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NOTES ON RADIAL AND QUASIRADIAL FOURIER MULTIPLIERS ANDREAS SEEGER This is an updated version of notes that were originally prepared for lectures at the Summer School on Nonlinear Analysis, Function Spaces and Applications (NAFSA 10), in Tˇreˇst',Czech Republic. They are based on material in various joint papers ([1], [20], [26], [27], [34], [36], [37]). The notes could not have been written without the contribu- tions of my coauthors Jong-Guk Bak, Gustavo Garrig´os,Yaryong Heo, Sanghyuk Lee, Fedya Nazarov, and Keith Rogers. Any deficiencies and errors in these notes are my responsibility. 1. Fourier multipliers Given a translation invariant operator T which is bounded from S(Rd) to some Lq(Rd) there is a tempered distribution K so that d q T f = K ∗ f for all f 2 S(R ) (see [53]). We denote by Convp the space of all K 2 S0(Rd) such that kK ∗ fkq ≤ Ckfkp for all f 2 S(Rd), and set q kKkConvp = supfkK ∗ fkq : f 2 S; kfkp = 1g The convolution operators can also be written as a multiplier transfor- mation K ∗ f = F −1[mfb] where the Fourier transform of a Schwartz function in Rd is given by Z Ff(ξ) = fb(ξ) = f(y)e−ihy,ξidy and F −1[g](x) = (2π)−d R g(ξ)eihx,ξidξ is the inverse Fourier transform q of g. The space of Fourier transforms of distributions in Convp is q denoted by Mp and the Fourier transform is an isometric isomorphism q q q −1 q F : Convp ! Mp , i.e. kmkMp = kF [m]kConvp . We also abbreviate Date: July 1, 2018. 1991 Mathematics Subject Classification. 42B15. Supported in part by an NSF grant. 1 2 ANDREAS SEEGER p p Convp = Convp and Mp = Mp . Effective characterizations of the spaces q q Convp or Mp are known only in a few cases, see [28], [53]: 1 • Conv1 is the space of finite Borel measures (with the total variation norm). q q • For 1 < p ≤ 1, Conv1 = L , with identifications of the norms. 1 • M2 = L , with identifications of the norms. q q p0 1 1 1 1 Moreover, Mp = f0g for p > q, and Mp = Mq0 , p + p0 = q + q0 = 1. Of particular interest are the radial Fourier multipliers on Rd m(ξ) = h(jξj) : We shall assume throughout that d ≥ 2. The radial multipliers cor- respond to operators commuting with translations and rotations, i.e. convolution operators with radial kernels, and also to spectral multi- plier operators for the Laplace operator, via p h( −∆)f = F −1[h(j · j)fb] : A great deal of research has been done on a class of radial model mul- tipliers which are now called of Bochner-Riesz type. Given a smooth bump function φ supported in (−1=2; 1=2) we define 1−s (1.1) hδ(s) = φ( δ ) so that hδ(j · j) is supported on an annulus of width ∼ δ. 2d Conjecture I: For 1 < p < d+1 , one has −λ(p) (1.2) khδ(j · j)kMp . δ ; λ(p) = d(1=p − 1=2) − 1=2 This is referred to as the Bochner-Riesz conjecture since a resolution would prove that the Bochner-Riesz means of the Fourier integral Z λ 1 jξjλ ihx,ξi R f(x) = d 1 − fb(ξ)e dξ : (2π) jξ|≤t t converge to f in Lp(Rd) if λ > λ(p). Inequality (1.2) is known in two dimensions for the full range, and various partial results have been proved in higher dimensions (see [17],[13], [19], [5], [6], [61], [33], [8]). For the relationships of various related conjectures see [56], [57]. A somewhat more general problem involves classical function spaces 2 2 measuring regularity, in particular the L type Sobolev spaces Lα or NOTES ON RADIAL AND QUASIRADIAL FOURIER MULTIPLIERS 3 2 the Besov spaces Bα,p. The following estimate would be optimal within this class of spaces. Conjecture II: Let h be supported in a compact subinterval J of (0; 1). The estimate (1.3) kh(j · j)k khk 2 ; α = d(1=p − 1=2) Mp . Bα,p 2d holds for 1 ≤ p < d+1 . Partial results with the condition α > d(1=p − 1=2) and related square-function estimates are in [10], [12], [14], [34], and a proof of the 2 2(d+1) endpoint Bd(1=p−1=2);p bound, in the range 1 < p < d+3 can be found in [35]. Inequality (1.3) is still far from a characterization of radial multipliers (even those supported in (1=2; 2)). Consider the multiplier ht(s) = −its 1 χ(s)e where χ 2 Cc (1=2; 2). It is closely related to the solution operator