Trigonometric multipliers on real periodic Hardy spaces

Abstract

In this paper we are concerned with the boundedness of Hörmander–Mihlin multipliers of order $\,r\, (1\le r<\infty )\,$ on the real periodic Hardy space $\,\mathcal H^p_{2\pi }, 0<p<1.$ The case $\,r=1\,$ corresponds to the classical Marcinkiewicz multiplier condition which is known to be sufficient for the boundedness of the trigonometric multipliers on the Lebesgue spaces $L_{2\pi }^p, 1<p<\infty ,$ but not on $L^1_{2\pi }.$ Daly (Can Math Bull 48:370–381, 2005) and the author showed among others that this is the situation for the Hörmander–Mihlin condition with $\,r>1.$ 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 $\,\mathcal H^p_{2\pi }\, (0<p<1)\,$ is more adequate than that of the Lebesgue spaces in this regard. Namely, for any $\,r>1\,$ we give a sharp bound $\,p_r<1\,$ such that if $\,p>p_r\,$ then the Hörmander–Mihlin condition of order $\,r\,$ is sufficient on $\mathcal H^p_{2\pi }$.