Relations between Schramm spaces and generalized Wiener classes

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by  Milad Moazami Goodarzi

RELATIONS BETWEEN SCHRAMM SPACES AND GENERALIZED
WIENER CLASSES
MILAD MOAZAMI GOODARZI,MAHDIHORMOZI ANDNACIMA MEMIC´
7
1 Abstract. WegivenecessaryandsufficientconditionsfortheembeddingsΛBV(p) ⊆ΓBV(qn↑q)
0 and ΦBV ⊆ BV(qn↑q). As a consequence, a number of results in the literature, including a
2 fundamental theorem of Perlman and Waterman, are simultaneously extended.

n
a
J

9 1. Introduction and main results
1
A] FolLloewtiΛng=[1{]λ,jw}∞je=1cablleΛa naonWdaetcerremasainngsseeqquueennccee. oLfeptosΦiti=ve {nφujm}b∞j=e1rsbseucahstehqautePnc∞je=o1fλ1jin=cre∞as-.
F ing convex functions on [0,∞) with φj(0) = 0. We say that Φ is a Schramm sequence if
h. 0 < φj+1(x) ≤ φj(x) for all j and P∞j=1φj(x) = ∞ for all x > 0. This terminology is used
t throughout.
a
We begin by recalling two generalizations of the concept of bounded variation which are
m
central to our work.
[
Definition 1.1. A real-valued function f on [a,b] is said to be of Φ-bounded variation if
2
v n
3 V (f)= V (f;[a,b]) = sup φ (|f(I )|) < ∞,
Φ Φ X j j
4
j=1
8
3 where the supremum is taken over all finite collections {Ij}nj=1 of nonoverlapping subintervals
0 of [a,b] and f(I )= f(supI )−f(infI ). We denote by ΦBV the linear space of all functions f
j j j
.
1 such that cf is of Φ-bounded variation for some c > 0.
0
7 If for every f ∈ ΦBV, we define
1
kfk:= |f(a)|+inf{c > 0 :V (f/c) ≤ 1},
: Φ
v
i then it is easily seen that k·k is a norm, and ΦBV endowed with this norm turns into a Banach
X
space. The space ΦBV is introduced in Schramm’s paper [15]. For more information about
r
ΦBV, the reader is referred to [1].
a
If φ is a strictly increasing convex function on [0,∞) with φ(0) = 0, and if Λ = {λ }∞ is a
j j=1
Waterman sequence, by taking φ (x) = φ(x)/λ for all j, we get the class φΛBV of functions of
j j
φΛ-bounded variation. This class was introduced by Schramm and Waterman in [16] (see also
[17] and [11]). More specifically, if φ(x) = xp (p ≥ 1), we get the Waterman-Shiba class ΛBV(p),
which was introduced by Shiba in [18]. When p = 1, we obtain the well-known Waterman class
ΛBV.
In the case λ = 1 for all j, we obtain the class φBV of functions of φ-bounded variation
j
introduced by Young [26]. More specifically, when φ(x) = xp (p ≥ 1), we obtain the Wiener
class BV (see [24]), and taking p = 1, we have the well-known Jordan class BV.
p
Date: January 20, 2017.
2010 Mathematics Subject Classification. Primary 46E35; Secondary 26A45.
Key words and phrases. generalized bounded variation, modulus of variation, Embedding, Banach space.
1
2 MILADMOAZAMIGOODARZI,MAHDIHORMOZIANDNACIMAMEMIC´
Remark 1.2. One can easily observe that functions of Φ-bounded variation are bounded and
canonlyhavesimplediscontinuities(countablymanyofthem,indeed). TheclassΦBVhasmany
applications in Fourier analysis as well as in treating topics such as convergence, summability,
etc. (see [24, 26, 21, 22, 23, 12, 15]).
