- Email: [email protected]
On the spectra of a Hardy kernel
JOURNAL
OF FUNCTIONAL
21, 187-194
ANALYSIS
On the Spectra
(1976)
of a Hardy
E. B. FABES,*
MAX
Kernel
JODEIT,
JR.*
University of Minnesotc2 AND
J. E.
I,E\VIS
of Illinois at Chicqo
University
Communicated by A. P. Cnlderdn Received December
19, 1974
A class of integral operators, K, on L“ spaces is studied. The operator I - K is invertible on L’ and L” but not on LJ’ for some p between r and s. This gives an example of an operator T densely defined and bounded in I,’ and L” norms but which is not bounded in Lp norm for some intermediate p.
1. INTRODUCTION
If K is a linear transformation given by a kernel, so that Kf(x) = J K(x, y)f(y) dy, the integral converging absolutely a.e. for f in an appropriate class of functions, then K may determine a bounded operator from X to Y for a variety of Banach spaces X, Y. M’e are interested here in the case X = Y. If the norm of K as an operator on X is strictly less than unity, the Neumann series C Kn then converges to the inverse of 1- K. It can happen that the iterated kernels K”(x, y) exist as absolutely convergent integrals, and that the series C K”(x, y) converges absolutely a.e. in x, y, even though the norm of K may be greater than or equal to unity as an operator on X. If we let L(x, y) = Cz==, K”(x, y), then formally, L(x, y) - K(x, y) -
1 If+-, z?)L(z, y) dx
= 0 = qx, y) - K(x,y) - f qx, 2) K(z,y) * Author
partially
supported
by N.S.F.-GP-43212.
187 Copyright All rights
0 1976 by Academic Press, Inc. of reproduction in any form reserved.
A,
188
FABES, JODEIT
AND
LEWIS
so that, again formally, (I -- K)(I + L) = I : (I + L)(I - K). We now raise the question: Is I + L the inverse of I - K? We will give examples of kernels (GO) for which the answer is yes, and a class of examples in which the answer is no even though I - K is invertible. Our examples are given by what we call Hardy kernels, acting on the spaces I,” :::I L”(O, co) with Lebesgue measure. If K(~,J) is defined for positive x and J, K is a Hay& kernel for 1~’ (where p t [I, 01) if (a) K is homogeneous for h > 0)
of degree -I
(K(hx, hq’) ~~=hmm’K(?z, 4’)
and
For a treatment of these kernels see [I, Chap. 91. If K is a Hardy kernel for L”, p,, -; p < p, , in Section 2 we will find the spectrum of K as an operator on I, I’, and express the resolvent of K, (d ~ K)-‘, as an operator (l/z) I + Lp,- where L,,,, is a Hardy kernel for Ll’, and give a formula by which L,,, , L,?., are seen to differ by a (possibly) nonzero degenerate kernel. Some examples are given in Section 3. The phenomenon we consider here gives rise to a subtle point in the theory of interpolation of linear operators. We will see in Section 2 that operators K exists for which (I)
I - K is invertible
on L’ and L” (P c; s);
(2) (I - K)pl = I + L, L” with L, -+ L,? ; and (3)
I - K is not invertible
on L’
and (I - K)-l
= I + I,,< on
on L” for some p, I’ < p < s.
If we let X :m (I - K)(L’ n L”), then X is dense in I,’ and in L”, and T :: I + I,,. = I + L,? on X. This is an example of an operator T defined on X for which
Clearly T does not extend to a continuous operator on Lj’. ‘I’he point is that the extensions of T to L’ and L” differ on L’ n L”. They must also differ on the class of simple functions (Riesz-Thorin) and on the “Marcinkiewicz cones” introduced by Sagher [3].
189
SPECTRA
Throughout this for Lp, p, < p < of functionsf with In the strip l/p,
section we assume that K(x, y) is a Hardy kernel p, . We will use the notation L.+l’ for the space Jr If(x)~~(d~/x) finite. < Re 5 < I/p,, , define k(c) by
THEOREM. If K is a Hardy kernel for Lfi), p, < p < p, , then the operator K given by Kf (x) = Jr K(x, y) f (y) dy is a bounded operatolc on L/J for p, < p < p, , and the spectrum of K as an operator on Lp consists of the origin in the plane and all points z of the form x = k(( l/p) + is) for some real e. If z is in the resolvent set of K, then (zI - K)-l = (1 /z) I + L,,,3 , where L,,,x is a Hardy kernel fey L”, given by the formula
zL&,
1) =
(I/274
+k(<):(z
1
- h(i)) d{.
