NEW ZEALAND JOURNAL OF MATHEMATICS Volume 28 (1999), 193-202

T W O REAL-V ALU ED FU N C T IO N S A SSOC IAT ED W IT H

C O N V O LU T IO N S W IT H M EA SU RES

R a y m o n d J. G r in n e l l

(Received June 1998)

Abstract. Let M (G ) denote the set of Borel measures on an infinite compact

abelian group G. For /x € M(G), the index function I(fi, ■) is the real-valued

function defined on the interval [1, oo) by J(/i,p) = sup{q > p \ n * Lp C Lq}.

Given p £ [l,oo), the measure function I(- ,p ) is defined on M (G ) also as

/(/x, p). In this paper we initiate a study of real-variable properties of these two

functions, both of which are naturally associated with Lp-improving measures.

1. Introduction and Notation

In [14], Stein posed the following problem concerning convolution operators

which are induced by measures: “... characterize ... the condition of / —> / * d/i

yielding a bounded operator from an Lp space to an Lr space.” Measures which ex

hibit the contraction, or improving property implicit in Stein’s problem have been

studied extensively and in a wide variety of settings (see [5], [6], and the references cited therein). In this paper we study real-variable properties of two functions

which are naturally associated with Lp-improving measures.

A Borel measure fi is called Lp-improving if for some p G [l,oc) there exists

q G (p, oo) such that p * / G L9 for all / G Lp. It is convenient to express this condition compactly as fi * Lp C Lq. Young’s inequality shows that each

absolutely continuous measure whose Radon-Nikodym derivative belongs to Ls for

some s G (l,oo) is Lp-improving. Certain Riesz product measures [2], [11] and

certain Cantor-Lebesgue measures on the circle group [4], [9], [12] are also known

to be Lp-improving. In contrast, the set of non-Lp-improving measures contains

all discrete measures and certain absolutely continuous measures [5].

Throughout this paper G represents an infinite non-discrete compact abelian

group, T its discrete dual group, and A its normalized Haar measure. Let M(G)

denote the space of finite complex Borel measures on G and let Mq(G) be the set

of measures p in M(G) such that p(7 ) —> 0 as 7 —> 00. Unless stated otherwise we abbreviate Lp(A) as Lp. All other notation is as in [5].

For each p G M(G) it is well-known that /i * Lp C Lp for all p G [1,00]. If fi is Lp-improving then the Riesz-Thorin interpolation theorem implies that for each

p G (1,00) there exists a q € (p, 00) such that fi * Lp C Lq. We are interested in studying the behavior of the index q as a function of p. Here is the main definition.

Definition 1.1. For p G M(G) and p G [1,00) let /(/i,p) = sup{(/ > p \ fi * Lp C

in -

1991 AMS Mathematics Subject Classification: Primary 43A05; Secondary 42A85, 26A99.

194 RAYMOND J. GRINNELL

For a given measure fi G M(G) the function I(fi, •) is called the index function.

For a fixed value of p G [1, oo) the function /(• ,p) is called the measure function.

The quantity I{fi,p) measures the maximum contraction of Lp when acted upon

by the convolution operator induced by fi. There are three instances where a

closed formula for I(fi,p) can be written explicitly in terms of fi, fi, and the index

p (but without reference to supremum). The trivial situation is when fi is not

Lp-improving; here we have that I{fi,p) = p for all p G [1, oo). A well-known and

important example is due to Young’s inequality; see Proposition 2.1. The third

instance is far less obvious and has its origins in [2]. The details of this third

example are developed in Example 2.7. With the exception of these three examples

it is a difficult fundamental problem to write a closed formula for I(fi,p).

2. The Index Function

For fi G M(G) we are primarily interested in studying I(fi, •) as a finite real

valued function. However, we write I(fi,p) = oo for p G [l,oo) to mean precisely

that fi * Lp C Lq for all q G [p, oo). Define the domain of I(fi, •) as the set D„ = {p e [1 , oo) | I(fi,p) < oo}. Note that = 0 if and only if fi* L1 C Lq for all q G [l,oo). The Riesz-Thorin interpolation theorem shows that fi is not

Lp-improving if and only if I(fi,p ) = p for all p G (1, oo). Therefore DM = [1, oo) if

fi is not Lp-improving. In contrast, if fi = hX where h G L°° then fi* L1 C L°° and thus = 0. The following important proposition is a consequence of the sharp

form of Young’s inequality [10, Theorem 1.1] and the standard Young’s inequality

for measures [8, (20.12)]. For q G (l,oo), q' denotes the index conjugate to q.

