# Nikol’skii–Type Inequalities for Trigonometric Polynomials for Lorentz–Zygmund Spaces

Authors: Leo R. Ya. Doktorski
Journal: Journal of Function Spaces
 (2020)

---

## Abstract

Nikol’skii–type inequalities, that is inequalities between different metrics of trigonometric polynomials on the torusTd for the Lorentz–Zygmund spaces, are obtained. The results of previous paper “Nikol’skii inequalities for Lorentz–Zygmund spaces” are extended. Applications to approximation spaces in Lorentz–Zygmund spaces and to Besov spaces are given.


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## Body

## 1. Introduction

The classical Nikol’skii (or Jackson–Nikol’skii) inequality for the trigonometric polynomialsTn on [0,1] of degree at most n can be written as [1, 2](1)Tnp≺n1/   q−1/   pTnq,where1≤q<p≤∞ and ∗p is the usual norm on the Lebesgue spaces Lp. For p=∞, this estimate has been proved by Jackson [3]. The proofs in [1–3] are based on Bernstein’s inequality.Inequalities between different (quasi-)norms of the same function are known as Nikol’skii–type (or Jackson–Nikol’skii–type) inequalities. They play a crucial role in many areas of mathematics, e.g., theory of approximation, theory of functions of several variables, and functional analysis (embedding theorems for Besov spaces).Nessel and Wilmes [4] extended inequality (1) for 0<q<p≤∞. They also observed that in inequality (1) one may in fact take into account the spectrum of the function involved for q≤2 or certain gaps in the spectrum for q>2. Sherstneva [5] extended inequality (1) in the Lorentz spaces Lp,b and showed that they are exact relative to the order n. Moreover, she investigated the limiting case when Lorentz spaces on both sides have the same value of the main parameter. She proved that if 0<p<∞ and 0<b<c≤∞, then(2)Tnp,b≺1+ln   n1/   b−1/   cTnp,c.Some Nikol’skii–type inequalities for the Lorentz–Zygmund spaces are considered in [6–8] and for the generalized Lorentz space in [9]. In other investigations, different sets of functions, domains, and measures were explored. For further information about these results and applications, we refer to [1–19] and references therein.In [19], the results of [5] were improved in two directions. First, the functions Tn of the form ∑k=1nckφk were considered, where φk is an orthonormal system in L2M uniformly bounded in L∞M (with μM<∞). No assumptions about smoothness of φk were made. Secondly, the inequalities (1) and (2) were extended to the Lorentz–Zygmund spaces. However, in [19], only the case 0<q≤2 is obtained.The principal aim of this paper is to extend the results of [19] for the case 2≤q<∞ for the trigonometric polynomials. The technique we apply relies on the observation that the power of a trigonometric polynomial is also a trigonometric polynomial [4, 10, 17, 18]. This paper is organized as follows. Section 2 contains necessary notations and definitions. Main results of this contribution are Theorems 4, 6, 8, and 9. They are formulated and proved in Section 3. Note that Theorems 8, and 9 deal with the limiting case. In Section 4, we reformulate Theorems 4, 6, 8, and 9 for trigonometric polynomials of degree at most n. In Section 5, we give some applications to embeddings between approximation spaces in Lorentz–Zygmund spaces and between Besov spaces.


## 2. Preliminaries

We writeX⊂Y for two quasi-normed spaces X and Y to indicate that X is continuously embedded in Y. The notation X≅Y means that X⊂Y and Y⊂X. If f and g are positive functions, we write f≺g if f≤Cg, where the constant C is independent on all significant quantities. We put f≈g if f≺g and g≺f. We adopt the convention that 1/∞=0. We use the abbreviation lx=1+lnx and llx=llx,0<x<∞.Throughout the paper, letTd=x=x1,…,xd∈Rd:0≤xi≤   1d∈N be the d-dimensional torus with Lebesgue measure. We consider (equivalence classes of) complex-valued measurable functions f on Td and bounded complex-valued sequences ck. As usual [20–23], f∗tt>0 and cn∗n∈N are the nonincreasing rearrangements of a function f and of a sequence ck, respectively. Because the measure of the torus Td equals 1, f∗t=0 if t>1.Definition 1.
Let0<p,b≤∞ and −∞<α<∞. The Lorentz–Zygmund space Lp,b;α≡Lp,b;αTd can be defined as follows:(3)
L
p
,
b
;
α
≔
f
:
f
p
,
b
;
α
≔
t  1p−1blαtf∗t
b
<
∞
,where∗b is the usual (quasi-)norm on the Lebesgue space Lb(0,1). Similarly,(4)lp,b;α≔ck:ckp,b;α≔n  1p−1blαncn∗b<   ∞,where∗b is the usual (quasi-)norm on the Lebesgue sequence space lb.We use the same notation∗p,b;α for both (quasi-)norms. If α=0, then Lp,b;α coincides with the Lorentz space Lp,b. If in addition p=b, then the space becomes the Lebesgue space Lp. The same is valid for sequence spaces, too. Note that L∞,b;α is not trivial if and only if either α<−1/b or b=∞ and α=0. For the detailed information about these spaces we refer to [21–24]. It is known that all spaces Lp,b;α are complete. Moreover [23, Theorem 7.4], on Lp,b;α there exists a norm, equivalent to ∗p,b;α, iff p,b;α∈Ω, where(5)Ω≔p,b;α∈1,∞×1,∞×−∞,∞p=b=1,α≥0;or1<p<∞;orp=∞,α<−1b;orp=b=∞,α=0.In this case, it is a Banach function space. For details see [20, 23]. In addition, let(6)Γ≔p,b;α0<p≤∞,0<b<min1,p,−∞<α<∞;or0<b=p<1,α≥0;orp=∞,0<b<1,α<−1b.The following statement must surely be known. (see Acknowledgments).Lemma 2.
Letp,b;α∈Γ. Then, the space Lp,b;α is b-normed, that is, there exists C=Cp,b;α>0 such that(7)
∑n=0∞gn
p
,
b
;
α
<
C
∑n=0∞gnp,b;αb
1
/
b
,for each sequencegn⊂Lp,b;αn=0,1,2,… such that the series ∑n=0∞gn converse in Lp,b;α.Proof.
It is enough to prove that (7) holds for finite sums ∑n=0N…. Because b<1, for x∈Td, we have(8)
∑n=0Ngnx
b
≤
∑n=0Ngnx
b
≤
∑
n
=
0
N
g
n
x
b
.Letf∗∗ be as usual the maximal function of f∗ given by f∗∗s=1/s∫0sf∗tdt. If b<p or b=p, α≥0, then [21, Theorem 9.1 and Theorem 9.3 (i)] Lp,b;α⊂Lb. Hence, for each function g∈Lp,b;α, it holds gb∈L1 and therefore gb∗∗ is well-defined. In virtue of the monotonicity and subadditivity of f∗∗ (see [20, Chapter 2, Proposition 3.2 and Theorem 3.4]), we conclude:(9)∑n=0Ngnb∗∗s≤∑n=0Ngnb∗∗s≤∑n=0Ngnb∗∗s,which together with [20, Chapter 2, (1.20)] yields that(10)∫0s∑n=0Ngn∗tbdt≤∑n=0N∫0sgn∗tbdt.Applying now Hardy’s lemma [20, Chapter 2, Proposition 3.6], we infer that(11)∫01ht∑n=0Ngn∗tbdt≤∑n=0N∫01htgn∗tbdt=∑n=0Nhgn∗bb,for any nonnegative decreasing functionh. Let 0<p≤∞, 0<b<min1,p, −∞<α<∞ or 0<b=p<1, α>0. In this case, we can find a decreasing function h0 such that h0t≈t1/p−1/blαt (see, e.g., [25]). Therefore, by (11),(12)∑n=0Ngnp,b;αb≈h0t∑n=0Ngn∗tbb≤∑n=0Nh0gn∗bb≈∑n=0Ngnp,b;αb.The case0<b=p<1, α=0 is trivial. This completes the proof.Lemma 3.
Let0<p,b≤∞, −∞<α<∞, f∈Lp,b;α, and k∈N. Then,(13)
fp,b;α
k
=
fk
p
/
k
,
b
/
k
;
α
k
.