'i,Iwi=1
P-4
Remark. The integral in (2.2) in general must be interpreted as the inverse Mellin transform of /z/(2 - A). The left-hand side is also the limit as n -+ cc of the integrals obtained by multiplying the integrand by e-c2ir1. If p, < Y < s < p, and 2 is in the resolvent set of K COROLLARY. as an operator on L’ and on L”, then L,.,,(x,
1) = I&x,
I) +
c
C,,,,,,,.(log
P.111.12
x)”
.y-l “1 ic,, )
(2.3)
where the sum is taken over the roots ill,, = (1 /p) + if, of k(5) - z Z= 0, with l/s < Re 5 = I/p < l/r, and over those m with m < m,,,, , where mr,lL is the multiplicity qf the root <,,,! . The sum is finite, and is identically zero if and only if x is in the resolvent set of K as an operatol on LP for each p between Y and s. Remark. The formula (2.3) appears in [5] in a different context. To prove these results we convert the operator K to a convolution operator on L.+p, with a kernel K,, depending on p (see [5]), by means of the isometry Lp + L,* given by
190
FABES,
Thus by the homogeneity
x’l”Kf(x)
JODEIT
AND
LEWIS
of K,
= s’ (x/y)‘:?) K(x;y, I)y”“f(y)(dy/y)
= K, * (~l;~f’f(y)),
0
with convolution relative to the multiplicative group of the positive real numbers. The condition that K be Hardy kernel is thus equivalent to the condition, K,, EL,]. Moreover, the Fourier transform for this group is the Mellin transform of K, , which is A(( l/p) + it), E E OX.Since &(1/p) + $) therefore tends to zero as 1 t 1 + 00, we will have 0 in the spectrum of K as soon as every . shown to be in it. Suppose 0 + z = nonzero value of L?r,(t) 1s A(( 1/$J) + if,,). Then for [ near f,, , 2 ~ WP)
+ Z) = (E - Eo)n’W/p)
+ X),
(2.4)
where h(c) is holomorphic near &, = ( 1/p) + $a , and h(<,) F 0. If q is a suitably chosen C” function with support near c,, , then J(t) : (e - f,,)m-l A(( l/p) + it) q(f) is the Mellin transform of a function g which belongs to every L .+,I’.But then the equation (I - K,)^P = 2 (interpreted in the sense of tempered distributions) implies that and thus, F does not belong to L,p for -Qt3 - l/(t - &J near to , any finite p. Thus the spectrum of K as it acts on L” contains 4(1/P) + w If z f 0 and k((l/p) -+ if) - s :: 0 has no roots for 5 E [w, the Wiener-Ievy theorem [2] implies that
where L,, EL, l. Then forf
EL,’
n L.+l’,
Tf = Ulz)(f, has Mellin
L,, -f)
transform 5) = (l/x)(1 -I- (fi/(z - l;))f^ = (l/(.2 - R))j,
and this shows that z is in the resolvent set of K as an operator on Lx’. If we now set Lp,z(x, 1) = x-l’pL,,(x), we get (2.2). To prove the corollary we use the residue theorem. The poles of fi(t)/(x - R(c)) with l/s < Re 5 < 1/ Y 1ie in a bounded set since, by a theorem of in the strip l/p, < l/s < Monte1 [4, 5.231, k(iJ + 0 uniformly Re 5 < 1/r < l/p,, . Then the integral of
191
SPECTRA
around the contour bounding [t: 1/s < Re c < l/r, 1Im c ! < Z?} is, for R sufficiently large, the sum of the residues, and converges to the difference between the integrals in (2.2) for p = r, s (with the convergence factor ec2in). The sum of the residues of F,(c) for l/s < Re 5 < l/r is a sum of terms of the following form, when c,, = l/p + if, is a root of A([) - 2 = 0 with multiplicity m, m-1 C (I/(m j=O
z.d
-
)! j!)(log
1 -j
s)j [(d;‘d<)“‘+-j
(~?“yh(())]~.,~,
,
are carried where h is defined by (2.4). Wh en the differentiations out, those terms involving a derivative of er”” tend to 0 as II + pa, leaving z.Y-$
(”
;
I) l/(m
-
l)!
