On Some Fourier Multipliers for $$H^p(\mathbb {R}^n)$$ H p ( R n ) with Restricted Smoothness Conditions
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On Some Fourier Multipliers for H p (Rn ) with Restricted Smoothness Conditions Jingren Qiang1 · Peng Chen2 · Shanlin Huang3
· Quan Zheng3
Received: 19 November 2018 © Mathematica Josephina, Inc. 2019
Abstract Given 0 < p < 2, we consider Mikhlin and Hörmander type multiplier theorems conditions. More precisely, we assume that on H p (Rn ) with restricted smoothness m ∈ C k (Rn \ {0}), where k = n| 1p − 21 | + 1, meanwhile, we have restrictions on the order of differentiation with respect to each coordinate. In particular, if p > 23 , we only need to differentiate at most once with respect to any single coordinate. Keywords Fourier multipliers · H p spaces · Hörmander multipliers · Interpolation Mathematics Subject Classification Primary 42B15; Secondary 42B30
1 Introduction We start by recalling two classical Fourier multiplier theorems. The first one is due to Mikhlin [12,13] (see also [14, p. 232]), who proved that if |ξ ||α|1 |D α m(ξ )| ≤ C, for ξ ∈ Rn \ {0} and all α ∈ Nn with |α|∞ ≤ 1,
B
(1.1)
Shanlin Huang [email protected] Jingren Qiang [email protected] Peng Chen [email protected] Quan Zheng [email protected]
1
School of Information, Wuhan College, Wuhan 430212, People’s Republic of China
2
Department of Mathematics, Sun Yat-sen (Zhongshan) University, Guangzhou 510275, People’s Republic of China
3
School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, People’s Republic of China
123
J. Qiang et al.
then m is a Fourier multiplier on L p (Rn ) for all 1 < p < ∞. Here and throughout n we denote nthe 1−norm and ∞−norm for a multi-index α = (α1 , . . . , αn ) ∈ N by |α|1 i=1 αi and |α|∞ sup1≤i≤n αi , respectively. The second one is due to Hörmander [7], who showed that the same result is true under the following n + 1. |D α m(ξ )|2 dξ ≤ C, for all α ∈ Nn with |α|1 ≤ sup R 2|α|1 −n 2 R 1, let a, b ≥ 0 such that b = n p · a and k = [n p ] + 1. Suppose that m(ξ ) ∈ C k (Rn ) and m(ξ ) = 0 for |ξ | ≤ 1. Furthermore suppose there exists a constant C such that |D α m(ξ )| ≤ C|ξ |−b |ξ |(a−1)|α|1 , for |α|1 ≤ k and |α|∞ ≤ 1.
(1.9)
Then m is a Fourier multiplier on L p (Rn ). (ii) Given p > 1, let c, d ≥ 0 such that c = n p · d and k = [n p ] + 1. Suppose that m is bounded, m(ξ ) ∈ C k (Rn \ {0}) and m(ξ ) = 0 for |ξ | ≥ 1. Furthermore, suppose there exists a constant C such that |D α m(ξ )| ≤ C|ξ |c |ξ |(−d−1)|α|1 , ξ = 0, for |α|1 ≤ k and |α|∞ ≤ 1.
(1.10)
Then m is a Fourier multiplier on L p (Rn ). Here we construct an explicit example to show that, for certain non-radial multipliers, Theorems 1.1 and 1.3 indeed imply better results than previous multiplier theorems of Miyachi [16] and Calderón–Torchinsky [1]. Take m ∈ C ∞ (R3 ) such that ⎧ 2 ·sin ξ3 ⎨ sin ξ1 ·sin |ξ |ξ1/2 |ξ |1/2 , if |ξ | > 1, m(ξ ) = |ξ |3/2 ⎩ 0, if |ξ | ≤ 21 . Then a direct computation yields |∂i m(ξ | ≤
C C , i = 1, 2, 3 and |∂i j m(ξ | ≤ , i = j. |ξ | |ξ |2
However, the following estimates are not true. 2 |∂11 m(ξ ) ≤
C , |ξ |2
3 |∂12
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