Definition 1.3. Let {q }∞ and {δ }∞ be sequences of positive real numbers such that
n n=1 n n=1
1 ≤ q ↑ q ≤ ∞ and 2 ≤ δ ↑∞. A real-valued function f on [a,b] is said to be of q -Λ-bounded
n n n
variation if
s |f(I )|qn 1
V (f)= V (f;q ↑ q;δ) := supsup j qn < ∞,
Λ Λ n (cid:16)X λ (cid:17)
n≥1{Ij} j=1 j
wherethe{I }s arecollections ofnonoverlappingsubintervalsof[a,b]suchthatinf |I | ≥ b−a.
j j=1 j j δn
The class of functions of q -Λ-bounded variation is denoted by ΛBV(qn↑q) (= ΛBV(qn↑q)). In the
n δ
sequel, we suppose that [a,b] = [0,1].
The class ΛBV(qn↑q) was introduced by Vyas in [19]. When λ = 1 for all j and δ = 2n for
j n
all n, we get the class BV(qn↑q)—inroduced by Kita and Yoneda (see [9])—which in turn recedes
to the Wiener class BV , when q = q for all n.
q n
A natural and important problem is to determine relations between the above-mentioned
classes; see [21], [12], [4], [9], [6], [13], [8] and [5] for some results in this direction. In particular,
Perlman and Waterman found the fundamental characterization of embeddings between ΛBV
classes in [12]. Ge and Wang characterized the embeddings ΛBV ⊆ φBV and φBV ⊆ ΛBV (see
[5]). It was shown by Kita and Yoneda in [9] that the embedding BV ⊆ BV(pn↑∞) is both
p
automatic and strict for all 1 ≤ p < ∞. Furthermore, Goginava characterized the embedding
ΛBV ⊆ BV(qn↑∞) in [6], and a characterization of the embedding ΛBV(p) ⊆ BV(qn↑q) (1 ≤ q ≤
∞) was given by Hormozi, Prus-Wi´sniowski and Rosengren in [8]. In this paper, we investigate
the embeddings ΛBV(p) ⊆ ΓBV(qn↑q) and ΦBV ⊆ BV(qn↑q) (1 ≤ q ≤ ∞). The problem as to
when the reverse embeddings hold is also considered, which turns out to have a simple answer
(see Remark (1.10)(ii) below).
Throughout this paper, the letters Λ and Γ are reserved for a typical Waterman sequence.
We associate to Λ a function which we still denote by Λ and define it as Λ(r) := [r] 1 for
Pj=1 λj
r ≥ 1. The function Λ(r) is clearly nondecreasing and Λ(r) → ∞ as r → ∞. Our first main
result reads as follows.
Theorem 1.4. Let 1 ≤ p ≤ q ↑ q ≤ ∞. Then, a necessary and sufficient condition for the
n
embedding ΛBV(p) ⊂ ΓBV(qn↑q) is
1 −1
(1.1) limsup max Γ(k)qnΛ(k) p < ∞.
n→∞ 1≤k≤δn
Moreover, if the hypothesis is replaced by the condition that {Γ(n)/Λ(n)}∞ be nondecreasing,
n=1
then the conclusion of the theorem still holds true.
An important consequence of Theorem (1.4) is the following corollary, which is indeed a
nontrivial extension of [12, Theorem 3].
RELATIONS BETWEEN SCHRAMM SPACES AND GENERALIZED WIENER CLASSES 3
Corollary 1.5. Let 1 ≤ p ≤ q < ∞. Then, a necessary and sufficient condition for the
embedding ΛBV(p) ⊆ ΓBV(q) is
1
Γ(n)q
sup < ∞.
1
1≤n<∞ Λ(n)p
Corollary 1.6. ([8, Theorem 1]) Let 1 ≤ p < ∞. Then, a necessary and sufficient condition
for the embedding ΛBV(p) ⊆ BV(qn↑q) is
1
kqn
limsup max < ∞.
n→∞ 1≤k≤δn k 1 p1
(cid:16)Pi=1 λi(cid:17)
Next corollary extends [9, Lemma 2.1].
Corollary 1.7. Let 1 < q ≤ ∞. Then, We have
ΛBV(p) ⊆ ΛBV(qn↑q).
