Optimal Regularity Properties of the Generalized Sobolev Spaces

Abstract

We prove optimal embeddings of the generalized Sobolev spaces 𝑊^𝑘𝐸, 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 𝑊^𝑘_𝑝, 1≤𝑝<∞, consists of all locally integrable functions 𝑓, defined on 𝐑^𝑛, 𝑛≥1, with the Lebesgue measure, such that the following norm is finite: ‖𝑓‖_𝑊𝑘𝑝=∑_{|𝛼|≤𝑘}‖𝐷^𝛼𝑓‖_𝑝, where ‖𝑓‖_𝑝 stands for the 𝐿^𝑝-norm. In investigating the regularity of the function 𝑓∈𝑊^𝑘_𝑝, we may assume, without any loss of generality, that 𝑓∈𝐿¹(Ω), Ω is a domain in 𝐑^𝑛, 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, 𝑊^𝑘_𝑝↪𝒞^{𝑘−𝑛/𝑝}𝑛,𝑘>𝑝, (1) where 𝒞^𝛾,𝛾>0, is the Hölder-Zygmund space (see [1]). In the critical case 𝑘=𝑛/𝑝 the function 𝑓∈𝑊^𝑘_𝑝 may not be even continuous. The result (1) is not optimal. We prove that the optimal one is obtained if in (1) 𝐿^𝑝 is replaced by the Marcinkiewicz space 𝐿^𝑝,∞. In this paper we prove similar optimal results, when 𝐿^𝑝,∞ is replaced by a more general rearrangement invariant space 𝐸. The Sobolev space 𝑊^𝑘𝐸 consists of all 𝑓∈𝑊^𝑘₁ with a finite quasinorm ‖𝑓‖_𝑊𝑘𝐸=∑_{|𝛼|≤𝑘}‖𝐷^𝛼𝑓‖_𝐸. More precisely, we consider quasinormed rearrangement invariant spaces 𝐸, consisting of functions 𝑓∈𝐿¹(Ω), such that the quasinorm ‖𝑓‖_𝐸≈𝜌_𝐸(𝑓^∗)<∞, where 𝜌_𝐸 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 𝜌_𝐸(𝑔₁)≤𝜌_𝐸(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∶𝜇_𝑓(𝜆)≤𝑡},𝑡>0, and 𝜇_𝑓 is the distribution function of 𝑓, defined by 𝜇_𝑓(𝜆)=|{𝑥∈𝐑^𝑛∶|𝑓(𝑥)|>𝜆}|_𝑛,|⋅|_𝑛 denoting Lebesgue 𝑛-measure. Note that 𝑓^∗(𝑡)=0 for 𝑡>1. Finally, 𝑓^∗∗(𝑡)∶=(1/𝑡)∫^𝑡₀𝑓^∗(𝑠)𝑑𝑠.

Let 𝛼_𝐸, 𝛽_𝐸 be the Boyd indices of 𝐸. For example, if 𝐸=𝐿^𝑝, then 𝛼_𝐸=𝛽_𝐸=1/𝑝 and the condition 𝑘/𝑛≥1/𝑝 means 𝑘/𝑛≥𝛼_𝐸>0. Note that for 𝑘>𝑛 this is always satisfied. For these reasons we suppose that for the general 𝐸, 0<𝛼_𝐸=𝛽_𝐸≤1 and the case min(𝑘,𝑛)/𝑛>𝛼_𝐸 is called super-critical, while the case min(𝑘,𝑛)/𝑛=𝛼_𝐸-critical. In the super-critical case the function 𝑓∈𝑊^𝑘𝐸 is always continuous, while the spaces in the critical case 𝛼_𝐸=𝑘/𝑛,𝑘<𝑛, can be divided into two subclasses: in the first subclass the functions 𝑓∈𝑊^𝑘𝐸 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 𝐿^𝑛/𝑘,1, 𝑘<𝑛. If 𝑘≥𝑛; then 𝑊^𝑘𝐸 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 𝑊^𝑘𝐸 into the generalized Hölder-Zygmund space 𝒞𝐻. First we prove that this embedding for 𝑘≤𝑛 is equivalent to the continuity of the operator 𝑅_𝑘𝑔(𝑡)=∫^𝑡₀𝑢^{𝑘/𝑛−1}𝑔(𝑢)𝑑𝑢. The case 𝑘>𝑛 is reduced to the continuity of 𝑅_𝑛 by using the lifting principle ([1]). Moreover, if, for example, 𝑘≤𝑛, then in the super-critical case, we can replace 𝑅_𝑘 by the operator of multiplication 𝑡^𝑘/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 𝜌_𝐸(𝑡^−𝑘/𝑛𝑔(𝑡)) and the quasinorm in the optimal domain space 𝐸(𝐻) is given by 𝜌_𝐻(𝑡^𝑘/𝑛𝑔(𝑡)). Note that we do not require 𝜌_𝐸 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 Σ^𝑞-method of extrapolation ([3]) from the super-critical case. As a byproduct, we also characterize the embedding 𝑊^𝑘𝐸↪𝐶^𝑗,𝑗<𝑘, where 𝐶^𝑗 consists of all functions with bounded and uniformly continuous derivatives up to order 𝑗. Namely, this is equivalent to the embedding 𝐸↪𝐿^{𝑛/(𝑘−𝑗),1} if 𝑘≤𝑛. The embedding 𝑊^𝑛+𝑗𝐸↪𝐶^𝑗 is always true since 𝑊^𝑛𝐸↪𝑊^𝑛₁↪𝐶⁰.

