Fefferman duality theorem

Remarks

The theorem is due to Charles Fefferman, and a full account appeared in the work of Fefferman and Stein on $H^p$ spaces.

With the atomic decomposition, part 1 is short. For an atom $a$ on a ball $B$, the cancellation $\int a = 0$ gives $\abs{\int a b} = \abs{\int_B a (b - b_B)} \le \frac{1}{\abs{B}} \int_B \abs{b - b_B} \le C \norm{b}_{\mathrm{BMO}}$, and the bound passes to sums of atoms. For part 2, the definition of $H^1$ through the Riesz transforms embeds $H^1$ isometrically into a closed subspace of $L^1 \times \cdots \times L^1$ ($n + 1$ factors). By the Hahn–Banach theorem, $\ell$ is given by bounded functions $b_0, \dots, b_n$, so $b = b_0 - \sum_j R_j b_j$, and each $R_j$ maps $L^\infty$ into BMO.

The pairing $\int f b$ need not converge absolutely for general $f \in H^1$ and $b \in \mathrm{BMO}$, which is why it is first defined on $H^1_0$. An analogous duality on the circle identifies the dual of the real $H^1$ of the circle with BMO of the circle.

Going deeper

For a student

The theorem explains why BMO appears so often as the endpoint of $L^p$ theory at $p = \infty$. Since Calderón–Zygmund operators map $H^1$ into $L^1$, their adjoints map $L^\infty$ into BMO by duality. It is also the basis for interpolation between $H^1$ and $L^p$, or between $L^p$ and BMO. Combined with the Carleson measure characterization of BMO, it links the real-variable theory with harmonic extensions to the half-space.