We study the behavior of Fourier integrals summed by the symbols of elliptic operators and pointwise convergence of Fourier inversion. We consider generalized localization principle which in classical Lp spaces was investigated by Sjölin (1983), Carbery and Soria (1988, 1997) and Alimov (1993). Proceeding these studies, in this paper, we establish sharp conditions for generalized localization in the class of finitely supported distributions.

1. Introduction

In this paper, we study the behavior of spherical Fourier integrals and pointwise convergence and summability of Fourier inversion.

Let
(1.1)A(D)=∑|α|=mcαDα
be a homogeneous elliptic differential operator of order m. Let us consider its symbol defined as polynomial:
(1.2)A(x)=∑|α|=mcαxα,
and assume that the Gaussian curvature of surface S={x∈Rn:A(x)=1} is always strictly positive.

We recall that for f∈L2(Rn) its Fourier transform is defined as
(1.3)f^(ξ)=∫f(y)e-iyξdy
and partial Fourier integral associated with elliptic operator (1.1):
(1.4)Eλf(x)=(2π)-n∫A(ξ)<λf^(ξ)dξ
(note that throughout the paper we consider only Lebesque measure on Rn and ∫=∫Rn). For some functions, Fourier integrals do not converge pointwisely and various summation techniques are applied to recover convergence property. In this paper, we consider the method of the Riesz means. The Riesz means of order s are defined as
(1.5)Eλsf(x)=(2π)-n∫A(ξ)≤λ(1-|ξ|2λ2)sf^(ξ)eiξxdξ.

As an example, one can consider Laplacian A(D)=∑i=1n(∂2/∂xi2), and note that the level surfaces of its symbol are Euclidean spheres. Thus, Fourier inversion associated with Laplace operator has the form:
(1.6)Eλf(x)=(2π)-n∫|ξ|2<λf^(ξ)eiξxdξ
and known as spherical partial Fourier integrals. The question of Eλf(x) convergence to f(x) almost everywhere is not solved in Rn,n≥2 even for classical L2 functions and presents one of the most challenging open problems of classical harmonic analysis, and even special cases of this problem are of particular interest. One of such special cases is the problem of generalized localization, which for the first time was formulated by V. Ii'in in [1]. For convenience, we give its definition for the Riesz means Eλs.

Definition 1.1.

We say that, for the Riesz means of order s, the generalized localization principle in function class 𝔉 is satisfied, if for any function f∈𝔉, the equality
(1.7)limλ→∞Eλsf(x)=0
is true for a.e. x∈Rn∖
supp
f.

This localization principle generalizes the classical Riemann localization principle and for Lp functions was intensively investigated by Sjölin [2], Carbery and Soria [3, 4], Bastis [5–7], and Ashurov et al. [8]. It was established that Rn localization holds true in Lp, where p∈[2,2n/(n-1)] and fails otherwise.

Over the last several years, a number of Fourier inversion studies considered distributions and investigated the behavior of their Fourier integrals (see, e.g., [9–12]). In particular, Alimov in [13] considered the classical Riemann localization principle for compactly supported distributions and established criteria for its validity (see also [14, 15]).

In this paper, we study generalized localization principle for compactly supported distributions and present conditions for its fulfillment.

2. Notation and Definitions

We define Schwartz space S(Rn) as the function class of all infinitely differentiable functions that are rapidly decreasing at infinity along with all partial derivatives. It is well known that S(Rn), being equipped with a family of seminorms
(2.1)dα,β(ϕ)=supx∈Rn|xαDβϕ(x)|,
is a Frechet space (here α,β are multi-indices and D is a partial derivative). As usual, we also consider class of tempered distributions S′ defined as dual to S.

Let ℰ be the space of infinitely differentiable functions with topology τE such that ϕn→0 in τE if and only if for each multiindex α and compact K(2.2)supx∈KDαϕn(x)→0.
As usual we denote its conjugate space by ℰ′.

It is known (see, e.g., [16]) that each f∈ℰ′ has finite support and equivalent to the class of finitely supported tempered distributions. Thus, it follows from the Paley-Wienner theorem that, for each f∈ℰ′, its Fourier transform f^∈C∞. Since f^ is locally integrable, it is natural to define Fourier integral of f∈ℰ′ and its Riesz means by (1.4) and (1.5), respectively.