The problem of the optimal target rearrangement invariant space for potential type operators is considered in [4] by using 𝐿^𝑝-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 𝑊^𝑘𝐸↪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 𝜌_𝑝≈𝜌_𝐸 that satisfies the triangle inequality 𝜌^𝑝_𝑝(𝑔₁+𝑔₂)≤𝜌^𝑝_𝑝(𝑔₁)+𝜌^𝑝_𝑝(𝑔₂) for some 𝑝∈(0,1] that depends only on the space 𝐸 (see [24]). We say that the quasinorm 𝜌_𝐸 satisfies Minkowski’s inequality if for the equivalent quasinorm 𝜌_𝑝, 𝜌^𝑝_𝑝𝑔_𝑗≲𝜌^𝑝_𝑝𝑔_𝑗,𝑔_𝑗∈𝑀⁺. (2) Usually we apply this inequality for functions 𝑔_𝑗∈𝑀⁺ with some kind of monotonicity.

Recall the definition of the lower and upper Boyd indices 𝛼_𝐸 and 𝛽_𝐸. Let 𝑔_𝑢(𝑡)=𝑔(𝑡/𝑢) if 𝑡<𝑢 and 𝑔_𝑢(𝑡)=0 if 𝑡≥𝑢, where 0<𝑡<1,𝑔∈𝑀⁺, and let ℎ_𝐸(𝑢)𝜌=sup^𝐸𝑔^∗_𝑢𝜌_𝐸𝑔^∗∶𝑔∈𝑀⁺,𝑔_𝑢(𝑡)𝑡∶=𝑔𝑢,𝑢>0, (3) be the dilation function generated by 𝜌_𝐸. Suppose that it is finite. Then 𝛼_𝐸∶=sup_0<𝑡<1logℎ_𝐸(𝑡)log𝑡,𝛽_𝐸∶=inf_1<𝑡<∞logℎ_𝐸(𝑡).log𝑡 (4) The function ℎ_𝐸 is submultiplicative, increasing, ℎ_𝐸(1)=1,ℎ_𝐸(𝑢)ℎ_𝐸(1/𝑢)≥1; hence 0≤𝛼_𝐸≤𝛽_𝐸. We suppose that 0<𝛼_𝐸=𝛽_𝐸≤1.

If 𝛽_𝐸<1 we have by using Minkowski’s inequality that 𝜌_𝐸(𝑓^∗)≈𝜌_𝐸(𝑓^∗∗). In particular, ‖𝑓‖_𝐸≈𝜌_𝐸(𝑓^∗∗) if 𝛽_𝐸<1. For example, consider the Gamma spaces 𝐸=Γ^𝑞(𝑤), 0<𝑞≤∞, 𝑤—positive weight, that is, a positive function from 𝑀⁺, with a quasinorm ‖𝑓‖_Γ𝑞(𝑤)∶=𝜌_𝐸(𝑓^∗),𝜌_𝐸(𝑔)∶=𝜌_𝑤,𝑞(∫¹₀𝑔(𝑡𝑢)𝑑𝑢), where 𝜌_𝑤,𝑞𝑔∶=¹₀𝑔(𝑡)𝑤(𝑡)^𝑞𝑑𝑡𝑡^1/𝑞,𝑔∈𝑀⁺,₁₀𝑤^𝑞(𝑡)𝑑𝑡𝑡^1/𝑞<∞. (5) Then 𝐿^∞(Ω)↪Γ^𝑞(𝑤)↪𝐿¹(Ω). If 𝑤(𝑡)=𝑡^1/𝑝,1<𝑝<∞, we write as usual 𝐿^𝑝,𝑞 instead of Γ^𝑞(𝑡^1/𝑝). Consider also the classical Lorentz spaces Λ^𝑞(𝑤), 0<𝑞≤∞; 𝑓∈Λ^𝑞(𝑤) if ‖𝑓‖_Λ^𝑞_𝑤∶=𝜌_𝑤,𝑞(𝑓^∗)<∞, 𝑤(2𝑡)≈𝑤(𝑡). We suppose that 𝐿^∞(Ω)↪Λ^𝑞(𝑤)↪𝐿¹(Ω).

