We prove optimal embeddings of the generalized Sobolev spaces WkE, where E is a rearrangement invariant function space, into the generalized Hölder-Zygmund space 𝒞H generated by a function space H.

1. Introduction

The classical Sobolev space Wpk, 1≤p<∞, consists of all locally integrable functions f, defined on Rn, n≥1, with the Lebesgue measure, such that the following norm is finite: ∥f∥Wpk=∑|α|≤k∥Dαf∥p, where ∥f∥p stands for the Lp-norm. In investigating the regularity of the function f∈Wpk, we may assume, without any loss of generality, that f∈L1(Ω), Ω is a domain in Rn, and f is zero outside Ω. For simplicity we suppose that the Lebesgue measure of Ω equals one and that the origin lies in Ω. It is well known that in the supercritical case k>n/p,
(1)Wpk↪𝒞k-n/p,k>np,
where 𝒞γ,γ>0, is the Hölder-Zygmund space (see [1]). In the critical case k=n/p the function f∈Wpk may not be even continuous. The result (1) is not optimal. We prove that the optimal one is obtained if in (1) Lp is replaced by the Marcinkiewicz space Lp,∞. In this paper we prove similar optimal results, when Lp,∞ is replaced by a more general rearrangement invariant space E. The Sobolev space WkE consists of all f∈W1k with a finite quasinorm ∥f∥WkE=∑|α|≤k∥Dαf∥E. More precisely, we consider quasinormed rearrangement invariant spaces E, consisting of functions f∈L1(Ω), such that the quasinorm ∥f∥E≈ρE(f*)<∞, where ρE is a monotone quasinorm, defined on M+ with values in [0,∞] and M+ is the cone of all locally integrable functions g≥0 on (0,1) with the Lebesgue measure. Monotonicity means that g1≤g2 implies ρE(g1)≤ρE(g2). We suppose that L∞(Ω)↪E↪L1(Ω), which means continuous embeddings. Here f* is the decreasing rearrangement of f, given by f*(t)=inf{λ>0:μf(λ)≤t},t>0, and μf is the distribution function of f, defined by μf(λ)=|{x∈Rn:|f(x)|>λ}|n,|·|n denoting Lebesgue n-measure. Note that f*(t)=0 for t>1. Finally, f**(t):=(1/t)∫0tf*(s)ds.

Let αE, βE be the Boyd indices of E. For example, if E=Lp, then αE=βE=1/p and the condition k/n≥1/p means k/n≥αE>0. Note that for k>n this is always satisfied. For these reasons we suppose that for the general E, 0<αE=βE≤1 and the case min(k,n)/n>αE is called super-critical, while the case min(k,n)/n=αE-critical. In the super-critical case the function f∈WkE is always continuous, while the spaces in the critical case αE=k/n,k<n, can be divided into two subclasses: in the first subclass the functions f∈WkE may not be continuous—then the target space is rearrangement invariant, while these functions in the second subclass are continuous and the target space is the generalized Hölder-Zygmund space 𝒞H (see Definition 1). The separating space for these two subclasses is given by the Lorentz space Ln/k,1, k<n. If k≥n; then WkE consists of continuous functions (see the classical result of Stein [2]).

The main goal of this paper is to prove optimal embeddings of the Sobolev space WkE into the generalized Hölder-Zygmund space 𝒞H. First we prove that this embedding for k≤n is equivalent to the continuity of the operator Rkg(t)=∫0tuk/n-1g(u)du. The case k>n is reduced to the continuity of Rn by using the lifting principle ([1]). Moreover, if, for example, k≤n, then in the super-critical case, we can replace Rk by the operator of multiplication tk/ng(t). This implies a very simple characterization of both optimal target space H and optimal domain space E. Namely, the quasinorm in the optimal target space H(E) is given by ρE(t-k/ng(t)) and the quasinorm in the optimal domain space E(H) is given by ρH(tk/ng(t)). Note that we do not require ρE to be rearrangement invariant. In the critical case, the formula for the optimal target space is more complicated. In some cases it can be simplified. To this end, we apply the Σq-method of extrapolation ([3]) from the super-critical case. As a byproduct, we also characterize the embedding WkE↪Cj,j<k, where Cj consists of all functions with bounded and uniformly continuous derivatives up to order j. Namely, this is equivalent to the embedding E↪Ln/(k-j),1 if k≤n. The embedding Wn+jE↪Cj is always true since WnE↪W1n↪C0.

The problem of the optimal target rearrangement invariant space for potential type operators is considered in [4] by using Lp-capacities. The problem of the mapping properties of the Riesz potential in optimal couples of rearrangement invariant spaces is treated in [5–7]. The optimal embeddings of generalized Sobolev type spaces into rearrangement invariant spaces are characterized in several papers [5, 8–21]. The characterization of the continuous embedding of the generalized Bessel potential spaces into the generalized Hölder-Zygmund spaces 𝒞H, when H is a weighted Lebesgue space, is given in [22]. The optimal embeddings of Calderón spaces into the generalized Hölder-Zygmund spaces are characterized in [23].

