Some new Fourier inequalities for unbounded orthogonal systems in Lorentz-Zygmund spaces

R E S E A R C H Open Access

Some new Fourier inequalities for unbounded orthogonal systems in Lorentz–Zygmund spaces

G. Akishev^1,2, D. Lukkassen³and L.E. Persson^3,4*

*Correspondence:

larserik6pers@gmail.com;

lars.e.persson@uit.no;

larserik.persson@kau.se

3Department of Computer Science and Computational Engineering, Campus Narvik, The Arctic University of Norway, Narvik, Norway

4Department of Mathematics and Computer Science, Karlstad University of Sweden, Karlstad, Sweden

Full list of author information is available at the end of the article

Abstract

In this paper we prove some essential complements of the paper (J. Inequal. Appl.

2019:171,2019) on the same theme. We prove some new Fourier inequalities in the case of the Lorentz–Zygmund function spacesLq,r(logL)^αinvolved and in the case with an unbounded orthonormal system. More exactly, in this paper we prove and discuss some new Fourier inequalities of this type for the limit caseL2,r(logL)^α, which could not be proved with the techniques used in the paper (J. Inequal. Appl.

2019:171,2019).

MSC: 42A16; 42B05; 26D15; 26D20; 46E30

Keywords: Inequalities; Fourier series; Fourier coefficients; Unbounded orthogonal systems; Lorentz–Zygmund spaces

1 Introduction

Let q ∈ (1, +∞), r∈ (0, +∞) and α ∈ R. As usual L_q,r(logL)^α denotes the Lorentz–

Zygmund space, which consists of all measurable functionsf on [0, 1] such that

fq,r,α:=

1 0

f^∗(t)r

1 +|lnt|αr

·t^qr–1dt ¹_r

< +∞,

wheref^∗denotes a nonincreasing rearrangement of the function|f|(see e.g. [21]).

Ifα= 0, then the Lorentz–Zygmund space coincides with the Lorentz spaceLq,r(logL)^α= L_q,rso that, if in additionr=q, thenL_q,r(logL)^αspace coincides with the Lebesgue space Lq[0, 1] (see e.g. [14]) with the norm

fq:=

1 0

f(x)^qdx

1q

, 1≤q< +∞.

Lets∈(2, +∞]. We consider an orthogonal system{ϕ_n}inL2[0, 1] such that

ϕns≤Mn, n∈N, (1)

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whereMn↑andMn≥1 (see [24] or [12]). Moreover, let

μ_n=μ^(s)_n :=sup

n k=1

ckϕ_k s

: n

k=1

c²_k= 1

, ρ_n= _∞

k=n

|ak|^{2 1}₂

. (2)

For one variable function Marcinkiewicz and Zygmund [12] proved the following theo- rems.

Theorem 1.1 Assume that the orthogonal system{ϕ_n}satisfies the condition(1)and2≤ p<s.If the real number sequence{a_n}satisfies the condition

∞ n=1

|an|^pM^(p–2)

s

n s–2n^{(p–2)s–1s–2}< +∞,

then the series ∞

n=1

anϕ_n(x)

converges in L_pto some function f ∈L_p[0, 1]and

fp≤C_{p,s ∞}

n=1

|a_n|^pM^(p–2)

s

n s–2n^{(p–2)s–1s–2 1}_p

.

Theorem 1.2 Let the orthogonal system{ϕ_n}satisfy the condition(1),and_s–1^s =μ<p≤2.

Then the Fourier coefficients an(f)of the function f ∈Lp[0, 1]with respect to the system{ϕ_n} satisfy the inequality

_∞

n=1

an(f)^pM^(p–2)

s

n s–2n^{(p–2)s–1s–2 1}_p

≤Cp,sfp.

There are several generalizations of Theorems1.1and1.2for different function spaces and systems (see e.g. [5–7,13] and the references therein).

In particular, Flett [5] generalized this to the case of Lorentz spaces and Maslov [13]

derived generalizations of both Theorem1.1and Theorem1.2to the case of Orlicz spaces.