Note that if 𝐸=Λ^𝑞(𝑡^𝛼𝑤),0<𝛼≤1, where 𝑤 is slowly varying, then 𝛼_𝐸=𝛽_𝐸=𝛼. Recall that 𝑤∈𝑀⁺ is slowly varying if for all 𝜀>0 the function 𝑡^𝜀𝑤(𝑡) is equivalent to an increasing function and the function 𝑡^−𝜀𝑤(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 𝜔^𝑘𝑡,𝑓=sup_|ℎ|≤𝑡sup_{𝑥∈𝐑𝑛}Δ^𝑘_ℎ𝑓(𝑥), (6) where Δ^𝑘_ℎ𝑓 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=𝜌_𝐻(|𝑔|), where 𝜌_𝐻 is a monotone quasinorm on 𝑀⁺ which satisfies Minkowski’s inequality. The dilation function generated by 𝜌_𝐻 is given by ℎ_𝐻(𝑢)𝜌=sup^𝐻𝑔_1/𝑢𝜌_𝐻𝑔∶𝑔∈𝑀_𝑚, (7) where 𝑀_𝑚∶=𝑔∈𝑀⁺(𝑡)𝑡∶𝑔isincreasing,_−𝑚/𝑛𝑔(𝑡)isdecreasingand𝑔(2𝑡)≈𝑔(𝑡). (8) The choice of the space 𝑀_𝑚 is motivated by the fact that 𝜔^𝑚(𝑡^1/𝑛,𝑓) is equivalent to a function 𝑔∈𝑀_𝑚. The function ℎ_𝐻(𝑢) is submultiplicative, increasing and 𝑢^−𝑚/𝑛ℎ_𝐻(𝑢) is decreasing and ℎ_𝐻(1)=1,ℎ_𝐻(𝑢)ℎ_𝐻(1/𝑢)≥1. Therefore the Boyd indices of 𝐻 are well defined 𝛼_𝐻=sup_0<𝑡<1logℎ_𝐻(𝑡)log𝑡,𝛽_𝐻=inf_1<𝑡<∞logℎ_𝐻(𝑡),log𝑡 (9) and they satisfy 𝛼_𝐻≤𝛽_𝐻≤𝑚/𝑛. In what follows, we suppose that 0≤𝛼_𝐻=𝛽_𝐻≤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)=𝜌_𝑤,𝑞(|𝑔|), where 𝜌_𝑤,𝑞 is given by (5). It turns out that 𝛼_𝐻=𝛽_𝐻=𝛾/𝑛.

Definition 1. Let 𝑗=0,1,… and let 𝐶^𝑗 stand for the space of all functions 𝑓, defined on 𝐑^𝑛, that have bounded and uniformly continuous derivatives up to the order 𝑗, normed by ‖𝑓‖_𝐶𝑗=sup∑^𝑗_𝑙=0|𝑃^𝑙𝑓(𝑥)|, where 𝑃^𝑙𝑓(𝑥)=∑_|𝜈|=𝑙𝐷^𝜈𝑓(𝑥). (i)If 𝑗/𝑛<𝛼_𝐻<(𝑗+1)/𝑛 for 𝑗≥1 or 0≤𝛼_𝐻<1/𝑛 for 𝑗=0, then 𝒞𝐻 is formed by all functions 𝑓 in 𝐶^𝑗 having a finite quasinorm 𝑓_𝒞𝐻=𝑓_𝐶𝑗+𝜌_𝐻𝜒_(0,1)(𝑡)𝑡^𝑗/𝑛𝜔𝑡^1/𝑛,𝑃^𝑗𝑓. (10) (ii)If 𝛼_𝐻=(𝑗+1)/𝑛, then 𝒞𝐻 consists of all functions 𝑓 in 𝐶^𝑗 having a finite quasinorm 𝑓_𝒞𝐻=𝑓_𝐶𝑗+𝜌_𝐻𝜒_(0,1)(𝑡)𝑡^𝑗/𝑛𝜔²𝑡^1/𝑛,𝑃^𝑗𝑓. (11)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 𝒞^𝛾 (see [1]). Also, if 𝐻=𝐿^∞, then 𝒞𝐻=𝐶⁰.

We will use the following equivalent quasinorm.

Theorem 2 (equivalence [23]). Let 0≤𝛼_𝐻=𝛽_𝐻<𝑚/𝑛. If 𝜌_𝐻(𝜒_(0,1)(𝑡)𝑡^𝛼)<∞ for 𝛼>𝛼_𝐻, then for all such 𝑚, 𝑓_𝒞𝐻≈𝑓_𝐶0+𝜌_𝐻𝜒_(0,1)(𝑡)𝜔^𝑚𝑡^1/𝑛,𝑓. (12)

