AbstractWe describe the idempotent Fourier multipliers that act contractively on $$H^p$$
H
p
spaces of the d-dimensional torus $$\mathbb {T}^d$$
T
d
for $$d\ge 1$$
d
≥
1
and $$1\le p \le \infty $$
1
≤
p
≤
∞
. When p is not an even integer, such multipliers are just restrictions of contractive idempotent multipliers on $$L^p$$
L
p
spaces, which in turn can be described by suitably combining results of Rudin and Andô. When $$p=2(n+1)$$
p
=
2
(
n
+
1
)
, with n a positive integer, contractivity depends in an interesting geometric way on n, d, and the dimension of the set of frequencies associated with the multiplier. Our results allow us to construct a linear operator that is densely defined on $$H^p(\mathbb {T}^\infty )$$
H
p
(
T
∞
)
for every $$1 \le p \le \infty $$
1
≤
p
≤
∞
and that extends to a bounded operator if and only if $$p=2,4,\ldots ,2(n+1)$$
p
=
2
,
4
,
…
,
2
(
n
+
1
)
.
Idempotent Fourier multipliers acting contractively on $$H^p$$ spaces