## 3. Main Results

LetΛ⊂Zd be a finite set of lattice points. We denote the number of elements of the set Λ by #Λ. Everywhere below TΛx=∑m∈Λcme2πimxx∈Td,cm⊂C,cm≠   0. So, for the mth m∈Zd Fourier coefficient of the function TΛ it holds T^Λm=cm. Hence, suppT^Λ≔m∈Zd:T^Λm≠   0=Λ.Our goal is to obtain Nikol’skii–type inequalities of the form(14)TΛp,b;α≤CGΛ;p,b,α,q,c,   βTΛq,c;β,where the constantC does not depend on Λ. Or, writing it shorter(15)TΛp,b;α≺GΛ;p,b,α,q,c,   βTΛq,c;β.Obviously, only the situationsLq,c;β⊄Lp,b;α are of interest, otherwise G=1.For any tripleq,c;β∈0,   ∞×0,∞×−∞,∞, we define a corresponding natural number ρ=ρq,c;β in the following way. Let M:=0,2×0,∞×−∞,∞∪2,2;0. For all triples q,c;   β∈M, we set ρ=1. For each other triple q,c;   β, we define ρ as the smallest integer such that q/   ρ,c/   ρ,ρ  β∈M. We set NTΛ,   ρ≔#suppTΛρ∧. Note that NTΛ,   1=#Λ.Theorem 4.
Let0<q<∞, 0<c≤∞, −∞<β<∞, and ρ=ρq,c;   β. For the triple p,b,   α, we assume that either q<p<∞, 0<b≤∞, −∞<α<∞, or p=b=∞, α=0. Then(16)TΛp,b;α≺NTΛ,ρ1/q−1/plNTΛ,ρα−βTΛq,c;β.Proof.
For the caseρ=1, from [19, Theorem 1] we immediately have(17)TΛp,b;α≺#Λ1/q−1/   pl#Λα−βTΛq,c;β.Consider the caseρ>1. Note that for the triple q/    ρ,c/   ρ,ρ   β the corresponding ρ-value is 1. Hence, due to (13) and (17) we have(18)TΛp,b;αρ=TΛρp/ρ,b/ρ;αρ≺NTΛ,    ρρ1/    q−1/    plNTΛ,    ρρα−   βTΛρq/ρ,c/ρ;βρ=NTΛ,ρ1/    q−1/    plNTΛ,ρα−    βTΛq,c;βρ.This completes the proof.Remark 5.
Theorem4 covers Theorems 1 and 2 in [4]. See also [10].Theorems6, 8, and 9 can be proved following similar approach as proof of Theorem 4, using [19, Theorems 2, 3, and 4] respectively.Theorem 6.
Let0<q<∞, 0<b,c≤∞, −∞<β<∞−∞<α<−1/b, and ρ=ρq,c;    β. Then(19)TΛ∞,b;α≺NTΛ,    ρ1/qlNTΛ,    ρα+1/    b−    βTΛq,c;β.Remark 7.
According to [21, Theorem 9.3], the conditions of Theorems 4 and 6 imply thatLq,c;β⊃Lp,b;α.Both Theorems4 and 6 deal with the case q<p. The next two theorems examine the limiting case q=p.Theorem 8.
Let0<q<∞, 0<b≤c≤∞, −∞<α,β<∞, and ρ=ρq,c;    β.(i) Ifα+1/b>β+1/c, then,
(20)
TΛ
q
,
b
;
α
≺
lNTΛ,    ρ
α+1/    b−β−1/  c
TΛ
q
,
c
;
β
.
(ii) Ifα+1/b=β+1/c, then
(21)
TΛ
q
,
b
;
α
≺
llNTΛ,    ρ
1/   b−1/    c
TΛ
q
,
c
;
β
.Theorem 9.
Let0<q<∞, 0<c≤b≤∞, −∞<β<α<∞, and ρ=ρq,c;β. Then(22)TΛq,b;α≺lNTΛ,ρα−βTΛq,c;β.Remark 10.
According to [22, Proposition 3.1], Theorem 8 (i) and Theorem 9 deal with comparable Lq,b;α⊂Lq,c;β as well as incomparable pairs of Lorentz–Zygmund spaces. Theorem 8(ii) only handles incomparable pairs.


## 4. Corollaries for Trigonometric Polynomials of Degree at Mostn

LetTn be the set of all trigonometric polynomials of degree at most n n=1,2,    …, i. e.(23)Tn≔Tnx=∑m≤ncme2πimx:cm∈C,m∈Zd,x∈Td,wherem=m1+⋯+md.Corollary 11.
Let0<q<∞, 0<c≤∞, and −∞<β<∞. For the triple p,b,    α, we assume that either q<p<∞, 0<b≤∞, −∞<α<∞, or p=b=∞, α=0. Then for all Tn∈Tn.(24)
Tn
p
,
b
;
α
≺
n
d
1/    q−1/p
ln
α−    β
Tn
q
,
c
;
β
.Proof.
Becauseρ in Theorem 4 depends only on q,c, and β, we have(25)
N
Tn,    ρ
=
#
s
u
p
p
Tnρ
∧
≤
2ρn+    1
d
≺
n
d
.Forν>0, −∞<γ<∞, the sequence kνlkγ is either increasing or equivalent to an increasing sequence (see, e.g., [25]). In both cases, because of (25)(26)NTn,    ρ1/    q−1/    plNTn,    ρα−    β≺nd1/    q−1/    plnα−    β.The last estimate and Theorem4 imply (24).Analogously, Theorems6, 8, and 9 imply next three corollaries.Corollary 12.
Let0<q<∞, 0<b,c≤∞, −∞<β<∞, and −∞<α<−1/b. Then for all Tn∈Tn.(27)
Tn
∞
,
b
;
α
≺
n
d
/
q
ln
α+1/b−β
Tn
q
,
c
;
β
.Corollary 13.
Let0<q<∞, 0<b≤c≤∞, and −∞<α,β<∞.(i) Ifα+1/b>β+1/c, then for all Tn∈Tn,
(28)
Tn
q
,
b
;
α
≺
ln
α+1/    b−β−1/    c
Tn
q
,
c
;
β
.
(ii) Ifα+1/b=β+1/c, then for all Tn∈Tn,
(29)
Tn
q
,
b
;
α
≺
lln
1/    b−1/    c
Tn
q
,
c
;
β
.Corollary 14.
Let0<q<∞, 0<c≤b≤∞, and −∞<β<α<∞. Then for all Tn∈Tn.(30)
Tn
q
,
b
;
α
≺
l
n
α−    β
Tn
q
,
c
;
β
.Remark 15.
Corollary13(i) covers [7, (3.16)], [8, Lemma 5.4], and [9, Corollary 2]. Corollary 14 covers [7, Lemma 3.4]. In [26, Lemma 3] a variant of (29) was obtained. Inequality given in [26, Lemma 2] provides a limiting counterpart (i.e., α=β) of (30). Both Lemmas in [26] involve generalized Lorentz–Zygmund spaces with two iterations of logarithm.