Note that if 𝜌_𝐻(𝜒_(0,1)(𝑡)𝑡^𝑚/𝑛)<∞, then 𝒞𝐻 is a 𝐾-interpolation space for the couple (𝐶⁰,𝐶^𝑚), namely, 𝒞𝐻=(𝐶⁰,𝐶^𝑚)_𝐻1, where 𝜌_𝐻1(𝑔)=𝜌_𝐻(𝑔(𝑡^𝑚/𝑛)). In particular, 𝒞𝐿¹_∗(𝑡^−𝑗)↪𝐶^𝑗↪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 𝐾𝑡,𝑓=𝐾𝑡,𝑓;𝐴₀,𝐴₁=inf_{𝑓=𝑓0+𝑓1}𝑓_0𝐴0+𝑡𝑓_1𝐴1,𝑓∈𝐴₀+𝐴₁, (13) be the 𝐾-functional of Peetre. By definition, the 𝐾-interpolation space 𝐴_Φ=(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↪𝐴_Φ↪𝐴₀+𝐴₁. If 𝑔_Φ=^∞₀𝑤(𝑡)𝑡^−𝜃𝑔(𝑡)^𝑞𝑑𝑡𝑡^1/𝑞,0≤𝜃≤1,0<𝑞≤∞,𝑤∈ℳ⁺, (14) 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, 𝑓_{𝐴0∩𝐴1}=𝑓_𝐴0+𝑓_𝐴1,𝑓_𝐴0+𝐴1=𝐾1,𝑓;𝐴₀,𝐴₁. (15)

Theorem 3 (lifting principle). Let 𝛼_𝐻>0 and let 𝜌_𝐻𝑗(𝑔)∶=𝜌_𝐻(𝑡^−𝑗/𝑛𝑔(𝑡)),𝑗≥1. Then 𝑓_𝒞𝐻𝑗≈𝒟^𝑗𝑓_𝒞𝐻,𝒟^𝑗=_{|𝛼|≤𝑗}𝐷^𝛼. (16)

Proof. Let 𝑚/𝑛>𝛼_𝐻+𝑗/𝑛. Since 𝜔^𝑚𝑡^1/𝑛,𝑓≲𝑡^𝑗/𝑛𝜔^𝑚−𝑗𝑡^1/𝑛,𝑃^𝑗𝑓, (17) it follows 𝜌_𝐻𝑗𝜔^𝑚𝑡^1/𝑛,𝑓≲𝜌_𝐻𝜔^𝑚−𝑗𝑡^1/𝑛,𝒟^𝑗𝑓. (18) Hence 𝑓_𝒞𝐻𝑗≲𝒟^𝑗𝑓_𝒞𝐻. (19) To prove the reverse, we use the formula (see [25, page  342]) 𝜔^𝑚𝑡^1/𝑛,𝐷^𝛼𝑓≲^𝑡₀𝑢^{−|𝛼|/𝑛}𝜔^𝑚𝑢^1/𝑛,𝑓𝑑𝑢𝑢. (20) Then applying Minkowski’s inequality and 𝛼_𝐻𝑗=𝛽_𝐻𝑗=𝑗/𝑛+𝛼_𝐻>𝑗/𝑛, we get 𝜌_𝐻𝜒_(0,1)(𝑡)𝜔^𝑚𝑡^1/𝑛,𝐷^𝛼𝑓≲𝜌_𝐻𝑗𝜒_(0,1)(𝑡)𝜔^𝑚𝑡^1/𝑛,|𝛼|,𝑓≤𝑗. (21) Since 𝒞𝐻_𝑗↪𝐶^𝑗, we derive 𝒟^𝑗𝑓_𝒞𝐻≲𝑓_𝒞𝐻𝑗. (22)

Remark 4. The relation (19) is always true. But if 𝛼_𝐻=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 𝑁_𝑑 consist of all domain quasinorms 𝜌_𝐸 that are monotone, satisfying Minkowski’s inequality, 0<𝛼_𝐸=𝛽_𝐸≤min(𝑘,𝑛)/𝑛, 𝜌_𝐸(𝜒_(0,1)𝑡^−𝛼)<∞ if 𝛼<𝛼_𝐸 and the condition (30) below for 𝑘≤𝑛, and 𝐸↪𝐿¹ for 𝑘>𝑛. Let 𝑁_𝑡 consist of all target quasinorms 𝜌_𝐻 that are monotone, satisfy Minkowski’s inequality, 0≤𝛼_𝐻=𝛽_𝐻<min(𝑘,𝑛)/𝑛, 𝜌_𝐻(𝜒_(0,1)(𝑡)𝑡^𝛼)<∞ if 𝛼>𝛼_𝐻 and sup_0<𝑡<1𝑔(𝑡)≲𝜌_𝐻𝑔. (23)

We use the following definitions.

Definition 5 (admissible couple). We say that the couple 𝜌_𝐸∈𝑁_𝑑,𝜌_𝐻∈𝑁_𝑡 is admissible if 𝑊^𝑘𝐸↪𝒞𝐻 when 𝑘≤𝑛, and if 𝒟^𝑗(𝑊^𝑛+𝑗𝐸)↪𝒞𝐻 for 𝑗≥1. Moreover, 𝜌_𝐸(𝐸) is called domain quasinorm (domain space), and 𝜌_𝐻(𝐻) is called target quasinorm (target space).