Proposition 2.1. Let q G (l,oo), let h G Lq be such that h £ Lr for any r > q, and set fi = hX. Then:

(i) i {v,p) = (pq)/(p + q ~ pq) i f p t [ W ) ;

(ii) I(fi,p) = oo if p e [g',oo);

(iii) = [W );

(iv) For all p G D I ( f i , p ) — max{r > p \ fi * Lp C Lr }.

The set of non-Lp-improving measures, the measures described in Proposition

2.1, and the measure studied in Example 2.7 appear to be the only known examples

for which I(fi,p) can be written explicitly as a closed formula in terms of p, fi, and fi.

Note that the measures in these three examples also have the interesting property

that I(fi,p) can be defined as max{r > p \ fi * Lp C Lr }, which is an improvement on Definition 1.1. The following result describes the structure of the domain of

Theorem 2.2. For each fi G M(G), either DM is empty or is an interval of the

form [l,po) for some po G (l,oo].

Proof. If fi is not Lp-improving then = [l,oo) which gives the result with

P o = oo. Consider the case where fi is Lp-improving. Proposition 2.1 and the

remarks preceding it show that may indeed be of the asserted form. Suppose

Dp 7̂ 0 and that p G D Then I(fi,s ) < oo for all s G (1 ,p) since Lp C Ls. It follows that must either be an interval of the form [l,po) where po G (l,oo], or of the form [l,po] where po G (l,oo). By way of contradiction we show that

TWO REAL-VALUED FUNCTIONS 195

the second form does not occur. Define sequences rn = and un = where

n > 2 and let N be a sufficiently large integer such that rn > 1 whenever n > N.

Note that rn, un —> po as n —> oo and that for n > N, 9n = satisfies the

equation — = -1-—a + For each n > N, let sn be a number in the intervalP O T n U n

(rn, I(p, rn)) and observe that p * Lrn C LSn. Since po < un, I(p ,u n) = oo, and

thus /i * LUn C Ln. By the Riesz-Thorin interpolation theorem we conclude that

H * Lpo C L9n where qn is defined by the equation = 1~°n + ^ and n > N. A

calculation shows that qn = (sn(n + 1))/(1 + sn) and hence qn —* oo as n —> oo. It

follows that I(/j,,po) = oo which is a contradiction. □

With reference to Theorem 2.2 above, if = [l,po) for some po G (l,oo], let

= (l,Po).

Theorem 2.3. Let p, G M (G ) and assume Z)° = (l,po)- Then:

(i) p is Lp -improving if and only if I{p,p) > p for all p G D °;

(ii) If Pi,P2 G £>£ and px < p2 then I{p,p2) > £7

(iii) /(/x, •) is strictly increasing on

Proof. Assertions (i) and (ii) are easy consequences of the Riesz-Thorin interpo

lation theorem and (iii) follows obviously from (ii). □

A consequence of Proposition 2.1 is that the inequality in Theorem 2.3 (i) is best

possible in the sense that there does not exist a universal constant c > 1 whereby I(p,p) > cp for all p G D°̂ and all Lp-improving measures p. The inequality in

Theorem 2.3 (ii) is also optimal; this is easily seen by taking a non-Lp-improving

measure.

Theorem 2.4. Let p be Lp-improving and assume = [l,po) for some

PoG(l,oo]. Then:

(i) The function I(p, •) is continuous on (l,po);

(ii) If I(p , 1) > 1 then I(p, •) is continuous on [l,Po)/

(iii) If po < oo then I(p,p) —> oo as p —> p$ ;

(iv) If po = oo then I(p,p) —► oo as p —► oo.

Proof. To prove (i), let x q G D°̂ . We first show that I(p, •) is left-continuous at

x q ■ Let p G (1, xo) and let {pn}^=i be a sequence in (p,xo) which increases to x q .

