Sharp weak-type inequalities for Hilbert transform and Riesz projection

Abstract

The paper is devoted to the study of the weak norms of the classical operators in the vector-valued setting.

• (i) Let S, H denote the singular integral involution operator and the Hilbert transform on $L^p \left( {\mathbb{T}, \ell _\mathbb{C}^2 } \right)$, respectively. Then for 1 ≤ p ≤ 2 and any f,

  $$\left\| {\mathcal{S}f} \right\|_{p,\infty } \leqslant \left( {\frac{1} {\pi }\int_{ - \infty }^\infty {\frac{{\left| {\tfrac{2} {\pi }\log \left| t \right|} \right|^p }} {{t^2 + 1}}dt} } \right)^{ - 1/p} \left\| f \right\|p,$$
  $$\left\| {\mathcal{H}f} \right\|_{p,\infty } \leqslant \left( {\frac{1} {\pi }\int_{ - \infty }^\infty {\frac{{\left| {\tfrac{2} {\pi }\log \left| t \right|} \right|^p }} {{t^2 + 1}}dt} } \right)^{ - 1/p} \left\| f \right\|p.$$

  Both inequalities are sharp.

• (ii) Let P ₊ and P ₋ stand for the Riesz projection and the co-analytic projection on $L^p \left( {\mathbb{T}, \ell _\mathbb{C}^2 } \right)$, respectively. Then for 1 ≤ p ≤ 2 and any f,

  $$\left\| {P + f} \right\|_{p,\infty } \leqslant \left\| f \right\|_p ,$$
  $$\left\| {P - f} \right\|_{p,\infty } \leqslant \left\| f \right\|_p .$$

  Both inequalities are sharp.

• (iii) We establish the sharp versions of the estimates above in the nonperiodic case.

The results are new even if the operators act on complex-valued functions. The proof rests on the construction of an appropriate plurisubharmonic function and probabilistic techniques.