Definition 6 (optimal target quasi-norm). Given the domain quasinorm 𝜌_𝐸∈𝑁_𝑑, the optimal target quasi-norm, denoted by 𝜌_𝐻(𝐸), is the strongest target quasi-norm; that is, 𝜌_𝐻𝑔≲𝜌_𝐻(𝐸)𝑔,𝑔∈𝑀_{min(𝑘,𝑛)} (24) for any target quasinorm 𝜌_𝐻∈𝑁_𝑡 such that the couple 𝜌_𝐸, 𝜌_𝐻 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 𝜌_𝐻∈𝑁_𝑡, the optimal domain quasi-norm, denoted by 𝜌_𝐸(𝐻), is the weakest domain quasi-norm; that is, 𝜌_𝐸(𝐻)𝑓^∗≲𝜌_𝐸𝑓^∗,𝑓∈𝐿¹(Ω), (25) for any domain quasinorm 𝜌_𝐸∈𝑁_𝑑 such that the couple 𝜌_𝐸, 𝜌_𝐻 is admissible.

Definition 8 (optimal couple). The admissible couple 𝜌_𝐸∈𝑁_𝑑, 𝜌_𝐻∈𝑁_𝑡 is said to be optimal if both 𝜌_𝐸 and 𝜌_𝐻 are optimal.

3. Admissible Couples

Here we give a characterization of all admissible couples 𝜌_𝐸∈𝑁_𝑑, 𝜌_𝐻∈𝑁_𝑡. We start with the main estimate. For 𝑘=1, see also [26].

Theorem 9. Let 𝑓∈𝑊^𝑘₁ and 𝑘≤𝑛. Then 𝜔^𝑘𝑡^1/𝑛,𝑓≲^𝑡₀𝑢^{𝑘/𝑛−1}𝒟^𝑘𝑓^∗(𝑢)𝑑𝑢, (26) where 𝒟^𝑘𝑓∶=∑_{|𝛼|≤𝑘}𝐷^𝛼𝑓.

Proof. We use the embedding 𝑊^𝑘𝐿^𝑛/𝑘,1↪𝐿^∞, (27) whence 𝜔^𝑘𝑡^1/𝑛,𝑓≲𝐾𝑡^𝑘/𝑛,𝑓;𝐿^∞,𝑊^𝑘_∞≲𝐾𝑡^𝑘/𝑛,𝑓;𝑊^𝑘𝐿^𝑛/𝑘,1,𝑊^𝑘_∞. (28) Then (26) follows from the basic formula [25, page 360] 𝐾𝑡,𝑓;𝑊^𝑘₁,𝑊^𝑘_∞≈^𝑡₀𝒟^𝑘𝑓^∗(𝑠)𝑑𝑠 (29) and the reiteration formula of Holmstedt for the 𝐾-functional [25, page 310].

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

Theorem 10. A necessary and sufficient condition for the embedding 𝑊^𝑘𝐸↪𝐶⁰,𝑘≤𝑛, is the following one ¹₀𝑡^{𝑘/𝑛−1}𝑔(𝑡)𝑑𝑡≲𝜌_𝐸𝑔,𝑔∈𝐷₁, (30) where 𝐷₁=𝑔∈𝑀⁺:𝑔𝑖𝑠𝑑𝑒𝑐𝑟𝑒𝑎𝑠𝑖𝑛𝑔𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛. (31)

Proof. The conditions (30) and (26), (27) imply the embedding 𝑊^𝑘𝐸↪𝐿^∞ and lim_𝑡→0𝜔^𝑘(𝑡^1/𝑛,𝑓)=0 if 𝑓∈𝑊^𝑘𝐸. On the other hand, by Marchaud’s inequality (see [25], Theorem 5.4.4), we have 𝜔𝑡^1/𝑛,𝑓≲𝑡^1/𝑛∞_𝑡𝑢^−1/𝑛𝜔^𝑘𝑢^1/𝑛,𝑓𝑑𝑢𝑢. (32) It is easy to see that lim_𝑡→0𝜔(𝑡^1/𝑛,𝑓)=0. Thus 𝑊^𝑘𝐸↪𝐶⁰.
Before proving the reverse, note that (30) is always satisfied if 𝑘/𝑛>𝛼_𝐸. Since 𝜌_𝐸𝜒_(0,1)𝑔(𝑢)≤𝜌_𝐸𝑔(𝑡𝑢)≤ℎ_𝐸1𝑢𝜌_𝐸𝑔, (33) we have 𝑔(𝑢)≲ℎ_𝐸1𝑢𝜌_𝐸𝑔,𝑔∈𝐷₁. (34) Hence for 0<𝜀<𝑘/𝑛−𝛼_𝐸, ¹₀𝑢^{𝑘/𝑛−1}𝑔(𝑢)𝑑𝑢≲𝜌_𝐸𝑔¹₀𝑢^{𝑘/𝑛−𝛼𝐸−𝜀}𝑑𝑢𝑢≲𝜌_𝐸𝑔. (35) It remains to prove that if 𝑊^𝑘𝐸↪𝐶⁰, 𝑘≤𝑛, then (30) is true for 𝛼_𝐸=𝑘/𝑛. To this end we choose a test function 𝑓 as follows: 𝑓(𝑥)=¹₀𝑔(𝑢)𝑢^𝑘/𝑛𝜓𝑥𝑢^−1/𝑛𝑑𝑢𝑢, (36) where 𝑔∈𝐷₁ and 𝜓 is in 𝐶^∞₀ 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)|≤∫¹_{𝑐|𝑥|𝑛}𝑔(𝑢)𝑢^𝑘/𝑛𝑑𝑢/𝑢, whence |𝑓(𝑥)|≲𝑔(𝑐|𝑥|^𝑛); therefore 𝑓^∗(𝑡)≲𝑔(𝑡). This implies 𝜌_𝐸𝑓^∗≲𝜌_𝐸𝑔. (37) Analogously, since |𝐷^𝛼𝑓(𝑥)|≲∫^{2𝑛𝑐|𝑥|𝑛}_{𝑐|𝑥|𝑛}𝑔(𝑢)𝑑𝑢/𝑢, 1≤|𝛼|≤𝑘, we have 𝑓_𝑊𝑘𝐸≲𝜌_𝐸𝑔. (38) Also 𝑓(0)=𝜓(0)/𝑛∫¹₀𝑢^{𝑘/𝑛−1}𝑔(𝑢)𝑑𝑢≲‖𝑓‖_𝑊𝑘𝐸≲𝜌_𝐸(𝑔). Thus (30) is proved.