The plan of the paper is as follows. In Section 2 we provide some basic definitions and known results. In Section 3 we characterize the embedding WkE↪𝒞H. The optimal quasinorms are constructed in Section 4.

2. Preliminaries

We use the notations a1≲a2 or a2≳a1 for nonnegative functions or functionals to mean that the quotient a1/a2 is bounded; also, a1≈a2 means that a1≲a2 and a1≳a2. We say that a1 is equivalent to a2 if a1≈a2.

Let E be a quasinormed rearrangement invariant space as in the Introduction. There is an equivalent quasinorm ρp≈ρE that satisfies the triangle inequality ρpp(g1+g2)≤ρpp(g1)+ρpp(g2) for some p∈(0,1] that depends only on the space E (see [24]). We say that the quasinorm ρE satisfies Minkowski’s inequality if for the equivalent quasinorm ρp,
(2)ρpp(∑gj)≲∑ρpp(gj),gj∈M+.
Usually we apply this inequality for functions gj∈M+ with some kind of monotonicity.

Recall the definition of the lower and upper Boyd indices αE and βE. Let gu(t)=g(t/u) if t<u and gu(t)=0 if t≥u, where 0<t<1,g∈M+, and let
(3)hE(u)=sup{ρE(gu*)ρE(g*):g∈M+},gu(t):=g(tu),u>0,
be the dilation function generated by ρE. Suppose that it is finite. Then
(4)αE:=sup0<t<1loghE(t)logt,βE:=inf1<t<∞loghE(t)logt.
The function hE is submultiplicative, increasing, hE(1)=1,hE(u)hE(1/u)≥1; hence 0≤αE≤βE. We suppose that 0<αE=βE≤1.

If βE<1 we have by using Minkowski’s inequality that ρE(f*)≈ρE(f**). In particular, ∥f∥E≈ρE(f**) if βE<1. For example, consider the Gamma spaces E=Γq(w), 0<q≤∞, w—positive weight, that is, a positive function from M+, with a quasinorm ∥f∥Γq(w):=ρE(f*),ρE(g):=ρw,q(∫01g(tu)du), where
(5)ρw,q(g):=(∫01[g(t)w(t)]qdtt)1/q,g∈M+,(∫01wq(t)dtt)1/q<∞.
Then L∞(Ω)↪Γq(w)↪L1(Ω). If w(t)=t1/p,1<p<∞, we write as usual Lp,q instead of Γq(t1/p). Consider also the classical Lorentz spaces Λq(w), 0<q≤∞; f∈Λq(w) if ∥f∥Λwq:=ρw,q(f*)<∞, w(2t)≈w(t). We suppose that L∞(Ω)↪Λq(w)↪L1(Ω).

Note that if E=Λq(tαw),0<α≤1, where w is slowly varying, then αE=βE=α. Recall that w∈M+ is slowly varying if for all ɛ>0 the function tɛw(t) is equivalent to an increasing function and the function t-ɛw(t) is equivalent to a decreasing function.

In order to introduce the Hölder-Zygmund class of spaces, we denote the modulus of continuity of order k by
(6)ωk(t,f)=sup|h|≤tsupx∈Rn|Δhkf(x)|,
where Δhkf are the usual iterated differences of f. When k=1 we simply write ω(t,f).

Let H be a quasinormed space of locally integrable functions on the interval (0,1) with the Lebesgue measure, continuously embedded in L∞(0,1) and ∥g∥H=ρH(|g|), where ρH is a monotone quasinorm on M+ which satisfies Minkowski’s inequality. The dilation function generated by ρH is given by
(7)hH(u)=sup{ρH(g1/u)ρH(g):g∈Mm},
where
(8)Mm:={t-m/ng∈M+:g(t)isincreasing,t-m/ng(t)isdecreasingandg(2t)≈g(t)}.
The choice of the space Mm is motivated by the fact that ωm(t1/n,f) is equivalent to a function g∈Mm. The function hH(u) is submultiplicative, increasing and u-m/nhH(u) is decreasing and hH(1)=1,hH(u)hH(1/u)≥1. Therefore the Boyd indices of H are well defined
(9)αH=sup0<t<1loghH(t)logt,βH=inf1<t<∞loghH(t)logt,
and they satisfy αH≤βH≤m/n. In what follows, we suppose that 0≤αH=βH≤m/n.

For example, let H=L*q(b(t)t-γ/n). Here 0≤γ≤m and b is a slowly varying function, and L*q(w), or simply L*q if w=1, is the weighted Lebesgue space with a quasinorm ∥g∥L*q(w)=ρw,q(|g|), where ρw,q is given by (5). It turns out that αH=βH=γ/n.

Definition 1.

Let j=0,1,… and let Cj stand for the space of all functions f, defined on Rn, that have bounded and uniformly continuous derivatives up to the order j, normed by ∥f∥Cj=sup∑l=0j|Plf(x)|, where Plf(x)=∑|ν|=lDνf(x).

