Idempotent Fourier multipliers acting contractively on $$H^p$$ spaces

Abstract

We 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 ) .

Contractive projections in Paley-Wiener spaces

“…Recently, O.F. Brevig, J. Ortega-Cerdà, and K. Seip [2] studied the contractivity of the similar idempotent Fourier multipliers in the case when X is a Hardy space, that is…”

“…They showed that if p / ∈ 2N then the only contractions are the same as in the result of Andô, while for p = 2k, k ∈ N there exist non-trivial examples if d ≥ 3. For the complete statement of their results, see [2], Theorem 1.2.…”

“…Since the Hardy spaces are the subsets of the Lebesgue spaces on the torus with some restrictions on the Fourier transform, it is natural to consider the analogue of the problem solved in [2] in the setting of Euclidean spaces. Specifically, in this note we consider the operators acting on the Paley-Wiener spaces.…”

“…The proof of the general case is done exactly in the same way and therefore is omitted. We follow the argument from [2] (see Lemma 3.1 there) and invoke the criterion due to H.S. Shapiro from [5].…”

“…Second, we assume that for the sets S 1 and S 2 equation (1) does not hold and prove that the projection P : P W p S1∪S2 → P W p S1 , is not a contraction. We use again Lemma 1 and reduce the problem to the construction of functions f and g that violate condition (2). Recall that p = 2k, k ∈ N, and set…”