[(d/ds)“l-1-j
(l!h(~))]c=r,
(log(l
which is a nontrivial linear combination of the form the term with j = m - 1 has a nonzero coefficient.
;s))‘,
(2.3), since
3. EXAMPLES In this section X+ denotes the characteristic real numbers. Let K(x,
y)
=-: -(
1/x) x+(x
-
function
of the positive
(3.1)
y).
K is a Hardy kernel for Lp if 1 < p < CO, and (since the kernel has constant sign) the norm of K, acting on Li' is
s
= (1 Is) xi (x 0
1) .X--1/1” dX z-7 p’
> I
for all
p <
co.
The iterated kernels K,(x, y) are given by K&&y) SO
= (-I)“/((?2
- I)!x)(log (dy))“-’
L(X,Y) = c," K,(x,y)
= -(y/x2)
.,;, I qx,
=
1)1 X- liD’dx
l/(l
X+(X
x+(x - .Y)>
-Y>, and
+ (l/p’))
= p’i( p’ +- 1).
Then(I-K)-l=I+L,oneachLp,p>l. This is a special case of the next example.
In passing we note
192
FABES, JODEIT
AND
LEWIS
that if K, L are nonpositive kernels homogeneous of degree -1 and -K + L - KL = 0, then L is a Hardy kernel for LP whenever K is (but not conversely). Now let K(T Y) =: (l/-4 x+(.2:- Y>.
(3.2)
We write (from (2.1)) h(5) = i,= (l/x) xc--l d&X= l/(1 - 0, so the spectrum of K as an operator on L” (1 < p < cx) is {x: (1 - (l/p) - it)-’ = 1 f or some f E [w), which is the circumference of the circle of radius *p’ centered at BP’ * ((l/p) + (l/p’) = 1). If s is not in the spectrum, Fz(i)/z - k(5) = l/(x(1 - 5) - 1) = (l/z) l/((l/p’) - a - i(f + ,B)) = E,,(t) where 5 = (l/p) + it, l/z = cy.+ ifi, and N + 1!p’. Case 1.
(l/p’)
- u: > 0. In this case,
-&(O = u/4 j”= e-t”’ IIf-a-i(5+6)) & - 1/z y- (1/x)-“-‘B+l P-1 dx. '1
Hence, on Lp(O, CO),
(I - K)-1 = I + I,,,,
where L&x,
1) = 0
X
= (l/x) &I, Case
2.
(1
.Y > 1.
/p’) - OL< 0. Here we have &to
= (1 id jy ,-tcU-(l/~‘)+i(E+B))dt == -1;‘x
[’
.ra”
io~l~b-1
&
‘0
On this set of p, we then have (I - K)-1 = (I + I,,,,)
x>l
where Lg,Jx, 1) = 0 = -(l/z)
Y-1
x < 1.
193
SPECTRA
The difference
across the critical p(( l/p) = 1 const( 1/z) .+r,
The “potential
kernel”
Re z) is thus
x > 0.
is defined by IQ, y) =: (l/V) X/.X++ y’.
(3.3)
AYfis
the restriction to the positive x-axis of the Poisson integral of a density with support on the positive y-axis. The kernel is a Hardy kernel for Lp if p > I, and has norm less than I if p > 3/2. k(i) == (1 4~) .r (X/(X’ + 1)) .+r dx = f set (n/2)5,
which is equal to one for 0 < Re 5 < 1 only when 5 = 213, and the root of /z(c) - 1 = 0 is simple. Hence, I - K is invertible for p # 312, and there are two kernels, differing by a multiple of X-~/S, which give the inverses. Thus (I - K)-l can be calculated forp < 312 by “correcting” the Neumann series. As the next example, we give a kernel for which a log x “correction” is required to cross a critical value of p. Let qx,
1) = x-0 z: .+J
0 < x < 1,
1 < x.
