On the Failure of Multilinear Multiplier Theorem with Endpoint Smoothness Conditions
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On the Failure of Multilinear Multiplier Theorem with Endpoint Smoothness Conditions Bae Jun Park1 Received: 27 March 2020 / Accepted: 4 September 2020 / © Springer Nature B.V. 2020
Abstract We study a multilinear version of the H¨ormander multiplier theorem, namely (n) 2 Tσ (f1 , . . . , fn )Lp sup σ (2k ·, . . . , 2k ·)φ L
(s1 ,...,sn )
k∈Z
f1 H p1 · · · fn H pn .
We show that the estimate does not hold in the limiting case min (s1 , . . . , sn ) = d/2 or k∈J (sk /d − 1/pk ) = −1/2 for some J ⊂ {1, . . . , n}. This provides the necessary and sufficient condition on (s1 , . . . , sn ) for the boundedness of Tσ . Keywords H¨ormander multiplier theorem · Multilinear operator · Sobolev space with sharp regularity conditions Mathematics Subject Classification (2010) 42B15 · 46E35
1 Introduction Let n be a positive integer and σ be a bounded function on (Rd )n . The n-linear multiplier operator Tσ associated with σ is defined by ⎛ ⎞ n n σ (ξ1 , . . . , ξn ) ⎝ f j (ξj )⎠ e2πix, j =1 ξj dξ1 · · · dξn Tσ f1 , . . . , fn (x) := (Rd )n
j =1
for Schwartz functions f1 , . . . , fn in where f (ξ ) := Rd f (x)e−2πix,ξ dx is the Fourier transform of f . The study on Lp1 × · · · × Lpn → Lp boundedness of Tσ , 1/p = 1/p1 + · · · + 1/pn , is one of principal questions in harmonic analysis as a multilinear extension of classical Fourier multiplier theorems and there have been many attempts to characterize σ for which the boundedness holds. A multilinear version of the H¨ormander S(Rd ),
The author is supported in part by NRF grant 2019R1F1A1044075 and by a KIAS Individual Grant MG070001 at Korea Institute for Advanced Study. Bae Jun Park
[email protected] 1
School of Mathematics, Korea Institute for Advanced Study, Seoul, Republic of Korea
B.J. Park
multiplier theorem was studied by Tomita [11] who obtained Lp1 × · · · × Lpn → Lp boundedness (1 < p1 , . . . , pn , p < ∞) under the condition (n) sup σ (2j ·1 , . . . , 2j ·n )φ < ∞, s > nd/2. (1.1) L2 ((Rd )n ) s
j ∈Z
Here L2s ((Rd )n ) denotes the standard fractional Sobolev space on (Rd )n and φ (n) is a (n) ) ⊂ {ξ := (ξ , . . . , ξ ) ∈ Schwartz function on (Rd )n having the properties that Supp(φ 1 n d n −1 k (n) (R ) : 2 ≤ |ξ | ≤ 2} and k∈Z φ (ξ /2 ) = 1 for ξ = 0. Grafakos and Si [6] extended the result of Tomita to the case p ≤ 1, using Lr -based Sobolev spaces with 1 < r ≤ 2. Fujita and Tomita [1] provided weighted extensions of these results for 1 < p1 , . . . , pn , p < ∞ with j j (n) L2,(n) < ∞, (1.2) (s1 ,...,sn ) [σ ] := sup σ (2 ·1 , . . . , 2 ·n )φ L2 ((Rd )n ) j ∈Z
(s1 ,...,sn )
L2(s1 ,...,sn ) ((Rd )n )
is a product-type Sobolev for s1 , . . . , sn > d/2, instead of (1.1), where space with the norm ⎛ ⎞ ⎞1/2 ⎛ n ⎝ (1 + 4π 2 |xj |2 )sj ⎠ |f (x1 , . . . , xn )|2 dx1 · · · dxn ⎠ . ⎝ f L2 ((Rd )n ) := (s1 ,...,sn )
(Rd )n
j =1
The range of p in this result was extended by Grafakos, Miyachi, and Tomita [2]. In [7] Miyachi and Tomita obtained minimal conditions of s1 , s2 in (1.2) for the H p1 ×
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