We also note that for f∈L2 the Riesz mean Eλsf can be considered as an integral operator:
(2.3)Eλsf(x)=(2π)-n∫f(y)θλs(x-y)dy,
with kernel θλs(y)=m^λs(y) where
(2.4)mλs(y)=(1-A(y)λ)+s,
where (1-A(y)/λ)+s=(1-A(y)/λ)s·χA(y)<λ(y).

Representation (2.3) has its natural analogue for f∈ℰ′. Let ψn be a sequence of Schwartz functions such that ψn(y)=0 as |y|>λ and ψn(y)→mλs(y) in L1 norm. Then:
(2.5)Eλsf(x)=limn→∞(2π)-n∫f^(ξ)ψn(ξ)eixξdξ=(2π)-nlimn→∞〈f^(ξ),ψn(ξ)eixξ〉=(2π)-nlimn→∞〈f(y),ψ^n(x-y)〉.
Note that inequality ∥g^∥∞≤∥g∥1 implies that ψ^n→m^λs in ℰ and since f is continuous on ℰ(2.6)Eλsf(x)=(2π)-n〈f(·),θλs(x-·)〉.

We will need Sobolev's classes which can be defined for l∈ℝ in the following way.

Definition 2.1.

We say that tempered distribution f belongs to Sobolev class Hl if f^ is a regular distribution such that
(2.7)∥f∥Hl2=∫|f^(ξ)|2(1+|ξ|2)ldξ<∞.

One can see that, in particular, H0=L2. We also remark that for every f∈ℰ' there is l∈ℝ such that f∈Hl (for proof see, e.g., [16]).

In other respects, we make the following conventions:

symbol Jν is used to denote Bessel function of the first kind and order ν≥0,

χE is preserved for an indicator function of E⊂Rn,

unless otherwise indicated, all functions are assumed to be defined on Rn and by definition Lp(Ω)≡{f∈Lp(Rn):suppf⊂Ω⊂Rn}.

3. Main Result

As has been mentioned above, every f∈ℰ′ belongs to some Sobolev classes Hl, in this paper, we use this fact to establish criterion of generalized localization for finitely supported distributions. The following theorems present major results of current study.

Theorem 3.1.

Let f∈ℰ′∩H-l, l≥0. Then, for integer s≥l, equality
(3.1)limλ→∞Eλsf(x)=0
holds true a.e. on Rn∖
supp f.

Our approach is based on the methods by Carbery and Soria [3] and in order to prove Theorem 3.1, we will follow his idea first proving some auxiliary facts in the following section.

4. Dual Sets

Let a(x)=[A(x)]1/m and K={x∈Rn:a(x)≤1}. Then, K is a symmetric body that is convex compact symmetric set. We recall that set K*={y:|x·y|≤1,∀x∈K} is called polar set with respect to K.

As it is done in [17], we will introduce the norm ∥·∥a generated by a(x) as
(4.1)∥x∥a=a(x)
and dual norm ∥·∥a* as
(4.2)∥y∥a*=sup∥x∥a≤1|x·y|=sup∥x∥a=1|x·y|.
Next, let S and S* be the boundaries of K and K*, respectively.

It is not difficult to show that S*={∇a(x),x∈S}. Indeed on the one hand a(λx)=λa(x) and, therefore, for x∈S(4.3)x·∇a(x)=dadr(x)=a(x)=1,(-x)·∇a(x)=-dadr(x)=a(x)=-1,
which means that ∥∇a(x)∥a*≥1. On the other hand, for any y∈S, one can consider F(y)=y·∇a(x) and examine its local extremums on the surface S. Since S is compact, F(y) reaches its extremum values and it is known that, at extremum points, ∇F(y) must be parallel to the normal to S at point y, which is parallel to ∇a(y). Since ∇F(y)=∇a(x), we can conclude that ∇a(x)∥∇a(y) at the extremum points. Since S is strictly convex, it is possible only for y=±x, that implies ∥∇a(x)∥a*≤1.

It is convenient for given x∈Rn to use the notation θ(x) to denote the point on S such that the outer normal to S at θ(x) is parallel to x. Similarly, we denote η(x) the point on S* such that the outer normal to S* at η(x) is parallel to x. One can remark that we have just seen that for y∈S*(4.4)y·θ(y)=1.