[
1≤p<q
If Φ= {φ }∞ is aSchramm sequence, we defineΦ (x) := k φ (x) for x ≥0. ThenΦ (x)
j j=1 k j=1 j k
P
is clearly an increasing convex function on [0,∞) such that Φ (0) = 0 and Φ (x) > 0 for x > 0.
k k
Without loss of generality we assume that Φ (x) is strictly increasing on [0,∞). Let Φ−1(x) be
k k
the inverse function of Φ (x). Our next main result can be formulated as follows.
k
Theorem 1.8. A necessary and sufficient condition for the embedding ΦBV ⊂ BV(qn↑q) is
(1.2) limsup max kq1nΦ−1(1) < ∞.
k
n→∞ 1≤k≤δn
Corollary 1.9. A necessary and sufficient condition for the embedding φΛBV ⊂ BV(qn↑q) is
limsup max kq1nφ−1(Λ(k)−1) <∞.
n→∞ 1≤k≤δn
Remark 1.10. (i) When φ(x) = xp, 1≤ p < ∞, Corollary 1.9 yields Corollary 1.6 as a special
case.
(ii) By [9, Theorem 3.3], the class BV(qn↑∞) always contains a function with nonsimple dis-
continuities. Since clearly BV(qn↑∞) ⊆ ΛBV(qn↑∞), this is also the case for the class ΛBV(qn↑∞).
Ontheotherhand,aspointedoutinRemark(1.2), thefunctionsintheclasses ΦBVandΛBV(p)
can only have simple discontinuities. Hence, the corresponding reverse embeddings can never
happen.
2. An auxiliary inequality
In this section we establish an inequality (see (2.1) below) which plays a crucial role in the
sufficiency part of the proof of Theorem (1.4). Also some applications of it are presented in
Corollary (2.2) and Remark (2.3). The following proposition is indeed a generalization of [10,
Lemma].
4 MILADMOAZAMIGOODARZI,MAHDIHORMOZIANDNACIMAMEMIC´
Proposition 2.1. Let 1 ≤ q < ∞ and n ∈ N. Then
n 1 n k 1 k −1
(2.1) xqz q ≤ x y max z q y ,
(cid:16)X j j(cid:17) X j j1≤k≤n(cid:16)X j(cid:17) (cid:16)X j(cid:17)
j=1 j=1 j=1 j=1
where {x }, {y } and {z } are positive nonincreasing sequences.
j j j
Proof. Without loss of generality we may assume that n x y = 1. With this in mind, it is
j=1 j j
n q P
enough to prove that the maximum value of x z under above assumptions is
j=1 j j
P
k k
−q
max z y .
1≤k≤n(cid:16)X j(cid:17)(cid:16)X j(cid:17)
j=1 j=1
We claim that the solution to this problem satisfies condition
(2.2) x = x = ··· = x > x = x = ··· = x = 0
1 2 k k+1 k+2 n
for some 1≤ k ≤ n. To prove our claim, we suppose to the contrary that there exists a solution
which does not satisfy condition (2.2). Then for some 1 ≤ k ≤ n, we have x > 0 and
k+1
x = x = ··· = x > x ≥ x ≥ ··· ≥ x ≥ 0.
1 2 k k+1 k+2 n
Put
k n
x
k+1
A:= x y , B := x y , C := ,
j j j j
X X x
k
j=1 j=k+1
and define
Aη(t)+Bt = 1.
Then the n-tuple
(η(t)x ,η(t)x ,··· ,η(t)x ,tx ,··· ,tx )
1 2 k k+1 n
satisfies conditions of the problem, whenever 0 ≤ t < 1/AC +B. Now define
k n
f(t):= η(t)q xqz +tq xqz
X j j X j j
j=1 j=k+1
and consider two possibilities:
1) If q > 1 then
k n
f′′(t) = q(q−1) η(t)q−2(η′(t))2 xqz +tq−2 xqz
(cid:16) X j j X j j(cid:17)
j=1 j=k+1
and hence f′′(1) > 0 which in turn implies that f has a local minimum at t = 1. This is a
contradiction.
2) If q = 1 then f(t) is linear. Consequently,
n k
q q
A x z −B x z = 0
X j j X j j
j=k+1 j=1
which implies that the problem has a solution satisfying condition (2.2). This completes the
proof. (cid:3)
RELATIONS BETWEEN SCHRAMM SPACES AND GENERALIZED WIENER CLASSES 5
Let f be a bounded function on [0,1]. The modulus of variation of f is the sequence ν and
f
is defined by
n
ν (n) := sup |f(I )|,
f X j
j=1
where the supremum is taken over all finite collections {I }n of nonoverlapping subintervals
j j=1
of [0,1]. The modulus of variation of f is nondecreasing and concave. A sequence ν with such
properties is called a modulus of variation. The symbol V[v] denotes the class of all functions
f for which there exists a constant C > 0 (depending on f) such that ν (n)/ν(n) ≤ C for all n
f
(see [3]). The following corollary is an immediate consequence of inequality (2.1).