Remark 11. Similar arguments show that 𝑊^𝑘𝐸↪𝐶^𝑗,𝑗<𝑘≤𝑛, if and only if 𝐸↪𝐿^{𝑛/(𝑘−𝑗),1}.

Theorem 12. The couple 𝜌_𝐸∈𝑁_𝑑, 𝜌_𝐻∈𝑁_𝑡 is admissible if and only if 𝜌_𝐻𝜒_(0,1)𝑅_{min(𝑘,𝑛)}𝑔≲𝜌_𝐸𝑔,𝑔∈𝐷₁, (39) where 𝑅_𝑘𝑔(𝑡)∶=^𝑡₀𝑢^{𝑘/𝑛−1}𝑔(𝑢)𝑑𝑢. (40)

Proof. Step 1 (sufficiency of (39)). If 𝑘≤𝑛 then it is clear that the embedding 𝑊^𝑘𝐸↪𝒞𝐻 follows from (39), (26), and (27). Let now 𝑘=𝑛+𝑗, 𝑗≥1. Then (39) for 𝑘=𝑛 implies 𝑊^𝑛𝐸↪𝒞𝐻. Hence 𝒟^𝑗(𝑊^𝑛+𝑗𝐸)↪𝒞𝐻 for 𝑗≥1.
Step 2  (necessity of (39) when 𝑘≤𝑛). Now we prove that the embedding 𝑊^𝑘𝐸↪𝒞𝐻 implies (39) for 𝑘≤𝑛. To this end we choose the test function 𝑓 as in (36).
Let |ℎ|=𝐶𝑡^1/𝑛. We split 𝑓=𝑓_1𝑡+𝑓_2𝑡, 𝑓_1𝑡(𝑥)=∫^𝑡₀𝑢^𝑘/𝑛𝑔(𝑢)𝜓(𝑥𝑢^−1/𝑛)𝑑𝑢/𝑢, 𝑓_2𝑡(𝑥)=∫¹_𝑡𝑢^𝑘/𝑛𝑔(𝑢)𝜓(𝑥𝑢^−1/𝑛)𝑑𝑢/𝑢, 0<𝑡<1. First we prove that for some large 𝐶>0, 𝜔^𝑚𝐶𝑡^1/𝑛,𝑓_1𝑡≥𝜓(0)𝑅_𝑘𝑔(𝑡),0<𝑡<1. (41) Indeed, we have 𝜔^𝑚(𝐶𝑡^1/𝑛,𝑓_1𝑡)≥|(Δ^𝑚_ℎ𝑓_1𝑡)(0)| and 𝜓(𝑗𝐶𝑡^1/𝑛𝑢^−1/𝑛)=0 for 𝑢<𝑡,1≤𝑗≤𝑚 if 𝐶>0 is large enough. Hence (41) follows. Further, 𝜔^𝑚𝑡^1/𝑛,𝑓≥𝜔^𝑚𝑡^1/𝑛,𝑓_1𝑡−𝜔^𝑚𝑡^1/𝑛,𝑓_2𝑡,0<𝑡<1. (42) Since 𝜔^𝑚𝑡^1/𝑛,𝑓_2𝑡≲𝑡^𝑚/𝑛𝑃^𝑚𝑓_2𝑡𝐿∞≲𝑡^𝑚/𝑛1_𝑡𝑢^{(𝑘−𝑚)/𝑛}𝑔(𝑢)𝑑𝑢𝑢 (43) and 𝑅_𝑘𝑔(𝑡)≳𝑡^𝑘/𝑛𝑔(𝑡) for 𝑔∈𝐷₁, we get 𝜔^𝑚𝑡^1/𝑛,𝑓_2𝑡≲𝑡^𝑚/𝑛1_𝑡𝑢^−𝑚/𝑛𝑅_𝑘𝑔(𝑢)𝑑𝑢𝑢. (44) Therefore 𝑅_𝑘𝑔(𝑡)≤𝑐₁𝜔^𝑚𝑡^1/𝑛,𝑓+𝑐𝑡^𝑚/𝑛1_𝑡𝑢^−𝑚/𝑛𝑅_𝑘𝑔(𝑢)𝑑𝑢𝑢,𝑅0<𝑡<1,_𝑘𝑔(𝑡)≤𝑐₁𝜔^𝑚𝑡^1/𝑛,𝑓+𝑐𝑡^𝑚/𝑛1_𝑡𝑢^−𝑚/𝑛𝑔(𝑢)𝑑𝑢𝑢,0<𝑡<1. (45)(46) To solve the integral inequality (45) for 