(3.4)
Then K is a Hardy kernel for Lp if Re a < 1/p < Re b and 0 < l/p < I. We then have /z(c) = (b - a)/(b - l)({ - a). When z = 2(b - CZ)/(CZ” + b2), R(c) - z = 0 has a double root at 5 = (a + b)/2, which corresponds to p = 2/(Re a + Re b). (3.5). Let K(x, y) = l/(x + y). Th en Kf is the Stieltjes transform [6] of J For 0 < Re < < I, A([) = r/sin(&J; hence, the spectrum of K as an operator on IY is the same as that on L@ if l/p + l/p’ = 1. For 0 < X < n, X - K is invertible on L” for p f 2, although the Neumann series X-r Cj”=, (A-lK)j diverges in operator norm; for 0 < X < n, the sum of the residues on Re 5 = l/2 is (~x-*/~/(T” - Az)lje) sin((l/rr) cash-l(x/X) log x); if h = n, the series converges residue on Re c is -xx-11P’(X2 -
residue is 7~-~2x-~/~ log x. If X > n the Neumann for sin n/p > n/h; if p > 2 and sin n/p = n/X, the = 1/p is x-lIp(X2 - ,2)1/2; the residue on Re 1: = 1/p’ $)lP.
194
FABES, JODEIT AND LEWIS REFERENCES
1. G. H. HARDY, J. E. LITTLEWOOD XND G. P~LYA, “Inequalities,” Cambridge Univ. Press, London/New York, 1952. 2. L. H. LOOMIS, “,4n Introduction to Abstract Harmonic Analysis,” Van Sostrand, Princeton, N. J., 1953. 3. Y. SAGHER, An application of interpolation theory to I:ourier series, Stt&z M&z.
41 (1972), 169-181. 4. E. C. TITCHMARSH, “The Theory of Functions,” Oxford Univ. Press, London, New York, 1948. to the Theory of Fourier Integrals,” Oxford 5. E. C. TITCHMARSH, “Introduction Univ. Press (Clarendon), London, 1948. 6. D. V. WIDDER, “An Introduction to Transform Theory,” Academic Press, London/ New York, 1971.
OF FUNCTIONAL
21, 187-194
ANALYSIS
On the Spectra
(1976)
of a Hardy
E. B. FABES,*
MAX
Kernel
JODEIT,
JR.*
University of Minnesotc2 AND
J. E.
I,E\VIS
of Illinois at Chicqo
University
Communicated by A. P. Cnlderdn Received December
19, 1974
A class of integral operators, K, on L“ spaces is studied. The operator I - K is invertible on L’ and L” but not on LJ’ for some p between r and s. This gives an example of an operator T densely defined and bounded in I,’ and L” norms but which is not bounded in Lp norm for some intermediate p.
1. INTRODUCTION
If K is a linear transformation given by a kernel, so that Kf(x) = J K(x, y)f(y) dy, the integral converging absolutely a.e. for f in an appropriate class of functions, then K may determine a bounded operator from X to Y for a variety of Banach spaces X, Y. M’e are interested here in the case X = Y. If the norm of K as an operator on X is strictly less than unity, the Neumann series C Kn then converges to the inverse of 1- K. It can happen that the iterated kernels K”(x, y) exist as absolutely convergent integrals, and that the series C K”(x, y) converges absolutely a.e. in x, y, even though the norm of K may be greater than or equal to unity as an operator on X. If we let L(x, y) = Cz==, K”(x, y), then formally, L(x, y) - K(x, y) -
1 If+-, z?)L(z, y) dx
= 0 = qx, y) - K(x,y) - f qx, 2) K(z,y) * Author
partially
supported
by N.S.F.-GP-43212.
187 Copyright All rights
0 1976 by Academic Press, Inc. of reproduction in any form reserved.