## 5. Applications

LetTn be the set of all trigonometric polynomials of degree at most n described above. The sequence Tnn=1,2,    … allows construction of an approximation family in all Lorentz–Zygmund spaces, which produces approximation spaces. In Subsection 5.1 we start with some necessary definitions and auxiliary results dealing with approximation spaces. In Subsection 5.2 we present some corollaries of the statements from Section 4 dealing with embeddings of the approximation spaces Lp,b;αuσ,    γ into Lorentz–Zygmund spaces and between these approximation spaces. In Subsections 5.3 and 5.4  these corollaries will be reformulated in terms of Besov spaces.


### 5.1. Basic Approximation Constructions

A quasi-norm on a quasi-Banach spaceX is denoted by ∗X. Let X be a quasi-Banach space and CX≥    1 be the constant from the triangle inequality in the space X. A sequence Gnn=0,1,2,    … of subsets of X is called an approximation family in X if the following conditions are satisfied:(1)
0=G0⊂G1⊂G2⊂…⊂X,(2)
λGn⊆Gn for all scalars λ and n=1,2,…,(3)
Gm+Gn⊆Gm+n for m, n=1,2,….Forf∈X and n=1,2,…, the nth approximation number is defined by(31)Enf≡Enf;    X=inff−    gX:g∈Gn−1.Letσ≥0, 0<u≤∞, and −∞<γ<∞. The approximation space Xuσ,    γ≡X,Gnuσ,    γ is formed by all those f∈X, for which Enf∈l1/σ,u;γ with the quasi-norm fXuσ,    γ≔Enf1/    σ,u;γ. Note that Xu0,γ coincides with X if γ<−1/u. Approximation spaces were investigated in many works. For more information about such spaces, we refer to [8, 27–30] and references therein. We will use some common statements from the approximation theory. We begin with the two representation theorems.Theorem 16 [8, 27, 28].
Letσ>0. An element f∈X belongs to Xuσ,    γ if and only if there is a representation(32)
f
=
∑
n
=
0
∞
g
n
,
convergencein    X
,
g
n
∈
G
2
n
,with(33)∑n=0∞2nσ1+    nγgnXu<∞.Moreover,(34)fXuσ,    γrep≔inf∑n=0∞2nσ1+    nγgnXu1/u,where the infimum is taken over all possible representations (32), defines an equivalent quasi-norm on Xuσ,    γ with equivalence constants depending only on σ, u, γ, and CX. The usual modification shall be made when u=∞.Theorem 17 ([29], Theorem 1).
Let0<u≤∞ and γ>−1/u. Denote μn=22nn=0,1,2,…. An element f∈X belongs to Xu0,γ if and only if there is a representation(35)
f
=
∑
n
=
0
∞
g
n
,
convergencein    X
,
g
n
∈
G
μ
n
,with(36)∑n=0∞2nγ+1/    ugnXu<∞.Moreover,(37)fXu0,    γrep≔inf∑n=0∞2nγ+1/    ugnXu1/u,where the infimum is taken over all possible representations (35), defines an equivalent quasi-norm on Xu0,    γ with equivalence constants depending only on u, γ, and CX. The usual modification shall be made when u=∞.These representation theorems are useful to prove the following two lemmas.Lemma 18 [8, 30].
LetX, Y be quasi-Banach spaces that are continuously embedded in a Hausdorff topological vector space. Let Gn be an approximation family such that Gn⊆X∩Yn=1,2,    …. Assume that there are constants c, β>0 such that(38)
g
Y
≤
c
ln
β
g
X
,
g
∈
G
n
,
n
=
1
,
2
,
…
.Then, for0<u≤∞ and γ>−1/u, we have(39)Xu0,γ+    β⊂Yu0,γ.Lemma 19 (Cf. [8, Lemma 2.1], [28, Theorem 4.3], and [27, Theorem 3.4]).
LetX,Y be quasi-Banach spaces which are continuously embedded in a Hausdorff topological vector space. Let Gn be an approximation family such that Gn⊆X∩Yn=1,2,    …. Assume that there are constants c>0, δ≥0, and −∞<β<∞ such that(40)
g
Y
≤
c
n
δ
ln
β
g
X
,
g
∈
G
n
,
n
=
1
,
2
,
…
.Then, forσ>0, 0<u≤∞, and −∞<γ<∞, we have(41)Xuσ+δ,γ+    β⊂Yuσ,    γ.We omit the proof since it can be carried out as in [8, Lemma 2.1]. We will also use some interpolation formulae for approximation spaces. Let 0≤θ≤1, 0<b≤∞, and −∞<α<∞. By ∗,∗θ,b, we denote the classical real interpolation functor [20, 31] and by ∗,∗θ,b;α the real interpolation functor involving logarithmic factor for ordered couples with integration over (0,1) (see, e.g., [30, 32, 33]). Note that ∗,∗θ,b=∗,∗θ,b;0.Lemma 20 ([30, Proposition 2.7]).
Suppose thatσ>0, 0<u, q≤∞, 0<θ<1, and −∞<γ<∞, then(42)
X,Xuσ,    0
θ
,
q
;
γ
≅
X
q
θσ,    γ
.The reiteration relation (see, e.g., [32]) leads to the following result.Corollary 21.
Suppose thatσ>0, 0<u,q≤∞, 0<θ<1, and −∞<γ,δ<∞, then(43)X,Xuσ,    δθ,q;γ≅Xqθσ,θδ+    γ.


