WEAK TYPE RADIAL CONVOLUTION OPERATORS
ON FREE GROUP
TADEUSZ PYTLIK AND RYSZARD SZWARC
Abstract. Radial convolution operators on free groups with non-
negative kernel of weak type (2, 2) and of restricted weak type (2, 2)
are characterized. Estimates of weak type (p, p) are obtained as
well for 1 < p < 2.
1. Introduction
A discrete group G is called amenable if there exists a linear func-
tional m on ℓ∞
(G) such that
(1) inf
f(x) ≤ m(f) ≤ sup
f(x),
(2) m(xf) = m(f), where xf(y) = f(x
−1y).
m is called a left invariant mean. Then the functionalM(f) = m(m(fx))
satisfies (1), (2) and is also right invariant, where fx(y) = f(yx).
Let G be a discrete group. Consider a symmetric probability measure
µ on G, i.e.
µ(x)δx, µ(x) ≥ 0,
µ(x) = 1, µ(x−1) = µ(x).
The left convolution operator λ(µ) with µ is bounded on ℓ2(G) and
‖λ(µ)(f)‖2 = ‖µ ∗ f‖2 ≤ ‖f‖2, f ∈ ℓ
2(G).
Indeed
‖µ ∗ f‖2 =
µ(x)[δx ∗ f ]
µ(x)‖δx ∗ f‖2 = ‖f‖2.
2000 Mathematics Subject Classification. Primary 43A15, Secondary 43A07,
43A15.
Key words and phrases. free group, convolution operators, weak type, restricted
weak type.
The second author was supported by European Commission Marie Curie Host
Fellowship for the Transfer of Knowledge “Harmonic Analysis, Nonlinear Analysis
and Probability” MTKD-CT-2004-013389 and by MNiSW Grant N201 054 32/4285.
http://arxiv.org/abs/0704.1859v1
2 TADEUSZ PYTLIK AND RYSZARD SZWARC
Thus ‖λ(µ)‖2→2 ≤ 1.
Kesten [5] showed that a discrete group G is amenable iff for any
symmetric probability measure µ on G we have ‖λ(µ)‖2→2 = 1. He
showed that G is amenable if condition is satisfied for one measure µ
such that supp µ generates G algebraically. In particular let G be gen-
erated by g1, g2, . . . , gk and µ =
(δgi + δg−1
). Then G is amenable
if and only if ‖λ(µ)‖2→2 = 1.
In [4] Følner came up with another property equivalent to amenabil-
ity. We say that a discrete group G satisfies the Følner condition if for
any number ε > 0 and any finite set K ⊂ G there exists a finite set
N ⊂ G such that
(1) |xN △N | < ε|N |, x ∈ K.
In other words N is almost K invariant. He showed that G is amenable
if and only if the Følner condition holds.
Assume that G is amenable. Let µ be a probability measure with
finite support K. For ε = η2 > 0 choose N so as to satisfy (1). Then
‖µ ∗ χN − χN‖2 =
µ(x)[χxN − χN ]
µ(x)‖χxN − χN‖2
µ(x)‖χxN△N‖2 =
µ(x)|xN △N |1/2 ≤ η|N |1/2 = η‖χN‖2.
Therefore
〈µ ∗ χN , χN〉ℓ2(G)
= 〈χN , χN〉ℓ2(G) + 〈µ ∗ χN − χN , χN〉ℓ2(G) ≥ (1− η)‖χN‖
which implies
(2) sup
N,M−finite
〈µ ∗ χN , χM〉
‖χN‖2‖χM‖2
= 1 = ‖λ(µ)‖2→2.
The same holds (with the same proof) for any 1 < p < +∞, i.e.
(3) sup
N,M−finite
〈µ ∗ χN , χM〉
‖χN‖p‖χM‖p′
= 1 = ‖λ(µ)‖p→p,
where p′ = p/(p− 1).
We will use the notion of Lp,q spaces, which have been introduced
by Lorentz (see [1]). Consider a general σ-finite measure space (Ω, ω)
WEAK TYPE RADIAL CONVOLUTION OPERATORS ON FREE GROUP 3
and 1 < p < +∞. For f ∈ Lp(Ω, ω) and t > 0 we have
tp ω{x : |f(x)| > t} ≤
|f(x)|pdω(x).
Functions for which the left hand side is bounded form a linear space
Lp,∞(Ω, ω) =
f : sup
tpω{x : |f(x)| > t} < +∞
called the weak Lp space. This space contains Lp(Ω, ω).
For p′ = p/(p − 1) the predual of Lp
′,∞(Ω, ω) with respect to the
standard inner product is denoted by Lp,1(Ω, ω). We have
Lp,1(Ω, ω) ⊂ Lp(Ω, ω) ⊂ Lp,∞(Ω, ω).
For p > 1 these spaces are normed.
Any linear operator mapping Lp into itself is called of strong type
(p, p). Linear operators T mapping Lp(Ω, ω) into Lp,∞(Ω, ω) are called
of weak type (p.p), while those which map Lp,1(Ω, ω) into Lp,∞(Ω, ω)
are called of restricted weak type (p, p).
We will use the following facts. A linear operator T is bounded from
Lp,1 into a Banach space X if and only if
(4) ‖T‖L(p,1)→X = sup
‖TχE‖X
‖χE‖p
< +∞.
