Best constants for certain multilinear integral operators

INTEGRAL OPERATORS

´ARP ´AD B ´ENYI AND CHOONGHONG (TADAHIRO) OH

Received 7 December 2004; Revised 8 March 2005; Accepted 27 March 2005

We provide explicit formulas in terms of the special function gamma for the best con-stants in nontensorial multilinear extensions of some classical integral inequalities due to Hilbert, Hardy, and Hardy-Littlewood-P ´olya.

Copyright © 2006 ´A. B´enyi and C. (Tadahiro) Oh. This is an open access article distrib-uted under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

1. Introduction and statement of results

In his lectures on integral equations, Hilbert initiated the study of maxima of bilinear and multilinear forms. The casep=2 in the following inequality is nowadays known as “Hilbert’s double series theorem.”

Ifp >1,p=p/(p−1),∞m=1a p

m≤A, and∞n=1b p

n ≤B, then ∞

m=1 ∞

n=1

ambn

m+n≤πcsc

π p

A1/ p_B1/ p

. (1.1)

Hilbert’s proof, apart from the determination of the best possible constantπcsc(π/ p), was published by Weyl [7]. The calculation of the constant, and the integral analogue of Hilbert’s double series theorem (for p=2) are due to Schur [6]. The generalizations to otherps of both the discrete and integral versions of this result were discovered later on by Hardy and Riesz and published by Hardy in [4]. The statement of the integral analogue of the theorem above is the following.

Ifp >1,p=p/(p−1),0∞fp(x)dx≤F,

_∞

0 gp

(y)d y≤G, then

∞

0

f(x)g(y)

x+y dx d y≤πcsc _π

p

F1/ p_G1/ p_{. (1.2)}

Other proofs and generalizations in different directions were given by several authors; see the book by Hardy et al. [5, page 227]. Applications of Hilbert’s theorem range from the

Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2006, Article ID 28582, Pages1–12

theory of analytic functions to results about the moments of real variable functions [5, pages 236–247].

Let nowK:Rm

+→Rbe a measurable kernel such that

Cm=

∞

0 ···

∞

0

_{K y1,. . .,ymy}−1/ p1

1 ···y −1/ pm

m d y1···d ym<∞; (1.3)

here, 1< p1,p2,. . .,pm<∞are some arbitrary (fixed) indices. Them-linear operatorTis defined via

T f1,. . .,fm

(x)=

_∞

0 ···

_∞

0 K y1,. . .,ym

f1 xy1

···fm xym

d y1···d ym, (1.4)

wherex >0 and f1,. . .,fmare measurable functions onR+=[0,∞). Note thatTis in fact an integral operator having a homogeneous kernelKof degree−m,

T f1,. . .,fm

(x)=

_∞

0 ···

_∞

0

K x,y1,. . .,ym

f1 y1

···fm ym

d y1···d ym, (1.5)

whereK(x,y1,. . .,ym)=x−m_K(x−1_{y1,. . .,x}−1_{ym). Condition (1.3) can be rewritten as}

Cm=

_∞

0 ···

_∞

0 K 1,y1,. . .,ym

_y−1/ p1

1 ···y− 1/ pm

m d y1···d ym<∞. (1.6)

We remark also that, ifp0>1 is such that 1/ p0+ 1/ p1+···+ 1/ pm=1, then

Cm=

_∞

0 ···

_∞

0 K x, 1,y2,. . .,ym

_x−1/ p0_y−1/ p2

2 ···y− 1/ pm

m dx d y2···d ym. (1.7)

In particular, (1.6) and (1.7) imply that, ifKis symmetric with respect to the variables

x,y1,. . .,ym, thenCmis a symmetric expression of the indicesp0,p1,. . .,pm.

The goal of this paper is twofold: to give a proof of the multilinear extension of the integral inequality (1.2) and to provide some interesting applications of this result.

Theorem1.1. Letm≥1and1< p,p1,. . .,pm<∞be such that1/ p1+ 1/ p2+···+ 1/ pm= 1/ p. Then

_{T f1,. . .,fm}

Lp_(R+)≤Cmf1_Lp1(R+)···fmLpm(R+), (1.8)

whereCm is the constant defined by (1.3) or (1.6). Moreover, ifK(y1,. . .,ym)≥0 for all

y1,. . .,ym≥0, then the constantCmis the best possible in (1.8).

seem to be in the literature. These formulas are presented in Section 3. We prove the main result,Theorem 1.1, in the following section. The elementary, yet nontrivial, proof is a nice application of the Dominated Convergence theorem.

