Internat. J. Math. & Math. Sci.
(1985) 29-35
29
Vol. 8 No.
A TRACE THEOREM FOR CALORIC FUNCTIONS
H.N. MHASKAR
Department of Mathematics and Computer Science
California State University
Los Angeles, California
90032
U.S.A.
(Received August 4, 1983)
ABSTRACT.
Given a solution of the heat equation in an open strip, we state necessary
and sufficient conditions for the existence of a boundary function in a given weighted
We then investigate the relationship between the smoothness of this
Banach space.
boundary function and the growth of the solution of the heat equation.
KEY WORDS AND PHRASES. Caloric functions, moduli of continuity.
1980 MATHEtTTCS SUBJECT CLASSIFICATION CODES. 35BG5, 35B30, 26A75
io
IN’[RODUCTION.
An interesting problem in classical analysis can be described as follows
We are
given a sufficiently nice function on an open domain and we wish to extend it to the
boundary of the domain while preserving continuity in some sense.
When this is possible,
we also wish to investigate the relationship between the smoothness of the boundary
function and the growth of the function on the domain.
ing well known theorems in the theory of H
THEOREM I.
M
P
(a)
([I],
Izl
unit disc {z
,
For example, we have the follow-
spaces’
p. 33) Let f be a complex valued harmonic function in the open
i} and set
If(re
p
max
0_< t _< 2H
If (re
(r,f) :=
If 1 < p
p
1 -< p
d
(I.i)
it
then f is the Poisson integral of a function in L
p
on the unit circle
if and only if
sup
0<r<l
(b)
For p
M (r,f)
P
(1.2)
i, f is the Poisson integral of a complex Baire measure on the unit circle
if and only if
sup
O<r<l
M
l(r,f)
(1.3)
The following theorem of Hardy and Littlewood relates the smoothness of the boundary
function with the growth of the function on the disc.
30
H.N. MHASKAR
i-< p<=, f be analytic in the open unit disc and(l.2)hold.
[2], p. 78) Let
THEOREM 2.
(Then f is the Poisson integral of a 2H-periodic function g in
LP[-N,
H].)
The follow-
ing are equivalent:
f
sup
0<lhl<-t
M
P
0
(r,f ])
Here, 0
2H
Ig( 0 +
=(I
r)
g(O) p dO
h)
o(t =p)
(1.4)
=-I
(1.5)
i.
It is well known that caloric functions (i.e. the solutions of the heat equation)
share quite a few properties with analytic functions.
[3], [4]).
(e.g
In 1954,
Czipszer [5] obtained an analogue of Theorem l(b) for caloric functions defined on the
R x
strip
(0,c)
where the boundary measure is not necessarily finite itself, but a
certain weight function is integrable with respect to the measure.
[6],
In my dissertation
it is demonstrated how the proof of Czipszer can be modified to yield an
of Theorem
l(a),
not just for L
p
analogue
spaces but even for more general weighted rearrangement
invariant Banach spaces.
In this paper, we wish to report a somewhat more interesting part of our work in
[6], namely,
an analogue of Theorem 2.
Unlike the case of analytic functions, the first
order modulus of continuity does not seem to be an adequate measurement of the smoothness
Introducing a second order modulus of continuity, we shall
of the boundary function.
obtain a relationship between the growth of the caloric functions and the smoothness of
its boundary values.
2.
MAIN RESULTS.
Let X be a rearrangement invariant Banach function space.
We assume the following:
(A)
X is isometrically isomorphic to the normed dual of its associate space.
(B)
If f is in X, then
lim
II
f(x+h)
II
f(x)
(2.1)
0
h+O
.
Both the conditions are satisfied for L
will not be true for p
I,
versions valid for the space
p
spaces if l<p<=.
With slight modifications,
Co(R).
In general, our theorems
it is possible to obtain the
A survey of some of the important properties of
the rearrangement invariant Banach function spaces can be found in [7
],
8 ].
We say that a function u is caloric on S := R x (O,c) if
c
2u
u
For a function u caloric on S
c
(x,t)
e
(2.2)
Sc
we define the following analogue of the norms M
p
defined
in (I.i):
N(u,c;t) :=
Nx(u,c;t)
:= (c
t)
1/2
If
w(c-t, x) u(x,t)
I,
O<t<c
(2.3)
where
w(h,x) := exp
In
[6],
x 2/4h)
we proved the following analogue of Theorem
Theorem l.l(b) being given in [5].
(2.4)
l.l(a), the analog,e of
A TRACE THEOREM FOR CALORIC FUNCTIONS
31
The following are equivalent:
(a) u is a caloric function on S and for each h e (O,c),
c
sup
N(u,h;t) <=
(245)
O<t<h
(b) There exists a function f such that w(h,x)f(x)e X for each h
u(x,t)
(O,c)
and
H(f,x,t)
(2.6)
where the operator H is defined by
I
H(f,xt,t) :=
w(t.x-y) f(y)dy
(2.7)
In (2.7) and in all the other formulae of the paper, all integrals are taken over the
whole real line unless otherwise specified.
We note that the boundary function f itself is not necessarily in X.
w(h,x),
the presence of the weight functions
Because of
we need to modify the expression for the
usual second order modulus of continuity to measure the smoothness of f.
In [9
],
Freud
introduced a certain modification which was found to be completely satisfactory for the
study of weighted polynomial approximation of such functions.
2(h,f,6)
:=
O.
Let h
defined as follows.
llF(x+2t)
sup
+ 6
With
-2F(x+t) + F(x)
If F6(x)
sup
This new modulus is
F(x) := w(h,x)f(x), put
[f(x+t)
II
F(x)
]If
=I I(x)F()II
+
(2.8)
where
6(x)
:= min {6
(l+x2) 1/2}
-1,
(2.9)
Then the modulus of smoothness of f is given by
2(h,f,6)
:=
inf
a,beR
2(h,f(x)
(2.10)
bx, 6)
a
This modulus measures not just the smoothness of f but also the growth of f near
.