Some inequalities related to the summability of the Fourier coefficients in bounded or- thonormal systems with functions from some Lorentz spaces were investigated e.g. by Stein [23], Bochkarev [3], Kopezhanova and Persson [9] and Kopezhanova [8].

Moreover, as a further generalization of a result of Kolyada [7] Kirillov proved an essen- tial generalization of Theorem 1 [6].

In [2] the authors recently generalized and complemented all statements mentioned above. In particular, we proved the following generalizations.

Theorem 1.3 Let2 <q<s≤+∞,α∈(–∞, +∞),r> 0andδ=^rs(q–2)_q(s–2).If{a_n} ∈l₂and

Ω_q,r,α= _∞

n=1

ρ_n^r–ρ_n+1^r μ^δ_n

1 + 2s

s– 2lnμ_n

αr¹_r

< +∞,

whereρ_nandμ_nare defined by(2),then the series ∞

n=1

a_nϕ_n(x)

with respect to an orthogonal system{ϕn}^∞n=1,which satisfies the condition(1),converges to some function f ∈Lq,r(logL)^αandfq,r,α≤CΩq,r,α.

Remark For the caseα= 0, Theorem1.3coincides with Theorem 1 in [6].

Theorem 1.4 Let s∈(2, +∞],_s–1^s <q< 2,r> 1,α∈R andδ=^r(q–2)s_q(s–2).If f ∈Lq,r(logL)^α,then the inequality

_∞

n=1

_ν

n+1–1 k=νn

ak(f)^2r₂

(1 +logμν_n)^rαμ^δ_ν

n

¹_r

≤Cfq,r,α

holds,whereμν_nare defined by(2)and ak(f)denote the Fourier coefficients of f with respect to an orthogonal system satisfying(1).

The methods of proofs of Theorems1.3and1.4presented in [2] are not sufficient to cover the caseq= 2 in both cases. In this paper we fill in this gap by proving complements of Theorem 1 in [6] (see Theorem2.1) and Theorem1.2(see Theorem3.1) for this case.

In Sect.4we include some concluding remarks with comparisons of other recent research of this type (see [8,15] and [17]) and suggestions of further possibilities for development of this area.

2 A complement of Theorem1.3. The caseq= 2 Our main result in this section reads as follows.

Theorem 2.1 Let{ϕ_n}^∞n=1 be an orthogonal system,which satisfies the condition(1)and s∈(2, +∞], 0 <θ≤2, 0≤α< +∞.If{an} ∈l2and

Λ_2,θ,α(a) = ∞

n=1

ln

1 +

n l=1

M²_l

1–^θ₂+αθ

ρ_n^θ–ρ_n+1^θ

< +∞,

then the series _∞

n=1anϕ_n(x) converges in the space L2,θ(logL)^α to some function f ∈ L2,θ(logL)^αandf2,θ,α≤C(Λ2,θ,α)^1/θ.

For the proof of this theorem we need the following well-known results of Kolyada [7].

Lemma 2.2 Let_∞

k=1α_kbe a convergent numerical series and0 <s<q<∞.Then

∞

k=1

αk

q

≤2^qsup

n

n k=1

αk

q–s

∞ k=n

αk

s

.

Lemma 2.3 Letα_n≥0andε_n> 0,and assume that for someβ∈(0, 1)ε_n+1≤βε_n (n= 1, 2, . . .).Then the following inequalities hold for all r> 0:

∞ n=1

ε_{n n}

k=1

α_k r

≤C_r,β ∞

n=1

ε_nα_n^r, ∞

n=1

_∞

k=n

α_k r

ε^–1_n ≤Cr,β

∞ n=1

α_n^rε_n^–1.

Here,as usual,Cr,βdenotes a positive constant depending only on r andβ.

Proof of Theorem2.1 Sinceρ_n↓0,n→+∞, we can choose a sequence of natural numbers {ν_k}^∞_k=1 as follows:ν₁= 1 andν_k+1=min{n∈N:ρ_n≤¹₂ρ_νk}, whereρ_nare defined by (2).