If j/n<αH<(j+1)/n for j≥1 or 0≤αH<1/n for j=0, then 𝒞H is formed by all functions f in Cj having a finite quasinorm
(10)∥f∥𝒞H=∥f∥Cj+ρH(χ(0,1)(t)tj/nω(t1/n,Pjf)).

If αH=(j+1)/n, then 𝒞H consists of all functions f in Cj having a finite quasinorm
(11)∥f∥𝒞H=∥f∥Cj+ρH(χ(0,1)(t)tj/nω2(t1/n,Pjf)).

Here χ(a,b),0<a<b<∞, is the characteristic function of the interval (a,b).

In particular, if H=L∞(t-γ/n),γ>0, then 𝒞H coincides with the usual Hölder-Zygmund space 𝒞γ (see [1]). Also, if H=L∞, then 𝒞H=C0.

Let 0≤αH=βH<m/n. If ρH(χ(0,1)(t)tα)<∞ for α>αH, then for all such m,
(12)∥f∥𝒞H≈∥f∥C0+ρH(χ(0,1)(t)ωm(t1/n,f)).

Note that if ρH(χ(0,1)(t)tm/n)<∞, then 𝒞H is a K-interpolation space for the couple (C0,Cm), namely, 𝒞H=(C0,Cm)H1, where ρH1(g)=ρH(g(tm/n)). In particular, 𝒞L*1(t-j)↪Cj↪𝒞L∞(t-j). By χ(a,b), 0≤a<b≤∞ we denote the characteristic function of the interval (a,b).

Recall some basic definitions from the theory of interpolation spaces [21]. Let (A0,A1) be a couple of two quasinormed spaces, such that both are continuously embedded in some quasinormed space and let
(13)K(t,f)=K(t,f;A0,A1)=inff=f0+f1{∥f0∥A0+t∥f1∥A1},f∈A0+A1,
be the K-functional of Peetre. By definition, the K-interpolation space AΦ=(A0,A1)Φ has a quasinorm ∥f∥AΦ=∥K(t,f)∥Φ, where Φ is a quasinormed function space with a monotone quasinorm on (0,∞) with the Lebesgue measure and such that min{1,t}∈Φ. Then A0∩A1↪AΦ↪A0+A1. If
(14)∥g∥Φ=(∫0∞[w(t)t-θg(t)]qdtt)1/q,ii0≤θ≤1,0<q≤∞,w∈ℳ+,
we write (A0,A1)wt-θ,q instead of (A0,A1)Φ. Also, if w=1 then we write (A0,A1)θ,q. By definition,
(15)∥f∥A0∩A1=∥f∥A0+∥f∥A1,∥f∥A0+A1=K(1,f;A0,A1).

Theorem 3 (lifting principle).

Let αH>0 and let ρHj(g):=ρH(t-j/ng(t)),j≥1. Then
(16)∥f∥𝒞Hj≈∥𝒟jf∥𝒞H,𝒟j=∑|α|≤jDα.

Proof.

Let m/n>αH+j/n. Since
(17)ωm(t1/n,f)≲tj/nωm-j(t1/n,Pjf),
it follows
(18)ρHj(ωm(t1/n,f))≲ρH(ωm-j(t1/n,𝒟jf)).
Hence
(19)∥f∥𝒞Hj≲∥𝒟jf∥𝒞H.
To prove the reverse, we use the formula (see [25, page 342])
(20)ωm(t1/n,Dαf)≲∫0tu-|α|/nωm(u1/n,f)duu.
Then applying Minkowski’s inequality and αHj=βHj=j/n+αH>j/n, we get
(21)ρH(χ(0,1)(t)ωm(t1/n,Dαf))≲ρHj(χ(0,1)(t)ωm(t1/n,f)),iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii|α|≤j.
Since 𝒞Hj↪Cj, we derive
(22)∥𝒟jf∥𝒞H≲∥f∥𝒞Hj.

Remark 4.

The relation (19) is always true. But if αH=0 then the reverse might not be true. For example, let n=1, H=L∞, j=1, and f(x)=xln|2x| if |x|≤1/2 and f(x)=0 if |x|>1/2. Then f∉C1, but ω2(t,f)=O(t).

It will be convenient to introduce the classes of the domain and target quasinorms, where the optimality is investigated. Let Nd consist of all domain quasinorms ρE that are monotone, satisfying Minkowski’s inequality, 0<αE=βE≤min(k,n)/n, ρE(χ(0,1)t-α)<∞ if α<αE and the condition (30) below for k≤n, and E↪L1 for k>n. Let Nt consist of all target quasinorms ρH that are monotone, satisfy Minkowski’s inequality, 0≤αH=βH<min(k,n)/n, ρH(χ(0,1)(t)tα)<∞ if α>αH and
(23)sup0<t<1g(t)≲ρH(g).

We use the following definitions.

Definition 5 (admissible couple).

We say that the couple ρE∈Nd,ρH∈Nt is admissible if WkE↪𝒞H when k≤n, and if 𝒟j(Wn+jE)↪𝒞H for j≥1. Moreover, ρE(E) is called domain quasinorm (domain space), and ρH(H) is called target quasinorm (target space).

Definition 6 (optimal target quasi-norm).