A linear operator T is bounded from Lp,1 into Lp,∞ if and only if
(5) ‖T‖(p,1)→(p,∞) = sup
E,F⊂Ω
|〈TχE, χF 〉|
‖χE‖p‖χF‖p′
< +∞.
Using this and duality between spaces L(p
′,1) and L(p,∞) we obtain
(6) ‖T‖p→(p,∞) = ‖T
∗‖(p′,1)→p′ = sup
‖T ∗χE‖p′
‖χE‖p′
The equalities (2) and (3) can be interpreted as follows. If the group
G is discrete and amenable and µ is a symmetric probability measure
on G, then
‖λ(µ)‖p→p = ‖λ(µ)‖(p′,1)→p′ = λ(µ)‖p→(p,∞) = ‖λ(µ)‖(p,1)→(p,∞) = 1.
Hence for these groups convolution operators with nonnegative func-
tions of strong type (p, p), of weak type (p, p) and of restricted weak
type (p, p) coincide for any 1 < p < ∞.
The situation is entirely different for nonamenable groups. Only
special examples have been studied. It has been shown [9] that for
p = 2 and G = Fk, the free group on k generators, k ≥ 2, there
exist nonnegative functions f on G such that ‖λ(f)‖2→(2,∞) is finite
4 TADEUSZ PYTLIK AND RYSZARD SZWARC
while ‖λ(f)‖2→2 is infinite, i.e. there exist convolution operators with
nonnegative functions of weak type (2,2) which are not of strong type
(2,2). The same has been shown for 1 < p < 2 [10]. These functions f
can be chosen to be radial, i.e. constant on elements of the group G of
the same length. It is an open problem if these results remain true for
any discrete nonamenable group.
In this work will focus on G = Fk. We are going to determine all non-
negative radial functions f on G such that λ(f) is of weak type (2,2),
as well those f for which λ(f) is of restricted weak type (2,2). In par-
ticular we prove that these spaces are different. Next we will turn our
attention to the case 1 < p < 2. By using interpolation machinery, du-
ality and the results for p = 2 we will be able to determine nonnegative
radial functions f for which λ(f) is of weak type (p, p). In this way we
obtain a simpler proof of the upper estimate of ‖λ(f)‖p→(p,∞) obtained
in [3]. Our method does not rely on deep theorems on representation
theory
2. Radial convolution operators of weak type (2, 2)
Let Fk = gp{g1, g2, . . . , gk} be a free group on k ≥ 2 generators.
The group consists of reduced words in generators and their inverses.
This representation is unique. The number of letters in reduced form
defines length function on Fk. Let χn denote the indicator function of
words of length n. There are 2k(2k − 1)n−1 such words. as we have
2k choices for the first letter and 2k − 1 choices for every consecutive
one. Let q = 2k − 1. The next theorem generalizes the estimate for
‖λ(χn)‖2→(2,∞) given in [9].
Theorem 1. Let f =
n=0 fnχn. The operator λ(f) is of weak type
(2, 2) if
A(f) :=
n,m=0
|fn||fm|q
−(n+m)/2{1 + min(n,m)} < +∞.
Moreover if fn ≥ 0 the condition is necessary and
A(f) ≤ ‖λ(f)‖22→(2,∞) ≤ 4A(f).
Proof. By (7), instead of estimating ‖λ(f)‖2→(2,∞) we may estimate
‖λ(f)‖(2,1)→2, which (see (4)) is equivalent to
‖f ∗ χE‖2
|E|1/2
WEAK TYPE RADIAL CONVOLUTION OPERATORS ON FREE GROUP 5
We have
‖f ∗ χE‖
2 = 〈f ∗ f ∗ χE∗, χE〉 =
n,m=0
fnfm〈χn ∗ χm ∗ χE , χE〉.
Simple calculation gives that for n ≥ 1 we have
χn ∗ χm = q
n−1δmn χ0 +
k=|n−m|
k≡n+m mod 2
q(n+m−k)/2 χk.
Clearly χ0 ∗ χ0 = χ0. Therefore
χn ∗ χm ≤ 2
k=|n−m|
k≡n+m mod 2
q(n+m−k)/2χk.
Hence
‖f ∗ χE‖
2 ≤ 2
n,m=0
fnfm q
(n+m)/2
k=|n−m|
k≡n+m mod 2
q−k/2 〈χk ∗ χE , χE〉.
Lemma 1.
〈χk ∗ χE, χE〉 ≤ 2q
[k/2]|E|.
Proof. Define an operator Pk by the rule
〈Pkδx, δy〉 =
〈χk ∗ δx, δy〉 if |x| ≥ |y|
0 if |x| < |y|.
〈χk ∗ δx, δy〉 ≤ 〈Pkδx, δy〉+ 〈δx, Pkδy〉.
This implies
〈χk ∗ χE , χE〉 ≤ 2〈PkχE , χE〉 ≤ 2‖PkχE‖1 ≤ 2|E| sup
‖Pkδx‖1
Pkδx =
|w|=k
|wx|≤|x|
Let w = w1w2 where |w1| ≤ |w2| ≤ (k + 1)/2. The conditions |w| = k
and |wx| ≤ |x| imply that w2 is determined by the first [(k + 1)/2]
letters of x. Hence we have as many terms in the sum as choices for
w1, i.e. at most q
[k/2]. Thus
‖Pkδx‖1 ≤ q
[k/2].
Therefore
〈χk ∗ χE, χE〉 ≤ 2q
[k/2]|E|.