5. Technical Lemmas for Theorem <xref ref-type="statement" rid="thm1">3.1</xref>

We will need the asymptotic representation of θλs(y), which can be derived by stationary-phase method (see, e.g., [18]):
(5.1)θλs(y)=λ(n-1)/2|y|-(n+1)/2·[R+s(y,λ)eiλy·θ(y)+R-s(y,λ)e-iλy·θ(y)],
where functions R±s(y,λ)∈C∞({y:|y|>ϵ}×[1,∞)) and
(5.2)DyαDλβR±s(y,λ)=O(λ-s-β),
uniformly on |y|>δ and λ>δ.

Now, let us consider positive numbers ɛ and R, ɛ<R and function ϕ(x)=ϕ(∥x∥a*)∈C0∞ vanishing on {x:(∥x∥a*<ɛ)∨(∥x∥a*>R)}. Then, for s≥0, we set by definition
(5.3)Θλs(x)=ϕ(x)θλs(x),
where θλs as in (2.3).

We will need some estimates for the Fourier transform of Θλs. With this aim, we will need the following lemmas.

Lemma 5.1.

Let t≥δ>0 and |ξ|<1. Then, for any α>0(5.4)|Θ^ts(ξ)|≤O(t-α).

Proof.

This estimate easily follows from the definition of Θts. Indeed,
(5.5)Θ^ts(ξ)=∫θts(x)ϕ(x)e-iξxdx=∫(1-A(y)t)+s^(x)ϕ(x)e-iξxdx=∫(1-A(x)t)+sϕ^(x+ξ)dx=∫A(x)<tϕ^(x+ξ)dx+∑k=1sCktk∫A(x)<tϕk(·,ξ)^(x)dx,
where ϕk(y,ξ)=Bk(Dy)[ϕ(y)e-iyξ] and B(D) is formally conjugate to operator A(D). Since ϕ(0)=ϕk(0,ξ)=0,
(5.6)Θ^ts(ξ)=∫A(x)>tϕ^(x+ξ)dx+∑k=1sCktk∫A(x)>tϕk(·,ξ)^(x)dx.
Further, we notice that since ϕk(y,ξ)∈C0∞ then for any α>0 there is Cα such that functions ϕ^k(x,ξ)=Cα/(1+A(x))α, uniformly for k=1,…,s and |ξ|<1. For the same reason, for any α>0, one has ϕ^(x)≤O((1+x)-α). Now substituting these estimates into (5.6), we complete the proof.

Lemma 5.2.

Let t≥δ>0 and |ξ|≥1. Then, for any α>0,
(5.7)|Θ^ts(ξ)|=O(1)t-s(1+|∥ξ∥a-t|)α.

Proof.

By definition,
(5.8)Θ^ts(ξ)=∫ϵ<∥y∥a*<Rϕ(y)θts(y)e-iξ·ydy.
Let us pass to a new coordinate system y→(r=∥y∥a*,η=η(y)). Then,
(5.9)Θ^ts(ξ)=∫ϵRϕ(r)rn-1∫η∈S*θts(rη)e-irξ·ηdσ(η)dr,
where dσ(η) is a Lebesgue surface measure of S*.

Using (5.1), we have
(5.10)Θ^ts(ξ)=t(n-1)/2∫ϵRϕ(r)r(n-3)/2∫η∈S*|η|-(n+1)/2eitr[η·θ(η)]R~+s(rη,t)e-irξ·ηdσ(η)dr+t(n-1)/2∫ϵRϕ(r)r(n-3)/2∫η∈S*|η|-(n+1)/2e-itr[η·θ(η)]R~-s(rη,t)e-irξ·ηdσ(η)dr.
We will focus on the first term since the second one can be handled alike
(5.11)Its(ξ)=t(n-1)/2∫ϵRϕ(r)r(n-3)/2∫η∈S*|η|-(n+1)/2eitr[η·θ(η)]R~+s(rη,t)e-irξ·ηdσ(η)dr
and note that due to (4.4) η·θ(η)=1, and thus
(5.12)Its(ξ)=t(n-1)/2∫ϵRϕ(r)r(n-3)/2eitr∫η∈S*|η|-(n+1)/2R~+s(rη,t)e-irξ·ηdσ(η)dr.
One can use the expression ξ=∥ξ∥aθ(ξ) and employ stationary phase method to obtain
(5.13)∫η∈S*|η|-(n+1)/2R~+s(rη,t)e-irξ·ηdσ(η)=∥ξ∥a-(n-1)/2eir∥ξ∥a[θ(ξ)·η(θ(ξ))]P+s(rξ,t)+∥ξ∥a-(n-1)/2e-ir∥ξ∥a[θ(ξ)·η(θ(ξ))]P-s(rξ,t),
where P±s are smooth functions such that DtαDzβP±s(z,t)=O(t-s-α). Using this expression, we have
(5.14)Its(ξ)=(t∥ξ∥a)(n-1)/2∫ϵRϕ(r)r(n-3)/2eir(t-∥ξ∥a)P+s(rξ,t)dr+(t∥ξ∥a)(n-1)/2∫ϵRϕ(r)r(n-3)/2e-ir(t-∥ξ∥a)P-s(rξ,t)dr.