Then

ρν_k+1≤1

2ρν_k, ρν_k+1–1>1

2ρν_k. (3)

We defineu_k(x) :=|ν_k+1–1

n=νk a_nϕ_n(x)|. Then, by using Parseval’s relation, we have

uk2= _νk+1–1

n=νk

|an|^{2 1}₂

≤ρν_k:=ε_k. (4)

For each numberl∈Nwe consider the functionfl(x) :=ν_l+1–1

n=1 anϕ_n(x). Next we show that{fl}is a fundamental sequence in the spaceL2,θ(logL)^α. For any natural numbersm,l by the property of the modulus of numbers we have

fl(x) –fm(x)≤ l

k=m

uk(x).

By using Lemma2.2withq= 1 we get fl(x) –fm–1(x)≤2 sup

n=m,...,l

n k=m

uk(x)

1–β

l k=n

uk(x)

β

, (5)

where the numberβ∈(0, 1] will be chosen later on in the proof. By the property of noni- creasing rearrangement of a function (see e.g. [10], p. 89) we know that

f^∗(t)≤1 t

_t

0

f^∗(y)dy=1 t sup

e⊂[0,1]

|e|=t

e

f(x)dx. (6)

Now, by using (5) and (6) we can conclude that

(fl–fm–1)^∗(t)(t)≤2 sup

n=m,...,l

1 t sup

e⊂[0.1]

|e|=t

e

n

k=m

uk(x)

1–β

l

k=n

uk(x)

β

dx

. (7)

Next we use Hölder’s inequality with exponentsp=_β¹ > 1 andp= 1/(1 –β) in (7) and find that

(fl–fm–1)^∗(t)≤2 sup

n=m,...,l

1 t sup

e⊂[0.1]

|e|=t

e

n k=m

uk(x) dx

1–β

e

l k=n

uk(x) dx

β

= 2 sup

n=m,...,l

1 t

t 0

_n

k=m

uk(y) ∗

dy 1–β

1 t

t 0

_l

k=n

uk(y) ∗

dy β

.

We raise both sides to the powerθ, multiply by^θ₂(1 +|lnt|)^αθt^θ2–1and integrate. Then

θ 2

₁

0

(f_l–f_m–1)^∗(t)θ

1 +|lnt|αθ

t^θ2–1dt

≤2^θθ 2

l n=m

1 0

1 t

t 0

_n

k=m

uk(y) _∗

dy θ(1–β)

× 1

t _t

0

_l

k=n

uk(y) ∗

dy θβ

1 +|lnt|αθ

t^θ2–1dt

= 2^θθ 2

l n=m

₁

0

t^θ2–1

1 +|lnt|αθ

F_m,n^θ(1–β)(t)Φ_n,l^θβ(t)dt. (8)

For simplicity we introduce the notations

Fm,n(t) = 1 t

_t

0

_n

k=m

uk(y) _∗

dy and Φ_n,l(t) =1 t

_t

0

_l

k=n

uk(y) _∗

dy.

Choose the numberrsuch that 1 <r<θ≤2 and note thats:=^2(θ–r)_2–r > 0 andβ=_θ^s. Then βθ=sand^{(r–2)(θ–s)}_2r =^θ₂– 1. Therefore,

₁

0

F_m,n^θ(1–β)(t)Φ_n,l^θβ(t)

1 +|lnt|αθ

t^θ2–1dt= ₁

0

F_m,n^θ–s(t)Φ_n,l^s (t)

1 +|lnt|αθ

t^{(r–2)(θ–s)2r} dt. (9)

By again using the Hölder inequality now with exponentsp=_θ–s^r andp=_r–θ+s^r on the integral on the right hand side of (9) we find that

₁

0

F_m,n^θ(1–β)(t)Φ_n,l^θβ(t)

1 +|lnt|αθ

t^θ2–1dt

≤ ₁

0

F_m,n^r (t)

1 +|lnt|αθ ν

t^r2–1dt

^θ–s_{r 1}

0

Φ_n,l^sν(t)dt ¹

ν

≤C 1 t

t 0

_n

k=m

u_k(y) ∗

dy

θ–s

2,r,^{αθ ν}_r

l k=n

u_k

s

2

. (10)

Next, by using the norm property ofl2spaces, the Parseval theorem and the definition of the numbersν_k, we obtain (see (4))

l k=n

u_k 2

≤ l

k=n

u_k2= l

k=n

_νk+1–1

j=νk

a²_j ¹₂

≤ l

k=n

ρν_k≤2ρνn.