Given the domain quasinorm ρE∈Nd, the optimal target quasi-norm, denoted by ρH(E), is the strongest target quasi-norm; that is,
(24)ρH(g)≲ρH(E)(g),g∈Mmin(k,n)
for any target quasinorm ρH∈Nt such that the couple ρE, ρH is admissible. Since 𝒞H(E)↪𝒞H, we call 𝒞H(E) the optimal Hölder-Zygmund space.

Definition 7 (optimal domain quasi-norm).

Given the target quasinorm ρH∈Nt, the optimal domain quasi-norm, denoted by ρE(H), is the weakest domain quasi-norm; that is,
(25)ρE(H)(f*)≲ρE(f*),f∈L1(Ω),
for any domain quasinorm ρE∈Nd such that the couple ρE, ρH is admissible.

Definition 8 (optimal couple).

The admissible couple ρE∈Nd, ρH∈Nt is said to be optimal if both ρE and ρH are optimal.

3. Admissible Couples

Here we give a characterization of all admissible couples ρE∈Nd, ρH∈Nt. We start with the main estimate. For k=1, see also [26].

Theorem 9.

Let f∈W1k and k≤n. Then
(26)ωk(t1/n,f)≲∫0tuk/n-1(𝒟kf)*(u)du,
where 𝒟kf:=∑|α|≤kDαf.

Proof.

We use the embedding
(27)WkLn/k,1↪L∞,
whence
(28)ωk(t1/n,f)≲K(tk/n,f;L∞,W∞k)≲K(tk/n,f;WkLn/k,1,W∞k).
Then (26) follows from the basic formula [25, page 360]
(29)K(t,f;W1k,W∞k)≈∫0t(𝒟kf)*(s)ds
and the reiteration formula of Holmstedt for the K-functional [25, page 310].

Now we discuss the embedding WkE↪C0. For k=1 more general results are proved in [27, Chapter 4].

Theorem 10.

A necessary and sufficient condition for the embedding WkE↪C0,k≤n, is the following one
(30)∫01tk/n-1g(t)dt≲ρE(g),g∈D1,
where
(31)D1={g∈M+:gisdecreasingfunction}.

Proof.

The conditions (30) and (26), (27) imply the embedding WkE↪L∞ and limt→0ωk(t1/n,f)=0 if f∈WkE. On the other hand, by Marchaud’s inequality (see [25], Theorem 5.4.4), we have
(32)ω(t1/n,f)≲t1/n∫t∞u-1/nωk(u1/n,f)duu.
It is easy to see that limt→0ω(t1/n,f)=0. Thus WkE↪C0.

Before proving the reverse, note that (30) is always satisfied if k/n>αE. Since
(33)ρE(χ(0,1))g(u)≤ρE(g(tu))≤hE(1u)ρE(g),
we have
(34)g(u)≲hE(1u)ρE(g),g∈D1.
Hence for 0<ɛ<k/n-αE,
(35)∫01uk/n-1g(u)du≲ρE(g)∫01uk/n-αE-ɛduu≲ρE(g).
It remains to prove that if WkE↪C0, k≤n, then (30) is true for αE=k/n. To this end we choose a test function f as follows:
(36)f(x)=∫01g(u)uk/nψ(xu-1/n)duu,
where g∈D1 and ψ is in C0∞ such that ψ(x)=1 if |x|≤2-1c-1/n and ψ(x)=0 if |x|≥c-1/n. Then f(x)=0 for |x|>c-1/n, whence f*(t)=0 for t>1 for appropriate c and |f(x)|≤∫c|x|n1g(u)uk/ndu/u, whence |f(x)|≲g(c|x|n); therefore f*(t)≲g(t). This implies
(37)ρE(f*)≲ρE(g).
Analogously, since |Dαf(x)|≲∫c|x|n2nc|x|ng(u)du/u, 1≤|α|≤k, we have
(38)∥f∥WkE≲ρE(g).
Also f(0)=ψ(0)/n∫01uk/n-1g(u)du≲∥f∥WkE≲ρE(g). Thus (30) is proved.

Remark 11.

Similar arguments show that WkE↪Cj,j<k≤n, if and only if E↪Ln/(k-j),1.

Theorem 12.

The couple ρE∈Nd, ρH∈Nt is admissible if and only if
(39)ρH(χ(0,1)Rmin(k,n)g)≲ρE(g),g∈D1,
where
(40)Rkg(t):=∫0tuk/n-1g(u)du.

Proof.

Step 1 (sufficiency of (39)). If k≤n then it is clear that the embedding WkE↪𝒞H follows from (39), (26), and (27). Let now k=n+j, j≥1. Then (39) for k=n implies WnE↪𝒞H. Hence 𝒟j(Wn+jE)↪𝒞H for j≥1.

Step 2 (necessity of (39) when k≤n). Now we prove that the embedding WkE↪𝒞H implies (39) for k≤n. To this end we choose the test function f as in (36).

