# Ulam Problem for the Cosine Addition Formula in Sato Hyperfunctions

Authors: Jaeyoung Chung; Soon-Yeong Chung
Journal: Journal of Function Spaces
 (2015)

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## Abstract

We prove the Ulam problem for the cosine addition formula in the spaces of Schwartz distributions and Sato hyperfunctions with respect to bounded distributions and bounded hyperfunctions.


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## Body

## 1. Introduction

The Ulam problem for functional equations goes back to 1940 when Ulam proposed the following [1].Let
f
be a mapping from a group
G
1
to a metric group
G
2
with metric
d
(
·
,
·
)
such that
(1)
d
f
x
y
,
f
x
f
y
≤
ɛ
.
Then does there exist a group homomorphism
h
and
δ
ϵ
>
0
such that
(2)
d
f
x
,
h
x
≤
δ
ϵ
for all
x
∈
G
1
?This problem was solved affirmatively by Hyers under the assumption thatG2 is a Banach space (see Hyers [2], Hyers et al. [3]). In 1949–1951, this result was generalized by Aoki [4] and Bourgin [5, 6]. Since then Ulam problems of many other functional equations have been investigated [7–16]. Among the results, Székelyhidi has developed his idea of using invariant subspaces of functions defined on a group or semigroup in connection with Ulam problem for cosine functional equations [17, 18]. As a direct consequence of the elegant results of Székelyhidi, it was obtained that if f,g:Rn→C satisfy
(3)gx+y-gxgy+fxfy≤M,x,y∈Rn
for some M>0, then either there exist λ1, λ2∈C not both zero (either λ1 or λ2 is not zero), and N>0 such that
(4)λ1fx-λ2gx≤N
for all x∈Rn, or
(5)gx+y-gxgy+fxfy=0
for all x, y∈Rn. Furthermore, the functions f and g satisfying both (3) and (4) are investigated.In 1950, Laurent Schwartz introduced the theory of distributions in his monographThéorie des distributions [19]. In this book Schwartz systematizes the theory of generalized functions, basing it on the theory of linear topological spaces, relates all the earlier approaches, and obtains many important results. After his elegant theory appeared, many important concepts and results on the classical spaces of functions have been generalized to the space of distributions. For example,positive functions and positive-definite functions have been generalized topositive distributions and positive-definite distributions, respectively, and it is shown that every positive distribution is a positive measure [20, page 38] and every positive-definite distribution is the Fourier transform of positive measure μ such that ∫(1+|x|)-pdμ<∞ for some p≥0 [21, page 157], which is calledBochner-Schwartz theorem and is a natural generalization of the famousBochner theorem stating that every positive-definite function is the Fourier transform of a positive finite measure. For other examples,the Paley-Wiener theorem has been generalized tothe Paley-Wiener-Schwartz theorem which characterizes the distributions with bounded supports [20, page 181]. The main purpose of this paper is to prove a Hyers-Ulam type stability problem for the cosine functional equation inSchwartz distributions andGelfand hyperfunctions. As a generalization of inequality (3), it is very natural to consider the inequality
(6)gx+y-gxgy+fxfy∈L∞R2n,
where f and g are Lebesgue measurable functions and L∞(R2n) is the space of all bounded measurable functions defined in R2n. Note that inequality (6) means that inequality (3) holds in almost everywhere sense. In [22–26], some stability problems of several functional equations including inequality (6) were considered in various spaces of generalized functions including Schwartz distributions. In [24–26], for example, replacing f and g by distributions u and v in inequality (6) we have considered
(7)v∘S-v⊗v+u⊗u∈L∞R2n,
where S(x,y)=x+y, x, y∈Rn, ∘ and ⊗ denote the pullback and tensor product of generalized functions. Inequality (7) cannot be considered as a complete formulation in the sense of generalized functions because the differences are assumed to be classical bounded measurable functions and all the previous results in [24–26] have the same formulations as in (7).Due to Schwartz [19] the space L∞(Rn) of bounded measurable functions was generalized to the space DL∞′(Rn) of bounded distributions. Taking the above generalizations into account, it is very natural and is a complete generalization to consider the stability problem for cosine functional equation in distributions and hyperfunctions u,v with respect to bounded distributions and bounded hyperfunctions, respectively:
(8)v∘S-v⊗v+u⊗u∈DL∞′R2n,(9)v∘S-v⊗v+u⊗u∈AL∞′R2n,
where DL∞′(R2n) and AL∞′(R2n) are the spaces of bounded distributions and bounded hyperfunctions, respectively, and S, ∘, and ⊗ are the same as in (7). For some related results in Schwartz distributions, we refer the reader to [19, 20, 22, 23, 27–29].The main tool of solving (8) and (9) isthe heat kernel method initiated by Matsuzawa which represents the generalized functions as the initial values of solutions of the heat equation with some growth conditions [30–33].