Therefore, from inequality (10) it follows that 1

0

F_m,n^θ(1–β)(t)Φ_n,l^θβ(t)

1 +|lnt|αθ

t^θ2–1dt

≤ 1 t

_t

0

_n

k=m

u_k(y) ∗

dy

θ–s

2,r,^{αθ ν}_r

ρ^s_νn≤C _n

k=m

1 t

_t

0

u^∗_k(y)dy 2,r,^{αθ ν}_r

θ–s

ρ_ν^s_n. (11)

By applying Lemma2.3and inequalities (8) and (11) we obtain 1

0

(f_l–f_m–1)^∗(t)θ

1 +|lnt|αθ

t^θ2–1dt

≤C l n=m

_n

k=m

u_k2,r,^{αθ ν}

r

θ–s

ρ_ν^s_n≤C l n=m

u_n^θ–s_2,r,αθ ν r

ρ_ν^s_n. (12)

Since 1 <r< 2, then, by the inequality of different metrics (see [1]), we get

un_2,r,^{αθ ν}

r ≤C

ln

1 +

νn+1–1 j=1

M_j²

¹_r–¹₂+^{αθ ν}_r

un2.

Therefore taking into account that (¹_r –¹₂+^{αθ ν}_r )(θ–s) = 1 –^θ₂+αθ we obtain

u_n^θ–s_2,r,αθ ν

r ≤C

ln

1 +

νn+1–1 j=1

M_j²

(¹_r–¹₂+^{αθ ν}_r )(θ–s)

u_n^θ–s2

=C

ln

1 +

ν_n+1–1 j=1

M²_j

1–^θ₂+αθ

un^θ–s₂

≤C

ln

1 +

ν_n+1–1 j=1

M_j²

1–^θ₂+αθ

ρ_ν^θ–s_n .

Hence, from (12) it follows that ₁

0

(fl–fm)^∗(t)θ

1 +|lnt|αθ

t^θ2–1dt≤C l n=m

ln

1 +

ν_n+1–1 j=1

M²_j

1–^θ₂+αθ

ρ_ν^θ_n. (13)

By definition of the numbers νn(see (3)) we have ρνn< 2ρνn+1–1 andρνn+2≤ ¹₂ρνn+1<

1

2ρ_νn+1–1. Thusρ_ν^θ

n+1–1–ρ^θ_ν

n+2≥(1 – 1/2)^θρ_ν^θ

n+1–1so that ρ_ν^θ

n+1–1≤ 2^θ 2^θ– 1

ρ_ν^θ

n+1–1–ρ_ν^θ

n+2

.

Therefore, l n=m

ln

1 +

νn+1–1 j=1

M_j²

1–^θ₂+αθ

ρ_ν^θ_n

≤2^θ l n=m

ln

1 +

ν_n+1–1 j=1

M²_j

1–^θ₂+αθ

ρ_ν^θ

n+1–1

≤2^θ 2^θ 2^θ– 1

l n=m

ln

1 +

νn+1–1 j=1

M²_j

1–^θ₂+αθ

ρ^θ_νn+1–1–ρ_ν^θ_n+2

≤ 2^2θ 2^θ– 1

l n=m

ν_n+2–1 k=νn+1–1

ρ_k^θ–ρ_k+1^θ ln

1 +

k j=1

M²_j

1–^θ₂+αθ

= 2^2θ 2^θ– 1

l n=m

ρ^θ_ν

n+1–1–ρ_ν^θ

n+1

ln

1 +

ν_n+1–1 j=1

M²_j

1–^θ₂+αθ

+

νn+2–1 k=νn+1

ρ_k^θ–ρ_k+1^θ ln

1 +

k j=1

M²_j

1–^θ₂+αθ

≤ 2^2θ 2^θ– 1

l n=m

_ν

n+2–1

k=νn+1

ρ_k^θ–ρ_k+1^θ ln

1 +

ν_n+1–1 j=1

M_j²

1–^θ₂+αθ

+

νn+2–1 k=νn+1

ρ_k^θ–ρ_k+1^θ ln

1 +

k j=1

M²_j

1–^θ₂+αθ

≤2 2^2θ 2^θ– 1

ν_l+2–1 n=νm

ρ_k^θ–ρ_k+1^θ ln

1 +

k j=1

M²_j

1–^θ₂+αθ

.

