Riesz transforms, Cauchy-Riemann systems, and Hardy-amalgam spaces

Abstract

In this paper, we study Hardy spaces Hp,q(R^d), 0 < p, q < ∞, modeled over amalgam spaces (Lp, lq)(R^d). We characterize Hp,q(R^d) by using first-order classical Riesz transforms and compositions of first-order Riesz transforms, depending on the values of the exponents p and q.

We also describe the distributions in Hp,q(R^d) as the boundary values of solutions to harmonic and caloric Cauchy–Riemann systems. We remark that caloric Cauchy–Riemann systems involve fractional derivatives in the time variable.

Finally, we characterize the functions in L2(R^d) ∩ Hp,q(R^d) by means of Fourier multipliers mθ with symbol θ(·/|·|), where θ ∈ C∞(Sd−1) and Sd−1 denotes the unit sphere in R^d.