $n = 1$ and for each Riesz transform the case of the corresponding kernel $c_{n} x_{j} / ∣ x ∣^{n + 1}$ .

2.1step 1.1F2

Applying [F2] to the Hilbert transform and to each Riesz transform gives, for every $b \in L^{\infty} (R^{d})$ in the corresponding dimension, a well-defined class $T b$ modulo constants with $∥ T b ∥_{B M O} \leq C_{d, 1} (A_{2}^{'} + 1) ∥ b ∥_{L^{\infty}}$ , and when $b \in L^{2}$ that class is the class of the $L^{2}$ function $T b$ ; the constants $A_{2}^{'}$ and hence $C_{d, 1} (A_{2}^{'} + 1)$ depend only on $d$ .

3.1step 2.1∎

The assertions of the statement are exactly those of step 2.1 for $T = H$ and $T = R_{j}$ , with $C_{d} : = C_{d, 1} (A_{2}^{'} + 1)$ . Countable Choice is inherited from both [F1] and [F2], including the endpoint gluing and $L^{2}$ consistency argument.

Depends on

Used by

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Sources

Mark Williams, Notes on Harmonic Analysis (January 11, 2022) (standard reference, not scraped)
Terence Tao, Math 247A Lecture Notes 4 (UCLA, Fall 2006) (standard reference, not scraped)