6 TADEUSZ PYTLIK AND RYSZARD SZWARC
Lemma 1 implies that
‖f ∗ χE‖
n,m=0
|fn||fm|q
(n+m)/2
k=|n−m|
k≡n+m mod 2
n,m=0
|fn||fm|q
(n+m)/2{1 + min(m,n)}.
We obtain the upper estimate
‖λ(f)‖22→(2,∞) ≤ 4
n,m=0
|fn||fm|q
(n+m)/2{1 + min(m,n)}.
On the other hand if fn ≥ 0 we have
‖λ(f)‖22→(2,∞) ≥
q + 1
q−2k‖f ∗ χ2k‖
fn(χn ∗ χ2k)
l=|n−2k|
l≡n mod 2
q(n+2k−l)/2 χl
q−l/2χl
n=|2k−l|
n≡l mod 2
n=|2k−l|
n≡l mod 2
n=2k−l
n≡l mod 2
n,m=0
fnfmq
(n+m)/2
l=max(2k−n,2k−m)
l≡n≡m mod 2
Considering even or odd values of m and n gives
‖λ(f)‖22→(2,∞) ≥
n,m=0
f2nf2mq
n+m{1 + min(n,m)},
‖λ(f)‖22→(2,∞) ≥
n,m=0
f2n+1f2m+1q
n+m+1{1 + min(n,m)}.
Since k is arbitrary
‖λ(f)‖22→(2,∞) ≥
n,m=0
n≡m mod 2
fnfmq
(n+m)/2{1 + min(n,m)}.
WEAK TYPE RADIAL CONVOLUTION OPERATORS ON FREE GROUP 7
This implies
‖λ(f)‖22→(2,∞) ≥
n,m=0
fnfmq
(n+m)/2{1 + min(n,m)},
because the matrix a(n,m) = 1 + min(n,m) is positive definite. �
Theorem 2. For n ≥ 0 there holds
‖λ(χn‖(2,1)→(2,∞) ≤ cq
Proof. We have
‖λ(χn‖(2,1)→(2,∞) = sup
E,F⊂Fr
〈χn ∗ χE, χF 〉
|E|1/2|F |1/2
The proof will be completed if we show
(8) 〈χn ∗ χE , χF 〉 ≤ cq
n/2|E|1/2|F |1/2.
We will prove (8) by modification of the argument used in the proof of
Lemma 1. Fix α ∈ R. Let Qαn denote the operator defined by the rule
〈Qαnδx, δy〉 =
〈χn ∗ δx, δy〉 if |x| ≥ q
0 if |x| < qα|y|.
〈χn ∗ δx, δy〉 ≤ 〈Q
nδx, δy〉+ 〈δx, Q
n δy〉.
This implies
〈χn ∗ χE , χF 〉 ≤ ‖Q
nχE‖1 + ‖Q
n χF‖1
≤ |E| sup
‖Qαnδx‖1 + |F | sup
‖Q−αn δx‖1(9)
Qαnδx =
|w|=n
|wx|≤q−α|x|
Let w = w2w1 where |w1| = [n/2] + [α] and |w2| = n− [n/2]− [α]. The
conditions |w| = n and |wx| ≤ q−α|x| imply that w1 is determined by
the first [n/2] + [α] letters of x. Hence we have as many terms in the
sum as choices for w2, i.e. at most q
n−[n/2]−[α]. Thus
(10) ‖Qαnδx‖1 ≤ q
3/2q−αqn/2.
Similarly
‖Q−αn δx‖1 ≤ q
3/2qαqn/2.
Hence by (9) we get
〈χn ∗ χE , χF 〉 ≤ q
3/2qn/2{q−α |E|+ qα |F |}.
8 TADEUSZ PYTLIK AND RYSZARD SZWARC
Choosing α = (log |E| − log |F |)/(2 log q) gives
〈χn ∗ χE , χF 〉 ≤ 2q
3/2qn/2|E|1/2|F |1/2.
Theorem 3. Let f =
n=0 fnχn and fn ≥ 0. The operator λ(f) is of
restricted weak type (2, 2) if and only if f ∈ L(2,1).
Proof. By Theorem 2 we have
‖λ(χn)‖(2,1)→(2,∞) ≤ Cq
for some constant C > 0. Let f =
n=0 fnχn. Then triangle inequality
yields
‖f‖(2,1)→(2,∞) ≤ C
By [8, Lemma 1]
n/2 ≈ ‖f‖(2,1).
On the other hand for fn ≥ 0 we have
‖f‖(2,1)→(2,∞) ≥ C sup
q−(n+m)/2 〈f ∗ χn, χm〉
= C sup
q−(n+m)/2 〈f, χm ∗ χn〉 ≥ C
k=|n−m|
k≡n+m mod 2
qk/2fk.
Taking m = n or m = n+ 1 and letting n tend to infinity gives
‖f‖(2,1)→(2,∞) ≥ C
q2k/2f2k,
‖f‖(2,1)→(2,∞) ≥ C
q(2k+1)/2f2k+1.
Therefore
k=0 q
k/2fk < +∞, i.e. f ∈ L
(2,1) by (11). �
3. Weak type (p, p) for 1 < p < 2
Part of the next theorem, namely the first inequality is known from
[3]. Actually it has been simply observed there that the inequality fol-
lows by applying multilinear interpolation theorem to Pytlik’s estimate
for ‖
fnλ(χn)‖p→p given in [8]. We will reprove the second inequal-
ity by applying the same interpolation theorem to restricted weak type
estimates given in the previous section. In this way we skip p → p
WEAK TYPE RADIAL CONVOLUTION OPERATORS ON FREE GROUP 9
estimates whose proof as given in [8] is tricky, and later proof given in
[3] makes use of advanced representation theory.
