Hörmander multipliers on two-dimensional dyadic Hardy spaces

Abstract

In this paper we are interested in conditions on the coefficients of a two-dimensional Walsh multiplier operator that imply the operator is bounded on certain of the Hardy type spaces H p , 0 < p < ∞ . We consider the classical coefficient conditions, the Marcinkiewicz–Hörmander–Mihlin conditions. They are known to be sufficient for the trigonometric system in the one and two-dimensional cases for the spaces L p , 1 < p < ∞ . This can be found in the original papers of Marcinkiewicz [J. Marcinkiewicz, Sur les multiplicateurs des series de Fourier, Studia Math. 8 (1939) 78–91], Hörmander [L. Hörmander, Estimates for translation invariant operators in L p spaces, Acta Math. 104 (1960) 93–140], and Mihlin [S.G. Mihlin, On the multipliers of Fourier integrals, Dokl. Akad. Nauk SSSR 109 (1956) 701–703; S.G. Mihlin, Multidimensional Singular Integrals and Integral Equations, Pergamon Press, 1965]. In this paper we extend these results to the two-dimensional dyadic Hardy spaces.