In [i0] we proved
THEOREM 2.
O, w(h,x)f(x)eX.
O,
Let h
Then there exist constants
Cl,
c
2 depending
upon h and X alone such that
ClK(h,f,6)
-<
oq2(h,f,6)
where the K-functional
K(h,f,6) :=
inf
K(h,f,6)
-<
is given by
{IIw(h,X)fl(x)II
the inf being taken over all
fl
(2.11)
c2K(h,f,6)
and
+
f2
621[w(h,x)f(x)ll}
such that f
a twice iterated integral of a function
fp
e
fl
+
(2.121
f2’ w(h’x)fl(x)eX’
and
f2
is
X.
We can now state our main theorem.
THEOREM 3.
(a)
<- i.
O, 0
Let c
Suppose for each h e(O,c), w(h,x)f(x)eX and
2(h’f’6)
Define u by (2.6).
6u
N(-,
h; t)
o(6 =)
(2.13)
Then u is a caloric function on S
o(t
=/2
i)
for each h
(O,c) and
satisfying (2.5) and further
e(0,h)
(2 ]4)
H.N. MHASKAR
32
(b) Let u be a caloric function satisfying (2.5) and (2.14) for each h e (O,c)
there exists a function f such that for every h e
(0,c), w(h,x)f(x)eX
Then
and further,
(2.6) and (2.13) hold.
3.
PROOFS.
We adopt the following notation.
A
B means that there exists a constant c
independent of the obvious variables such that A -< cB.
B
A.
AB means that A
B and
Thus, in Theorem 2.2,
K(h,f,6)
If s
(h,f,6)
2
(3.1)
O, l(s) will denote the indicator function of X; thus,
l(s) :=
f(st)
sup
It is known [7] that
I(s) -< max (l,s
-I)
(3.3)
Further for convenience, we shall often write
w
h
instead of
w(h,’).
PROOF OF THEOREM 2.3
(a)
We shall prove that 0
N(u
g +
Let f
a locally integrable function
u(x,t)
were
f2
f2
c, then
I
-)
f;
where
with
whf2
(3 4)
whge X,
f2
e X.
Then
is a twice iterated integral of
v(x,t) +
v(x,t) :=H(g,x,t)
8u2
t
Hence,
h
t-l 2(hl,
t) <<
h
We use Theorem 2.2.
h
u2(x,t
u2(x,t) := H(f2,
2u2
H(f
(3.5)
x,t).
x, t)
Note that
(3 6)
in view of the easily verified identity
(3.7)
(3.8)
Thu s,
N
8u2
(--
h t)
<<II whf211
-<
t
-I
t
II
w
h
f"
I
2
II
(3.9)
Part (a) will be proved if we now show that
N(--{Sv
h,t)
t-Ill whl flll
(3.10)
A TRACE THEOREM FOR CALORIC FUNCTIONS
Let
33
Fr om
O<t<h<hl<C.
ex
-
we get
4t z
t
I ex[’I (Y- -t x)2
-
+
g(y)dy
4t
y)
exp
+ 2
g(y)dy
(h-t).ty(y_h h-’--h x)
4h
(hzt
+
y2
]
(3.11)
We shall estimate each term in (3.11) separately.
Y
exp
’i-{
R
y
h--
wh(Y)g(y)dy
x
(h--t);Y2 (- whgll
Y2) II
-<
exp
h-t
dY
R
3
<<\-fFI
If
y(y
II Whlg
h
R
x) xp
(3.12)
I_h-t(
4-
y
h(y)g(y)dy [I
w
dy
y2g(y)wh(y) ll
R
(3.13)
(3.14)
H.N. MHASKAR
34
[[
4- (y-
exp
ht x2
wh(Y)g(y)dyll
R
-< I
h-t
exp
whg I[
dy
y
4-
\h-t]
lWh i g I[
(3.15)
R
(3.14) and (3.15).
Inequality (3.10) follows from (3.11), (3.12), (3.13),
completes the proof of part (a).
2d
c.
(b) Choose d small enough so that we can choose
This
0<hl<h
I[exp
(x
ox
=Z
n=O
d/2n1
N
<Z
n=0 IId/2n
<<
n=0
+1
d/2n
t
2n+l
N
a/2
n=0
2na/2
lxp
(-
x2
exp
I
-
d)
2n+l
n=0
2n
u
--
t)[
(x
)
x
d
2
dt
[l
u
n
(3.16)
Thus,
(3.17)
(,
o(d
u(x,d
2-)
and the series
exp
I u, (x,d)+n=
converges in X to say exp(
d___._)_2
(x,
n+l
u(x,
vd)
(3.18)
x2/4hl)f(x).
A standard argument then shows that
w
for all 0<t<h
I
t
(x-y) f (y)dy
and hence for all (x t) in S
C"
(3.19)
Since a subsequence of (3 18) can be
chosen to converge almost everywhere, fs not dependent upon h as it might appear to
I
be. Further, from (3.17)
lWhl(X)[f(x)
u(x,d) ]I
l<<d
a/2
(3.20)
A1 so,
2
lwh (x)-x
I
u(x,d)
I[-<l lexp
x
4(h-d)
2
x u(x,d) II<<
The proof of part (b) is now complete in view of
d
a/2
i
(3.20), (3.21)
(3.21)
and Theorem 2.2.
A TRACE THEOREM FOR CALORIC FUNCTIONS
35
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I.
2.
3.
4.
5.
6.
7.
8.
9.
I0.
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