Theorem 4. For 1 < p < 2 and for f =
n=0 fnχn we have
‖λ(f)‖p→(p,∞) ≤ C‖f‖(p,p′).
Moreover if f ≥ 0 then
c‖f‖(p,p′) ≤ ‖λ(f)‖p→(p,∞).
Proof. The subscript r will denote the subspace of radial functions,
i.e. functions of the form
n=0 fnχn, where fn are complex coefficients.
By Theorem 3 we have L
(2,1)
r ∗ L
(2,1) ⊂ L(2,∞). On the other hand
L1r ∗ L
1 ⊂ L1. By multilinear interpolation theorem [1, 3.13.5, p. 76]
we get L
(p,s)
r ∗ L
(p,t) ⊂ L(p,u) where 1 ≤ p < 2 and 1 + 1/u = 1/s+ 1/t.
Taking u = ∞, t = p and s = p′ gives L
(p,p′)
r ∗ L
p ⊂ L(p,∞). This gives
the first inequality.
On the other hand for f =
n=0 fnχn by (4) and by duality (6) we
‖λ(f)‖p→(p,∞) = ‖λ(f)‖(p′,1)→p′ ≥ c sup
q−n/p
‖f ∗ χn‖p′.
Similarly as in the proof of Theorem 4 we obtain
f ∗ χn ≥
q(n−l)/2
m=|n−l|
m≡l+n mod 2
qm/2fm
Hence
q−n‖f ∗ χn‖
′(n−l)/2ql−n
m=n−l
m≡l+n mod 2
qm/2fm
q(n−l)(p
n−l =
Taking supremum with respect to n and raising to the power 1/p′ give
‖λ(f)‖p→(p,∞) ≥ c
np′/p
)1/p′
Since the norm of f =
n=0 fnχn in L
(p,p′)
r is equivalent to
n=0 f
np′/p
)1/p′
the second inequality is proved. �
10 TADEUSZ PYTLIK AND RYSZARD SZWARC
4. Other estimates
Theorem 5. For 1 ≤ s ≤ 2 ≤ t ≤ ∞ we have
cn1−1/s+1/tqn/2 ≤ ‖λ(χn)‖(2,s)→(2,t) ≤ Cn
1−1/s+1/tqn/2.
Proof. In order to get the second inequality we use only interpolation.
First observe that the inequality is valid for s = 2, t = ∞ by Theorem
4 and for s = t = 2 by [2, 7]. Hence by complex interpolation of the
Lorentz spaces it is valid for s = 2, t ≥ 2.
Next it is valid for s = 1, t = ∞ by Theorem 3 and for s = t = 2.
Hence by complex interpolation it is valid for 1 ≤ s ≤ 2, t = s′.
Now we can use again complex interpolation to get the conclusion
for 1 ≤ s ≤ 2 ≤ t ≤ ∞.
The estimate from below can be obtained from
‖λ(χn)‖(2,s)→(2,t) ≥
‖χn ∗ f‖(2,t)
‖f‖(2,s)
where f =
k=0 q
−k/2χk. �
Theorems 1, 2 and 5 suggest the following.
Conjecture. Let f =
n=0 fnχn ≥ 0. Then for 1 ≤ s ≤ 2 the
operator λ(f) maps L(2,s) into L(2,∞) if and only if
n,m=0
fnfmq
−(n+m)/2{1 + min(n1/s
, m1/s
)} < +∞.
References
[1] J. Bergh, J. Löfström, Interpolation spaces. An introduction. Grundlehren der
mathematischen Wissenschaften 223, Springer-Verlag, Berlin-Heidelberg-New
York (1976).
[2] J. M. Cohen, Operator norms on free groups, Boll. Unione Mat. Ital. VI. Ser.
B 1(1982), 1055-1065.
[3] M. Cowling, S. Meda, A. G. Setti, Alberto, Invariant operators on function
spaces on homogeneous trees, Colloq. Math. 80 (1999), 53–61.
[4] E. Følner, On groups with full Banach mean value, Math. Scand. 3 (1955),
243–254.
[5] H. Kesten, Full Banach mean values on countable groups, Math. Scand. 7
(1959), 146–156.
[6] H. Kesten, Symmetric random walks on groups, Trans. Amer. Math. Soc. 92
(1959), 336–354.
[7] T. Pytlik, Radial functions on free groups and a decomposition of the regu-
lar representation into irreducible components, J. Reine Angew. Math. 326
(1981),124-135.
[8] T. Pytlik, Radial convolutors on free groups, Stud. Math. 78 (1984), 179–183.
[9] R. Szwarc, Convolution operators of weak type (2,2) which are not of strong
type (2,2), Proc. Am. Math. Soc. 87 (1983), 695–698.
WEAK TYPE RADIAL CONVOLUTION OPERATORS ON FREE GROUP 11
[10] R. Szwarc, Convolution operators of weak type (p, p) which are not of strong
type (p, p), Proc. Am. Math. Soc. 89 (1983), 184-185.