Remark 1.2. An alternate way of finding an upper bound for the norm of a positive mul-tilinear operator is via the so called mulmul-tilinear Schur test in the work of Grafakos and Torres [2, Theorem 1]. In particular, for anm-linear operatorT with positive and sym-metric kernel onRm

+, the multilinear Schur test gives the following implication.

If for all B > A >0, there exist measurable functions u1,. . .,um,w on R+ with 0<

u1,. . .,um,w <∞a.e. such that

T up1

1 ,. . .,u p

m

m≤Bwp a.e., (1.9)

thenTis a bounded operator fromLp1_(R+)× ··· ×_Lpm_{(R+) intoL}p_{(R+), with norm less} than or equal toA.

In the case of the multilinear nontensorial extension of the Hilbert operatorP¬⊗(see

Section 3.1for its definition), it was shown in [2] that an appropriate choice of the func-tionsu1,. . .,um,wgives in fact equality in (1.9). Furthermore, it is easy to see that the same choice of weightsuj(yj)=y

−1/ pjpj

j ,w(x)=x−1/ pp

, 1≤j≤m, gives equality in (1.9) for allm-linear operatorsT of the form (1.4). In this caseB=Cm, whereCm is defined by (1.3). Therefore, by the implication above, we see that the operator normT ≤Cm. We would like to point out, however, that in order to showT =Cm one needs to either trace back where equality holds in the inequalities proving the main result of [2] or go through a similar computation to the one presented here in the next section.

2. Proof ofTheorem 1.1

We letg∈Lp

(R+), 1/ p+ 1/ p=1 and denote by·,·the dual (Lp_,Lp

) pairing. For simplicity, we will write Lp _{for L}p_{(R+). Note that, for i=1,. . .,m,} ∞

0 |fi(xyi)|pidx=

yi−1

∞

0 |fi(x)|pidx. Using this fact and H¨older’s inequality, we obtain the following se-quence of inequalities:

_{T f1,. . .,fm}

,g

≤

∞

0 ···

∞

0

_{K y1,. . .,ym}∞ 0

_{g(x)f1 xy1···fm xymdx d y1···d ym}

≤

_∞

0 ···

_∞

0

_{K y1,. . .,ym)gL}p m

i=1

∞

0

_{fi xyi}pi

dx 1/ pi

d y1···d ym

=CmgLpf1_Lp1···fm_Lpm.

(2.1)

This proves the first part of our theorem.

sequences of functionsgn,fi,nby

gn(x)=x−1/ p+1/ pn_{χ[0,1](x), fi,n(x)=x}−1/ pi+1/ pin_{χ[0,1](x). (2.2)}

Clearly,gnp

Lp= fi,nLpipi= gnLpf_1,nLp1···f_m,nLpm=n. We have

T f1,n,. . .,fm,n

,gn

=

1

0x

−1/ p+1/ pn

_∞

0 ···

_∞

0 K y1,. . .,ym

f1,n xy1

···fm,n xym

d y1···d ymdx

=

1

0x

−1/ p_{+1/ pn}1/x 0 ···

1/x

0 K y1,. . .,ym

m

i=1

xyi

−1/ pi+1/ pin

d y1···d ymdx

=

1

0x −1+1/n

1/x

0 ···

1/x

0 K y1,. . .,ym

m

i=1

y−1/ pi+1/ pin

i d y1···d ymdx

= −n _∞

1 x

−1/nx 0 ···

x

0K y1,. . .,ym

m

i=1

y−1/ pi+1/ pin

i d y1···d ym

dx

=n 1

0···

1

0K y1,. . .,ym

m

i=1

y−1/ pi+1/ pin

i d y1···d ym+ m

i=1

Ii,

(2.3)

or

T f1,n,. . .,fm,n,gn

_gn

Lpf1,nLp1···fm,n_Lpm =

1

0···

1

0K y1,. . .,ym

m

i=1

y−1/ pi+1/ pin

i d y1···d ym+ m

i=1

Ii/n.