𝑝(𝑡)∶=𝜒_(0,1)(𝑡)𝑡^−𝑚/𝑛𝑅_𝑘𝑔(𝑡), we set 𝑞(𝑡)=𝑐₁𝑡^−𝑚/𝑛𝜔^𝑚(𝑡^1/𝑛,𝑓) and rewrite it as 𝑝(𝑡)≤𝑞(𝑡)+𝑐∫¹_𝑡𝑝(𝑢)𝑑𝑢/𝑢. If 𝑟(𝑡)=∫¹_𝑡𝑝(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)+𝑡^𝑐−1𝑞(𝑡), whence 𝑣(𝑡)≤∫¹_𝑡𝑢^𝑐−1𝑞(𝑢)𝑑𝑢. Therefore 𝜒_(0,1/2)(𝑡)𝑅_𝑘𝑔(𝑡)≲𝑡^{𝑚/𝑛−𝑐1}_𝑡𝑢^{𝑐−𝑚/𝑛}𝜔^𝑚𝑢^1/𝑛,𝑓𝑑𝑢𝑢. (47) Hence by using Minkowski’s inequality and choosing 𝑚 large enough, we obtain 𝜌_𝐻𝜒_(0,1/2)𝑅_𝑘𝑔≲𝜌_𝐻𝜔^𝑚𝑡^1/𝑛,𝑓. (48) On the other hand, from (46), it follows that 𝜌_𝐻𝜒_(1/2,1)𝑅_𝑘𝑔≤𝜌_𝐻𝜔^𝑚𝑡^1/𝑛,𝑓+¹₀𝑔(𝑢)𝑑𝑢. (49) Hence, using also (30), we get 𝜌_𝐻𝜒_(0,1)𝑅_𝑘𝑔≲𝜌_𝐻𝜔^𝑚𝑡^1/𝑛,𝑓+𝜌_𝐸𝑔. (50) Thus, if 𝑊^𝑘𝐸↪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 𝒟^𝑗(𝑊^𝑛+𝑗𝐸)↪𝒞𝐻, 𝑗≥1, implies (39) for 𝑘=𝑛. To this end we choose the test function 𝑓 in the form 𝑓(𝑥)=¹₀𝑔(𝑢)𝑢^𝑗/𝑛𝜓𝑥𝑢^−1/𝑛𝑑𝑢, (51) where 𝑔∈𝐷₁ and 𝜓 is the same as in (36). Note that 𝒟^𝑗𝜓(0)=1. Then as before we get (37), and 𝐷^𝛼𝑓(𝑥)≲𝑔𝑐|𝑥|^𝑛 (52) for 1≤|𝛼|≤𝑛+𝑗. Hence 𝑓_{𝑊𝑛+𝑗𝐸}≲𝜌_𝐸𝑔. (53)
On the other hand, using the arguments from Step 2 but for the function 𝒟^𝑗𝑓, 𝑗≥1, we obtain 𝜌_𝐻𝜒_(0,1)𝑅_𝑛𝑔≲𝜌_𝐻𝜔^𝑚𝑡^1/𝑛,𝒟^𝑗𝑓+𝜌_𝐸𝑔,𝑗≥1. (54) Thus, if 𝒟^𝑗(𝑊^𝑛+𝑗𝐸)↪𝒞𝐻, 𝑗≥1 is given, then (54), (53) imply (39) for 𝑘=𝑛.

Theorem 13. Let 𝛼_𝐻>0. Then the couple 𝜌_𝐸∈𝑁_𝑑, 𝜌_𝐻∈𝑁_𝑡 is admissible for 𝑘>𝑛 if and only if 𝑊^𝑛+𝑗𝐸↪𝒞𝐻_𝑗,𝜌_𝐻𝑗𝑔∶=𝜌_𝐻𝑡^−𝑗/𝑛𝑔(𝑡),𝑗≥1. (55) Moreover, (55) is equivalent to 𝜌_𝐻𝑗𝑅_𝑛+𝑗𝑔≲𝜌_𝐸𝑔,𝑔∈𝐷₁. (56)

