Bounded double square functions

Abstract

We extend some recent work of S. Y. Chang, J. M. Wilson and T. Wolff to the bidisc. For f∈L loc 1 (R 2 ), we determine the sharp order of local integrability obtained when the square function of f is in L ∞ . The Calderón-Torchinsky decomposition reduces the problem to the case of double dyadic martingales. Here we prove a vector-valued form of an inequality for dyadic martingales that yields the sharp dependence on p of C p in ∥f∥ p ≤C p ∥Sf∥ p .

Locations

• French digital mathematics library (Numdam) - View - PDF
• Annales de l’institut Fourier - View - PDF