(2.4)

Fori=1,. . .,m, we have denoted

Ii=n

_∞

1 y −1/n i

yi

0 ···

yi

0 K y1,. . .,ym

m

j=1

y−1/ pj+1/ pjn

j d y1···d yi···d ymd yi; (2.5)

here,d yimeans that we do not integrate with respect to the variableyi. In the transition from the fourth to the fifth line in the sequence of equalities above we made the change of variablesx→1/x. The last equality follows from integration by parts and the observation that, if we let

S z1,. . .,zm

=

z1

0 ···

zm

0 K y1,. . .,ym

m

i=1

y−1/ pi+1/ pin

then

d

dxS(x,. . .,x)=

m

i=1

∂S

∂zi(x,. . .,x)= m

i=1

x

0···

x

0K y1,. . ., (i)

x,. . .,ym

x−1/ pi+1/ pin

× j=i

y−1/ pj+1/ pjn

j d y1···d yi···d ym, (2.7)

where the upper index (i) means thatx replaces the variableyi in theith position. Let nowDidenote the domain of integration in the integralIiabove, that is,

Di= y1,. . .,ym

∈Rm+: 1≤yi<∞, 0≤yj≤yi, j=i

. (2.8)

Taking into account that 1/ p1+···+ 1/ pm=1/ p, we can bound the integrand ofIi/non

Dias follows:

yi−1/nK y1,. . .,ym

m

j=1

y−j1/ pj+1/ pjn≤y−1/n+1/ p

1n+···+1/ pmn

i K y1,. . .,ym

m

j=1

y−j1/ pj

=y−i1/ pnK y1,. . .,ym

m

j=1

y−j1/ pj≤K y1,. . .,ym

m

j=1

y−j1/ pj.

(2.9)

Also, on [0, 1]m_{we obviously have}

K y1,. . .,ym

m

j=1

y−1/ pj+1/ pjn

j ≤K y1,. . .,ym

m

j=1

y−1/ pj

j . (2.10)

Now, assumption (1.3) on the kernel K allows us to use the Dominated Convergence theorem to infer that

lim n→∞Ii/n=

_∞

1

yi

0 ···

yi

0 K y1,. . .,ym

m

j=1

y−j1/ pjd y1···d ym,

lim n→∞

1

0···

1

0K y1,. . .,ym

m

j=1

y−1/ pj+1/ pjn

j d y1···d ym

=

1

0···

1

0K y1,. . .,ym

m

j=1

y−j1/ pjd y1···d ym.

(2.11)

Furthermore, we have

[0, 1]m

_m

i=1

Di

=Rm+, (2.12)

and fori,j=1,. . .,m, any of the intersection sets [0, 1]m_Di,D

iDj,i=j, has Lebesgue measure 0 inRm_{. Consequently, (2.4), (2.11) imply that}

T =lim n→∞

T f1,n,. . .,fm,n,gn

_gn

Lpf1,nLp1···fm,n_Lpm

=Cm. (2.13)

3. Applications

In this section we wish to revisit some important inequalities due to Hilbert, Hardy-Littlewood-P ´olya, and Hardy. We will discuss the possible multilinear extensions of these inequalities and provide formulas for the best constants in a closed form via the gamma function. The formulas we present here do not seem to be in the literature. Let us first recall a few basic facts about the gamma and beta functions. For more details and further references on special functions, see the book by Andrews et al. [1]. For a complex number

zwith Rez >0 define

Γ(z)=

_∞

0 t

z−1_e−t_{dt. (3.1)}

It follows from the definition that the gamma functionΓ(z) is analytic in the right half-plane Rez >0. Two fundamental properties of the gamma function are that

Γ(z+ 1)=zΓ(z), Γ(z)Γ(1−z)= π

sin(πz). (3.2)

In particular, we also haveΓ(n)=(n−1)! for all positive integersn. The second formula above is known under the name of Euler’s reflection formula. Next we define the beta function. Forzandwcomplex numbers with positive real parts,

B(z,w)=

1

0t

z−1_(1−t)w−1_{dt. (3.3)}

We have the following relationship between the gamma and beta functions:

B(z,w)=Γ(_Γ(z_z)Γ(w)

+w). (3.4)

3.1. Hilbert’s operator. Consider the linear operator

P f(x)=

_∞

0

f(y)

x+yd y (3.5)

with the kernelK(1)_{(x,y)=1/(x+y) which is symmetric in the variablesx,yand} homo-geneous of degree−1. By duality, we see that the integral analogue of Hilbert’s theorem (1.2) is equivalent to the best constant inequality