## 2. Spaces of Distributions and Hyperfunctions and Some Preliminary Results

We first introduce the spacesS′ of Schwartz tempered distributions and the space G′ of Gelfand hyperfunctions (see [19–21, 33, 34] for more details of these spaces). We use the notations |α|=α1+⋯+αn, α!=α1!⋯αn!, |x|=x12+⋯xn2, xα=x1α1⋯xnαn, and ∂α=∂1α1⋯∂nαn, for x=(x1,…,xn)∈Rn, α=(α1,…,αn)∈N0n, where N0 is the set of nonnegative integers and ∂j=∂/∂xj.Definition 1 (see [19]).
One denotes byS or S(Rn) the Schwartz space of all infinitely differentiable functions φ in Rn such that
(10)φα,β=sup⁡xxα∂βφx<∞
for all α, β∈N0n, equipped with the topology defined by the seminorms ·α,β. The elements of S are called rapidly decreasing functions and the elements of the dual space S′ are called tempered distributions.Definition 2 (see [21, 34]).
One denotes byG or G(Rn) the Gelfand space of all infinitely differentiable functions φ in Rn such that
(11)φh,k=sup⁡x∈Rn,α,β∈N0n|xα∂βφ(x)|h|α|k|β|α!1/2β!1/2<∞
for some h, k>0. One says that φj→0 as j→∞ if φjh,k→0 as j→∞ for some h, k and denotes by G′ the strong dual space of G and calls its elements Gelfand hyperfunctions.As a generalization of the spaceL∞ of bounded measurable functions, Schwartz introduced the space DL∞′ of bounded distributions as a subspace of tempered distributions.Definition 3 (see [19]).
One denotes byDL1(Rn) the space of smooth functions on Rn such that ∂αφ∈L1(Rn) for all α∈N0n equipped with the topology defined by the countable family of seminorms
(12)φm=∑|α|≤m∂αφL1,m∈N0.
One denotes by DL∞′ the strong dual space of DL1 and calls its elements bounded distributions.Generalizing bounded distributions the spaceAL∞′ of bounded hyperfunctions has been introduced as a subspace of G′.Definition 4 (see [32]).
One denotes byAL1 the space of smooth functions on Rn satisfying
(13)φh=sup⁡α∂αφL1h|α|α!<∞
for some constant h>0. One says that φj→0 in AL1 as j→∞ if there is a positive constant h such that
(14)sup⁡α∂αφjL1h|α|α!⟶0asj⟶∞.
One denotes by AL∞′ the strong dual space of AL1.It is well known that the following topological inclusions hold:(15)G↪S↪DL1,DL∞′↪S′↪G′,G↪AL1↪DL1,DL∞′↪AL∞′↪G′.It is known that the spaceG(Rn) consists of all infinitely differentiable functions φ(x) on Rn which can be extended to an entire function on Cn satisfying
(16)φx+iy≤Cexp⁡-ax2+by2,x,y∈Rn
for some a, b, C>0 (see [34]).Definition 5.
Letuj∈G′(Rnj) for j=1,2. Then the tensor product u1⊗u2 of u1 and u2, defined by
(17)u1⊗u2,φx1,x2=u1,u2,φx1,x2
for φ(x1,x2)∈G(Rn1×Rn2), belongs to G′(Rn1×Rn2).For the proof of our theorems we employ then-dimensional heat kernel Et(x) given by
(18)Etx=4πt-n/2exp⁡-x24t,t>0.
In view of (16), we can see that the heat kernel Et belongs to the Gelfand space G(Rn) for each t>0. Thus, for each u∈G′(Rn), the convolution (u*Et)(x)∶=〈uy,Et(x-y)〉 is well defined. We call (u*Et)(x)the Gauss transform of u. From now on we denote by u~(x,t) the Gauss transform of u. It is well known that the Gauss transform u~(x,t) is a smooth solution of the heat equation such that u~(x,t)→u in weak star topology as t→0+; that is,
(19)u,φ=lim⁡t→0+∫‍u~x,tφxdx
for all φ∈G.For the proof of our main result the following results are very useful. The proof of Theorem 2.3 of [35] works even when p=∞; that is, we obtain the following.Lemma 6 (see [35]).