Let |h|=Ct1/n. We split f=f1t+f2t, f1t(x)=∫0tuk/ng(u)ψ(xu-1/n)du/u, f2t(x)=∫t1uk/ng(u)ψ(xu-1/n)du/u, 0<t<1. First we prove that for some large C>0,
(41)ωm(Ct1/n,f1t)≥ψ(0)Rkg(t),0<t<1.
Indeed, we have ωm(Ct1/n,f1t)≥|(Δhmf1t)(0)| and ψ(jCt1/nu-1/n)=0 for u<t,1≤j≤m if C>0 is large enough. Hence (41) follows. Further,
(42)ωm(t1/n,f)≥ωm(t1/n,f1t)-ωm(t1/n,f2t),0<t<1.
Since
(43)ωm(t1/n,f2t)≲tm/n∥Pmf2t∥L∞≲tm/n∫t1u(k-m)/ng(u)duu
and Rkg(t)≳tk/ng(t) for g∈D1, we get
(44)ωm(t1/n,f2t)≲tm/n∫t1u-m/nRkg(u)duu.
Therefore
(45)Rkg(t)≤c1ωm(t1/n,f)+ctm/n∫t1u-m/nRkg(u)duu,Rkg(t)≤c1ωm(t1n,f)+===ctm/n∫t10<t<1,(46)Rkg(t)≤c1ωm(t1/n,f)+ctm/n∫t1u-m/ng(u)duu,Rkg(t)≤c1ωm(t1n,f)+==ctm/n∫t10<t<1.
To solve the integral inequality (45) for p(t):=χ(0,1)(t)t-m/nRkg(t), we set q(t)=c1t-m/nωm(t1/n,f) and rewrite it as p(t)≤q(t)+c∫t1p(u)du/u. If r(t)=∫t1p(u)du/u, then we get the differential inequality 0≤tr′(t)+cr(t)+q(t). If r(t)=t-cv(t), then 0≤v′(t)+tc-1q(t), whence v(t)≤∫t1uc-1q(u)du. Therefore
(47)χ(0,1/2)(t)Rkg(t)≲tm/n-c∫t1uc-m/nωm(u1/n,f)duu.
Hence by using Minkowski’s inequality and choosing m large enough, we obtain
(48)ρH(χ(0,1/2)Rkg)≲ρH(ωm(t1/n,f)).
On the other hand, from (46), it follows that
(49)ρH(χ(1/2,1)Rkg)≤ρH(ωm(t1/n,f))+∫01g(u)du.
Hence, using also (30), we get
(50)ρH(χ(0,1)Rkg)≲ρH(ωm(t1/n,f))+ρE(g).
Thus, if WkE↪𝒞H is given, then (50), (38) imply (39).

Step 3 (necessity of (39) when k=n+j, j≥1). Now we prove that the embedding 𝒟j(Wn+jE)↪𝒞H, j≥1, implies (39) for k=n. To this end we choose the test function f in the form
(51)f(x)=∫01g(u)uj/nψ(xu-1/n)du,
where g∈D1 and ψ is the same as in (36). Note that 𝒟jψ(0)=1. Then as before we get (37), and
(52)|Dαf(x)|≲g(c|x|n)
for 1≤|α|≤n+j. Hence
(53)∥f∥Wn+jE≲ρE(g).

On the other hand, using the arguments from Step 2 but for the function 𝒟jf, j≥1, we obtain
(54)ρH(χ(0,1)Rng)≲ρH(ωm(t1/n,𝒟jf))+ρE(g),j≥1.
Thus, if 𝒟j(Wn+jE)↪𝒞H, j≥1 is given, then (54), (53) imply (39) for k=n.

Theorem 13.

Let αH>0. Then the couple ρE∈Nd, ρH∈Nt is admissible for k>n if and only if
(55)Wn+jE↪𝒞Hj,ρHj(g):=ρH(t-j/ng(t)),j≥1.
Moreover, (55) is equivalent to
(56)ρHj(Rn+jg)≲ρE(g),g∈D1.

Proof.

Let ρE∈Nd,ρH∈Nt be an admissible couple for k>n. Then 𝒟j(Wn+jE)↪𝒞H. Since
(57)ωm(t1/n,𝒟jf)≲∫0t(𝒟n+jf)*(u)du,m≥n,
we have, by applying (39) for k=n,
(58)ρH(ωm(t1/n,𝒟jf))≲ρE((𝒟n+jf)*).
Using also (16) and Wn+jE↪Cj, we obtain Wn+jE↪𝒞Hj. Further, as in the proof of the previous theorem, this embedding implies (56). Finally,
(59)ρHj(Rn+jg)≈ρH(Rng),αH>0.
Indeed, ρH(Rng)=ρHj(tj/nRng(t)) and
(60)ρHj(tj/nRng(t))≲ρHj(tj/n∫0tu(n+j)/n(Rn+jg)(u)du).
Applying Minkowski’s inequality, we get, since αHj>j/n,
(61)ρH(Rng)≲ρHj(Rn+jg).
For the reverse, we notice that Rn+jg(t)≲tj/nRng(t). Then
(62)ρHj(Rn+jg)≲ρH(Rng).

4. Optimal Quasinorms

Here we give a characterization of the optimal domain and optimal target quasinorms.