The sequence { / ( / i , ^ ) } ^ ! is increasing and thus bounded above by I(p,xo)- It

follows that s < I(p ,xo) where s = sup{I(p ,pn)}^=1. Let r G (xo,I{p,xo)) and

t G (p,I(p,p))', then p * Lx° C Lr and p * Lp C Ll . Since p < pn < x q for all

n, there exists a number 9n G (0,1) which satisfies the equation = 1̂ n +

By the Riesz-Thorin interpolation theorem, p * LPn C Lqn where qn is defined by

the equation — = + %*-. Since pn —* xq as n —> oo, it follows that 9n —* 1Q n t r

and hence qn —> r. Now s > qn for all n, and thus s > r which implies that

s > I(p ,x o). We see that s = I(p,po) which establishes the left continuity at x q .

The right continuity of I (p , •) at Xq is proved similarly by taking the infimum of a

decreasing sequence in an open interval (xo,po) contained in D°.

196 RAYMOND J. GRINNELL

For the proof of (ii), it follows from (i) above that we need only verify the right

continuity of I(fi, •) at 1. Let q = 7(^,1). Since [i * L 1 C Ls for all s G (1,

TWO REAL-VALUED FUNCTIONS 197

It is asserted in [5] that the formula in Example 2.7 is known to be true for

all tame //-improving measures on (Z 2)00. However, it is not known whether the tameness requirement can be omitted. The inequality in Example 2.7 has its

origins in [2] where it was first established for certain Riesz product measures whose

Fourier-Stieltjes coefficients do not tend to zero. It is also interesting to note in

Example 2.7 that /(/i,p) coincides with max{(? > p \ /i * Lp C Lq}~.

The next example displays a bound for the index function for a certain Cantor-

Lebesgue measure on the circle group. The main idea for the example is found in

the multivariable inequality in [9]. Related results are found in [1] and [4].

Example 2.8. Let /j denote the Cantor-Lebesgue measure on the circle group

T which is supported by the classical middle-third Cantor set. Lemma 1 in [9]

establishes a symmetric inequality in five real variables which is then used to prove

that fi is Z/p-improving. Furthermore, [9, p. 117] shows that i f l < p < g < o o and

/i * Lp C Lq then ^ + (l — ) (l — ^) < 1. This inequality determines a finite

bound for q if and only if p E (l, ). When p is in this small interval, we have

u j j ( ̂ ^ p( l°g(3)—log(2))the bound 7(/i,p) < fog(3)_p;og(2/ •

Regarding Example 2.8 above, Beckner (see [9]) has shown that /(/i, = 2;

it is also known that 7(/i, |) > 3 [15]. If fi is any non-zero Lp-improving measure

then J(/i, 2) < 2(J^J) where c(/i) is as in Example 2.7 [5, Corollary 3.2(ii)] and

||/j|| is the norm of [i.

Recall from Proposition 2.1 (iv) that max{r > p \ fi * Lp C Lr} coincides with

I(n,p) if n = hX and h E Lq for some q E (1,00) and h £ Lr for any r > q. The following example shows that it is indeed necessary to incorporate supremum in

Definition 1.1.

Example 2.9. Let p € (1,2). By [5, Theorem 1.6] there exists a non-negative

measure ^ = hX with h E L 1(T), and a function g E flte[i,oo) (-^(aO such that if v = g î then n * Lp C L2 and v * Lp % L2. We show that I(v,p) = 2 by demonstrating that v * U 3 C Lq for all q E (p, 2). For each t > the function g E (Ll (n) P|L*(T)). Following the proof of [5, Theorem 1.1] we see that

v * Lps C L2 for all s E (l, -) where s denotes the index conjugate to t. Since u * L 1 C L1, the Riesz-Thorin interpolation theorem implies that v * Lp C Lq where q is defined by the equation ^ = 1 — | and 9 is defined by the equation

- = 1 — 9 + ^ . Simplification shows that 9 assumes every value in the interval

(2 — l) and hence u * Lp C Lq for all q E (p , 2).

Let /i E M (G ) and assume = [l,po) f°r some po E (1,00]. Denote the range of I(fi, ■) by = {I(n,p) | p E D^}. If fi is not Lp-improving then = [1, 00). If n = hX where h E Lq for some q E (l,oo) and h Lr for any r > q, then

Proposition 2.1 shows that RM = [q, 00).

Theorem 2.10. Let n E M(G), assume D M = [l,po) for some po E (l,oo], and

assume I(fi, ■) is continuous on Then RM = [/(//, 1), 00).