The Gauss transformu~(x,t)∶=(u*E)(x,t) of u∈DL∞′(Rn) is a smooth solution of the heat equation (Δ-∂/∂t)u~=0 satisfying the following.(i)
There exist constantsC>0, N≥0 such that
(20)u~x,t≤Ct-N∀x∈Rn,t>0.(ii)
u~(x,t)→u as t→0+ in the sense that, for every φ∈DL1,
(21)u,φ=lim⁡t→0+∫‍u~x,tφxdx.Conversely, every smooth solution u~(x,t) of the heat equation satisfying the estimate (20) can be uniquely expressed as u~(x,t)=(u*E)(x,t) for some u∈DL∞′(Rn).The following lemma is a special case of Theorem 3.5 of [32] when p=∞ where the space AL∞′(Rn) is denoted by BL∞(Rn).Lemma 7 (see [32]).
The Gauss transformu~(x,t)∶=(u*E)(x,t) of u∈AL∞′(Rn) is a smooth solution of the heat equation (Δ-∂/∂t)u~=0 satisfying the following.(i)
For everyϵ>0 there exists a constant Cϵ>0 such that
(22)u~x,t≤Cϵeϵ/t∀x∈Rn,t>0.(ii)
u~(x,t)→u as t→0+ in the sense that, for every φ∈AL1,
(23)u,φ=lim⁡t→0+∫‍u~x,tφxdx.Conversely, every smooth solution u~(x,t) of the heat equation satisfying the estimate (22) can be uniquely expressed as u~(x,t)=(u*E)(x,t) for some u∈DL∞′(Rn).The following structure theorem for bounded distributions is well known.Lemma 8 (see [19]).
Everyu∈DL∞′(Rn) can be expressed as
(24)u=∑|α|≤p‍∂αfα
for some p∈N0, where fα are bounded measurable functions on Rn. Equality (24) implies that
(25)u,φ=∑α≤p‍-1α∫‍fαx∂αφxdx
for all φ∈DL1.As a special case of Theorem 3.4 of [32] when p=∞ where the space AL∞′(Rn) is denoted by BL∞(Rn) we obtain the following.Lemma 9 (see [32]).
Everyu∈AL∞′(Rn) can be expressed by
(26)u=∑k=0∞‍akΔkg+h,
where Δ denotes the Laplacian, g, h are bounded continuous functions on Rn, and ak, k=0,1,2,…, satisfy the following estimates; for every L>0 there exists C>0 such that
(27)ak≤CLkk!2
for all k=0,1,2,….The following properties of the heat kernel will be useful which can be found in [33].Proposition 10 (see [33]).
For eacht>0, Et(·) is an entire function and the following estimate holds; there exists C>0 such that
(28)∂xαEtx≤Cαt-n+α/2α!1/2exp⁡-x28t.
Also for each t, s>0 one has
(29)Et*Esx∶=∫‍Etx-yEsydy=Et+sx.


## 3. Stability Problems with Time Variable

The main tools of the proof of our main result are based on theheat kernel method initiated by Matsuzawa [33] (see Lemmas 6 and 7 in Section 2) andthe structure theorems (see Lemmas 8 and 9). Making use of the heat kernel and the structure theorems we convert (8) and (9) to the following classical stability problems for the Gauss transforms u~, v~:Rn×(0,∞)→C of u, v, respectively:
(30)thereexistC>0,N>0,d>0suchthatv~x+y,t+s-v~x,tv~y,s+u~x,tu~y,s≤C1t+1sN+d,∀x,y∈G,t,s>0;foreveryϵ>0,thereexistsCϵ>0suchthatv~x+y,t+s-v~x,tv~y,s+u~x,tu~y,s≤Cϵeϵ(1/t+1/s),∀x,y∈G,t,s>0.In this section we prove the stability problems of inequality (30) with more general settings: let f, g:G×(0,∞)→C with 〈G,+〉 an Abelian group and let N be a nonnegative real number. We consider the following stability problems, respectively:
(31)thereexistC>0,d>0suchthatgx+y,t+s-gx,tgy,s+fx,tfy,s≤C1t+1sN+d,∀x,y∈G,t,s>0;(32)foreveryϵ>0,thereexistsCϵ>0suchthatgx+y,t+s-gx,tgy,s+fx,tfy,s≤Cϵeϵ(1/t+1/s),∀x,y∈G,t,s>0.From now on, a functionA from a semigroup 〈S,+〉 to the field C of complex numbers is said to bean additive function provided A(x+y)=A(x)+A(y) for all x, y∈S and m:S→C is said to bean exponential function provided m(x+y)=m(x)m(y) for all x,y∈S. We introduce the following conditions on f:G×(0,∞)→C and N≥0:
(33)thereexistC>0,d>0suchthatfx,t≤Ct-N+d,∀x∈G,t>0;(34)foreveryϵ>0,thereexistsCϵ>0whichdependson⁡ϵsuchthatfx,t≤Cϵeϵ/t,∀x∈G,t>0.For the proof of the following lemma we refer to [36, Lemma 2.2].Lemma 11.
