Trigonometric multipliers on real periodic Hardy spaces

In this paper we are concerned with the boundedness of Hormander–Mihlin multipliers of order $$\,r\, (1\le r1.$$ On the other hand the boundedness extends to $$\,\mathcal H_{2\pi }$$ if $$\,r>1,$$ but not if $$\,r=1.$$ Generalizing this result in the present paper we show that the scale of Hardy spaces $$\,\mathcal H^p_{2\pi }\, (01\,$$ we give a sharp bound $$\,p_rp_r\,$$ then the Hormander–Mihlin condition of order $$\,r\,$$ is sufficient on $$\mathcal H^p_{2\pi }$$ .