Institute of Mathematics, University of Wroc law, pl. Grunwaldzki
2/4, 50-384 Wroc law, Poland
Institute of Mathematics and Computer Science, University of Opole,
ul. Oleska 48, 45-052 Opole, Poland
E-mail address : szwarc2@gmail.com
	1. Introduction
	2. Radial convolution operators of weak type (2,2)
	3. Weak type (p,p) for 1<p<2
	4. Other estimates
	References

  Radial convolution operators on free groups with nonnegative kernel of weak
type $(2,2)$ and of restricted weak type $(2,2)$ are characterized. Estimates
of weak type $(p,p)$ are obtained as well for $1<p<2.$

Introduction
A discrete group G is called amenable if there exists a linear func-
tional m on ℓ∞
(G) such that
(1) inf
f(x) ≤ m(f) ≤ sup
f(x),
(2) m(xf) = m(f), where xf(y) = f(x
−1y).
m is called a left invariant mean. Then the functionalM(f) = m(m(fx))
satisfies (1), (2) and is also right invariant, where fx(y) = f(yx).
Let G be a discrete group. Consider a symmetric probability measure
µ on G, i.e.
µ(x)δx, µ(x) ≥ 0,
µ(x) = 1, µ(x−1) = µ(x).
The left convolution operator λ(µ) with µ is bounded on ℓ2(G) and
‖λ(µ)(f)‖2 = ‖µ ∗ f‖2 ≤ ‖f‖2, f ∈ ℓ
2(G).
Indeed
‖µ ∗ f‖2 =
µ(x)[δx ∗ f ]
µ(x)‖δx ∗ f‖2 = ‖f‖2.
2000 Mathematics Subject Classification. Primary 43A15, Secondary 43A07,
43A15.
Key words and phrases. free group, convolution operators, weak type, restricted
weak type.
The second author was supported by European Commission Marie Curie Host
Fellowship for the Transfer of Knowledge “Harmonic Analysis, Nonlinear Analysis
and Probability” MTKD-CT-2004-013389 and by MNiSW Grant N201 054 32/4285.
http://arxiv.org/abs/0704.1859v1
2 TADEUSZ PYTLIK AND RYSZARD SZWARC
Thus ‖λ(µ)‖2→2 ≤ 1.
Kesten [5] showed that a discrete group G is amenable iff for any
symmetric probability measure µ on G we have ‖λ(µ)‖2→2 = 1. He
showed that G is amenable if condition is satisfied for one measure µ
such that supp µ generates G algebraically. In particular let G be gen-
erated by g1, g2, . . . , gk and µ =
(δgi + δg−1
). Then G is amenable
if and only if ‖λ(µ)‖2→2 = 1.
In [4] Følner came up with another property equivalent to amenabil-
ity. We say that a discrete group G satisfies the Følner condition if for
any number ε > 0 and any finite set K ⊂ G there exists a finite set
N ⊂ G such that
(1) |xN △N | < ε|N |, x ∈ K.
In other words N is almost K invariant. He showed that G is amenable
if and only if the Følner condition holds.
Assume that G is amenable. Let µ be a probability measure with
finite support K. For ε = η2 > 0 choose N so as to satisfy (1). Then
‖µ ∗ χN − χN‖2 =
µ(x)[χxN − χN ]
µ(x)‖χxN − χN‖2
µ(x)‖χxN△N‖2 =
µ(x)|xN △N |1/2 ≤ η|N |1/2 = η‖χN‖2.
Therefore
〈µ ∗ χN , χN〉ℓ2(G)
= 〈χN , χN〉ℓ2(G) + 〈µ ∗ χN − χN , χN〉ℓ2(G) ≥ (1− η)‖χN‖
which implies
(2) sup
N,M−finite
〈µ ∗ χN , χM〉
‖χN‖2‖χM‖2
= 1 = ‖λ(µ)‖2→2.
The same holds (with the same proof) for any 1 < p < +∞, i.e.
(3) sup
N,M−finite
〈µ ∗ χN , χM〉
‖χN‖p‖χM‖p′
= 1 = ‖λ(µ)‖p→p,
where p′ = p/(p− 1).
We will use the notion of Lp,q spaces, which have been introduced
by Lorentz (see [1]). Consider a general σ-finite measure space (Ω, ω)
WEAK TYPE RADIAL CONVOLUTION OPERATORS ON FREE GROUP 3
and 1 < p < +∞. For f ∈ Lp(Ω, ω) and t > 0 we have
tp ω{x : |f(x)| > t} ≤
|f(x)|pdω(x).
Functions for which the left hand side is bounded form a linear space
Lp,∞(Ω, ω) =
f : sup
tpω{x : |f(x)| > t} < +∞
called the weak Lp space. This space contains Lp(Ω, ω).
For p′ = p/(p − 1) the predual of Lp
′,∞(Ω, ω) with respect to the
standard inner product is denoted by Lp,1(Ω, ω). We have
Lp,1(Ω, ω) ⊂ Lp(Ω, ω) ⊂ Lp,∞(Ω, ω).
For p > 1 these spaces are normed.
Any linear operator mapping Lp into itself is called of strong type
(p, p). Linear operators T mapping Lp(Ω, ω) into Lp,∞(Ω, ω) are called
of weak type (p.p), while those which map Lp,1(Ω, ω) into Lp,∞(Ω, ω)
are called of restricted weak type (p, p).
We will use the following facts. A linear operator T is bounded from
Lp,1 into a Banach space X if and only if
(4) ‖T‖L(p,1)→X = sup
‖TχE‖X
‖χE‖p
< +∞.