Corollary 2.2. [2, Theorem 1] The following inclusion holds.
ΛBV ⊆ V[nΛ(n)−1].
Proof. Let {I }n be a collection of nonoverlapping subintervals of [0,1]. If f ∈ ΛBV, q = 1,
j j=1
x = |f(I )|, y = 1/λ and z = 1, from (2.1) we obtain
j j j j j
n n
|f(I )|
|f(I )| ≤ j max kΛ(k)−1 ≤ V (f)nΛ(n)−1,
X j X λj 1≤k≤n Λ
j=1 j=1
which means that f ∈ V[nΛ(n)−1]. (cid:3)
Remark 2.3. Let Λ = {λ } and Γ = {γ } be Waterman sequeneces. As stated on page 181 of
j j
[14], Perlman and Waterman have shown, in the course of the proof of [12, Theorem 3], that if
there is a constant C such that
n n
1 1
≤ C for all n,
Xγ Xλ
j j
j=1 j=1
then, given any nonincreasing sequence {a } of nonnegative numbers,
j
n n
a a
j j
≤ C .
Xγ X λ
j j
j=1 j=1
It is worth mentioning that one can easily see that this is a simple consequence of inequality
(2.1) above.
3. Proofs of main results
Proof of Theorem (1.4). Necessity. We proceed by contraposition. If (1.1) does not hold,
using the fact that Γ(r)→ ∞ as r → ∞, we may, without loss of generality, assume that γ = 1
1
and for each n
(3.1) Γ(δ ) ≥ 2n+2,
n
and
(3.2) Γ(rn)q1nΛ(rn)−1p > 24n
for some integer r , 1 ≤ r ≤ δ .
n n n
6 MILADMOAZAMIGOODARZI,MAHDIHORMOZIANDNACIMAMEMIC´
We are going to construct a function f in ΛBV(p) that does not belong to ΓBV(qn↑q). To this
end, let s be the greatest integer such that 2s −1 ≤ 2−nΓ(δ ) and put t = min{r ,s }. We
n n n n n n
define a sequence of functions {f }∞ on [0,1] as follows:
n n=1
f (x) := 2−nΛ(rn)−p1 , x ∈ [2−n + 2jδ−n2,2−n + 2jδ−n1); 1 ≤ j ≤ tn,
n 
0 otherwise.

Thefunctionsf ,definedinthisfashion,havedisjointsupportsandthereforef(x):= ∞ f (x)
n n=1 n
P
is a well-defined function on [0,1]. In addition, we have
V (f)≤ ∞ V (f ) = ∞ 2tn (2−nΛ(rn)−p1)p p1
Λ X Λ n X(cid:16)X λ (cid:17)
j
n=1 n=1 j=1
∞ rn Λ(r )−1 1 ∞ Λ(r ) 1
≤ 2−n+1 n p = 2−n+1 n p < ∞,
X (cid:16)X λ (cid:17) X (cid:16)Λ(r )(cid:17)
j n
n=1 j=1 n=1
since the sequence {Λ(r )−1}∞ is nonincreasing and t ≤ r . This means that f ∈ ΛBV(p).
n n=1 n n
On the other hand, f ∈/ ΓBV(qn↑q). To see this, note that the definition of s implies 2(s +
n n
1)−1 > 2−nΓ(δ ).Combiningthiswith(3.1), weobtainΓ(2s −1) ≥ 2−n−1Γ(δ ).Consequently,
n n n
if t = s , then the preceding inequality means that
n n
Γ(2t −1) ≥ 2−n−1Γ(δ ) ≥ 2−n−1Γ(r ),
n n n
since r ≤ δ . Also, if t = r , clearly 2t −1 ≥ r and hence Γ(2t −1) ≥ Γ(r ), since Γ(r) is
n n n n n n n n
increasing. Thus, we have shown
(3.3) Γ(2t −1) ≥ 2−n−1Γ(r ), for all n.