A,
188
FABES, JODEIT
AND
LEWIS
so that, again formally, (I -- K)(I + L) = I : (I + L)(I - K). We now raise the question: Is I + L the inverse of I - K? We will give examples of kernels (GO) for which the answer is yes, and a class of examples in which the answer is no even though I - K is invertible. Our examples are given by what we call Hardy kernels, acting on the spaces I,” :::I L”(O, co) with Lebesgue measure. If K(~,J) is defined for positive x and J, K is a Hay& kernel for 1~’ (where p t [I, 01) if (a) K is homogeneous for h > 0)
of degree -I
(K(hx, hq’) ~~=hmm’K(?z, 4’)
and
For a treatment of these kernels see [I, Chap. 91. If K is a Hardy kernel for L”, p,, -; p < p, , in Section 2 we will find the spectrum of K as an operator on I, I’, and express the resolvent of K, (d ~ K)-‘, as an operator (l/z) I + Lp,- where L,,,, is a Hardy kernel for Ll’, and give a formula by which L,,, , L,?., are seen to differ by a (possibly) nonzero degenerate kernel. Some examples are given in Section 3. The phenomenon we consider here gives rise to a subtle point in the theory of interpolation of linear operators. We will see in Section 2 that operators K exists for which (I)
I - K is invertible
on L’ and L” (P c; s);
(2) (I - K)pl = I + L, L” with L, -+ L,? ; and (3)
I - K is not invertible
on L’
and (I - K)-l
= I + I,,< on
on L” for some p, I’ < p < s.
If we let X :m (I - K)(L’ n L”), then X is dense in I,’ and in L”, and T :: I + I,,. = I + L,? on X. This is an example of an operator T defined on X for which
Clearly T does not extend to a continuous operator on Lj’. ‘I’he point is that the extensions of T to L’ and L” differ on L’ n L”. They must also differ on the class of simple functions (Riesz-Thorin) and on the “Marcinkiewicz cones” introduced by Sagher [3].
189
SPECTRA
Throughout this for Lp, p, < p < of functionsf with In the strip l/p,
section we assume that K(x, y) is a Hardy kernel p, . We will use the notation L.+l’ for the space Jr If(x)~~(d~/x) finite. < Re 5 < I/p,, , define k(c) by
THEOREM. If K is a Hardy kernel for Lfi), p, < p < p, , then the operator K given by Kf (x) = Jr K(x, y) f (y) dy is a bounded operatolc on L/J for p, < p < p, , and the spectrum of K as an operator on Lp consists of the origin in the plane and all points z of the form x = k(( l/p) + is) for some real e. If z is in the resolvent set of K, then (zI - K)-l = (1 /z) I + L,,,3 , where L,,,x is a Hardy kernel fey L”, given by the formula
zL&,
1) =
(I/274
+k(<):(z
1
- h(i)) d{.
'i,Iwi=1
P-4
Remark. The integral in (2.2) in general must be interpreted as the inverse Mellin transform of /z/(2 - A). The left-hand side is also the limit as n -+ cc of the integrals obtained by multiplying the integrand by e-c2ir1. If p, < Y < s < p, and 2 is in the resolvent set of K COROLLARY. as an operator on L’ and on L”, then L,.,,(x,
1) = I&x,
I) +
c
C,,,,,,,.(log
P.111.12
x)”
.y-l “1 ic,, )
(2.3)
where the sum is taken over the roots ill,, = (1 /p) + if, of k(5) - z Z= 0, with l/s < Re 5 = I/p < l/r, and over those m with m < m,,,, , where mr,lL is the multiplicity qf the root <,,,! . The sum is finite, and is identically zero if and only if x is in the resolvent set of K as an operatol on LP for each p between Y and s. Remark. The formula (2.3) appears in [5] in a different context. To prove these results we convert the operator K to a convolution operator on L.+p, with a kernel K,, depending on p (see [5]), by means of the isometry Lp + L,* given by
190
FABES,
Thus by the homogeneity
x’l”Kf(x)
JODEIT
AND
LEWIS
of K,
= s’ (x/y)‘:?) K(x;y, I)y”“f(y)(dy/y)
= K, * (~l;~f’f(y)),
0