We conclude that l

n=m

ln

1 +

ν_n+1–1 j=1

M_j²

1–^θ₂+αθ

ρ_ν^θ

n

≤2 2^2θ 2^θ– 1

ν_l+2–1 n=νm

ρ_k^θ–ρ_k+1^θ ln

1 +

k j=1

M²_j

1–^θ₂+αθ

.

Hence, from (13) it follows that ₁

0

(fl–fm)^∗(t)θ

1 +|lnt|αθ

t^θ2–1dt

≤C

ν_l+2–1 n=νm

ρ^θ_k–ρ_k+1^θ ln

1 +

k j=1

M²_j

1–^θ₂+αθ

. (14)

We use the assumptions in the theorem and conclude that the sequence{fl} ⊂L2,θ(logL)^α is fundamental in the spaceL2,θ(logL)^α. Hence, since the spaceL2,θ(logL)^αis complete (see

[21]) there exists a functionf ∈L2,θ(logL)^αsuch thatf–fl2,θ,α→0 forl→ ∞and

f(x)∼ ∞

n=1

anϕn(x).

By now taking the limitl→ ∞in (14) we get ₁

0

(f –fm)^∗(t)θ

1 +|lnt|αθ

t^θ2–1dt≤C ∞ n=νm

ρ_k^θ–ρ_k+1^θ ln

1 +

k j=1

M²_j

1–^θ₂+αθ

.

Finally, in this inequality we putm= 1 and use the norm property to conclude that

f2,θ,α≤C _∞

k=1

ρ_k^θ–ρ_k+1^θ ln

1 +

k j=1

M²_j

1–^θ₂+αθ¹_θ .

The proof is complete.

3 A complement of Theorem1.4. The caseq= 2 Our main result in this section reads as follows.

Theorem 3.1 Let{ϕ_n}^∞_n=1be an orthogonal system,which satisfies the condition(1),s∈ (2, +∞], 2 <θ< +∞andα< 0.If the function f ∈L_2,θ(logL)^α,then

_∞

n=1

ln

1 +

νn+1–1 l=1

M_l²

1–^θ₂+αθ_νn+1–1

k=νn

a²_k(f) ^θ₂¹_θ

≤Cf2,θ,α,

where ak(f)as usual denote the Fourier coefficients with respect to the system{ϕ_n}^∞₁ . Proof It is well known that for any functionf ∈Lq,θ(logL)^αthe following relation holds (see e.g. [21]):

fq,θ,α sup

g∈L_q,θ(logL)^–α gq,θ,–α≤1

1 0

f(x)g(x)dx

, 1/q+ 1/q= 1, 1/θ+ 1/θ= 1. (15)

Consider the functiong(x) with Fourier coefficients

b_n(g) = _∞

k=1

ln

1 +

ν_k+1–1 l=1

M²_l

1–^θ₂+αθ_νk+1–1

n=ν_k

a²_n(f) ^θ₂–_θ¹

×

ln

1 +

ν_k+1–1 l=1

M²_l

1–^θ₂+αθ_νk+1–1

n=ν_k

a²_n(f) ^θ–2₂

a_n(f),

forn=ν_k, . . . ,ν_k+1– 1,k∈N.