4.1. Optimal Domain Quasinorms

We can construct an optimal domain quasinorm ρE(H) by Theorem 9 as follows.

Definition 14 (construction of an optimal domain quasi-norm).

For a given target quasinorm ρH∈Nt, we set
(63)ρE(H)(g):=ρH(Rmin(k,n)g),g∈M+.

Note that Rmin(k,n)(gu)=umin(k,n)/n(Rmin(k,n)g~)1/u and Sg∈L if g∈D1. Hence αE(H)=βE(H)=min(k,n)/n-αH.

Theorem 15.

The quasinorm ρE(H) belongs to Nd, the couple ρE(H), ρH is admissible, and the domain quasinorm ρE(H) is optimal. Moreover, the target quasinorm ρH is also optimal and
(64)ρE(H)(g)≈ρH(tmin(k,n)/ng),g∈D1ifαH>0.

Proof.

It is easy to check that ρE(H)∈Nd. Further, the couple ρE(H), ρH is admissible since ρH(Rmin(k,n)g)=ρE(H)(g), g∈D1. Moreover, ρE(H) is optimal, since for any admissible couple ρE1∈Nd,ρH, we have ρH(χ(0,1)Rmin(k,n)g)≲ρE1(g), where g∈D1. Therefore for f∈L1(Ω),
(65)ρE(H)(f*)=ρH(Rmin(k,n)f*)≲ρH(χ(0,1)Rmin(k,n)f*)+ρE1(f*)≲ρE1(f*).
To prove that ρH is also optimal, let ρE(H),ρH1∈Nt be an arbitrary admissible couple. Then
(66)ρH1(χ(0,1)Rmin(k,n)g)≲ρE(H)(g),g∈D1.
It is enough to check that
(67)ρH1(χ(0,1)g)≲ρH(g),g∈Mmin(k,n).
Let k<n. (The case k≥n is easier.) We introduce a better function g1(t)=g(tn/k). Then g1 is quasiconcave; therefore g1(t)≈∫0th1(u)du and h1∈D1. By changing the variables, we get g≈Rkh with h∈D1. Then
(68)ρH1(χ(0,1)g)≲ρH1(χ(0,1)Rkh)≲ρE(H)(h)≈ρH(Rkh)≈ρH(g).
Thus (67) is proved. To prove the equivalence (64), we use tmin(k,n)/ng(t)≲Rmin(k,n)g(t),g∈D1 and Minkowski’s inequality as follows:
(69)ρHp(Rmin(k,n)g)≲∑j=-∞0hHp(2j)ρHp(tmin(k,n)/ng(t)),iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiig∈D1,αH>0,
whence ρE(H)(g)≲ρH(tmin(k,n)/ng(t)),g∈D1.

Remark 16.

Let k≥n and χ(0,1)(t)t∈H. Then the couple E=(L1,L∞)H, H is optimal.

Example 17.

Consider the space H=L*1(v), where ρH(g)=∫0∞v(t)g(t)dt/t and ρH∈Nt. Using Theorem 15, we can construct an optimal domain E, where
(70)ρE(g)=ρH(Rmin(k,n)g)=∫0∞tmin(k,n)/nw(t)g(t)dtt
and w(t)=∫t∞v(u)du/u. Hence E=Λ1(tmin(k,n)/nw), and this couple is optimal. Also αE=βE=min(k,n)/n if v is slowly varying. Note that if k≥n, then E=Λ1(tw)=Γ1(tv)=(L1,L∞)H.

Example 18.

Let H=L∞(v), where ρH(g)=supv(t)g(t) and ρH∈Nt and let
(71)ρE(g)=supv(t)∫0tumin(k,n)/ng(u)duu.
Then by Theorem 15, the domain E is optimal and the couple is optimal. In particular, the couple Ln/min(k,n),1,C0 is optimal. If k=n+j, j≥0, this means that the embedding W1n+j↪Cj is optimal.

Example 19.

Let H be as in the previous example. Since
(72)ρE(g)≤suptmin(k,n)/nw(t)g(t),1v(t)=∫0t1w(u)duu,
it follows that the couple E=Λ∞(tmin(k,n)/nw),H=L*∞(v) is admissible. In order to prove that ρH is optimal, take any g∈M0, and define h from tmin(k,n)/nw(t)h(t)=supu≥tv(u)g(u). Then h∈D1 and ρE1(h)≲ρH(g). On the other hand
(73)Rmin(k,n)h(t)=∫0tsupx≥uv(x)g(x)1w(u)duu≥supu≥tv(u)g(u)1v(t)≥g(t).
Hence ρH(E)(g)≤ρE1(h)≲ρH(g); therefore ρH is optimal.

Example 20 (case <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M534"><mml:mi>k</mml:mi><mml:mo>≥</mml:mo><mml:mi>n</mml:mi></mml:math></inline-formula>).

Let H=L*q(v), 0<q≤∞, where ρH(g)=(∫0∞[v(t)g(t)]qdt/t)1/q and ρH∈Nt and let k≥n. Using Remark 16, we can construct an optimal domain E=Γq(tv) and this couple is optimal. Also αE=βE=1 if v is slowly varying.