### 5.2. Approximation Spaces in Lorentz–Zygmund Spaces

Everywhere below, we consider the following sequence of subsets:G0=0, Gn=Tn be the set of all trigonometric polynomials of degree at mostnn=1,2,    …. It builds an approximation family in all nontrivial Lorentz–Zygmund spaces. Now we will apply Corollaries 11–14 to the approximation spaces Lp,b;αuσ,    γ. First, we will investigate embeddings of these approximation spaces into Lorentz–Zygmund spaces. Next, we will investigate embeddings between different approximation spaces L∗,∗;∗∗σ,    ∗ for σ>0 and between different limiting approximation spaces L∗,∗;∗∗0,    ∗. From Corollary 11, we get the following result.Corollary 22.
Let0<q<p<∞, 0<b, c≤∞, −∞<α, β<∞. Then(44)
Lq,c;β
b
d1/    q−1/    p,α−    β
⊂
L
p
,
b
;
α
.Proof.
By Theorem16, given any f∈Lq,c;β1d/    q,−    β, we can find a representation f=∑n=0∞gn, (convergence in Lq,c;β), gn∈T2n such that(45)
∑
n
=
0
∞
2
n
d
/
q
1+    n
−
β
gn
q
,
c
;
β
≺
f
Lq,c;β
1
d/    q,−β
.Because allgn∈T2n⊂Lq,c;β⋂L∞, it is not hard to show that the series ∑n=0∞gn converges to f in L∞. Since L∞ is a Banach space, using Corollary 11 and (45), we derive that(46)f∞≤∑n=0∞gn∞≺∑n=0∞2nd/q1+    n−βgnq,c;β≺fLq,c;β1d/    q,−    β.Therefore,(47)Lq,c;β1d/    q,−    β⊂L∞.Since0<q<p<∞, we can define θ and γ by formulae(48)1p=1−θqandα=1−    θβ+γ.From Corollary21, we have(49)Lq,c;β,Lq,c;β1d/    q,−    βθ,b;γ≅Lq,c;βbd1/    q−1/    p,α−    β.Furthermore, it is known [32, Theorem 7] that(50)Lq,c;β,L∞θ,b;γ≅Lp,b;α.Using now (47) and [32, Theorem 2], we obtain (44).From Corollary12, we immediately get the following result.Corollary 23.
Let0<q<∞, 0<c≤∞, −∞<β<∞, 0<b≤∞, and α<−1/b. Then(51)
Lq,c;β
min
b,    1
d/    q,α+1/    b−β
⊂
L
∞
,
b
;
α
.Proof.
Consider the case1≤b≤∞. By Theorem 16, given any f∈Lq,c;β1d/    q,α+1/    b−    β, we can find a representation f=∑n=0∞gn, (convergence in Lq,c;β), gn∈T2n such that(52)
∑
n
=
0
∞
2
n
d
/
q
1+n
α+1/    b−    β
gn
q
,
c
;
β
≺
f
Lq,c;β
1
d/    q,α+1/    b−    β
.Because allgn∈T2n⊂Lq,c;β⋂L∞,b;α, it is not hard to show that the series ∑n=0∞gn converges to f in L∞,b;α. Since ∞,b,α∈Ω, using Corollary 12 and (52), we derive that(53)f∞,b;α≺∑n=0∞gn∞,b;α≺∑n=0∞2nd/q1+    nα+1/    b−    βgnq,c;β≺fLq,c;β11/    q,α+1/    b−    β.For the case0<b<1, by Theorem 16, given any f∈Lq,c;βbd/    q,α+1/    b−    β, we can find a representation f=∑n=0∞gn, (convergence in Lq,c;β), gn∈T2n such that(54)∑n=0∞2nd/q1+    nα+1/    b−    βgnq,c;βb1/b≺fLq,c;βbd/    q,α+1/b−    β.Due to Lemma2, Corollary 12 and (54), we obtain(55)f∞,b;α≺∑n=0∞gn∞,b;αb1/b≺∑n=0∞2nd/q1+    nα+1/    b−    βgnq,c;βb1/b≺fLq,c;βb1/    q,α+1/    b−    β.This completes the proof.Using some real interpolation technique formulae we obtain the following result, which complements Corollary23.Corollary 24.
Let0<q<∞, 0<c, b≤∞, −∞<β<∞, and α<−1/b. Then(56)
Lq,c;β
b
d/    q,α+1/minb.    1−    β
⊂
L
∞
,
b
;
α
.Proof.
Using Lemma20, we get(57)
Lq,c;β,Lq,c;β12d/    q,    0
1/    2
,
1
;
−
β
≅
Lq,c;β
1
d/    q,−β
.We considerLq,c;β,Lq,c;β1d/    q,−    β1,b;α. From (57), [32, Theorem 6 ,and Lemma 4], and Lemma 20, we have(58)Lq,c;β,Lq,c;β1d/    q,−    β1,b;α≅Lq,c;β,Lq,c;β,Lq,c;β12d/    q,    01/    2,1;−β1,b;α⊃Lq,c;β,Lq,c;β12d/    q,    01/    2,b;α+1/minb.    1−β≅Lq,c;βbd/q,α+1/minb.    1−    β.To complete the proof, we have only to use (47) and [32, Corollary 7]:(59)Lq,c;β,Lq,c;β1d/    q,−    β1,b;α⊂Lq,c;β,L∞1,b;α≅L∞,b;α.Remark 25.
Recall that the spacesX1d/    q,α+1/    b−    β and Xbd/    q,α+1−    β are formed by all those f∈X for which Enf belongs to lq/d,1;α+1/b−β and lq/d,b;α+1−β, respectively. However, for b>1, these sequence spaces are incomparable [24]. Note that for b≤1, formulae (51) and (56) coincide.Corollary 26.
Letp,b,α∈Ω∪Γ, p<∞, b≤c≤∞, and −∞<β<∞. If α+1/b>β+1/c then(60)
Lp,c;β
min
b,    1
0,α+1/    b−β−1/    c−1/minb.    1
⊂
L
p
,
b
;
α
.Proof.
Letp,b,α∈Ω. In this case b≥1. By Theorem 17, given any f∈Lp,c;β10,α+1/    b−β−1/    c−    1, we can find a representation f=∑n=0∞gn, (convergence in Lp,c;β), gn∈T22n such that(61)
∑
n
=
0
∞
2
n
α+1/    b−β−1/    c
gn
p
,
c
;
β
≺
f
Lp,c;β
1
0
,
α
+
1
/  b
−
β
−
1
/  c
−  1
.Because allgn∈T22n⊂Lp,b;α⋂Lp,c;β, it is not hard to show that the series ∑n=0∞gn converges to f in Lp,b;α. Sincep,b,α∈Ω, using Corollary 13(i), we derive that(62)fp,b;α≺∑n=0∞gnp,b;α≺∑n=0∞2nα+1/  b−β−1/  cgnp,c;β≺fLp,c;β10,α+1/  b−β−1/  c−  1.Let nowp,b,α∈Γ. In this case, b<1. By Theorem 17, given any f∈Lp,c;βb0,α−β−1/  c, we can find a representation f=∑n=0∞gn, (convergence in Lp,c;β), gn∈T22n such that(63)∑n=0∞2nα+1/  b−β−1/  cgnp,c;βb1/b≺fLq,c;βb0,α−β−1/c.Due to Lemma2 and Corollary 13(i), we obtain(64)fp,b;α≺∑n=0∞gnp,b;αb1/b≺∑n=0∞2nα+1/b−β−1/cgnq,c;βb1/b≺fLq,c;βb0,α−β−1/c.This completes the proof.The following result can be proved analogously based on Corollary14.Corollary 27.
Letp,b,α∈Ω∪Γ, p<∞, 0<c≤b, and −∞<β<α<∞. Then(65)
L
p
,
c
;
β
min
b
,
1
0
,
α
−
β
−
1
/
m
i
n
b
.
1
⊂
L
p
,
b
;
α
.The next four corollaries investigate embeddings between different approximation spacesL∗,∗;∗∗σ,∗ for σ>0. They follow immediately from Corollaries 11, 12, 13(i), and 14 and from Lemma 19.Corollary 28.
Let0<q<∞, 0<c≤∞, −∞<β<∞. For the triple p,b,α, we assume that either q<p<∞, 0<b≤∞, −∞<α<∞, or p=b=∞, α=0. Then, for σ>0, 0<u≤∞, and −∞<γ<∞, we have(66)
L
q
,
c
;
β
u
σ
+
d
1
/
q
−
1
/
p
,
γ
+
α
−
β
⊂
L
p
,
b
;
α
u
σ
,
γ
.Corollary 29.
Let0<q<∞, 0<c,b≤∞, −∞<β<∞, and α<−1/b. Then, for σ>0, 0<u≤∞, and −∞<γ<∞, we have(67)
L
q
,
c
;
β
u
σ
+
d
/
q
,
γ
+
α
+
1
/
b
−
β
⊂
L
∞
,
b
;
α
u
σ
,
γ
.Corollary 30.
Let0<p<∞, 0<b<c≤∞, −∞<α, β<∞, and α+1/b>β+1/c. Then, for σ>0, 0<u≤∞, and −∞<γ<∞, we have(68)
L
p
,
c
;
β
u
σ
,
γ
+
α
+
1
/
b
−
β
−
1/c
⊂
L
p
,
b
;
α
u
σ
,
γ
.Corollary 31.
Let0<p<∞, 0<c≤b≤∞, and −∞<β<α<∞. Then, for σ>0, 0<u≤∞, and −∞<γ<∞, we have(69)
L
p
,
c
;
β
u
σ
,
γ
+
α
−
β
⊂
L
p
,
b
;
α
u
σ
,
γ
.The two next corollaries deal with embeddings between different limiting approximation spacesL∗,∗;∗∗0,∗. They follow immediately from Lemma 18 and Corollaries 13(i) and 14, respectively.Corollary 32.
Let0<p<∞, 0<b<c≤∞, −∞<α, β<∞, and α+1/b>β+1/c. Then, for 0<u≤∞ and γ>−1/u, we have(70)
L
p
,
c
;
β
u
0
,
γ
+
α
+
1
/
b
−
β
−
1
/
c
⊂
L
p
,
b
;
α
u
0
,
γ
.Corollary 33.
Let0<p<∞, 0<c≤b≤∞, and −∞<β<α<∞. Then, for 0<u≤∞ and γ>−1/u, we have(71)
L
p
,
c
;
β
u
0
,
γ
+
α
−
β
⊂
L
p
,
b
;
α
u
0
,
γ
.