A linear operator T is bounded from Lp,1 into Lp,∞ if and only if
(5) ‖T‖(p,1)→(p,∞) = sup
E,F⊂Ω
|〈TχE, χF 〉|
‖χE‖p‖χF‖p′
< +∞.
Using this and duality between spaces L(p
′,1) and L(p,∞) we obtain
(6) ‖T‖p→(p,∞) = ‖T
∗‖(p′,1)→p′ = sup
‖T ∗χE‖p′
‖χE‖p′
The equalities (2) and (3) can be interpreted as follows. If the group
G is discrete and amenable and µ is a symmetric probability measure
on G, then
‖λ(µ)‖p→p = ‖λ(µ)‖(p′,1)→p′ = λ(µ)‖p→(p,∞) = ‖λ(µ)‖(p,1)→(p,∞) = 1.
Hence for these groups convolution operators with nonnegative func-
tions of strong type (p, p), of weak type (p, p) and of restricted weak
type (p, p) coincide for any 1 < p < ∞.
The situation is entirely different for nonamenable groups. Only
special examples have been studied. It has been shown [9] that for
p = 2 and G = Fk, the free group on k generators, k ≥ 2, there
exist nonnegative functions f on G such that ‖λ(f)‖2→(2,∞) is finite
4 TADEUSZ PYTLIK AND RYSZARD SZWARC
while ‖λ(f)‖2→2 is infinite, i.e. there exist convolution operators with
nonnegative functions of weak type (2,2) which are not of strong type
(2,2). The same has been shown for 1 < p < 2 [10]. These functions f
can be chosen to be radial, i.e. constant on elements of the group G of
the same length. It is an open problem if these results remain true for
any discrete nonamenable group.
In this work will focus on G = Fk. We are going to determine all non-
negative radial functions f on G such that λ(f) is of weak type (2,2),
as well those f for which λ(f) is of restricted weak type (2,2). In par-
ticular we prove that these spaces are different. Next we will turn our
attention to the case 1 < p < 2. By using interpolation machinery, du-
ality and the results for p = 2 we will be able to determine nonnegative
radial functions f for which λ(f) is of weak type (p, p). In this way we
obtain a simpler proof of the upper estimate of ‖λ(f)‖p→(p,∞) obtained
in [3]. Our method does not rely on deep theorems on representation
theory
2. Radial convolution operators of weak type (2, 2)
Let Fk = gp{g1, g2, . . . , gk} be a free group on k ≥ 2 generators.
The group consists of reduced words in generators and their inverses.
This representation is unique. The number of letters in reduced form
defines length function on Fk. Let χn denote the indicator function of
words of length n. There are 2k(2k − 1)n−1 such words. as we have
2k choices for the first letter and 2k − 1 choices for every consecutive
one. Let q = 2k − 1. The next theorem generalizes the estimate for
‖λ(χn)‖2→(2,∞) given in [9].
Theorem 1. Let f =
n=0 fnχn. The operator λ(f) is of weak type
(2, 2) if
A(f) :=
n,m=0
|fn||fm|q
−(n+m)/2{1 + min(n,m)} < +∞.
Moreover if fn ≥ 0 the condition is necessary and
A(f) ≤ ‖λ(f)‖22→(2,∞) ≤ 4A(f).
Proof. By (7), instead of estimating ‖λ(f)‖2→(2,∞) we may estimate
‖λ(f)‖(2,1)→2, which (see (4)) is equivalent to
‖f ∗ χE‖2
|E|1/2
WEAK TYPE RADIAL CONVOLUTION OPERATORS ON FREE GROUP 5
We have
‖f ∗ χE‖
2 = 〈f ∗ f ∗ χE∗, χE〉 =
n,m=0
fnfm〈χn ∗ χm ∗ χE , χE〉.
Simple calculation gives that for n ≥ 1 we have
χn ∗ χm = q
n−1δmn χ0 +
k=|n−m|
k≡n+m mod 2
q(n+m−k)/2 χk.
Clearly χ0 ∗ χ0 = χ0. Therefore
χn ∗ χm ≤ 2
k=|n−m|
k≡n+m mod 2
q(n+m−k)/2χk.
Hence
‖f ∗ χE‖
2 ≤ 2
n,m=0
fnfm q
(n+m)/2
k=|n−m|
k≡n+m mod 2
q−k/2 〈χk ∗ χE , χE〉.
Lemma 1.
〈χk ∗ χE, χE〉 ≤ 2q
[k/2]|E|.
Proof. Define an operator Pk by the rule
〈Pkδx, δy〉 =
〈χk ∗ δx, δy〉 if |x| ≥ |y|
0 if |x| < |y|.
〈χk ∗ δx, δy〉 ≤ 〈Pkδx, δy〉+ 〈δx, Pkδy〉.
This implies
〈χk ∗ χE , χE〉 ≤ 2〈PkχE , χE〉 ≤ 2‖PkχE‖1 ≤ 2|E| sup
‖Pkδx‖1
Pkδx =
|w|=k
|wx|≤|x|
Let w = w1w2 where |w1| ≤ |w2| ≤ (k + 1)/2. The conditions |w| = k
and |wx| ≤ |x| imply that w2 is determined by the first [(k + 1)/2]
letters of x. Hence we have as many terms in the sum as choices for
w1, i.e. at most q
[k/2]. Thus
‖Pkδx‖1 ≤ q
[k/2].