198 RAYMOND J. GRINNELL

Proof. The result is obvious if fi is not Lp-improving. If fi is Lp-improving, then

Theorem 2.4 shows that /(/i,p) is continuous on DM = [l,po) and I(fi,p) —> oo as

p —> Po. It follows that is an interval in M and thus R^ = [/(/i, 1), oo). □

We now define a function which is related to I(/i, •). For fi G M (G ) and

r G (1, oo) let J(/i, r) = inf{p G (1, r] | fi * Lp C Lr}. Clearly J{fi,r) = r for

all r G (1, oo) precisely when fi is not Lp-improving. The following result is analo

gous to Proposition 2.1.

Proposition 2.11. Let q G (l,oo), let h G Lq be such that h £ Ls for any s > q,

and set fi — hX. Then:

(i) J(fi, r) = (rq)/(q + rq - r ) if r G [q, oo);

(ii) J(/i,r) = 1 if r e (1 ,q);

(iii) For each r G [q, oo), J(fi, r ) = min{p G (1, r] \ fi* Lp C Lr}.

The next result shows that the functions I(fi, •) and J(fi, •) are inverse functions.

Theorem 2.12. Let fi G M (G ) and assume D M = [l,po) for some po G (l,oo].

Then:

(i) J(fi,I(fi,p)) = p for all p G D^;

(ii) /(/i, J(fi,q)) = q for all q G (l,oo) such that J(fi,q) G D

Proof. We assume fi is Lp-improving; otherwise, both (i) and (ii) are obvious. To

prove (i), let p G Z)M and let a = I(p,p). Since fi * Lp C Lr for all r G [p, a), the

Riesz-Thorin interpolation theorem implies that fi* Ls C La for all s G (p,a], and

thus J(fi,a) < p. If J(p, a) < p then there is some pi G (J(/x,a),p) such that

fi * LPl C La. Since /(//, •) is strictly increasing, fi * Lp C Ls for some s > a which

is a contradiction. This shows that J(/i,a) = p which proves (i).

To prove (ii), let q G (l,oo), assume J(fi,q) G D and let (3 = J(fi,q). Since fi

is Lp-improving, (3 < q. Let A = {s > (3 \ fi * L@ C L3}. We claim that q is an

upper bound for the set A. If this false, then there is some s G A with s > q such

that fi * L 13 C Ls. Define 6 G (0,1) by the equation ^ | and define w bythe equation ^ Then 1 < w < (3 and the Riesz-Thorin interpolation

theorem shows that fi * Lw C Lq, which contradicts the fact that w < (3. This

establishes the claim. We now show that q is the least upper bound of A. If this

is false, then < q and thus fi * Lr C Lq for all r G (/?,

TWO REAL-VALUED FUNCTIONS 199

3. The Measure Function

For p G [l,oo) let Ep = {fi e M (G ) | I{fi,p) < 00}. Each set Ep contains all

discrete measures since discrete measures are not Lp-improving [5, Corollary 3.2].

The following example shows that, in contrast with Theorem 2.4, /(•,£>) is not

continuous on Ep.

Example 3.1. Let p G (1, 00) and let h and fi be as in Example 2.6. Let {hn

be a sequence of trigonometric polynomials converging to h in L l and set fin = hnX.

Then I(fin,p) = 00 for each n, however I{fi,p) = p for all p G (1,00).

Let fi, v G M(G) and let p G (l,oo). If fi G Ep then I(kfi,p) = I(fi,p ) for each

complex number k ^ 0. If u G Ep then I(fi + v,p) > mm{I(fi,p), I(v,p)}. Equality

can attain, for example, if fi is Lp-improving but v is not. Strict inequality can

also attain by letting fi = — v where v is not Lp-improving.

The following example shows there is no simple relationship between I(fi,p ) and

I(v ,p ) if v < fi.

Example 3.2. Let q G (1,00), let hi G Lq be such that hi ^ Lr for any r > q, and define the measure vi = hi\. As in Example 2.6, let h2 G L 1 so that v2 = h2X is not Lp-improving. Then i'l, v2 p for all p G (1, q'), however I{v2 ,p) = p.

This example is a special case of [5, Theorem 1.4] where it is shown that given

any non-zero Lp-improving measure fi, there exists a measure v such that fi and

v are mutually absolutely continuous and v is not Lp-improving. There is also no

clear relationship between I(fi,p ) and I{v,p) if v is singular with respect to fi.

Example 3.3. Consider a discrete measure ui on the Walsh group (Z 2)°°. Then

is not Lp-improving so I(u i,p) = p for all p G (1,00). From the comments following

Example 2.7, if v2 is a tame Lp-improving measure on (Z 2)°° then I(u2 ,p) > p for all p G (1,00). Note however that both Vi and v2 are singular with respect to A.