### 5.3. Embeddings of Besov SpacesBp,uσ,γ

Here we give some applications of previous results to embeddings of Besov spaces into Lorentz–Zygmund spaces and between Besov spaces.There are several definitions of Besov spaces. The Besov spaceBp,uσ,γ≡Bp,uσ,γTdσ≥0,0<p,u≤∞,−∞<γ<∞ is based on Lp and has classical smoothness σ and additional logarithmic smoothness with exponent γ. It is formed by all those f∈Lp such that(72)fBp,uσ,γ=fp+∫01t−σlγtωkf,tpudtt1/u<∞,with an obvious modification whenu=∞. Here ωkf,tp k>σ,k∈N is the modulus of smoothness of order k with respect to the quasi-norm on Lp. It makes sense to consider the spaces Bp,u0,γ with zero classical smoothness only for γ≥−1/u. All we need for our application is the characterization of the Besov spaces by approximation. An important result in approximation theory states that(73)Bp,uσ,γ≅Lpuσ,γ.For details see, e.g., [8, 30] and the references given there. Using this characterization with Corollaries 22–24 and 26–28, we obtain the following embeddings. Due to similarity, we will give only the proof of  Corollary 37.Corollary 34 (Cf. [8, Theorem 5.2]).
Let0<q<p<∞, 0<b≤∞, and −∞<α<∞. Then(74)
B
q
,
b
d
1
/
q
−
1
/
p
,
α
⊂
L
p
,
b
;
α
.Corollary 35.
Let0<q<∞, 0<b≤∞, and α<−1/b. Then(75)
B
q
,
min
b
,
1
d
/
q
,
α
+
1
/
b
⊂
L
∞
,
b
;
α
.Corollary 36 (Cf. [8, Theorem 5.3]).
Let 0<q<∞, 0<b≤∞, and α <−1/b. Then(76)
B
q
,
b
d
/
q
,
α
+
1
/
m
i
n
b
,
1
⊂
L
∞
,
b
;
α
.Corollary 37 (Cf. [8, Theorems 5.5, 5.6, and 5.7]).
Letp,b,α∈Ω∪Γ, b≤p<∞, and α>1/p−1/b. Then(77)
B
p
,
min
b
,
1
0
,
α
+
1
/
b
−
1
/
p
−
1
/
m
i
n
b
,
1
⊂
L
p
,
b
;
α
.Proof.
Using the characterization (73) and Corollary 26, we obtain(78)
B
p
,
min
b
,
1
0
,
α
+
1
/
b
−
1
/
p
−
1
/
m
i
n
b
,
1
≅
L
p
min
b
,
1
0
,
α
+
1
/
b
−
1
/
p
−
1
/
m
i
n
b
,
1
⊂
L
p
,
b
;
α
.Corollary 38.
Letα>0. In addition, let 0<p=b≤1, or 1<p<∞ and p≤b≤∞. Then(79)
B
p
,
min
b
,
1
0
,
α
−
1
/
m
i
n
b
,
1
⊂
L
p
,
b
;
α
.Corollary 39 (Cf. [34, Corollary 5.3 (i)] and [31, Theorem 2.8.1]).
Let0<q<p≤∞. Then, for σ>0, 0<u≤∞, and −∞<γ<∞, we have(80)
B
q
,
u
σ
+
d
1
/
q
−
1
/
p
,
γ
⊂
B
p
,
u
σ
,
γ
.


### 5.4. Embeddings of Besov–Type SpacesBp,b;α,u0,γ

In [7], the Besov–type spaces Bp,b;α,u0,γ≡Bp,b;α,u0,γTd (with zero classical smoothness) based on Lp,b;α are introduced, so that Bp,p;0,u0,γ=Bp,u0,γ. The space Bp,b;α,u0,γ0<u<∞,1<p<∞,1≤b≤∞,−∞<α<∞,γ>−1/u. is formed by all those f∈Lp,b;α such that(81)fBp,b;α,u0,γ=fp,b;α+∫01lγtωkf,tp,b;αudtt1/u<∞,with an obvious modification whenu=∞. Here ωkf,tp,b;αk∈N is the modulus of smoothness of order k with respect to the quasi-norm on Lp,b;α. The definition is independent of k. The spaces Bp,b;α,u0,γ have the following characterization by approximation [7, (3.7)]:(82)Bp,b;α,u0,γ≅Lp,b;αu0,γ.Using this characterization and Corollaries26, 27, 32, and 33, we obtain the following embeddings.Corollary 40 (Cf. [7, (3.9)]).
Let1<p<∞, 1≤c≤∞, 0<b≤c, and −∞<α, β<∞. If α+1/b>β+1/c, then(83)
B
p
,
c
;
β
,
min
b
,
1
0
,
α
+
1
/
b
−
β
−
1
/
c
−
1
/
m
i
n
b
,
1
⊂
L
p
,
b
;
α
.Corollary 41 (Cf. [7, (3.8)]).
Let1<p<∞, 1≤c≤b≤∞, and −∞<β<α<∞. Then(84)
B
p
,
c
;
β
,
1
0
,
α
−
β
−
1
⊂
L
p
,
b
;
α
.Corollary 42.
Let1<p<∞, 1≤b≤c≤∞, −∞<α, β<∞, andα+1/b>β+1/c. Then, for 0<u≤∞ and γ>−1/u, we have(85)
B
p
,
c
;
β
,
u
0
,
γ
+
α
+
1
/
b
−
β
−
1
/
c
⊂
B
p
,
b
;
α
,
u
0
,
γ
.Corollary 43.
Let1<p<∞, 1≤c≤b≤∞, and −∞<β<α<∞. Then, for 0<u≤∞ and γ>−1/u, we have(86)
B
p
,
c
;
β
,
u
0
,
γ
+
α
−
β
⊂
B
p
,
b
;
α
,
u
0
,
γ
.Remark 44.
Some other limiting embeddings between Besov spaces based on generalized Lorentz–Zygmund spaces with two iterations of logarithm were obtained in [26, Theorem 2].


