Endpoint map for Hilbert and Riesz transforms

This page proves the strict-range facts for the periodic conjugate operator and the $L^2$ facts for the line Hilbert transform and the Euclidean Riesz transforms: The Marcel Riesz conjugate-function theorem on the circle bounds the conjugate operator on $L^p(\mathbb{T})$ for $1<p<\mathrm{\infty }$ and records the failure of compatible strong-type $L^1$ and $L^{\mathrm{\infty }}$ extensions; The Hilbert transform is an L2 isometry and squares to minus the identity identifies the line Hilbert transform as the $L^2$ multiplier by $-i\mathrm{sgn}$ with $H^2=-I$; and Riesz transforms are L2 contractions and square to minus the identity in sum gives the Euclidean $L^2$ contractions with ${\sum }_j{R}_j^2=-I$.

Three distinct endpoints are deliberately not settled here, and the reader should not read this page as a negative statement about them. First, weak $(1,1)$ bounds, the real-line strict-range $L^p$ theory for the line and Riesz transforms, and the almost-everywhere convergence of truncated integrals are deferred to the later Calderón–Zygmund decomposition and singular-integrals material, which supplies the covering and Calderón–Zygmund kernel estimates this page stops short of. Second, the real Hardy space endpoint belongs to the later real Hardy space and maximal-function material. Third, the bounded mean-oscillation endpoint belongs to the later BMO material. Each of those later pages is named here by title only; no result from them is used as a premise anywhere on this page.

Two further distinctions are recorded. The periodic conjugate operator is presented through the circle's zero mode: constants lie in its kernel and the multiplier vanishes at frequency $0$, whereas on the line the corresponding signum multiplier vanishes on a Lebesgue-null singleton and the $L^2$ square identity holds with no zero-mode exception. The line Hilbert and Riesz endpoint questions are distinct from the circle's partial-sum operator norms. For the periodic conjugate operator itself, however, the Lebesgue-constant lower bound above is used to rule out compatible strong $L^1$ and $L^{\mathrm{\infty }}$ extensions.

Finally, the companion examples page constructs the interval indicator whose Hilbert transform is ${\pi }^{-1}\log \mathrm{\mid }x\mathrm{/}(x-1)\mathrm{\mid }$ and uses it to refute a bounded strong-type $L^1\to L^1$ action and a bounded $L^{\mathrm{\infty }}\to L^{\mathrm{\infty }}$ action compatible with the $L^2$ transform. Those computations refute strong-type mapping only: they are consistent with a weak $(1,1)$ bound and with a bounded BMO-valued endpoint, and they say nothing against the deferred results named above.