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An Inequality on Solutions of Heat Equation*

DU-WONBYUN

DepartmentofMathematics,InhaUniversity,253Yong Hyun-Dong, Nam-ku,In-chon402-751,Korea

(Received28 April 1997;Revised17 October1997)

Letv(x,t)bethesolutionof the initial valueproblem forthendimensional heat equation.

Then,for anyaand foranyto>0,^aninequality aboutv(a, t)andv(x, to)is obtained.

Keywords: Heatequation; Integraltransform;Positivematrix; Reproducingkernel 1991 Mathematics subjectClassification. ^{Primary35K05;Secondary30C40}

1.

INTRODUCTION

Forapositive integern, we considerthendimensionalheat equation

Xv(x, t) Otv(x, t), v(x, o) F(x),

X ]1ⁿ and

>

O;

xERn (1.1)

where AisthendimensionalLaplacian andFis amemberinthe space

Lz(R n)

for the Lebesgue measure on

R’.

Then, the solution is representedby

v(x, t)

2v ^F()exp

^4t

^{d. (1.2)}

Thisresearch is partiallysupported by KOSEF-GARC.

E-mail: [email protected].

269

Furthermore,fromtheexpression

(1.2)

^weknowthatthesolutionv(x,

t)

canbeholomorphically extendedonthendimensionalcomplex space C with respect to the space variable x. For the time variable also,

v(x,t)

^can be holomorphically extended on the right half plane D

{z Iz ^{> 0}}

of the complexplane C.These facts arefoundin[4,6].

In this paper, for anya E

n

and for a fixed time to, we derive an inequality which expresses the relation of v(a,

t)

and v(x,

to).

Our inequality is the generalization of an inequality in

[6]

for the n dimension.

THEOREM ForaninitialvaluesFin

L2( n)

letv(x,

t)

be thesolution

of

the ndimensionalheat equation

(1.1).

Then,

for

^anya

n

^and

for

^any

to >

O, thefollowing inequalityis valid."

2t0 J

^dAdr,

wherez x

+

^iy

^(x,

^y6]),^w A

+

ⁱ⁷

^(A,

^7-

^).

^Moreover,the equality holds

if

^{and only}

if

^{F is a member in}

^M(n,a).

^Here,

^{C(n, to)--}

nE(n/Z)/(Z2"+Tr"-lto/2)

^and

^M(n,a)

^{is theclosure}

of

^thespace spanned linearlyby

{f()-

^e

^-l-al2

in

L2().

2.

SOME HOLOMORPHIC FUNCTION SPACES

WeletK(z, u) bethe

Bergman

kernelonthedomainDwithrespectto themeasuredxdy/Tr. Itisexplicitlyrepresentedby

K(z, u) / (z + )2.

Foranyq

>

1,weconsidertheHilbertspace

Hq {f."

holomorphicin

D]

JJ ¹²

^-qdxdy

^<

^o,

[[fl[ZH" _{7rF(2q 1)} ^{If(z) K(z, z)}

z- x

+ iy}.

Then, the kernelfunction

Kq(z, u)= P(2q)K(z, bt) q, (z, bt)

EO D,

is the reproducing kernel of

Hq

ⁱⁿ the following sense: for anyzED, K(.,

z)

^isthe memberin

Hq

and every member

fin Hq

^isrepresented by

f(z) (f, Kq(.,z))i4

_q, ^{z D,}

where

{.,.)/_/q

^{istheinnerproductintheHilbertspace}

^{Hq (refer}

^to[2,3]).

Meanwhile, the kernelfunction

Kq

^canberepresented by

Kq(z, u) e-Ze-

^2q-1d,

^{z,uD, (2.1)}

andthe right handsideof

(2.1)

converges for allq

>

^0.

^Hence,

^{for any}q with0

<

q

<

^{1, thefunction}

Kq

^{also determines the}

Hq

^{thatadmits the}

reproducing kernel

Kq(z, u) (see [1,7]).

^Foranyq

>

^0,wedenote

Kq(z, u) r(2q)K(z, b/) q,

Z,bt

D,

andwealsoconsidertheHilbertspace

Aq {g:

holomorphicin

D]

2__

ffD ¹²

[Igl[Aq _{rrP(2q + 1)} [g’(z) K(z, Z)

^-qdxdy

<

lim

g(x) 0}.

Since the mapping

f---f’

^{is the} isometry from

Hq

^onto

Hq_+_l, Hq- Aq,

^and

Kq(z, u)

^isthe

reproducing

kernel of

Aq (see [3]).

3.

PROOF OF THEOREM

Following the theoryofgeneralized integral transforms [5], we prove ourtheorem. First, for a---0,weconsider theintegral transform

7-{F(z)-

2x/_ ,,F()exp ^d-v

^zD.

Forany

to >

0,wecalculatethe kernel form

(-- (--

K(z, U)

^n/4

Since the function

T,,(z, u)

is positive matrix on D, it determines the reproducing kernelHilbert space

Sn (see

[1,7]). Ontheotherhand, the space

Sn

^ischaracterizedby

Sn {f."

holomorphicin

D[

23n/2+Tr"/2-1f fz ^}

Ilfll

²

^If’(z) 12x

^n/2dxdy

^<

^oc

Hencewe havethenorminequality

IIv(O,z)ll ^< f ^IF()I

²

^{ds c. (3.1)}

Fortheorthogonal complementN^+/-of the null space N

R{F

^E

L)(R")IT-[F(z) 0},

zED

the equalityin

(3.1)

holdsifand onlyifFis a memberinN

+/-.

In fact, N^+/- is the closure of the space ⁱⁿ

L2(I n)

^{which is}linearly spanned by membersof the family

{G() exp(-cl12)

^c

D},

and soN^+/--M(n,

0).

From

[4],

thenormequality

( )

^n/2

ffc" ^{Iv(w’ to) 12}

^exp

( _--oJ ^[’r12

^dA67-

^{J,, IF(sC)l}

²

^{d. (3.2)}

holds, andfrom

(3.1)

and

(3.2)

^ourinequalityis obtainedfora-0.

Foranya ER

n,

^since

(1)n ( .a_12)

^d

v(a, t) 2v/_ ^F()exp

4t

2v/ .F( ^{+ a)}

^{exp d,}

wehave

Ilv(a, t)][]. ^_< f. ^{]F( + a)12d-} f. ^(3.3)

From

(3.2)

and

(3.3),

the inequality

(1.3)

^{is valid.} Meanwhile, the equality in

(3.3)

^{holds ifand onlyif}

F( + ^a)

^E

^{M(n, 0).}

Therefore the proof has beencompleted.

References

[1] N. Aronszajn, Theoryofreproducing kernels, Trans. Amer.Math. Soc. 68(1950), 337-404.

[2] J. Burbea,Totalpositivityof certainreproducing kernelsPacific^{J.Math. 67(1976),}

101-130.

[3] D.-W. Byun Isometrical mappings between the Szeg6 and the Bergman-Selberg spaces,ComplexVariables20(1992),13-17.

[4] D.-W.Byun,Relationshipbetween theanalyticsolutionsofthe heat equation, Math.

Japonica(1993)477-481.

[5] S. Saitoh,HibertspacesinducedbyHilbertspacevaluedfunctions,Proc.Amer.Math.

Soc.89(1983),74-78.

[6] S. Saitoh, Inequalitiesfor the solutions of the heatequation,inGeneralInequalities6, BirkhfiuserVerlag,BaselBoston (1992)^139-149.

[7] S. Saitoh, TheoryofReproducing Kernelsand itsApplications,PitmanRes. Notesin Math.Series189,LongmanScientific&Technical,England,1988.