The following example shows that the mapping f i ^ I (fi, ■) is not injective.

Example 3.4. Let a i,a2 be distinct points in G and let ntij be the point-mass at the point a,j. Then mi / m2 but I(m i,p ) = I(m 2 ,p) = p for all p G [1, 00).

The next result establishes a functional-inequality for I in terms of fi, v, and

fl*U.

Theorem 3.5. For fi, u G M(G) we have l(fi, I(u,p)) < I(fi*u,p) for all p G A ,.

Proof. Assume first that l(fi,I(v ,p )) = 00 for some p G D u. Suppose = . Then for all r G [p, 00), fi * Lp C Lr and thus f i * v * L p C f i * L p C Lr. This

shows that I(fi * v,p) = 00. Now suppose = [l,po) for some po G (l,oo]. Let

w = I(v,p). Since I(fi,w) = 00 it follows that w qL and hence 1 < po < w.

Let r G [p, 00). By Theorem 2.4, there exists s G (l,Po) such that I(fi,s ) > r and

thus fi * Ls C Lr. We also have that v * Lp C Ls since s G (l,w ). Therefore

f i* v * Lp C Lr which implies that I ( f i*u ,p ) = 00.

Assume next that / ( / i , I(u,p)) < 00 for some p G D v. If v is not Lp-improving, then I{v,p) = p and thus l(fi, I(is,p)) = p. But p < I(fi*v,p) since f i*v*Lp C Lp.

200 RAYMOND J. GRINNELL

We now assume v is Lp-improving. Let w = I(u, p). Then w G (p, oo) and /(/x, s)

is finite for each s G (1, w]. Consider r > w such that * Lw C Lr. If v * Lp C Lw,

then p ,*u *L P C / j ,*L w C Lr and thus r e { q > p \ p ,* v * L p C Lq}. This implies

Z(/x, I(u,p)) < /(// * v,p) as required. Suppose now that p < w, that v * Lp C L 1 for each t E [p, w), but i/ * Lp Lw. Since the function I(n, •) is continuous on

(1 ,w), given e G (0, r — 1) there exists S G (0, w — 1) such that p, * Lw~s C Lr~£.

Since u * IP C Lw~s we have that fi * v * Lp C ^ * Lw~6 C Lr~e. Therefore, sup{

TWO REAL-VALUED FUNCTIONS 201

[4], [9], [12] are all examples of finitely improving measures whose Fourier-Stieltjes

coefficients do not tend to zero.

It is not known whether the assumption in Theorem 2.4 that /(p, 1) > 1 is

necessary to prove I(n , •) is continuous on [l,po)> The function I(/i, •) is strictly

increasing and continuous on DM hence it follows that I(fi, •) is differentiable almost

everywhere on D In Proposition 2.1 and Example 2.7 it is obvious that function

/(/i, •) is convex. The Riesz-Thorin interpolation theorem suggests that I(p, •)

is a convex function for all ji G M (G ), but this has yet to be verified. It is also

not known whether /(//, •) is differentiable on all of in the case where fi is

Lp-improving. A short calculation shows that if p\,p2 £ with p2 > p\, then Q{h,P2) > Q(fJ>iPi) where Q(fi,p) = . It is not clear whether the function Q(p, •) is strictly increasing on D^.

Let M denote the set of measures in M (G ) such that — [l,po) for some

Po G (l,oo] and let F denote the set of continuous real-valued strictly increasing

functions each having domain of the form (l,po) for some po G (1, oo]. Consider the

mapping (j) defined on M as (p(fi) = /(//, •). Theorems 2.3 and 2.4 show that the

image of M under is contained in F. A natural question is that of characterizing

the image of M in F.

Finally, the discussion following Example 3.1 entails the inequality I(n + v,p) >

min{/(/x,p), I(v,p)}. It is of interest to establish a characterization or an alternative

description of those measures for which strict inequality or equality attains.

Acknowledgement. The author thanks the referee for the helpful comments and

thanks Douglas Bridges for the kind editorial assistance.

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Raymond J. Grinnell

Department of Mathematics and Computer Science

Mount Allison University

67 York Street

Sackville

NB CANADA E4L 1B3

rgrinnell@mta.ca

mailto:rgrinnell@mta.ca