Letf, g:G×(0,∞)→C satisfy the inequality; for each y∈G and s>0 there exist positive constants C=C(y,s) and d=d(y,s) (resp., for each y∈G, s>0, and ϵ>0 there exists a positive constant Cϵ=Cϵ(y,s)) such that
(35)fx+y,t+s-fx,tgy,s≤Ct-N+dresp.,Cϵeϵ/t
for all x∈G, t>0. Then either f satisfies (33) (resp., (34)) or g is an exponential function.For the proof of the following lemma we refer to [36, Lemma 2.3].Lemma 12.
Letg:G×(0,∞)→C be a nonzero exponential function satisfying (33) (resp., (34)). Then g can be written in the form
(36)gx,t=m1xm2t,
where m1 is an exponential function on G satisfying |m1(x)|=1 for all x∈G and m2 is an exponential function on (0,∞) satisfying 0<|m2(t)|≤1 for all t>0.For the proof of the following lemma we refer to [36, Lemma 2.4].Lemma 13.
Letm:G×(0,∞)→C be a nonzero exponential function satisfying (33) (resp., (34)). Suppose that g:G×(0,∞)→C satisfies the inequality; there exist C>0 and d>0 (resp., for every ϵ>0 there exists Cϵ>0) such that
(37)gx+y,t+s-gx,tmy,s-mx,tgy,s≤C1t+1sN+dresp.,Cϵeϵ1/t+1/s
for all x, y∈G, t, s>0. Then one has
(38)mx,t=m1xm2t,gx,t=Axm1xm2t+2g0,t2m2t2m1x+Rx,t,
where m1 and m2 are exponential functions on G and (0,∞), respectively, such that |m1(x)|=1 for all x∈G and 0<|m2(t)|≤1 for all t>0, and A(x) is an additive function on G and R is a function satisfying (33) (resp., (34)).Theorem 14.
Letg, f:G×(0,∞)→C satisfy (31) (resp., (32)). Then (g,f) satisfies one of the following:(i)
bothg and f satisfy (33) (resp., (34));(ii)
g is an exponential function and f satisfies (33) (resp., (34));(iii)
g(x,t)=m(x,t)+λR(x,t)/1-λ2, f(x,t)=λm(x,t)+R(x,t)/1-λ2, where λ(≠±1)∈C, m is a nonzero exponential function, and R is a function satisfying (33) (resp., (34));(iv)
g(x,t)=(A(x)+1)m1(x)m2(t)+2g(0,t/2)m2(t/2)m1(x)+R(x,t), ±f(x,t)=A(x)m1(x)m2(t)+2g(0,t)m2(t/2)m1(x)+R(x,t), where A(x) is an additive function, m1 and m2 are exponential functions on G and (0,∞), respectively, such that |m1(x)|=1 for all x∈G and 0<|m2(t)|≤1 for all t>0, and R is a function satisfying (33) (resp., (34));(v)
g(x+y,t+s)-g(x,t)g(y,s)+f(x,t)f(y,s)=0 for all x, y∈G, t, s>0.Proof.
We first prove that either there existλ1, λ2∈C, not both zero, and C>0, d>0 (resp., for every ϵ>0, there exists Cϵ>0) such that
(39)λ1fx,t-λ2gx,t≤Ct-N+dresp.,Cϵeϵ/t
or
(40)gx+y,t+s-gx,tgy,s+fx,tfy,s=0
for all x, y∈G, t,s>0.
Suppose that inequality (39) holds only when λ1=λ2=0. Then we can choose y1∈G and s1>0 satisfying f(y1,s1)≠0. Let
(41)Fx,y,t,s=gx+y,t+s-gx,tgy,s+f(x,t)f(y,s).