P fLp_(R+)≤πcsc

π p

fLp_(R+). (3.6)

by them-linear operator

P⊗ f1,. . .,fm(x)=

_∞

0 ···

_∞

0

f1 y1

···fm ym

x+y1)··· x+ymd y1···d ym (3.7)

having the kernel

K⊗(m)= _x 1

+y1

··· x+ym

= m

i=1

K(1) _x,y

i

. (3.8)

AlthoughK⊗(m) is symmetricx,y1,. . .,ym and homogeneous of degree−m, and thus it preserves the properties satisfied by its one dimensional counterpartK(1)_{, the} multilin-ear extensionP⊗ is not very interesting due to its tensorial character. Throughout the remaining of this paper, we let 1< p,p1,. . .,pm<∞be such that 1/ p=1/ p1+···1/ pm, and denote byp0=pthe dual exponent ofp. We will also writeLpforLp(R+). Since

P⊗ f1,. . .,fm

=(P⊗ ··· ⊗P) f1,. . .,fm

=P(f1)···P fm

, (3.9)

using H¨older’s inequality and (3.6), we obviously have

_{P⊗ f1,. . .,fmLp≤π}mm i=1

csc

π pi

_f1

Lp1···fmLpm. (3.10)

The constant on the right (the operator normP⊗ofP⊗) is the best possible.

A much more interesting situation arises when we consider the nontensorial extension

P¬⊗ f1,. . .,fm

(x)=

_∞

0 ···

_∞

0

f1 y1

···fm ym

x+y1+···+ym

md y1···d ym (3.11)

having the (symmetric in variablesx,y1,. . .,ymand homogeneous of degree−m) kernel

K¬⊗(m)= 1

x+y1+···+ym

m. (3.12)

Noting thatK¬⊗(m)≤K⊗(m), we know that the operatorP¬⊗is bounded fromLp1× ··· ×_Lpm intoLp_{. Nevertheless, the a priori constantP⊗is not the best possible in this case.}

Claim 3.1. The following best constant inequality holds:

_{P¬⊗ f1,. . .,fm}

Lp≤ 1 (m−1)!

m

i=0 Γ

1

pi

_f1

Lp1···fmLpm. (3.13)

Proof. ByTheorem 1.1, the best constant is given by

Cm=

_∞

0 ···

_∞

0 1 +y1+···+ym

−m

y−1/ p1

1 ···y−m1/ pmd y1···d ym. (3.14)

Let us denote the integral on the right byIm(m, 1/ p1,. . ., 1/ pm). By making the change of variablesym=(1 +y1+···+ym−1)tand integrating first with respect todt, we get

Im m, 1/ p1,. . ., 1/ pm

=

_∞

0 (1 +t)

−m_t−1/ pm_dtI

m−1 m−1/ pm, 1/ p1,. . ., 1/ pm−1

. (3.15)

Observe that, if we make the change of variablest+ 1=1/s,

∞

0 (1 +t) α_tβ_dt=

1

0s

−α−β−2_(1−s)β_{ds=B(−α−β−1,β+ 1). (3.16)}

Therefore, if we recall the relationship between the beta and gamma functions, we obtain

_∞

0 (1 +t)

−m_t−1/ pm_dt=Γ m−1/ p m

Γ 1/ pm

Γ(m) ,

Im m, 1/ p1,. . ., 1/ pm

=Γ m−1/ pm

Γ 1/ pm

Γ(m) Im−1 m−1/ p

m, 1/ p1,. . ., 1/ pm−1

.

(3.17)

By a simple induction argument, we obtain from this recurrence that

Im m, 1/ p1,. . ., 1/ pm

=Γ m−1/ pm− ··· −1/ p1

Γ 1/ p_m···Γ 1/ p1

Γ(m) . (3.18)

Due to the relation on the exponents, the latter equality simplifies exactly to

Cm=_(m−1_1)! m

i=0 Γ

1

p_i

. (3.19)

Remark 3.2. One can construct other multilinear extensions ofP. For example, consider them-linear operator

Pm f1,. . .,fm

(x)=

_∞

0 ···

_∞

0

f1 y1

···fm ym

x+y1···ym

m d y1···d ym. (3.20)

3.2. Hardy-Littlewood-P ´olya’s operator. We letQdenote Hardy-Littlewood-P ´olya’s lin-ear operator defined by

Q f(x)=

∞

0

f(y)

max(x,y)d y. (3.21)