for the wave equation. It is known (see [45]) that kht(j·j)kMp . (d−1)(1=p−1=2) α t for t > 1, 1 ≤ p ≤ 2, but kh k 2 t and thus the t Bα,q & optimal Mp bound cannot be derived from (1.3). An affirmative answer to the following conjecture would close this gap. 2d Conjecture III: Let 1 < p < d+1 , and let h be supported in a compact subinterval J of (0; 1). Then it −1 (1.4) kh(j · j)kMp ≈ F [h(j · j)] p (with the implicit constants depending on J)? Given that the weaker conjectures I and II are not completely re- solved, we may have, for Conjecture III, to settle for smaller ranges of 2(d−1) p. It has been shown to hold in [26] for the range d ≥ 4, p < d+1 , and no such result is currently known in dimensions 2 and 3. As in [26] the proof of this conjecture would likely lead to a simple characterization p 2d of all radial FL multipliers for p < d+1 , as in: Conjecture IV: Let m = h(j · j) be a bounded radial function on Rd and d define the convolution operator Th on R by Tdhf(ξ) = h(jξj)fb(ξ): 2d d Let 1 < p < d+1 . Let η be any nontrivial Schwartz function on R . p Then Th is bounded on L if and only if d=p sup t Th[η(t·)] p < 1 t>0 4 ANDREAS SEEGER The resolution of this conjecture would imply that in the above range p p Th is bounded on L if and only if Th is bounded on the subspace Lrad of radial Lp functions. By Fefferman’s theorem [18] on the ball multiplier 2d (or some variant of it) this is clearly false for d+1 ≤ p ≤ 2. Similar 2d arguments show that the condition p < d+1 is necessary in the three previous conjectures. Finally one can formulate a version of Conjecture IV for Lp ! Lq estimates. Here we focus on the case q ≤ 2. Conjecture V: Let m = h(j · j) be a radial function on Rd which is in 2 d 2d Lloc(R n f0g) and let Tdhf(ξ) = h(jξj)fb(ξ): Let 1 < p < d+1 , p ≤ q ≤ 2, d−1 0 d and q ≤ d+1 p . Let η be any nontrivial Schwartz function on R . Then p q Th is bounded from L to L if and only if d=p sup t Th[η(t·)] q < 1 t>0 d−1 0 2(d+1) 2d The condition q ≤ d+1 p is relevant for the range d+3 ≤ p < d+1 and cannot be dispensed with in general, by a Knapp example. One considers hδ(j · j) as in (1.1) and tests the operatorp on functionsp fδ, for which fbδ is a bump function supported in a δ × δ ×· · ·× δ rectangle addapted to the support of hδ(j · j). These notes. In x2 we consider the case of general quasiradial multipli- 2 ers in Mp and discuss a characterization for the case when the under a decay assumption on the Fourier transform of the surface measure of Σρ. When ρ is the Minkowski functional of a set with smooth bound- q ary (then P = I) we examine further the M1 classes and express the condition F −1[m] 2 Lq in terms of the one dimensional Fourier trans- form of h. This is done in x3. In x4 we discuss a sharp theorem from [20] on the Lp boundedness of convolution operators with radial kernels when acting on radial functions, and in x5 we prove the same result for radial multipliers acting on general functions. We shall focus on the case where the multipliers are supported in (1=2; 2). NOTES ON RADIAL AND QUASIRADIAL FOURIER MULTIPLIERS 5 2 2 2. L Fourier restriction theorems and quasiradial Mp multipliers In this section we shall first discuss a general version of the Stein- Tomas restriction theorem and then deduce as a consequence a straight- 2 forward characterization of radial and quasi radial multipliers in Mp , for the p-range of the L2 restriction theorem. 2.1. L2 Fourier restriction theorems. In the 1960's Stein observed that the Fourier transform of an Lp function can, for a nontrivial range of p > 1 be restricted to some compact hypersurfaces with suitable curva- ture assumptions. An almost sharp theorem for the sphere was proved by Tomas [62], with an endpoint version due to Stein [51]. Further results are in Greenleaf [23] and many other papers. Here we present the rather general setup by Mockenhaupt [43] and by Mitsis [42], with the endpoint version in a joint paper with Bak [1]. We are given a compactly supported Borel measure µ satisfying the Fourier decay bound −β=2 (2.1) jµb(ξ)j .
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