4.2. Optimal Target QuasinormsDefinition 21 (construction of the optimal target quasi-norm).

For a given domain quasinorm ρE∈Nd, we set
(74)ρH(E)(g):=inf{ρE(h):g≤Rmin(k,n)h,h∈D1},iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiig∈Mmin(k,n).
Note that αH(E)=βH(E)=min(k,n)/n-αE.

Theorem 22.

The target quasinorm ρH(E) belongs to Nt, the couple ρE,ρH(E) is admissible, and the target quasinorm is optimal.

Proof.

The property “ρH(E)(g)=0 implies g=0” follows from (30). Also, since ρE∈Nd it is easy to check that ρH(E)∈Nt. The couple is admissible since ρH(E)(Rmin(k,n)h)≤ρE(h), h∈D1. Suppose that the couple ρE, ρH1∈Nt is admissible. Then ρH1(χ(0,1)Rmin(k,n)h)≲ρE(h), h∈D1. Therefore if χ(0,1)g≤Rmin(k,n)h, h∈D1, then ρH1(χ(0,1)g)≤ρH1(χ(0,1)Rmin(k,n)h)≲ρE(h), whence ρH1(χ(0,1)g)≲ρH(E)(g), g∈Mmin(k,n). Hence ρH(E) is optimal.

Theorem 23 (supercritical case).

If αE<min(k,n)/n and ρE(χ(1,∞)(t)t-min(k,n)/n)<∞, then
(75)ρH(E)(g)≈ρE(t-min(k,n)/ng(t)),g∈Mmin(k,n).
Moreover, the couple ρE,ρH(E) is optimal.

Proof.

If g≤Rmin(k,n)h, h∈D1, then, by Minkowski’s inequality and since min(k,n)/n>αE, it follows
(76)ρE(t-min(k,n)/ng(t))≤ρE(t-min(k,n)/nRmin(k,n)h(t))≲ρE(h).
Hence, taking the infimum, we get ρE(t-min(k,n)/ng(t))≲ρH(E)(g).

On the other hand, for g∈Mmin(k,n), we have g≲Rmin(k,n)h, h(t)=t-min(k,n)/ng(t). Since h∈D1 it follows ρH(E)(g)≲ρE(h)≲ρE(t-mink,n/ng(t)).

The domain quasinorm ρE is also optimal since for f∈L1(Ω),
(77)ρE(H(E))(f*)=ρH(E)(Rmin(k,n)f*)≈ρE(t-min(k,n)/nRmin(k,n)f*(t))≳ρE(f*).

Example 24.

Consider the space E=Λq(w), 0<q≤∞, min(k,n)/n>βE=αE>0. Then by Theorem 23 the couple E,H, H=L*q(t-min(k,n)/nw) is optimal. In particular, using also Theorem 13, the embedding WkLp,∞↪𝒞k-n/p, k>n/p, 1<p<∞, is optimal.

In the critical case we do not know how to simplify the optimal target quasi-norm, defined in (74). Instead, we can construct a large class of domain quasinorms and the corresponding optimal target quasinorms by using extrapolation from the super-critical case. Recall some basic definitions and results from the extrapolation theory [3]. Let (A0,A1) be a couple of quasi-Banach spaces. The sigma extrapolation space Σq(M(σ)(A0,A1)a(t)t-σ,q), a-positive weight, 0<σ<σ0, 0<q≤∞, M-positive decreasing weight, consists of all f∈A0+A1 such that f=∑j=l∞gj, gj∈Aj, Aj:=(A0,A1)a(t)t-1/2j,q, with a quasinorm
(78)∥f∥Σq(M(σ)(A0,A1)a(t)t-σ,q)=inf(∑j=l∞[M(2-j)∥gj∥Aj]q)1/q,
where the infimum is taken with respect to all representations f=∑j=l∞gj.

This space can be characterized as an interpolation space.

Theorem 25 (see [<xref ref-type="bibr" rid="B15">3</xref>]).

Let a(t)=t-θb(t), b-slowly varying, 0<θ<1. Then
(79)Σq(M(σ)(A0,A1)a(t)t-σ,q)=(A0,A1)w,q,
where
(80)1w(t)=1a(t)(∫0σ0[tσM(σ)]rdσσ)1/r
and 1/r+1/q=1 if q>1, r=∞ if 0<q≤1.

Our main result is the following one.

Theorem 26.

Let E=Λq(tk/nc(t)(1-lnt)), k<n, c-slowly varying weight, c(+0)=∞, c(t2)≈c(t), 0<q≤∞, H=L*q(c). We suppose that ρE∈Nd and ρH∈Nt. Then this couple is admissible and the target quasinorm is optimal.

Proof.

Step 1 (admissibility). Since αE=βE=k/n<1, it will be enough to check that
(81)ρH(Rk(g**))≲ρE(g**),
where
(82)ρE(g)=(∫01[tk/nc(t)(1-lnt)g(t)]qdtt)1/q,ρH(g)=(∫01[c(t)g(t)]qdtt)1/q.