Itsm-linear nontensorial extension is

Q¬⊗ f1,. . .,fm

(x)=

_∞

0 ···

_∞

0

f1 y1

···fm ym

max x,y1,. . .,ym

md y1···d ym. (3.22)

Claim 3.3. The following best constant inequality holds:

_{Q¬⊗ f1,. . .,fmL}p≤

_m

i=0 m

j=0,j=i

p_j

_f1

Lp1···fm_Lpm. (3.23)

Proof. With the notation in the proof ofTheorem 1.1, we can decomposeRm

+ into the union [0, 1]m₍m

i=1Di) so that the interiors of the sets in this union are pariwise disjoint. Furthermore, the best constant in the inequality we want to prove is

Cm=

∞

0 ···

∞

0

max 1,y1,. . .,ym−my1−1/ p1···y −1/ pm

m d y1···d ym

=

[0,1]my −1/ p1

1 ···y− 1/ pm

m d y1···d ym+ m

i=1

Ji,

(3.24)

where

Ji=

_∞

1

yi

0 ···

yi

0 y −1/ p1

1 ···y− m−1/ pi i ···y−

1/ pm

m d y1···d yi···d ymd yi. (3.25)

Now,

[0,1]my −1/ p1

1 ···ym−1/ pmd y1···d ym= m

j=1

y1−1/ pj j 1−1/ pj|

1 0=

m

j=1

p_j (3.26)

and (recall thatp0=p)

Ji=

_∞

1 y −m−1/ pi i

m

j=1,j=i

y1−1/ pj j 1−1/ pj|

yi 0d yi=

m

j=1,j=i

p_{j ∞}

1 y

−1−1/ p1−···−1/ pm

i d yi

= m

j=1,j=i

pj

y−1/ p1−···−1/ pm i

−1/ p1− ··· −1/ pm| ∞ 1 =

m

j=0,j=i

pj.

(3.27)

In particular, form=1, we recover the best constant in the (Lp_,Lp_{) inequality satisfied} by the operatorQ,C1=p+p=p2/(p−1); see [5, page 254]. Furthermore, for them -linear tensorial extension ofQ,

Q⊗ f1,. . .,fm

(x)=

_∞

0 ···

_∞

0

f1 y1

···fm ym

max x,y1

···max x,ym

d y1···d ym (3.28)

the best constant inequality is

_{Q⊗ f1,. . .,fm}

Lp≤ m

i=1

p2 i

pi−1

_f1

Lp1···fm_Lpm. (3.29)

3.3. Hardy’s operator. In his attempts to simplify the proofs known at the time of Hilbert’s double series theorem, Hardy introduced in [3] the operator

R f(x)=1

x _x

0 f(y)d y (3.30)

and proved that it is bounded fromLp _intoLp_{with best constant p/(p−1). Unlike the} kernels of the operatorsP,Qconsidered before, the kernelk(1)_{(x,y)=(1/x)χ[0,x](y) ofR} is not symmetric with respect to the variablesx,y(to emphasize this difference, we use now the lower case letterk). Here,χIdenotes the characteristic function of the setI. The

m-linear tensorial extension

R⊗ f1,. . .,fm(x)= 1

xm

x

0···

x

0 f1 y1

···fm ymd y1···d ym (3.31)

is then bounded from Lp1× ··· ×_Lpm _intoLp_{, and the best constant is}m

i=1pi/(pi− 1). Note that the kernel k⊗(m) of R⊗ is still positive and homogeneous of degree−m, as required in the hypothesis of Theorem 1.1. In our search for an appropriate non-tensorial multilinear extension of R, we observed that we could also writek(1)_(x,y)= (1/x)χ[0,1](y/x)=(1/x)χ[1+y/x,∞)(2). This simple observation has suggested that we de-fine the nontensorial extension via

R¬⊗ f1,. . .,fm(x)= 1

xm

∞

0 ···

∞

0 χ[0,x] y1+···+ym

f1 y1···fm ymd y1···d ym. (3.32)

It is worth noting that, as in the previous examples, the kernel of the nontensorial ex-tension satisfiesk(m)¬⊗ ≤k(m)⊗ . We therefore have the a priori boundedness of R¬⊗ from

Lp1× ··· ×_Lpm _intoLp_{. However, sincek}(m)_{¬⊗ is symmetric in the variablesy1,. . .,ymbut} not inx, we expect the operator norm of the nontensorial extension to be symmetric in

p1,. . . pmbut not inp0=p(the index that “corresponds” tox).