## 5.1. Basic Approximation Constructions

A quasi-norm on a quasi-Banach spaceX is denoted by ∗X. Let X be a quasi-Banach space and CX≥    1 be the constant from the triangle inequality in the space X. A sequence Gnn=0,1,2,    … of subsets of X is called an approximation family in X if the following conditions are satisfied:(1)
0=G0⊂G1⊂G2⊂…⊂X,(2)
λGn⊆Gn for all scalars λ and n=1,2,…,(3)
Gm+Gn⊆Gm+n for m, n=1,2,….Forf∈X and n=1,2,…, the nth approximation number is defined by(31)Enf≡Enf;    X=inff−    gX:g∈Gn−1.Letσ≥0, 0<u≤∞, and −∞<γ<∞. The approximation space Xuσ,    γ≡X,Gnuσ,    γ is formed by all those f∈X, for which Enf∈l1/σ,u;γ with the quasi-norm fXuσ,    γ≔Enf1/    σ,u;γ. Note that Xu0,γ coincides with X if γ<−1/u. Approximation spaces were investigated in many works. For more information about such spaces, we refer to [8, 27–30] and references therein. We will use some common statements from the approximation theory. We begin with the two representation theorems.Theorem 16 [8, 27, 28].
Letσ>0. An element f∈X belongs to Xuσ,    γ if and only if there is a representation(32)
f
=
∑
n
=
0
∞
g
n
,
convergencein    X
,
g
n
∈
G
2
n
,with(33)∑n=0∞2nσ1+    nγgnXu<∞.Moreover,(34)fXuσ,    γrep≔inf∑n=0∞2nσ1+    nγgnXu1/u,where the infimum is taken over all possible representations (32), defines an equivalent quasi-norm on Xuσ,    γ with equivalence constants depending only on σ, u, γ, and CX. The usual modification shall be made when u=∞.Theorem 17 ([29], Theorem 1).
Let0<u≤∞ and γ>−1/u. Denote μn=22nn=0,1,2,…. An element f∈X belongs to Xu0,γ if and only if there is a representation(35)
f
=
∑
n
=
0
∞
g
n
,
convergencein    X
,
g
n
∈
G
μ
n
,with(36)∑n=0∞2nγ+1/    ugnXu<∞.Moreover,(37)fXu0,    γrep≔inf∑n=0∞2nγ+1/    ugnXu1/u,where the infimum is taken over all possible representations (35), defines an equivalent quasi-norm on Xu0,    γ with equivalence constants depending only on u, γ, and CX. The usual modification shall be made when u=∞.These representation theorems are useful to prove the following two lemmas.Lemma 18 [8, 30].
LetX, Y be quasi-Banach spaces that are continuously embedded in a Hausdorff topological vector space. Let Gn be an approximation family such that Gn⊆X∩Yn=1,2,    …. Assume that there are constants c, β>0 such that(38)
g
Y
≤
c
ln
β
g
X
,
g
∈
G
n
,
n
=
1
,
2
,
…
.Then, for0<u≤∞ and γ>−1/u, we have(39)Xu0,γ+    β⊂Yu0,γ.Lemma 19 (Cf. [8, Lemma 2.1], [28, Theorem 4.3], and [27, Theorem 3.4]).
LetX,Y be quasi-Banach spaces which are continuously embedded in a Hausdorff topological vector space. Let Gn be an approximation family such that Gn⊆X∩Yn=1,2,    …. Assume that there are constants c>0, δ≥0, and −∞<β<∞ such that(40)
g
Y
≤
c
n
δ
ln
β
g
X
,
g
∈
G
n
,
n
=
1
,
2
,
…
.Then, forσ>0, 0<u≤∞, and −∞<γ<∞, we have(41)Xuσ+δ,γ+    β⊂Yuσ,    γ.We omit the proof since it can be carried out as in [8, Lemma 2.1]. We will also use some interpolation formulae for approximation spaces. Let 0≤θ≤1, 0<b≤∞, and −∞<α<∞. By ∗,∗θ,b, we denote the classical real interpolation functor [20, 31] and by ∗,∗θ,b;α the real interpolation functor involving logarithmic factor for ordered couples with integration over (0,1) (see, e.g., [30, 32, 33]). Note that ∗,∗θ,b=∗,∗θ,b;0.Lemma 20 ([30, Proposition 2.7]).
Suppose thatσ>0, 0<u, q≤∞, 0<θ<1, and −∞<γ<∞, then(42)
X,Xuσ,    0
θ
,
q
;
γ
≅
X
q
θσ,    γ
.The reiteration relation (see, e.g., [32]) leads to the following result.Corollary 21.
Suppose thatσ>0, 0<u,q≤∞, 0<θ<1, and −∞<γ,δ<∞, then(43)X,Xuσ,    δθ,q;γ≅Xqθσ,θδ+    γ.