Then we can write
(42)fx,t=k1gx,t+k2gx+y1,t+s1-k2F(x,y1,t,s1),
where k1=g(y1,s1)/f(y1,s1) and k2=-1/f(y1,s1). Using (41) and (42) we have
(43)gx+y+z,t+s+r=gx+y,t+sgz,r-fx+y,t+sfz,r+Fx+y,z,t+s,r=gx,tgy,s-fx,tfy,siiiiiiiii+Fx,y,t,sgz,r-k1gx+y,t+siiiiiiiiiiii+k2gx+y+y1,t+s+s1iiiiiiiiiiii-k2Fx+y,y1,t+s,s1fz,r+Fx+y,z,t+s,r=gx,tgy,s-fx,tfy,siiiiiiiii+Fx,y,t,sgz,r-k1gx,tgy,s-fx,tfy,siiiiiiiiiiiiiiii+Fx,y,t,sfz,r-k2gx,tgy+y1,s+s1iiiiiiiiiiiiiiii-fx,tfy+y1,s+s1iiiiiiiiiiiiiiii+Fx,y+y1,t,s+s1fz,r+k2Fx+y,y1,t+s,s1fz,r+Fx+y,z,t+s,r.
On the other hand, we can write(44)gx+y+z,t+s+r=gx,tgy+z,s+r-fx,tfy+z,s+r+Fx,y+z,t,s+r.
By equating the right hand sides of (43) and (44) we have
(45)gx,tgy,sgz,r-k1gy,sfz,riiiiiiiiiiii-k2gy+y1,s+s1fz,r-gy+z,s+r+f(x,t)-fy,sgz,r+k1fy,sfz,riiiiiiiiiiiiiiiiiiiiii+k2fy+y1,s+s1fz,riiiiiiiiiiiiiiiiiiiiii+fy+z,s+r=-Fx,y,t,sgz,r+k1Fx,y,t,sfz,r+k2Fx,y+y1,t,s+s1fz,r-k2Fx+y,y1,t+s,s1fz,r-Fx+y,z,t+s,r+Fx,y+z,t,s+r.
If we fixy, s, z, r in (45), the right hand side of (45) satisfies (33) (resp., (34)). Thus, our assumption implies
(46)Fx,y+z,t,s+r-Fx+y,z,t+s,r=Fx,y,t,sgz,r+-k1Fx,y,t,s-k2Fx,y+y1,t,s+s1iiiiiiiiiiii+k2Fx+y,y1,t+s,s1fz,r.
Now if we fix x, y, t, s in (46), the left hand side of (46) satisfies (33) (resp., (34)) as a function of z and r. Thus our assumption implies F≡0. Thus we have case (v).
From now on, we assume that inequality (39) holds: if f satisfies (33) (resp., (34)), then cases (i) and (ii) follow immediately from Lemma 11. It is easy to see that if f does not satisfy (33) (resp., (34)), then neither does g. For this case, inequality (39) implies λ1λ2≠0 and we can write
(47)fx,t=λgx,t+Rx,t
for some λ(≠0)∈C and a function R satisfying (33) (resp., (34)).
Putting (47) in (31) (resp., (32)) using the triangle inequality and fixing y, s we have
(48)gx+y,t+s-gx,t1-λ2gy,s-λRy,s≤Rx,tRy,s+λgy,s+C1t+1sN+d≤C′t-N+d′resp.,Cϵ′eϵ/t
for all x∈G, t>0 and for some positive constants C′ and d′ (resp., for every ϵ>0 there exists a positive constant Cϵ′). Using Lemma 11, we have
(49)1-λ2gy,s-λRy,s=my,s
for all y∈G, s>0, where m is an exponential function on G×(0,∞).
Now ifλ≠±1, case (iii) follows. If λ=±1, then we have
(50)mx,t=∓Rx,t,(51)±fx,t=gx,t-mx,t
for all x∈G, t>0. Viewing (50) and using Lemma 12 we have
(52)mx,t=m1xm2t,
where |m1(x)|=1, 0<|m2(t)|≤1 for all x∈G, t>0. Putting (51) in (31) (resp., (32)) and using the triangle inequality we have
(53)gx+y,t+s-gx,tmy,s-gy,smx,t≤mx,tmy,s+C1t+1sN+d≤C1t+1sN+d+1resp.,Cϵ′eϵ1/t+1/s
for all x,y∈G, t,s>0. By Lemma 13 we have
(54)gx,t=Axm1xm2t+2g0,t2m2t2m1x+R1x,t,
where A(x) is an additive function and R1(x,t) is a function satisfying (33) (resp., (34)). Thus it follows from (51) and (54) that
(55)±fx,t=Ax-1m1xm2t+2g0,t2m2t2m1(x)+R1x,t
for all x∈G, t>0. Replacing R1-m by R we obtain (iv). This completes the proof.From Theorem14 we have the following.Corollary 15.