Claim 3.4. The following best constant inequality holds:

_{R¬⊗ f1,. . .,fm}

Lp≤

m i=1Γ 1/ pi

Γ m+ 1/ p0

Proof. ByTheorem 1.1, the best constant is given by

Cm=

∞

0 ···

∞

0 χ[0,1] y1+···+ym

y−1/ p1

1 ···y −1/ pm

m d y1···d ym

=

1

0

1−y1

0 ···

1−y1−···−ym−1

0 y

−1/ p1

1 ···y −1/ pm

m d ym···d y1

=pm

1

0

1−y1

0 ···

1−m−2

k=1yk

0 y

−1/ p1

1 ···y− 1/ pm−1

m−1

1− m−1

k=1

yk

1/ p m

d ym−1···d y1.

(3.34)

In the last integral, we make the substitutionym−1=tm−1(1−mk=−12yk) and integrate first with respect todtm−1. This allows us to simplify to an integral over onlym−2 variables multiplied by the integral in tm−1, 01t

−1/ pm−1

m−1 (1−tm−1)1/ pm_dtm−1 =_{B(1/ p}

m, 1 + 1/ pm). More precisely, we have

Cm=pmΓ 1/ p_m−1

Γ 1 + 1/ p_m

Γ 1 + 1/ p_m−1+ 1/ pm

×

1

0

1−y1

0 ···

1−m−3

k=1yk

0 y

−1/ p1

1 ···y −1/ pm−2

m−2

1− m−2

k=1

yk

1/ pm−1+1/ pm

d ym−2···d y1.

(3.35)

Since, pmΓ(1 + 1/ p_m)=Γ(1/ p_m), by letting successively ym−2=tm−2(1−km=−13yk),. . .,

y2=t1(1−y1) and integrating with respect todtm−2,. . .,dt1respectively, we obtain

Cm=

m

i=1Γ 1/ pi

Γ1 +mi=1 1/ pi

. (3.36)

Since,m_i=11/ pi=m−1/ p, the desired result follows.

Remark 3.5. The integral inequalities we considered have corresponding discrete ver-sions. For example, the m-linear discrete versions of Hilbert’s and Hardy-Littlewood-P ´olya’s inequalities are, respectively,

∞

k=1 ∞

k1=1

··· ∞

km=1

akak1···akm

k+k1+···+kmm ≤

Cm( ∞

k=1

a_kp)1/ p m

j=1

_∞

kj=1

apj kj

1/ pj ,

∞

k=1 ∞

k1=1

··· ∞ km=1

akak1···akj max k,k1,···,km

m ≤Cm

_∞

k=1

a_kp 1/ p

m

j=1

_∞

kj=1

apj kj

1/ pj

.

(3.37)

Acknowledgment

We are indebted to the anonymous referees for the many valuable suggestions that im-proved the quality of our presentation.

References

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[2] L. Grafakos and R. H. Torres,A multilinear Schur test and multiplier operators, Journal of Func-tional Analysis187(2001), no. 1, 1–24.

[3] G. H. Hardy,Note on a theorem of Hilbert, Mathematische Zeitschrift6(1920), no. 3-4, 314–317. [4] ,Note on a theorem of Hilbert concerning series of positive terms, Proceedings of the

Lon-don Mathematical Society23(1925), 45–46.

[5] G. H. Hardy, J. E. Littlewood, and G. P ´olya,Inequalities, 2nd ed., Cambridge University Press, Cambridge, 1952.

[6] I. Schur, Bemerkungen zur Theorie der beschr¨ankten Bilinearformen mit unendlich vielen ver¨anderlichen, Journal f¨ur die Reine und Angewandte Mathematik140(1911), 1–28 (German). [7] H. Weyl,Sing¨ulare Integralgleichungen mit besonderer Ber¨ucksichtigung des Fourierschen

Inte-graltheorems, Inaugural-Dissertation, Gottingen, 1908.

´Arp´ad B´enyi: Department of Mathematics, 516 High Street, Western Washington University, Bellingham, WA 98225-9063, USA

E-mail address:arpad.benyi@wwu.edu

Choonghong (Tadahiro) Oh: Department of Mathematics and Statistics, Lederle GRT, University of Massachusetts, Amherst, MA 01003-9305, USA