## 5.2. Approximation Spaces in Lorentz–Zygmund Spaces

Everywhere below, we consider the following sequence of subsets:G0=0, Gn=Tn be the set of all trigonometric polynomials of degree at mostnn=1,2,    …. It builds an approximation family in all nontrivial Lorentz–Zygmund spaces. Now we will apply Corollaries 11–14 to the approximation spaces Lp,b;αuσ,    γ. First, we will investigate embeddings of these approximation spaces into Lorentz–Zygmund spaces. Next, we will investigate embeddings between different approximation spaces L∗,∗;∗∗σ,    ∗ for σ>0 and between different limiting approximation spaces L∗,∗;∗∗0,    ∗. From Corollary 11, we get the following result.Corollary 22.
Let0<q<p<∞, 0<b, c≤∞, −∞<α, β<∞. Then(44)
Lq,c;β
b
d1/    q−1/    p,α−    β
⊂
L
p
,
b
;
α
.Proof.
By Theorem16, given any f∈Lq,c;β1d/    q,−    β, we can find a representation f=∑n=0∞gn, (convergence in Lq,c;β), gn∈T2n such that(45)
∑
n
=
0
∞
2
n
d
/
q
1+    n
−
β
gn
q
,
c
;
β
≺
f
Lq,c;β
1
d/    q,−β
.Because allgn∈T2n⊂Lq,c;β⋂L∞, it is not hard to show that the series ∑n=0∞gn converges to f in L∞. Since L∞ is a Banach space, using Corollary 11 and (45), we derive that(46)f∞≤∑n=0∞gn∞≺∑n=0∞2nd/q1+    n−βgnq,c;β≺fLq,c;β1d/    q,−    β.Therefore,(47)Lq,c;β1d/    q,−    β⊂L∞.Since0<q<p<∞, we can define θ and γ by formulae(48)1p=1−θqandα=1−    θβ+γ.From Corollary21, we have(49)Lq,c;β,Lq,c;β1d/    q,−    βθ,b;γ≅Lq,c;βbd1/    q−1/    p,α−    β.Furthermore, it is known [32, Theorem 7] that(50)Lq,c;β,L∞θ,b;γ≅Lp,b;α.Using now (47) and [32, Theorem 2], we obtain (44).From Corollary12, we immediately get the following result.Corollary 23.
Let0<q<∞, 0<c≤∞, −∞<β<∞, 0<b≤∞, and α<−1/b. Then(51)
Lq,c;β
min
b,    1
d/    q,α+1/    b−β
⊂
L
∞
,
b
;
α
.Proof.
Consider the case1≤b≤∞. By Theorem 16, given any f∈Lq,c;β1d/    q,α+1/    b−    β, we can find a representation f=∑n=0∞gn, (convergence in Lq,c;β), gn∈T2n such that(52)
∑
n
=
0
∞
2
n
d
/
q
1+n
α+1/    b−    β
gn
q
,
c
;
β
≺
f
Lq,c;β
1
d/    q,α+1/    b−    β
.Because allgn∈T2n⊂Lq,c;β⋂L∞,b;α, it is not hard to show that the series ∑n=0∞gn converges to f in L∞,b;α. Since ∞,b,α∈Ω, using Corollary 12 and (52), we derive that(53)f∞,b;α≺∑n=0∞gn∞,b;α≺∑n=0∞2nd/q1+    nα+1/    b−    βgnq,c;β≺fLq,c;β11/    q,α+1/    b−    β.For the case0<b<1, by Theorem 16, given any f∈Lq,c;βbd/    q,α+1/    b−    β, we can find a representation f=∑n=0∞gn, (convergence in Lq,c;β), gn∈T2n such that(54)∑n=0∞2nd/q1+    nα+1/    b−    βgnq,c;βb1/b≺fLq,c;βbd/    q,α+1/b−    β.Due to Lemma2, Corollary 12 and (54), we obtain(55)f∞,b;α≺∑n=0∞gn∞,b;αb1/b≺∑n=0∞2nd/q1+    nα+1/    b−    βgnq,c;βb1/b≺fLq,c;βb1/    q,α+1/    b−    β.This completes the proof.Using some real interpolation technique formulae we obtain the following result, which complements Corollary23.Corollary 24.
Let0<q<∞, 0<c, b≤∞, −∞<β<∞, and α<−1/b. Then(56)
Lq,c;β
b
d/    q,α+1/minb.    1−    β
⊂
L
∞
,
b
;
α
.Proof.
Using Lemma20, we get(57)
Lq,c;β,Lq,c;β12d/    q,    0
1/    2
,
1
;
−
β
≅
Lq,c;β
1
d/    q,−β
.We considerLq,c;β,Lq,c;β1d/    q,−    β1,b;α. From (57), [32, Theorem 6 ,and Lemma 4], and Lemma 20, we have(58)Lq,c;β,Lq,c;β1d/    q,−    β1,b;α≅Lq,c;β,Lq,c;β,Lq,c;β12d/    q,    01/    2,1;−β1,b;α⊃Lq,c;β,Lq,c;β12d/    q,    01/    2,b;α+1/minb.    1−β≅Lq,c;βbd/q,α+1/minb.    1−    β.To complete the proof, we have only to use (47) and [32, Corollary 7]:(59)Lq,c;β,Lq,c;β1d/    q,−    β1,b;α⊂Lq,c;β,L∞1,b;α≅L∞,b;α.Remark 25.
Recall that the spacesX1d/    q,α+1/    b−    β and Xbd/    q,α+1−    β are formed by all those f∈X for which Enf belongs to lq/d,1;α+1/b−β and lq/d,b;α+1−β, respectively. However, for b>1, these sequence spaces are incomparable [24]. Note that for b≤1, formulae (51) and (56) coincide.Corollary 26.
Letp,b,α∈Ω∪Γ, p<∞, b≤c≤∞, and −∞<β<∞. If α+1/b>β+1/c then(60)
Lp,c;β
min
b,    1
0,α+1/    b−β−1/    c−1/minb.    1
⊂
L
p
,
b
;
α
.Proof.
Letp,b,α∈Ω. In this case b≥1. By Theorem 17, given any f∈Lp,c;β10,α+1/    b−β−1/    c−    1, we can find a representation f=∑n=0∞gn, (convergence in Lp,c;β), gn∈T22n such that(61)
∑
n
=
0
∞
2
n
α+1/    b−β−1/    c
gn
p
,
c
;
β
≺
f
Lp,c;β
1
0
,
α
+
1
/  b
−
β
−
1
/  c
−  1
.Because allgn∈T22n⊂Lp,b;α⋂Lp,c;β, it is not hard to show that the series ∑n=0∞gn converges to f in Lp,b;α. Sincep,b,α∈Ω, using Corollary 13(i), we derive that(62)fp,b;α≺∑n=0∞gnp,b;α≺∑n=0∞2nα+1/  b−β−1/  cgnp,c;β≺fLp,c;β10,α+1/  b−β−1/  c−  1.Let nowp,b,α∈Γ. In this case, b<1. By Theorem 17, given any f∈Lp,c;βb0,α−β−1/  c, we can find a representation f=∑n=0∞gn, (convergence in Lp,c;β), gn∈T22n such that(63)∑n=0∞2nα+1/  b−β−1/  cgnp,c;βb1/b≺fLq,c;βb0,α−β−1/c.Due to Lemma2 and Corollary 13(i), we obtain(64)fp,b;α≺∑n=0∞gnp,b;αb1/b≺∑n=0∞2nα+1/b−β−1/cgnq,c;βb1/b≺fLq,c;βb0,α−β−1/c.This completes the proof.The following result can be proved analogously based on Corollary14.Corollary 27.
Letp,b,α∈Ω∪Γ, p<∞, 0<c≤b, and −∞<β<α<∞. Then(65)
L
p
,
c
;
β
min
b
,
1
0
,
α
−
β
−
1
/
m
i
n
b
.
1
⊂
L
p
,
b
;
α
.The next four corollaries investigate embeddings between different approximation spacesL∗,∗;∗∗σ,∗ for σ>0. They follow immediately from Corollaries 11, 12, 13(i), and 14 and from Lemma 19.Corollary 28.
Let0<q<∞, 0<c≤∞, −∞<β<∞. For the triple p,b,α, we assume that either q<p<∞, 0<b≤∞, −∞<α<∞, or p=b=∞, α=0. Then, for σ>0, 0<u≤∞, and −∞<γ<∞, we have(66)
L
q
,
c
;
β
u
σ
+
d
1
/
q
−
1
/
p
,
γ
+
α
−
β
⊂
L
p
,
b
;
α
u
σ
,
γ
.Corollary 29.
Let0<q<∞, 0<c,b≤∞, −∞<β<∞, and α<−1/b. Then, for σ>0, 0<u≤∞, and −∞<γ<∞, we have(67)
L
q
,
c
;
β
u
σ
+
d
/
q
,
γ
+
α
+
1
/
b
−
β
⊂
L
∞
,
b
;
α
u
σ
,
γ
.Corollary 30.
Let0<p<∞, 0<b<c≤∞, −∞<α, β<∞, and α+1/b>β+1/c. Then, for σ>0, 0<u≤∞, and −∞<γ<∞, we have(68)
L
p
,
c
;
β
u
σ
,
γ
+
α
+
1
/
b
−
β
−
1/c
⊂
L
p
,
b
;
α
u
σ
,
γ
.Corollary 31.
Let0<p<∞, 0<c≤b≤∞, and −∞<β<α<∞. Then, for σ>0, 0<u≤∞, and −∞<γ<∞, we have(69)
L
p
,
c
;
β
u
σ
,
γ
+
α
−
β
⊂
L
p
,
b
;
α
u
σ
,
γ
.The two next corollaries deal with embeddings between different limiting approximation spacesL∗,∗;∗∗0,∗. They follow immediately from Lemma 18 and Corollaries 13(i) and 14, respectively.Corollary 32.
Let0<p<∞, 0<b<c≤∞, −∞<α, β<∞, and α+1/b>β+1/c. Then, for 0<u≤∞ and γ>−1/u, we have(70)
L
p
,
c
;
β
u
0
,
γ
+
α
+
1
/
b
−
β
−
1
/
c
⊂
L
p
,
b
;
α
u
0
,
γ
.Corollary 33.
Let0<p<∞, 0<c≤b≤∞, and −∞<β<α<∞. Then, for 0<u≤∞ and γ>−1/u, we have(71)
L
p
,
c
;
β
u
0
,
γ
+
α
−
β
⊂
L
p
,
b
;
α
u
0
,
γ
.