with convolution relative to the multiplicative group of the positive real numbers. The condition that K be Hardy kernel is thus equivalent to the condition, K,, EL,]. Moreover, the Fourier transform for this group is the Mellin transform of K, , which is A(( l/p) + it), E E OX.Since &(1/p) + $) therefore tends to zero as 1 t 1 + 00, we will have 0 in the spectrum of K as soon as every . shown to be in it. Suppose 0 + z = nonzero value of L?r,(t) 1s A(( 1/$J) + if,,). Then for [ near f,, , 2 ~ WP)
+ Z) = (E - Eo)n’W/p)
+ X),
(2.4)
where h(c) is holomorphic near &, = ( 1/p) + $a , and h(<,) F 0. If q is a suitably chosen C” function with support near c,, , then J(t) : (e - f,,)m-l A(( l/p) + it) q(f) is the Mellin transform of a function g which belongs to every L .+,I’.But then the equation (I - K,)^P = 2 (interpreted in the sense of tempered distributions) implies that and thus, F does not belong to L,p for -Qt3 - l/(t - &J near to , any finite p. Thus the spectrum of K as it acts on L” contains 4(1/P) + w If z f 0 and k((l/p) -+ if) - s :: 0 has no roots for 5 E [w, the Wiener-Ievy theorem [2] implies that
where L,, EL, l. Then forf
EL,’
n L.+l’,
Tf = Ulz)(f, has Mellin
L,, -f)
transform 5) = (l/x)(1 -I- (fi/(z - l;))f^ = (l/(.2 - R))j,
and this shows that z is in the resolvent set of K as an operator on Lx’. If we now set Lp,z(x, 1) = x-l’pL,,(x), we get (2.2). To prove the corollary we use the residue theorem. The poles of fi(t)/(x - R(c)) with l/s < Re 5 < 1/ Y 1ie in a bounded set since, by a theorem of in the strip l/p, < l/s < Monte1 [4, 5.231, k(iJ + 0 uniformly Re 5 < 1/r < l/p,, . Then the integral of
191
SPECTRA
around the contour bounding [t: 1/s < Re c < l/r, 1Im c ! < Z?} is, for R sufficiently large, the sum of the residues, and converges to the difference between the integrals in (2.2) for p = r, s (with the convergence factor ec2in). The sum of the residues of F,(c) for l/s < Re 5 < l/r is a sum of terms of the following form, when c,, = l/p + if, is a root of A([) - 2 = 0 with multiplicity m, m-1 C (I/(m j=O
z.d
-
)! j!)(log
1 -j
s)j [(d;‘d<)“‘+-j
(~?“yh(())]~.,~,
,
are carried where h is defined by (2.4). Wh en the differentiations out, those terms involving a derivative of er”” tend to 0 as II + pa, leaving z.Y-$
(”
;
I) l/(m
-
l)!
[(d/ds)“l-1-j
(l!h(~))]c=r,
(log(l
which is a nontrivial linear combination of the form the term with j = m - 1 has a nonzero coefficient.
;s))‘,
(2.3), since
3. EXAMPLES In this section X+ denotes the characteristic real numbers. Let K(x,
y)
=-: -(
1/x) x+(x
-
function
of the positive
(3.1)
y).
K is a Hardy kernel for Lp if 1 < p < CO, and (since the kernel has constant sign) the norm of K, acting on Li' is
s
= (1 Is) xi (x 0
1) .X--1/1” dX z-7 p’
> I
for all
p <
co.
The iterated kernels K,(x, y) are given by K&&y) SO
= (-I)“/((?2
- I)!x)(log (dy))“-’
L(X,Y) = c," K,(x,y)
= -(y/x2)
.,;, I qx,
=
1)1 X- liD’dx
l/(l
X+(X
x+(x - .Y)>
-Y>, and
+ (l/p’))
= p’i( p’ +- 1).
Then(I-K)-l=I+L,oneachLp,p>l. This is a special case of the next example.
In passing we note
192
FABES, JODEIT
AND
LEWIS
that if K, L are nonpositive kernels homogeneous of degree -1 and -K + L - KL = 0, then L is a Hardy kernel for LP whenever K is (but not conversely). Now let K(T Y) =: (l/-4 x+(.2:- Y>.
(3.2)
We write (from (2.1)) h(5) = i,= (l/x) xc--l d&X= l/(1 - 0, so the spectrum of K as an operator on L” (1 < p < cx) is {x: (1 - (l/p) - it)-’ = 1 f or some f E [w), which is the circumference of the circle of radius *p’ centered at BP’ * ((l/p) + (l/p’) = 1). If s is not in the spectrum, Fz(i)/z - k(5) = l/(x(1 - 5) - 1) = (l/z) l/((l/p’) - a - i(f + ,B)) = E,,(t) where 5 = (l/p) + it, l/z = cy.+ ifi, and N + 1!p’. Case 1.