Applying Minkowski’s inequality we obtain for 0<σ<σ0<k/n, b-slowly varying weight,
(83)σ∥Rk(g**)∥L*q(b(t)t-σ)≲∥g∥Γq(tk/n-σb(t)).
In order to extrapolate these inequalities, we write
(84)Γq(tk/n-σb(t))=(L1,L∞)b(t)tk/n-1-σ,q,L*q(t-σb(t))=(L*q(t1/2b(t)),L*q(t-1/2b(t)))1/2+σ,qiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiσ0<12.
This is true since
(85)K(t,g;L*q(w0),L*q(w1))≈(∫01[g(u)min(w0(u),tw1(u))]qduu)1/q,0<t<1.
Let σ=2-j and g=∑gj (convergence in L1), where gj∈L∞. Then g**≤∑gj**, whence Rk(g**)≤∑Rk(gj**) and for M(σ)=σ-2,p=min(q,1), we have
(86)Kp(t,Rk(g**);B0,B1)≤Cν:=∑j≥lKp(t,Rk(gj**);B0,B1),
where B0=L*q(t1/2b(t)),B1=L*q(t-1/2b(t)). We can write
(87)Cν=∑j≥l[t-1/2-2-j2-jM(2-j)K(t,Rk(gj**);B0,B1)]p×[t1/2+2-j2-jM(2-j)]p
and using also Hölder’s inequality if q>1, we get
(88)[v(t)]pCν≤∑j≥l[t-1/2-2-j2-jM(2-j)K(t,Rk(gj**);B0,B1)]p,
where
(89)1v(t)=(∑j≥l[t1/2+2-j2-jM(2-j)]r)1/r.
Hence
(90)∥Rk(g**)∥(B0,B1)v,q≲(∑j≥l[2-jM(2-j)∥Rk(gj**)∥(B0,B1)1/2+2-j,q]q)1/q.
Since
(91)2-j∥Rk(gj**)∥(B0,B1)1/2+2-j,q≲∥gj∥Γq(tk/n-2-jb(t)),
we get
(92)∥Rk(g**)∥(B0,B1)v,q≲∥g∥Σq(M(σ)(L1,L∞)b(t)tk/n-1-σ,q),
whence
(93)Rk:(L1,L∞)w,q↦(L*q(t1/2b(t)),L*q(t-1/2b(t)))v,q,
where w is given by (80) with a(t)=b(t)tk/n-1 and M(σ)=σ-2. It is easy to calculate these weights, see [3]. We have
(94)w(t)≈b(t)tk/n-1(1-lnt)2,v(t)≈t-1/2(1-lnt),w(t)≈b(t)tk/n-1(1-lnt)2,v(t)≈t-1/20<t<1.
Then for b(t)=c(t)(1-lnt)-1 we get
(95)Γq(tk/nc(t)(1-lnt))↪(L1,L∞)w,q,(L*q(t1/2b(t)),L*q(t-1/2b(t)))v,q↪L*q(c).
Hence (81) is proved.

Step 2 (optimality of the target quasi-norm). We want to prove that ρH is an optimal target quasi-norm. It is sufficient to see that
(96)ρH(E)(χ(0,1)g)≲ρH(g),g∈Mk,
where ρH(E) is defined by (74). To this end for any such g we construct an h∈D1 such that χ(0,1)g≲Rkh and ρE(h)≲ρH(g). Let 0<q<∞. (The case q=∞ is analogous, see Example 19.) Let hq(t)=χ(0,2)(t)∫t2(1-lnu)-qu-qgq(ue)du/u. Then h∈D1 and ρE(h)≲ρH(g). On the other hand, for 0<t<1,
(97)Rkh(t)≥∫t2/etg(eu)1-lnuduu≳χ(0,1)(t)g(t),
since ∫t2/et(1-lnu)-1du/u=ln2. Then by the definition of ρH(E) we get
(98)ρH(E)(χ(0,1)g)≲ρE(h)≲ρH(g).

In the limiting case k=n we can use the fact that the weights b(t)=c(t)(1-lnt) and c(t), where c is decreasing and slowly varying, are Muckenhoupt’s weights; that is,
(99)(∫t1cq(u)duu)1/q(∫0t[b(u)]-rduu)1/r≲1,1q+1r=1,0<t<1.
Then the operator Rn is bounded from L*q(b) to L*q(c), 1≤q≤∞ (see [28]). In this way we have the following result, with optimality being proved as in the case k<n.

Theorem 27.

Let E=Λq(tc(t)(1-lnt)), c-decreasing slowly varying weight, c(+0)=∞, c(t2)≈c(t), 1≤q≤∞, H=L*q(c). We suppose that ρE∈Nd and ρH∈Nt. Then this couple is admissible and the target quasinorm is optimal.

Authors’ Contribution

G. E. Karadzhov research is partially supported by the Abdus Salam School of Mathematical Sciences, GC University Lahore, by a grant from HEC, Pakistan.

Acknowledgment

The authors are grateful to the reviewers for the useful remarks that improved the paper.

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