Therefore
〈χk ∗ χE, χE〉 ≤ 2q
[k/2]|E|.
6 TADEUSZ PYTLIK AND RYSZARD SZWARC
Lemma 1 implies that
‖f ∗ χE‖
n,m=0
|fn||fm|q
(n+m)/2
k=|n−m|
k≡n+m mod 2
n,m=0
|fn||fm|q
(n+m)/2{1 + min(m,n)}.
We obtain the upper estimate
‖λ(f)‖22→(2,∞) ≤ 4
n,m=0
|fn||fm|q
(n+m)/2{1 + min(m,n)}.
On the other hand if fn ≥ 0 we have
‖λ(f)‖22→(2,∞) ≥
q + 1
q−2k‖f ∗ χ2k‖
fn(χn ∗ χ2k)
l=|n−2k|
l≡n mod 2
q(n+2k−l)/2 χl
q−l/2χl
n=|2k−l|
n≡l mod 2
n=|2k−l|
n≡l mod 2
n=2k−l
n≡l mod 2
n,m=0
fnfmq
(n+m)/2
l=max(2k−n,2k−m)
l≡n≡m mod 2
Considering even or odd values of m and n gives
‖λ(f)‖22→(2,∞) ≥
n,m=0
f2nf2mq
n+m{1 + min(n,m)},
‖λ(f)‖22→(2,∞) ≥
n,m=0
f2n+1f2m+1q
n+m+1{1 + min(n,m)}.
Since k is arbitrary
‖λ(f)‖22→(2,∞) ≥
n,m=0
n≡m mod 2
fnfmq
(n+m)/2{1 + min(n,m)}.
WEAK TYPE RADIAL CONVOLUTION OPERATORS ON FREE GROUP 7
This implies
‖λ(f)‖22→(2,∞) ≥
n,m=0
fnfmq
(n+m)/2{1 + min(n,m)},
because the matrix a(n,m) = 1 + min(n,m) is positive definite. �
Theorem 2. For n ≥ 0 there holds
‖λ(χn‖(2,1)→(2,∞) ≤ cq
Proof. We have
‖λ(χn‖(2,1)→(2,∞) = sup
E,F⊂Fr
〈χn ∗ χE, χF 〉
|E|1/2|F |1/2
The proof will be completed if we show
(8) 〈χn ∗ χE , χF 〉 ≤ cq
n/2|E|1/2|F |1/2.
We will prove (8) by modification of the argument used in the proof of
Lemma 1. Fix α ∈ R. Let Qαn denote the operator defined by the rule
〈Qαnδx, δy〉 =
〈χn ∗ δx, δy〉 if |x| ≥ q
0 if |x| < qα|y|.
〈χn ∗ δx, δy〉 ≤ 〈Q
nδx, δy〉+ 〈δx, Q
n δy〉.
This implies
〈χn ∗ χE , χF 〉 ≤ ‖Q
nχE‖1 + ‖Q
n χF‖1
≤ |E| sup
‖Qαnδx‖1 + |F | sup
‖Q−αn δx‖1(9)
Qαnδx =
|w|=n
|wx|≤q−α|x|
Let w = w2w1 where |w1| = [n/2] + [α] and |w2| = n− [n/2]− [α]. The
conditions |w| = n and |wx| ≤ q−α|x| imply that w1 is determined by
the first [n/2] + [α] letters of x. Hence we have as many terms in the
sum as choices for w2, i.e. at most q
n−[n/2]−[α]. Thus
(10) ‖Qαnδx‖1 ≤ q
3/2q−αqn/2.
Similarly
‖Q−αn δx‖1 ≤ q
3/2qαqn/2.
Hence by (9) we get
〈χn ∗ χE , χF 〉 ≤ q
3/2qn/2{q−α |E|+ qα |F |}.
8 TADEUSZ PYTLIK AND RYSZARD SZWARC
Choosing α = (log |E| − log |F |)/(2 log q) gives
〈χn ∗ χE , χF 〉 ≤ 2q
3/2qn/2|E|1/2|F |1/2.
Theorem 3. Let f =
n=0 fnχn and fn ≥ 0. The operator λ(f) is of
restricted weak type (2, 2) if and only if f ∈ L(2,1).
Proof. By Theorem 2 we have
‖λ(χn)‖(2,1)→(2,∞) ≤ Cq
for some constant C > 0. Let f =
n=0 fnχn. Then triangle inequality
yields
‖f‖(2,1)→(2,∞) ≤ C
By [8, Lemma 1]
n/2 ≈ ‖f‖(2,1).
On the other hand for fn ≥ 0 we have
‖f‖(2,1)→(2,∞) ≥ C sup
q−(n+m)/2 〈f ∗ χn, χm〉
= C sup
q−(n+m)/2 〈f, χm ∗ χn〉 ≥ C
k=|n−m|
k≡n+m mod 2
qk/2fk.
Taking m = n or m = n+ 1 and letting n tend to infinity gives
‖f‖(2,1)→(2,∞) ≥ C
q2k/2f2k,
‖f‖(2,1)→(2,∞) ≥ C
q(2k+1)/2f2k+1.