## 5.3. Embeddings of Besov SpacesBp,uσ,γ

Here we give some applications of previous results to embeddings of Besov spaces into Lorentz–Zygmund spaces and between Besov spaces.There are several definitions of Besov spaces. The Besov spaceBp,uσ,γ≡Bp,uσ,γTdσ≥0,0<p,u≤∞,−∞<γ<∞ is based on Lp and has classical smoothness σ and additional logarithmic smoothness with exponent γ. It is formed by all those f∈Lp such that(72)fBp,uσ,γ=fp+∫01t−σlγtωkf,tpudtt1/u<∞,with an obvious modification whenu=∞. Here ωkf,tp k>σ,k∈N is the modulus of smoothness of order k with respect to the quasi-norm on Lp. It makes sense to consider the spaces Bp,u0,γ with zero classical smoothness only for γ≥−1/u. All we need for our application is the characterization of the Besov spaces by approximation. An important result in approximation theory states that(73)Bp,uσ,γ≅Lpuσ,γ.For details see, e.g., [8, 30] and the references given there. Using this characterization with Corollaries 22–24 and 26–28, we obtain the following embeddings. Due to similarity, we will give only the proof of  Corollary 37.Corollary 34 (Cf. [8, Theorem 5.2]).
Let0<q<p<∞, 0<b≤∞, and −∞<α<∞. Then(74)
B
q
,
b
d
1
/
q
−
1
/
p
,
α
⊂
L
p
,
b
;
α
.Corollary 35.
Let0<q<∞, 0<b≤∞, and α<−1/b. Then(75)
B
q
,
min
b
,
1
d
/
q
,
α
+
1
/
b
⊂
L
∞
,
b
;
α
.Corollary 36 (Cf. [8, Theorem 5.3]).
Let 0<q<∞, 0<b≤∞, and α <−1/b. Then(76)
B
q
,
b
d
/
q
,
α
+
1
/
m
i
n
b
,
1
⊂
L
∞
,
b
;
α
.Corollary 37 (Cf. [8, Theorems 5.5, 5.6, and 5.7]).
Letp,b,α∈Ω∪Γ, b≤p<∞, and α>1/p−1/b. Then(77)
B
p
,
min
b
,
1
0
,
α
+
1
/
b
−
1
/
p
−
1
/
m
i
n
b
,
1
⊂
L
p
,
b
;
α
.Proof.
Using the characterization (73) and Corollary 26, we obtain(78)
B
p
,
min
b
,
1
0
,
α
+
1
/
b
−
1
/
p
−
1
/
m
i
n
b
,
1
≅
L
p
min
b
,
1
0
,
α
+
1
/
b
−
1
/
p
−
1
/
m
i
n
b
,
1
⊂
L
p
,
b
;
α
.Corollary 38.
Letα>0. In addition, let 0<p=b≤1, or 1<p<∞ and p≤b≤∞. Then(79)
B
p
,
min
b
,
1
0
,
α
−
1
/
m
i
n
b
,
1
⊂
L
p
,
b
;
α
.Corollary 39 (Cf. [34, Corollary 5.3 (i)] and [31, Theorem 2.8.1]).
Let0<q<p≤∞. Then, for σ>0, 0<u≤∞, and −∞<γ<∞, we have(80)
B
q
,
u
σ
+
d
1
/
q
−
1
/
p
,
γ
⊂
B
p
,
u
σ
,
γ
.


## 5.4. Embeddings of Besov–Type SpacesBp,b;α,u0,γ

In [7], the Besov–type spaces Bp,b;α,u0,γ≡Bp,b;α,u0,γTd (with zero classical smoothness) based on Lp,b;α are introduced, so that Bp,p;0,u0,γ=Bp,u0,γ. The space Bp,b;α,u0,γ0<u<∞,1<p<∞,1≤b≤∞,−∞<α<∞,γ>−1/u. is formed by all those f∈Lp,b;α such that(81)fBp,b;α,u0,γ=fp,b;α+∫01lγtωkf,tp,b;αudtt1/u<∞,with an obvious modification whenu=∞. Here ωkf,tp,b;αk∈N is the modulus of smoothness of order k with respect to the quasi-norm on Lp,b;α. The definition is independent of k. The spaces Bp,b;α,u0,γ have the following characterization by approximation [7, (3.7)]:(82)Bp,b;α,u0,γ≅Lp,b;αu0,γ.Using this characterization and Corollaries26, 27, 32, and 33, we obtain the following embeddings.Corollary 40 (Cf. [7, (3.9)]).
Let1<p<∞, 1≤c≤∞, 0<b≤c, and −∞<α, β<∞. If α+1/b>β+1/c, then(83)
B
p
,
c
;
β
,
min
b
,
1
0
,
α
+
1
/
b
−
β
−
1
/
c
−
1
/
m
i
n
b
,
1
⊂
L
p
,
b
;
α
.Corollary 41 (Cf. [7, (3.8)]).
Let1<p<∞, 1≤c≤b≤∞, and −∞<β<α<∞. Then(84)
B
p
,
c
;
β
,
1
0
,
α
−
β
−
1
⊂
L
p
,
b
;
α
.Corollary 42.
Let1<p<∞, 1≤b≤c≤∞, −∞<α, β<∞, andα+1/b>β+1/c. Then, for 0<u≤∞ and γ>−1/u, we have(85)
B
p
,
c
;
β
,
u
0
,
γ
+
α
+
1
/
b
−
β
−
1
/
c
⊂
B
p
,
b
;
α
,
u
0
,
γ
.Corollary 43.
Let1<p<∞, 1≤c≤b≤∞, and −∞<β<α<∞. Then, for 0<u≤∞ and γ>−1/u, we have(86)
B
p
,
c
;
β
,
u
0
,
γ
+
α
−
β
⊂
B
p
,
b
;
α
,
u
0
,
γ
.Remark 44.
Some other limiting embeddings between Besov spaces based on generalized Lorentz–Zygmund spaces with two iterations of logarithm were obtained in [26, Theorem 2].



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