n n
Finally, the intervals
I := 2−n + j−1,2−n + j , j = 1,...,2t −1,
j h δn δni n
have length 1 for each n, and thus
δn
VΓ(f)≥ (cid:16)2Xtn−1|f(Iγj)|qn(cid:17)q1n = (cid:16)Γ(2tn −1)(2−nΛ(rn)−1p)qn(cid:17)q1n
j
j=1
1
≥ 2−n(cid:16)2−n−1Γ(rn)(Λ(rn)−p1)qn(cid:17)qn ≥ 2n,
where the last two inequalities are due to (3.3) and (3.2), respectively. As a result, V (f) is not
Γ
finite.
Sufficiency. Assume (1.1) and let f ∈ ΛBV(p). Let {I }s be a nonoverlapping collection
j j=1
of subintervals of [0,1] with inf|I | ≥ 1/δ , and let q = q /p ≥ 1, x = |f(I )|p, y = 1/λ ,
j n n j j j j
z = 1/γ . By [7, Theorem 368], we may also assume that the x ’s are arranged in descending
j j j
order. Now, we can apply (2.1) and get
(cid:16)Xs |f(Iγjj)|qn(cid:17)q1n ≤ (cid:16)Xs |f(λIjj)|p(cid:17)p1 1m≤ka≤xsΓ(k)q1nΛ(k)−p1
j=1 j=1
≤ (cid:16)Xs |f(λIjj)|p(cid:17)p1 1≤mka≤xδnΓ(k)q1nΛ(k)−p1,
j=1
RELATIONS BETWEEN SCHRAMM SPACES AND GENERALIZED WIENER CLASSES 7
where the second inequality is a consequence of s ≤ δ . Taking suprema over all collections
n
{I }s as above, and over all n yields
j j=1
1 −1
VΓ(f) ≤VΛ(f)sup max Γ(k)qnΛ(k) p < ∞.
n 1≤k≤δn
Hence f ∈ ΓBV(qn↑q) and the first part of the theorem is proved.
To prove the second part, let us assume that {Γ(n)/Λ(n)}∞ is nondecreasing. Observe that
n=1
the proof of necessity is identical to that given in the first part. For sufficiency, note that the
only case which needs to bejustified is when q < p for some n. If this is the case, we first apply
n
(2.1) with q = 1 to obtain
s |f(I )|p s |f(I )|p
(3.4) j ≤ j max Γ(k)Λ(k)−1.
X γj X λj 1≤k≤s
j=1 j=1
Then an application of H¨older’s inequality yields
s |f(Ij)|qn = s |f(Ij)|p qpnγqpn−1
X γ X(cid:16) γ (cid:17) j
j j
j=1 j=1
≤ s |f(Ij)|p qpnΓ(s)1−qpn
(cid:16)X γ (cid:17)
j
j=1
≤ s |f(Ij)|p qpnΓ(s)1−qpn max Γ(k)qpnΛ(k)−qpn
(cid:16)X λj (cid:17) 1≤k≤s
j=1
≤ s |f(Ij)|p qpn max Γ(k)Λ(k)−qpn,
(cid:16)X λj (cid:17) 1≤k≤δn
j=1
where the last two inequalities are due, respectively, to (3.4) and the fact that {Γ(n)/Λ(n)}∞
n=1
is nondecreasing. (cid:3)
Proof of Theorem (1.8). Necessity. Suppose (1.2) does not hold. Then, without loss of
generality, we may assume that for each n
δ ≥ 2n+2,
n
and
1
(3.5) rqnΦ−1(1) > 24n
n rn
for some integer r , 1 ≤ r ≤ δ .
n n n
We will now construct a function f ∈ ΦBV such that f ∈/ BV(qn↑q). To do so, let s be the
n
greatest integer such that 2s − 1 ≤ 2−nδ , let t = min{r ,s } and consider the sequence
n n n n n
{f }∞ of functions on [0,1] defined in the following way:
n n=1
2−nΦ−1(1) , x ∈ [2−n + 2j−2,2−n + 2j−1); 1 ≤ j ≤t ,
f (x) :=  rn δn δn n
n 
0 otherwise.