Therefore
k=0 q
k/2fk < +∞, i.e. f ∈ L
(2,1) by (11). �
3. Weak type (p, p) for 1 < p < 2
Part of the next theorem, namely the first inequality is known from
[3]. Actually it has been simply observed there that the inequality fol-
lows by applying multilinear interpolation theorem to Pytlik’s estimate
for ‖
fnλ(χn)‖p→p given in [8]. We will reprove the second inequal-
ity by applying the same interpolation theorem to restricted weak type
estimates given in the previous section. In this way we skip p → p
WEAK TYPE RADIAL CONVOLUTION OPERATORS ON FREE GROUP 9
estimates whose proof as given in [8] is tricky, and later proof given in
[3] makes use of advanced representation theory.
Theorem 4. For 1 < p < 2 and for f =
n=0 fnχn we have
‖λ(f)‖p→(p,∞) ≤ C‖f‖(p,p′).
Moreover if f ≥ 0 then
c‖f‖(p,p′) ≤ ‖λ(f)‖p→(p,∞).
Proof. The subscript r will denote the subspace of radial functions,
i.e. functions of the form
n=0 fnχn, where fn are complex coefficients.
By Theorem 3 we have L
(2,1)
r ∗ L
(2,1) ⊂ L(2,∞). On the other hand
L1r ∗ L
1 ⊂ L1. By multilinear interpolation theorem [1, 3.13.5, p. 76]
we get L
(p,s)
r ∗ L
(p,t) ⊂ L(p,u) where 1 ≤ p < 2 and 1 + 1/u = 1/s+ 1/t.
Taking u = ∞, t = p and s = p′ gives L
(p,p′)
r ∗ L
p ⊂ L(p,∞). This gives
the first inequality.
On the other hand for f =
n=0 fnχn by (4) and by duality (6) we
‖λ(f)‖p→(p,∞) = ‖λ(f)‖(p′,1)→p′ ≥ c sup
q−n/p
‖f ∗ χn‖p′.
Similarly as in the proof of Theorem 4 we obtain
f ∗ χn ≥
q(n−l)/2
m=|n−l|
m≡l+n mod 2
qm/2fm
Hence
q−n‖f ∗ χn‖
′(n−l)/2ql−n
m=n−l
m≡l+n mod 2
qm/2fm
q(n−l)(p
n−l =
Taking supremum with respect to n and raising to the power 1/p′ give
‖λ(f)‖p→(p,∞) ≥ c
np′/p
)1/p′
Since the norm of f =
n=0 fnχn in L
(p,p′)
r is equivalent to
n=0 f
np′/p
)1/p′
the second inequality is proved. �
10 TADEUSZ PYTLIK AND RYSZARD SZWARC
4. Other estimates
Theorem 5. For 1 ≤ s ≤ 2 ≤ t ≤ ∞ we have
cn1−1/s+1/tqn/2 ≤ ‖λ(χn)‖(2,s)→(2,t) ≤ Cn
1−1/s+1/tqn/2.
Proof. In order to get the second inequality we use only interpolation.
First observe that the inequality is valid for s = 2, t = ∞ by Theorem
4 and for s = t = 2 by [2, 7]. Hence by complex interpolation of the
Lorentz spaces it is valid for s = 2, t ≥ 2.
Next it is valid for s = 1, t = ∞ by Theorem 3 and for s = t = 2.
Hence by complex interpolation it is valid for 1 ≤ s ≤ 2, t = s′.
Now we can use again complex interpolation to get the conclusion
for 1 ≤ s ≤ 2 ≤ t ≤ ∞.
The estimate from below can be obtained from
‖λ(χn)‖(2,s)→(2,t) ≥
‖χn ∗ f‖(2,t)
‖f‖(2,s)
where f =
k=0 q
−k/2χk. �
Theorems 1, 2 and 5 suggest the following.
Conjecture. Let f =
n=0 fnχn ≥ 0. Then for 1 ≤ s ≤ 2 the
operator λ(f) maps L(2,s) into L(2,∞) if and only if
n,m=0
fnfmq
−(n+m)/2{1 + min(n1/s
, m1/s
)} < +∞.
References
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mathematischen Wissenschaften 223, Springer-Verlag, Berlin-Heidelberg-New
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B 1(1982), 1055-1065.
[3] M. Cowling, S. Meda, A. G. Setti, Alberto, Invariant operators on function
spaces on homogeneous trees, Colloq. Math. 80 (1999), 53–61.
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[5] H. Kesten, Full Banach mean values on countable groups, Math. Scand. 7
(1959), 146–156.
[6] H. Kesten, Symmetric random walks on groups, Trans. Amer. Math. Soc. 92
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[7] T. Pytlik, Radial functions on free groups and a decomposition of the regu-
lar representation into irreducible components, J. Reine Angew. Math. 326
(1981),124-135.
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[9] R. Szwarc, Convolution operators of weak type (2,2) which are not of strong
type (2,2), Proc. Am. Math. Soc. 87 (1983), 695–698.
WEAK TYPE RADIAL CONVOLUTION OPERATORS ON FREE GROUP 11
[10] R. Szwarc, Convolution operators of weak type (p, p) which are not of strong
type (p, p), Proc. Am. Math. Soc. 89 (1983), 184-185.
Institute of Mathematics, University of Wroc law, pl. Grunwaldzki
2/4, 50-384 Wroc law, Poland
Institute of Mathematics and Computer Science, University of Opole,
ul. Oleska 48, 45-052 Opole, Poland
E-mail address : szwarc2@gmail.com
	1. Introduction
	2. Radial convolution operators of weak type (2,2)
	3. Weak type (p,p) for 1<p<2
	4. Other estimates
	References