Letg,f:Rn×(0,∞)→C be continuous functions satisfying (31) (resp., (32)). Then (g,f) satisfies one of the following:(i)
bothg and f satisfy (33) (resp., (34));(ii)
g(x,t)=ec·x+bt and f satisfies (33) (resp., (34)), where c∈Cn, b∈C;(iii)
g(x,t)=ec·x+bt+λR(x,t)/1-λ2, f(x,t)=λec·x+bt+R(x,t)/1-λ2, where c∈Cn, b,λ(≠±1)∈C, and R is a function satisfying (33) (resp., (34));(iv)
g(x,t)=(a·x+1)eic·x+bt+2g(0,t/2)eic·x+1/2bt+R(x,t), ±f(x,t)=a·xeic·x+bt+2g(0,t/2)eic·x+1/2bt+R(x,t), where a∈Cn, c∈Rn, b∈C with Rb<0, and R is a function satisfying (33) (resp., (34));(v)
g(x+y,t+s)-g(x,t)g(y,s)+f(x,t)f(y,s)=0 for all x,y∈Rn, t,s>0.


## 4. Stability in Distributions and Hyperfunctions

In this section we consider stability problems (8) and (9). Recall that the pullback u∘S in (8) and (9) is defined by
(56)u∘S,φx,y=u,∫‍φx-y,ydy,φ∈GR2n.In view of Definition4, it is easy to see that if φj(x,y)∈G(R2n) is a sequence such that φj(x,y)→0 as j→∞, then ∫‍φj(x-y,y)dy→0 in G(Rn) as j→∞. Thus, we have u∘S∈G′(R2n).Theorem 16.
Letu,v∈G′(Rn). Then (v,u) satisfies (8) (resp., (9)) if and only if (v,u) satisfies one of the following:(i)
v,u: bounded distributions (resp., bounded hyperfunction);(ii)
v=ec·x, c∈Cn, and u: bounded distributions (resp., bounded hyperfunction);(iii)
v=ec·x+λw0/1-λ2, u=λec·x+w0/1-λ2 for some c∈Cn, λ∈C, and w0∈DL∞′(Rn) (resp., AL∞′(Rn));(iv)
v=(a·x+1)eic·x+w0, u=±a·xeic·x+w0 for some a∈Cn, c∈Rn, and w0∈DL∞′(Rn) (resp., AL∞′(Rn));(v)
v=(a·x+1)ec·x, u=±a·xec·x for some a,c∈Cn;(vi)
v=ec·x[cos⁡(a·x)+λsin(a·x)], u=±ec·xλ2+1sin(a·x) for some a,c∈Cn, λ∈C.Proof.
We first prove the inequality; there existC>0, N>0, and d>0 (resp., for every ϵ>0 there exists Cϵ>0) such that
(57)v~x+y,t+s-v~x,tv~y,s+u~x,tu~y,s≤C1t+1sN+dresp.,Cϵeϵ1/t+1/s,
where u~,v~ are the Gauss transforms of u,v, respectively, given in Lemmas 6 and 7. Convolving the tensor product Et(x)Es(y) of n-dimensional heat kernels in the left hand side of (8), in view of the semigroup property (Et*Es)(x)=Et+s(x) of the heat kernel we have
(58)v∘S*EtξEsηx,y=vξ,∫‍Etx-ξ+ηEsy-ηdη=vξ,Et*Esx+y-ξ=v~x+y,t+s.
Similarly we have
(59)v⊗v*EtξEsηx,y=v~x,tv~y,s,u⊗u*EtξEsηx,y=u~x,tu~y,s,
where u~(x,t), v~(x,t) are the Gauss transforms of u,v, respectively.
Letw∶=v∘S-v⊗v-u⊗u. Then w∈DL∞′(R2n) (resp., AL∞′(R2n)). First, we suppose that w∈DL∞′(R2n). Using Lemma 8 and Proposition 10 we have
(60)w*EtξEsηx,y≤∑α≤p‍∂αfα*EtξEsηx,y≤∑α≤p‍fα*∂ξ,ηαEtξEsηx,y≤∑|α|≤p‍fαL∞∂ξ,ηαEtξEsηL1≤C1∑|β|+|γ|≤p‍∂ξβEtξL1∂ηγEsηL1≤C2∑|β|+|γ|≤p‍t-(n+|β|)/2s-(n+|γ|)/2≤C1t+1sN+d,
where N=n+p/2 and the constants C and d depend only on p. Secondly we suppose that w∈AL∞′(R2n). Then, by Proposition 10 we have
(61)ΔkEtξEsηL1≤∑|α|=k‍k!α!∂2αEtξEsηL1≤∑|β|+|γ|=k‍k!β!γ!∂ξ2βEtξL1∂η2γEsηL1≤∑|β|+|γ|=k‍k!(2β)!1/2(2γ)!1/2M2kβ!γ!t-n/2-|β|s-n/2-|γ|≤∑|β|+|γ|=k‍k!2M2kt-n/2-|β|s-n/2-|γ|≤k!2nM2k1t+1sn+k.