Further integrating by parts the integrals, one can see that for any N>0 both integrals are controlled by (CNt-s)/(1+|t-∥ξ∥a|)N. As a result, we have
(5.15)|Its(ξ)|≤(t∥ξ∥a)(n-1)/2CNt-s(1+|t-∥ξ∥a|)N≤DNt-s(1+|t-∥ξ∥a|)N-((n-1)/2),
uniformly for |ξ|>1 and t>δ. Finally, substituting into (5.10), we obtain (5.7).

Now, combining Lemmas 5.1 and 5.2, we can claim that, in fact, for t>δ and any ξ∈Rn,
(5.16)|Θ^ts(ξ)|≤O(1)t-s(1+|∥ξ∥a-t|)α.

Lemma 5.3.

Let Θλs(x) be defined by (5.3). Then, for any δ>0 there is Cδ>0 such that
(5.17)∫δ∞|Θ^ts(ξ)|2dt≤Cδ(1+|ξ|)2s.

Proof.

As it follows from (5.16),
(5.18)∫δ∞|Θ^ts(ξ)|2dt≤O(1)∫δ∞t-2sdt(1+|∥ξ∥a-t|)2α,
where α>0 can be chosen arbitrary large. Changing the variables u=ξ-t, one has
(5.19)∫δ∞|t|-2sdt(1+|∥ξ∥a-t|)2α=∫δ<|∥ξ∥a-u|<∥ξ∥a/2|∥ξ∥a-u|-2sdt(1+|u|)2α+∫max(δ,∥ξ∥a/2)<|∥ξ∥a-u||∥ξ∥a-u|-2sdt(1+|u|)2α.
It is not difficult to see that for the values u in the first integral |u|>|∥ξ∥a-∣u-∥ξ∥a|>∥ξ∥a/2, and thus choosing α>max(2s,1)(5.20)∫δ<|∥ξ∥a-u|<∥ξ∥a/2|∥ξ∥a-u|-2sdt(1+|u|)2α≤O(1)(1+∥ξ∥a)α≤O(1)(1+∥ξ∥a)2s.
Moreover, it is clear that for such α(5.21)∫δ<|∥ξ∥a-u|<∥ξ∥a/2|∥ξ∥a-u|-2sdt(1+|u|)2α≤O(1)(1+∥ξ∥a)2s.

Therefore,
(5.22)∫δ∞|Θ^ts(ξ)|2dt≤O(1)(1+∥ξ∥a)2s.

Since all norms in Rn are equivalent, the lemma is proved.

Lemma 5.4.

Let Θλs be defined by (5.3). Then, for any δ>0, there is Cδ such that
(5.23)∫δ∞|ddtΘ^ts(ξ)|2dt≤Cδ(1+|ξ|)2s.

Proof.

For any t>1, using the Fubini theorem, one has
(5.24)∫1t∫dduΘus(y)e-iξydydu=∫e-iξy∫1tdduΘus(y)dudy=Θ^ts(ξ)-Θ^1s(ξ),
which implies (d/dt)Θ^ts(ξ)=(d/dt)Θts^(ξ).

If s>0(5.25)ddtΘts(x)=ϕ(x)ddtθts(x)=ϕ(x)ddt∫0t(1-u2t2)sdθu(x)=2sϕ(x)t∫0t(1-u2t2)s-1u2t2dθu(x)=2st(Θts-1(x)-Θts(x)).
Thus, using inequality (a+b)2≤2a2+2b2, one has
(5.26)∫δ∞|ddtΘ^ts(ξ)|2dt≤C∫δ∞t-2|Θ^ts-1(ξ)|2dt+C∫δ∞t-2|Θ^ts(ξ)|2dt.
Now, one can use estimate (5.16) to each integral on the right side and complete the proof.

