On a Problem of Pichorides
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On a Problem of Pichorides Odysseas Bakas1 Received: 24 May 2020 / Accepted: 15 October 2020 © The Author(s) 2020
Abstract Let S () denote the classical Littlewood–Paley operator formed with respect to a lacunary sequence of positive integers. Motivated by a remark of Pichorides, we obtain sharp asymptotic estimates of the behaviour of the operator norm of S () from p the analytic Hardy space H A (T) to L p (T) and of the behaviour of the L p (T) → L p (T) () operator norm of S (1 < p < 2) in terms of the ratio of the lacunary sequence . Namely, if ρ denotes the ratio of , then we prove that sup
f L p (T) =1 p f ∈H A (T)
() S ( f )
L p (T)
1 (ρ − 1)−1/2 (1 < p < 2) p−1
and () S
L p (T)→L p (T)
1 (ρ − 1)−1/2 (1 < p < 2) ( p − 1)3/2
and that these results are optimal as p → 1+ . Variants in higher dimensions and in the Euclidean setting are also obtained. Keywords Littlewood–Paley square function · Hardy spaces · Orlicz spaces · lacunary sequences Mathematics Subject Classification Primary 42B25 · 42B30 · 30H10 · Secondary 42A45 · 42B15
B 1
Odysseas Bakas [email protected] Centre for Mathematical Sciences, Lund University, 221 00, Lund, Sweden
123
O. Bakas
1 Introduction Given a strictly increasing sequence = (λ j ) j∈N0 of positive integers, consider the given by corresponding Littlewood–Paley projections () j j∈N 0
() j
:=
Sλ0 , if j = 0 Sλ j − Sλ j−1 , if j ∈ N
where, for N ∈ N, S N denotes the multiplier operator acting on functions over T with symbol χ{−N +1,...,N −1} . Define the Littlewood–Paley square function S () (g) of a trigonometric polynomial g by ⎛
⎞1/2 () 2 (g) ⎠ . S () (g) := ⎝ j
j∈N0
It is well known that in the case where = (λ j ) j∈N0 is a lacunary sequence in N, namely the ratio ρ := inf j∈N0 (λ j+1 /λ j ) is greater than 1, then S () can be extended as a sublinear L p (T) bounded operator for 1 < p < ∞; see [19] or [45] for the periodic case and [40] for the Euclidean case. In 1989, in [13], Bourgain proved that if D := (2 j ) j∈N0 , then the L p (T) → L p (T) operator norm of the Littlewood–Paley operator S ( D ) behaves like ( p − 1)−3/2 as p → 1+ , namely ( ) S D
L p (T)→L p (T)
∼ ( p − 1)−3/2 (1 < p ≤ 2).
(1.1)
Other proofs of the aforementioned theorem of Bourgain were obtained by the author in [2] and by Lerner in [30]. In 1992, in [35], Pichorides showed that if we restrict ourselves to the analytic Hardy spaces, then one has the improved behaviour ( p − 1)−1 as p → 1+ . More specifically, Pichorides proved in [35] that if = (λ j ) j∈N0 is a lacunary sequence of positive integers, then one has sup
f L p (T) =1 p f ∈H A (T)
() S ( f )
L p (T)
∼
1 (1 < p ≤ 2), p−1
(1.2)
where the implied constants in (1.2) depend only on the lacunary sequence and not on p. As remarked by Pichorides, see Remark (i) in [35, Sect. 3], if = (λ j ) j∈N0 is a lacunary sequence in N with ratio ρ ∈ (1, 2), then the argument in [35] yields that for fixed 1 < p < 2 the implied constant in
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