Riesz transforms, Cauchy-Riemann systems and amalgam Hardy spaces

In this article we study Hardy spaces ℋ^p,q(ℝ^d), 0 < p, q < ∞, modeled over amalgam spaces (L_p, ℓ^q)(ℝ^d). We characterize ℍ^p,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. Also, we describe the distributions in H^{p, q}(ℝ^d) as the boundary values of solutions of 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) ∪ H^{p, q}(ℝ^d) by means of Fourier multipliers m_θ with symbol θ(·=/| · |), where θ ∈ C∞ (S^d-1) and S^d-1 denotes the unit sphere in ℝ^d