Now, by Lemma 9 we have
(62)w*EtξEsηx,y≤∑k=0∞‍akΔkg*EtξEsηL∞+h*EtξEsηL∞≤gL∞∑k=0∞‍akΔkEtξEsηL1+hL∞EtξEsηL1≤C1∑k=0∞‍1k!4nM2Lk1t+1sn+k+hL∞≤C2∑k=0∞‍1k!ϵk1t+1sn+k+hL∞≤Cϵeϵ(1/t+1/s),
where L is taken so that 4nM2L<ϵ and the constant Cϵ depends only on w and ϵ. Thus, we have inequality (57). Using Corollary 15 we obtain one of the following:(I)
bothv~ and u~ satisfy (33) (resp., (34));(II)
v~(x,t)=ec·x+bt and u~ satisfies (33) (resp., (34)), where c∈Cn, b∈C;(III)
v~(x,t)=ec·x+bt+λR(x,t)/1-λ2, u~(x,t)=λec·x+bt+R(x,t)/1-λ2, where c∈Cn, b,λ∈C, and R is a function satisfying (33) (resp., (34));(IV)
v~(x,t)=(a·x+1)eic·x+bt+2v~(0,t)eic·x+1/2bt+R(x,t), ±u~(x,t)=a·xeic·x+bt+2v~(0,t)eic·x+1/2bt+R(x,t), where a∈Cn, c∈Rn, b∈C with Rb<0, and R is a function satisfying (33) (resp., (34));(V)
v~(x+y,t+s)-v~(x,t)v~(y,s)+u~(x,t)u~(y,s)=0.
Lettingt→0+ in (I), (II) we get (i) and (ii), respectively. From (III) we have
(63)u~x,t-λv~x,t=Rx,t.
Thus R(x,t) is a solution of the heat equation. Letting t→0+ in (III) and using Lemma 6 (resp., Lemma 7) we have R(x,t)→w0 for some w0∈DL∞′(Rn) (resp., AL∞′(Rn)), which gives case (iii). Now we prove (iv). From (IV), since
(64)v~x,t±u~x,t=eic·x+bt,eic·x+bt is a solution of the heat equation. Thus we have b=-(c12+⋯+cn2)∶=-|c|2, where c=(c1,…,cn). Let
(65)R1x,t=-2ia·cteic·x-c2t+2v~0,teic·x+1/2bt+Rx,t.
Then, from (IV) we have(66)v~x,t=a·x+2ia·ct+1eic·x-c2t+R1x,t,±u~x,t=a·x+2ia·cteic·x-c2t+R1x,t.
Thus R1(x,t) is a solution of the heat equation. Putting ±u~(x,t)=v~(x,t)-eic·x-|c|2t in (57) and using the triangle inequality we have
(67)v~x+y,t+s-v~x,teic·y-c2s-v~y,seic·x-c2t≤C1t+1sN+d+1resp.,Cϵ′eϵ1/t+1/s.
By the continuity ofv~, there exists M>0 such that |v~(0,t+1)|≤M for all t∈[0,1]. Putting x=y=0, s=1 in (67) and using the triangle inequality we have
(68)v~0,t≤ec2v~0,t+1+v~0,1e-c2t1t+1Niiiiiiiiiiiii+C1t+1N+d≤ec22M+C1t+1N+d≤C′t-Nresp.,Cϵ′eϵ/t
for all t∈(0,1). From (65) and (68) we can see that R1 satisfies (20) (resp., (22)). Letting t→0+ in (66) and using Lemma 6 (resp., Lemma 7) we get (iv). Finally, we consider case (V). Letting t, s→0+ in (V) we obtain
(69)v∘S-v⊗v+u⊗u=0.