Since{ϕ_n}is an orthogonal system and by the definition of the coefficientsbn(g) we have ₁

0

f(x)g(x)dx= _∞

n=1

ln

1 +

ν_n+1–1 l=1

M²_l

1–^θ₂+αθ_ν

n+1–1

k=νn

a²_k(f) ^θ₂–¹

θ

× ∞

k=1

ln

1 +

ν_k+1–1 l=1

M²_l

1–^θ₂+αθ_ν

k+1–1 n=νk

a²_n(f) ^θ–2_{2 ν}

k+1–1

n=νk

a²_n(f)

= _∞

n=1

ln

1 +

ν_n+1–1 l=1

M²_l

1–^θ₂+αθ_ν

n+1–1

k=νn

a²_k(f) ^θ₂–_θ¹

. (16)

Hence, according to Theorem2.1, we find that

g2,θ,–α≤C _∞

n=1

ln

1 +

n l=1

M²_l

1–^θ₂–αθ

ρ_n^θ–ρ_n+1^{θ 1}

θ

≤C _∞

k=1

ln

1 +

ν_k+1–1 l=1

M²_l

1–^θ₂–αθ

ρ_ν^θ

k–ρ_ν^θ

k+1

¹

θ

,

whereρ_n= (_∞

l=n|bl(g)|²)^1/2,n∈Nand –α> 0.

Ifa>b> 0, 0 <β≤1, thena^β–b^β≤(a–b)^β. Sinceθ/2 < 1, by this inequality we obtain

ρ_ν^θ

k–ρ_ν^θ

k+1= _∞

l=νk

b_l(g)²θ/2

– _∞

l=νk+1

b_l(g)²θ/2

≤ _∞

l=ν_k

b_l(g)²– ∞ l=ν_k+1

b_l(g)²θ/2

= _νk+1–1

l=ν_k

b_l(g)^2θ₂ .

Therefore,

g2,θ,–α≤C _∞

k=1

ln

1 +

ν_k+1–1 l=1

M²_l

1–^θ₂–αθ_νk+1–1

l=ν_k

b_l(g)^2θ₂¹

θ

.

By again using the definition of the coefficientsb_n(g) we obtain _νk+1–1

n=νk

b²_n(g) ¹₂

= _∞

k=1

ln

1 +

ν_k+1–1 l=1

M_l²

1–^θ₂+αθ_νk+1–1

n=νk

a²_n(f) ^θ₂–_θ¹

×

ln

1 +

ν_k+1–1 l=1

M²_l

1–^θ₂+αθ_νk+1–1

n=νk

a²_n(f) ^θ–1₂

.

Then g2,θ–α ≤C. Thus, the function g0 :=C^–1g ∈L2,θ(logL)^–α and g02,θ,–α ≤1.

Hence, by using (15), from (16) it follows that f2,θ,α≥

1 0

f(x)g₀(x)dx

≥C^–1 _∞

k=1

ln

1 +

ν_k+1–1 l=1

M_l²

1–^θ₂+αθ_νk+1–1

n=ν_k

a²_n(f) ^θ₂¹_θ

.

The proof is complete.

4 Concluding remarks

We say that a functionf on (0, 1) or (0,∞) is quasi-increasing (quasi-decreasing) if, for all x≤yand someC> 0,f(x)≤Cf(y) (f(y)≤Cf(x)). Moreover, we say that a positive function on (a,b), 0≤a<b<∞, is a quasi-monotone weight ifλ(x)x^cis quasi-increasing or quasi-decreasing for somec∈R. It is then natural to define the more general Lorentz spacesΛ_q(λ) than the usual oneL_p,qwhereλ(t) =t^1/p. In particular, ifλ(t) = (1 +|lnt|)^α, 0 <t≤1,λ(t) = 0,t≥1, then the spacesΛq(λ) andLp,q(logL)^αcoincide.

Remark4.1 In Refs. [9] and [8] these more general Lorentz spacesΛ_q(λ) were defined and investigated in a similar way but only for bounded systems. Hereλ(t) is a quasi-monotone weight considered early in Ref. [16] by Persson but then used only for Fourier inequalities related to the trigonometric system.

Remark4.2 Quasi-monotone weights are very useful and possible to handle in various situations in analysis since we have good control of the growth both up and down ast→0 ort→ ∞. For example the method of “interpolation with a parameter function” heavily depends on this idea (see [18]). The close relation to Matuszewska– Orlicz indices, the Bari–Stechkin condition and other remarkable properties were investigated in [19].