(l/p’)
- u: > 0. In this case,
-&(O = u/4 j”= e-t”’ IIf-a-i(5+6)) & - 1/z y- (1/x)-“-‘B+l P-1 dx. '1
Hence, on Lp(O, CO),
(I - K)-1 = I + I,,,,
where L&x,
1) = 0
X
= (l/x) &I, Case
2.
(1
.Y > 1.
/p’) - OL< 0. Here we have &to
= (1 id jy ,-tcU-(l/~‘)+i(E+B))dt == -1;‘x
[’
.ra”
io~l~b-1
&
‘0
On this set of p, we then have (I - K)-1 = (I + I,,,,)
x>l
where Lg,Jx, 1) = 0 = -(l/z)
Y-1
x < 1.
193
SPECTRA
The difference
across the critical p(( l/p) = 1 const( 1/z) .+r,
The “potential
kernel”
Re z) is thus
x > 0.
is defined by IQ, y) =: (l/V) X/.X++ y’.
(3.3)
AYfis
the restriction to the positive x-axis of the Poisson integral of a density with support on the positive y-axis. The kernel is a Hardy kernel for Lp if p > I, and has norm less than I if p > 3/2. k(i) == (1 4~) .r (X/(X’ + 1)) .+r dx = f set (n/2)5,
which is equal to one for 0 < Re 5 < 1 only when 5 = 213, and the root of /z(c) - 1 = 0 is simple. Hence, I - K is invertible for p # 312, and there are two kernels, differing by a multiple of X-~/S, which give the inverses. Thus (I - K)-l can be calculated forp < 312 by “correcting” the Neumann series. As the next example, we give a kernel for which a log x “correction” is required to cross a critical value of p. Let qx,
1) = x-0 z: .+J
0 < x < 1,
1 < x.
(3.4)
Then K is a Hardy kernel for Lp if Re a < 1/p < Re b and 0 < l/p < I. We then have /z(c) = (b - a)/(b - l)({ - a). When z = 2(b - CZ)/(CZ” + b2), R(c) - z = 0 has a double root at 5 = (a + b)/2, which corresponds to p = 2/(Re a + Re b). (3.5). Let K(x, y) = l/(x + y). Th en Kf is the Stieltjes transform [6] of J For 0 < Re < < I, A([) = r/sin(&J; hence, the spectrum of K as an operator on IY is the same as that on L@ if l/p + l/p’ = 1. For 0 < X < n, X - K is invertible on L” for p f 2, although the Neumann series X-r Cj”=, (A-lK)j diverges in operator norm; for 0 < X < n, the sum of the residues on Re 5 = l/2 is (~x-*/~/(T” - Az)lje) sin((l/rr) cash-l(x/X) log x); if h = n, the series converges residue on Re c is -xx-11P’(X2 -
residue is 7~-~2x-~/~ log x. If X > n the Neumann for sin n/p > n/h; if p > 2 and sin n/p = n/X, the = 1/p is x-lIp(X2 - ,2)1/2; the residue on Re 1: = 1/p’ $)lP.
194
FABES, JODEIT AND LEWIS REFERENCES
1. G. H. HARDY, J. E. LITTLEWOOD XND G. P~LYA, “Inequalities,” Cambridge Univ. Press, London/New York, 1952. 2. L. H. LOOMIS, “,4n Introduction to Abstract Harmonic Analysis,” Van Sostrand, Princeton, N. J., 1953. 3. Y. SAGHER, An application of interpolation theory to I:ourier series, Stt&z M&z.
41 (1972), 169-181. 4. E. C. TITCHMARSH, “The Theory of Functions,” Oxford Univ. Press, London, New York, 1948. to the Theory of Fourier Integrals,” Oxford 5. E. C. TITCHMARSH, “Introduction Univ. Press (Clarendon), London, 1948. 6. D. V. WIDDER, “An Introduction to Transform Theory,” Academic Press, London/ New York, 1971.