If s=0, then for any ξ∈Rn,
(5.27)|ddtΘ^t(ξ)|=|ddt∫A(y)≤tϕ^(ξ+y)dy|=|∫A(y)=tϕ^(ξ+y)n(y)dσ(y)|≤O(1)(1+|∥ξ∥a-t|)α,∀α>0.
Using this estimate and the reasoning presented in the previous lemma, we obtain the required estimate.

6. Proof of Theorem <xref ref-type="statement" rid="thm1">3.1</xref>

Let f∈H-l∩ℰ' be such that suppf⊂Ω. For 0<ϵ<1/2, we set
(6.1)Eϵ={x:2ϵ<dist(x,Ω)<(2ϵ)-1}
and consider an arbitrary radial function ϕϵ∈C0∞ such that
(6.2)ϕϵ(x)={1,3ϵ2≤|x|≤1ϵ+diamΩ;0,|x|≤ϵ.

It is clear that to prove the theorem it is sufficient to show that for any ϵ>0, limλ→∞Eλsf(x)=0, a.e. x∈Eϵ

In this case, as x∈Eϵ due to (2.6)
(6.3)Eλsf(x)=∫f^(ξ)[χΩ(·)θλs(x-·)].^(-ξ)dξ=∫f^(ξ)[ϕϵ(x-·)θλs(x-·)].^(-ξ)dξ=∫f^(ξ)[ϕϵθλs].^(ξ)eiξxdξ,
or using notation (5.3)
(6.4)Eλsf(x)=∫f^(ξ)Θ^λs(ξ)eiξxdξ=(f^(ξ)Θ^λs(ξ))^(-x).

Further, we consider maximal operator:
(6.5)E*sf(x)=supλ>1|Eλsf(x)|.
We recall that to prove a.e. convergence on Eϵ one can use the standard technique of Banach principle (see, e.g., [19]) according to which it is sufficient to estimate maximal operator on Eϵ⊂Rn∖supp f as
(6.6)∥E*sf(x)∥L2(Eϵ)≤C∥f∥H-l.

Let γ(t):ℝ→ℝ+ be a C∞ function such that
(6.7)γ(t)={0,t≤13;1,t≥23.
If we set E~λsf(x)=γ(λ)Eλsf(x), then by (6.4),
(6.8)E~λsf(x)=γ(λ)<f(·),Θλs(x-·)>=γ(λ)(f^(ξ)Θ^λs(ξ))^(-x).
According to Sobolev's embedding theorem (see, e.g., [20]) for any f∈H1(R1),
(6.9)∥f∥L∞≤C∥f∥H1.
Using this fact, we have
(6.10)E*sf(x)≤∥E~λsf(x)∥L∞(R)≤∥E~λsf(x)∥H1(R).
And, therefore, in order to obtain (6.6), it is sufficient to show that there are constants C1,C2 such that the following estimates are true:
(6.11)∫∥E~λsf(x)∥L2(R)2dx≤C1∥f∥H-l,∫∥ddλE~λsf(x)∥L2(R)2dx≤C2∥f∥H-l.

First, we note that estimate (5.16) and f∈H-l imply that f^Θ^λs∈L2 which in turn with (6.8) implies the fact E~λsf∈L2.

Further, using the Plancherel theorem, we have
(6.12)∫∫|E~λsf(x)|2dλdx≤∫γ2(λ)∫|f^(ξ)|2|Θ^λs(ξ)|2dξdλ≤∫1/3∞γ2(t)∫(1+|ξ|2)l|Θ^ts(ξ)|2(1+|ξ|2)-l|f^(ξ)|2dξdt≤supξ∈Rn(1+|ξ|2)l∫1/3∞|Θ^ts(ξ)|2dt×∥f∥H-l2≤C∥f∥H-l2
(the last inequality follows from Lemma 5.3).

For the same reason, (6.11) can be proved using Lemmas 5.3 and 5.4:
(6.13)∫∫1/3∞|ddt[γ(t)Θts]*f(x)|2dtdx≤∥γ'2(λ)∥∞∫∫1/3∞|Θts*f(x)|2dtdx+∫∫1/3∞|ddλΘλs*f(x)|2dλdx≤C∥f∥H-l2.

Acknowledgments

The authors are thankful to the University Putra Malaysia for the support under RUGS (Grant no. 05-03-11-1450RU). A. Butaev would also like to expresses his gratitude to for the support under IGRF scheme.

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