8 MILADMOAZAMIGOODARZI,MAHDIHORMOZIANDNACIMAMEMIC´
Since the f ’s have disjoint supports, f(x) := ∞ f (x) is a well-defined function on [0,1].
n n=1 n
P
Thus, using convexity of the Φ ’s we have
rn
∞ ∞ 2tn ∞
V (f) ≤ V (f ) = φ (2−nΦ−1(1)) = Φ (2−nΦ−1(1))
Φ X Φ n XX j rn X 2tn rn
n=1 n=1j=1 n=1
∞ ∞
≤ Φ (2−nΦ−1(1)) ≤ 2Φ (2−nΦ−1(1)) < ∞,
X 2rn rn X rn rn
n=1 n=1
that is, f ∈ ΦBV.
In conclusion, let us show that f ∈/ BV(qn↑q). To this end, proceeding in the same way as in
the proof of Theorem (1.4), we obtain
(3.6) 2t −1 ≥ 2−n−1r , for all n.
n n
Since for every n, all intervals
I := 2−n + j−1,2−n + j , j = 1,...,2t −1,
j h δn δni n
have length 1 , we get
δn
2tn−1 1 1
V(f;q ↑ q,δ) ≥ |f(I )|qn qn = (2t −1)(2−nΦ−1(1))qn qn
n (cid:16) X j (cid:17) (cid:16) n rn (cid:17)
j=1
1
≥ 2−n 2−n−1r (Φ−1(1))qn qn ≥ 2n,
(cid:16) n rn (cid:17)
where the last two inequalities are results of (3.6) and (3.5), respectively. Therefore, f ∈/
BV(qn↑q).
Sufficiency. Let f ∈ ΦBV. To show that f ∈ BV(qn↑q), it suffices to prove the inequality
(3.7) V(f;qn ↑ q;δ) ≤ Csup max kq1nΦ−k1(1),
n 1≤k≤δn
where C is a positive constant depending solely on f.
n q 1
In the course of the proof of Theorem 2.1 in [25], the author proceeds to estimate ( x )q
j=1 j
P
under the restriction
n
φ (x )≤ V (f),
X j τ(j) Φ
j=1
where the x ’s are arranged in descending order and τ is any permutation of n letters. Using
j
Wang’s approach in [20], he finds the following:
n 1
(3.8) (cid:16)Xxqj(cid:17)q ≤ 161m≤ka≤xnkq1Φ−k1(VΦ(f)).
j=1
Toprove(3.7),consideranon-overlappingcollection{I }s ofsubintervalsof[0,1]withinf|I |≥
j j=1 j
1/δ . If we put q = q , x = |f(I )|, and if the x ’s are rearranged in descending order, then we
n n j j j
may apply (3.8) to obtain
s 1
(cid:16)X|f(Ij)|qn(cid:17)qn ≤ 161m≤ka≤xskq1nΦ−k1(VΦ(f))
j=1
RELATIONS BETWEEN SCHRAMM SPACES AND GENERALIZED WIENER CLASSES 9
≤ 16 max kq1nΦ−k1(VΦ(f)).
1≤k≤δn
Taking suprema and using concavity of the Φ−1’s yields (3.7) with C = 16(1+V (f)). (cid:3)
k Φ
Acknowledgement. TheauthorswouldliketothankProfessorG.H.Esslamzadehforkindly
reading the manuscript of this paper and making valuable remarks. The second author is
supportedbyagrantfromIran’sNationalElitesFoundation.
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(M. Moazami Goodarzi) Department of Mathematics, Faculty of Sciences, Shiraz University, Shi-
raz 71454, Iran
E-mail address: [email protected]
(M. Hormozi) Department of Mathematical Sciences, Division of Mathematics, University of
Gothenburg, Gothenburg 41296, Sweden & Department of Mathematics, Faculty of Sciences, Shi-
raz University, Shiraz 71454, Iran
E-mail address: [email protected]
(N.Memi´c) Department of Mathematics, Faculty of Natural Sciencesand Mathematics, Univer-
sity of Sarajevo, Zmaja od Bosne 33-35, Sarajevo, Bosnia and Herzegovina
E-mail address: [email protected]