Now in view of [23, Theorem 3.1], all solutions (v,u) of (69) are of the same form as the smooth solutions (g,f) of the classical functional equation
(70)gx+y-gxgy+fxfy=0
and are given by (iii), (v), or (vi) (see [37, page 180], [23, Theorem 4.2]). This completes the proof.Now we consider Ulam problem (8) in the space of Schwartz tempered distributions. Recall that the following topological inclusions hold:
(71)G↪S↪DL1,DL∞′↪S′↪G′,G↪AL1↪DL1,DL∞′↪AL∞′↪G′.In view of the above topological inclusions, ifu,v∈S′(Rn) it is natural to consider the Ulam problem
(72)v∘S-v⊗v+u⊗u∈DL∞′R2n.Theorem 17.
Letu,v∈S′(Rn). Then (v,u) satisfies (72) if and only if (v,u) satisfies one of the following:(i)
v,u: bounded distributions;(ii)
v=(a·x+1)eic·x+w0, u=±a·xeic·x+w0 for some a∈Cn, c∈Rn, and w0∈DL∞′(Rn).Proof.
Sinceu, v∈S′(Rn), we have c=ia for some a∈Rn and cases (ii), (iii), and (vi) in Theorem 16 are reduced to case (i) and case (v) is contained in case (iv). This completes the proof.Finally we discuss the stability of the following stability problem (see [24]):
(73)v∘S-v⊗v+u⊗u∈L∞R2n,
where L∞(R2n) denotes the space of bounded measurable functions on R2n. For the proof we use the following lemma instead of Lemmas 6 and 7.Lemma 18 (see [38, page 122]).
Letf(x,t) be a solution of the heat equation. Then f(x,t) satisfies
(74)fx,t≤M,x∈Rn,t∈0,1
for some M>0, if and only if
(75)fx,t=f0*Etx=∫‍f0yEtx-ydy
for some bounded measurable function f0 defined in Rn. In particular, f(x,t)→f0(x) in almost everywhere x∈Rn as t→0+.Following the same approach in the proof of Theorem16 we have
(76)v~x+y,t+s-v~x,tv~y,s+u~x,tu~y,s≤C,
where C>0 and u~, v~ are the Gauss transforms of u,v. Now using Corollary 15 for N=0 and Lemma 18 we have the following.Theorem 19.
Letu, v∈G′(Rn). Then (v,u) satisfies (76) if and only if (v,u) satisfies one of the following:(i)
v,u: bounded measurable functions;(ii)
v=ec·x for some c∈Cn, u: bounded measurable functions;(iii)
v=ec·x+λB(x)/1-λ2, u=λec·x+B(x)/1-λ2 for some c∈Cn, λ∈C, and B∈L∞(Rn);(iv)
v=(a·x+1)eic·x+B(x), u=±a·xeic·x+B(x) for some a∈Cn, c∈Rn, and B∈Ł∞(Rn);(v)
v=(a·x+1)ec·x, u=±a·xec·x, where a,c∈Cn;(vi)
v=ec·x[cos⁡(a·x)+λsin(a·x)], u=±ec·xλ2+1sin(a·x) for some a,c∈Cn, λ∈C.Letf be a Lebesgue measurable function on Rn satisfying the inequality; for every ϵ>0 there exists Cϵ>0 such that
(77)fx≤Cϵeϵx2
for all x∈Rn. The function satisfying (73) is said to be aninfraexponential function of order 2. It is easy to see that every infraexponential function f of order 2 defines an element of G′(Rn) via the correspondence
(78)f,φ=∫‍fxφxdx
for φ∈G. Thus as a direct consequence of Theorem 19 we have the following.Theorem 20.
Letg, f be infraexponential functions of order 2. Then (g,f) satisfies the inequality; there exists C>0 such that
(79)gx+y-gxgy+fxfyL∞(R2n)≤C
if and only if (g,f) satisfies one of the following inalmost everywhere sense:(i)
g,f: bounded measurable functions;(ii)
g(x)=ec·x for some c∈Cn, f: bounded measurable functions;(iii)
g(x)=ec·x+λB(x)/1-λ2, f(x)=λec·x+B(x)/1-λ2 for some c∈Cn, λ∈C, and B∈L∞(Rn);(iv)
g(x)=(a·x+1)eic·x+B(x), f(x)=±a·xeic·x+B(x) for some a∈Cn, c∈Rn, and B∈Ł∞(Rn);(v)
g(x)=(a·x+1)ec·x, f(x)=±a·xec·x for some a,c∈Cn;(vi)
g(x)=ec·x[cos⁡(a·x)+λsin(a·x)], f(x)=±ec·xλ2+1sin(a·x) for some a,c∈Cn, λ∈C.



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*Source: 615167-2015-02-12.xml*