Remark4.3 In [8] (see Theorem 2.1, Theorem 2.3), theorems on the convergence of series of the Fourier coefficients of a function from the generalized Lorentz space Λ_q(λ) with respect to regular systems are proved. It is well known that a regular system is uniformly bounded (see [15], p. 117). Therefore, the assertions of Theorem2.1and Theorem3.1of this paper cannot follow from the results of [8].

Remark4.4 For the type of problems considered in this paper and [2] it is natural to con- sider the following slight generalizations of the classesA andBconsidered in [8] and [17]:A^∗=

s>0AδandB^∗=

s>0Bδ, whereAδconsists of positive functionsλ(t) such that λ(t)t^–δ is quasi-increasing andλ(t)t^{–(1/2–δ)} quasi-decreasing andBδ consists of positive functionsω(t) such thatω(t)t^–1/2–δis quasi-increasing andω(t)t^–1+δis quasi-decreasing.

Example4.5 It is well known that any concave functionψ(t) is quasi-monotone. More exactly,ψ(t) is nondecreasing andψ(t)/tis nonincreasing. A simple proof can be found on page 142 Ref. [11].

Inspired by the discussions above and in order to be able to compare with a recent result of Doktorski [4] we introduce the generalized Lorentz spaceL_ψ,θas follows: Forψ(t) quasi- monotone andθ> 0 we say that the measurable functionsf ∈Lψ,θwhenever

fψ,θ= 1

0

f^∗θ(t)ψ^θ(t)dt t

1 θ

<∞.

For the functionψwe set

αψ=lim_t→0ψ(2t)

ψ(t), βψ=lim_t→0ψ(2t) ψ(t). It is well known that 1≤αψ≤βψ≤2 (see e.g. [20]).

Consider the set of all non-negative functions on [0, 1],ψfor which (log2/t)^εψ(t)↑+∞

and (log2/t)^–εψ(t)↓0 fort↓0,∀ε> 0 (cf. [22]) and this set is also denoted bySVL.

By making modifications of the proof of Theorem2.1it is possible to prove the following generalization of this theorem.

Theorem 4.6 Let{ϕ_n}^∞_n=1be an orthogonal system,which satisfies the condition(1)and s∈(2, +∞].Moreover,assume thatψis a quasi-monotone function,which satisfy the con- ditionsαψ=βψ= 2^1/2,

sup

t∈(0,1]

t^1/2 ψ(t)<∞ and_ψ^t1/2_(t)∈SVL.

If1 <θ≤2,{an} ∈l2and

Λψ,θ(a) = ∞

n=1

ψ((1 +_n

l=1M²_l)^–1) (1 +_n

l=1M²_l)^–1/2

θ ln

1 +

n l=1

M_l²

(¹_θ–¹₂)θ

ρ_n^θ–ρ_n+1^θ

< +∞,

then the series _∞

n=1a_nϕ_n(x)converges in the space L_ψ,θ to some function f ∈L_ψ,θ and fψ,θ≤C(Λψ,θ)^1/θ.

Remark4.7 In the caseψ(t) =t^1/2(1 +ln|t|)^αTheorem4.6implies Theorem2.1.

Remark4.8 For a uniformly bounded system{ϕ_n}, Theorem4.6was recently proved dif- ferent way and in a slightly different form by Doktorski [4].

Remark4.9 The remarks above open the possibility of generalizing and unifying all the results in [2,4,8,9] and this paper. The present authors hope to investigate this in a forth- coming paper.

Acknowledgements

The first author is grateful for the support of this work given by the Russian Academic Excellence Project (agreement no.

02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University). We thank the referees for some good suggestions, which have improved this final version of our paper.

Funding Not applicable.

Availability of data and materials Not applicable.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

The authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

Author details

1Department of Fundamental Mathematics, L.N. Gumilyov Eurasian National University, Nur-Sultan, Republic of Kazakhstan. ²Institute of Mathematics and Computer Science, Ural Federal University, Yekaterinburg, Russia.

3Department of Computer Science and Computational Engineering, Campus Narvik, The Arctic University of Norway, Narvik, Norway.⁴Department of Mathematics and Computer Science, Karlstad University of Sweden, Karlstad, Sweden.

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Received: 1 October 2019 Accepted: 11 March 2020

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