Proof. Let 𝜌_𝐸∈𝑁_𝑑,𝜌_𝐻∈𝑁_𝑡 be an admissible couple for 𝑘>𝑛. Then 𝒟^𝑗(𝑊^𝑛+𝑗𝐸)↪𝒞𝐻. Since 𝜔^𝑚𝑡^1/𝑛,𝒟^𝑗𝑓≲^𝑡₀𝒟^𝑛+𝑗𝑓^∗(𝑢)𝑑𝑢,𝑚≥𝑛, (57) we have, by applying (39) for 𝑘=𝑛, 𝜌_𝐻𝜔^𝑚𝑡^1/𝑛,𝒟^𝑗𝑓≲𝜌_𝐸𝒟^𝑛+𝑗𝑓^∗. (58) Using also (16) and 𝑊^𝑛+𝑗𝐸↪𝐶^𝑗, we obtain 𝑊^𝑛+𝑗𝐸↪𝒞𝐻_𝑗. Further, as in the proof of the previous theorem, this embedding implies (56). Finally, 𝜌_𝐻𝑗𝑅_𝑛+𝑗𝑔≈𝜌_𝐻𝑅_𝑛𝑔,𝛼_𝐻>0. (59) Indeed, 𝜌_𝐻(𝑅_𝑛𝑔)=𝜌_𝐻𝑗(𝑡^𝑗/𝑛𝑅_𝑛𝑔(𝑡)) and 𝜌_𝐻𝑗𝑡^𝑗/𝑛𝑅_𝑛𝑔(𝑡)≲𝜌_𝐻𝑗𝑡^{𝑗/𝑛𝑡}₀𝑢^{(𝑛+𝑗)/𝑛}𝑅_𝑛+𝑗𝑔(𝑢)𝑑𝑢. (60) Applying Minkowski’s inequality, we get, since 𝛼_𝐻𝑗>𝑗/𝑛, 𝜌_𝐻𝑅_𝑛𝑔≲𝜌_𝐻𝑗𝑅_𝑛+𝑗𝑔. (61) For the reverse, we notice that 𝑅_𝑛+𝑗𝑔(𝑡)≲𝑡^𝑗/𝑛𝑅_𝑛𝑔(𝑡). Then 𝜌_𝐻𝑗𝑅_𝑛+𝑗𝑔≲𝜌_𝐻𝑅_𝑛𝑔. (62)

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 𝜌_𝐸(𝐻) by Theorem 9 as follows.

Definition 14 (construction of an optimal domain quasi-norm). For a given target quasinorm 𝜌_𝐻∈𝑁_𝑡, we set 𝜌_𝐸(𝐻)𝑔∶=𝜌_𝐻𝑅_{min(𝑘,𝑛)}𝑔,𝑔∈𝑀⁺. (63)

Note that 𝑅_{min(𝑘,𝑛)}(𝑔_𝑢)=𝑢^{min(𝑘,𝑛)/𝑛}(𝑅_{min(𝑘,𝑛)}𝑔)_1/𝑢 and 𝑆𝑔∈𝐿 if 𝑔∈𝐷₁. Hence 𝛼_𝐸(𝐻)=𝛽_𝐸(𝐻)=min(𝑘,𝑛)/𝑛−𝛼_𝐻.

Theorem 15. The quasinorm 𝜌_𝐸(𝐻) belongs to 𝑁_𝑑, the couple 𝜌_𝐸(𝐻), 𝜌_𝐻 is admissible, and the domain quasinorm 𝜌_𝐸(𝐻) is optimal. Moreover, the target quasinorm 𝜌_𝐻 is also optimal and 𝜌_𝐸(𝐻)𝑔≈𝜌_𝐻𝑡^{min(𝑘,𝑛)/𝑛}𝑔,𝑔∈𝐷₁𝑖𝑓𝛼_𝐻>0. (64)

Proof. It is easy to check that 𝜌_𝐸(𝐻)∈𝑁_𝑑. Further, the couple 𝜌_𝐸(𝐻), 𝜌_𝐻 is admissible since 𝜌_𝐻(𝑅_{min(𝑘,𝑛)}𝑔)=𝜌_𝐸(𝐻)(𝑔), 𝑔∈𝐷₁. Moreover, 𝜌_𝐸(𝐻) is optimal, since for any admissible couple 𝜌_𝐸1∈𝑁_𝑑,𝜌_𝐻, we have 𝜌_𝐻(𝜒_(0,1)𝑅_{min(𝑘,𝑛)}𝑔)≲𝜌_𝐸1(𝑔), where 𝑔∈𝐷₁. Therefore for 𝑓∈𝐿¹(Ω), 𝜌_𝐸(𝐻)𝑓^∗=𝜌_𝐻𝑅_{min(𝑘,𝑛)}𝑓^∗≲𝜌_𝐻𝜒_(0,1)𝑅_{min(𝑘,𝑛)}𝑓^∗+𝜌_𝐸1𝑓^∗≲𝜌_𝐸1𝑓^∗. (65) To prove that 𝜌_𝐻 is also optimal, let 𝜌_𝐸(𝐻),𝜌_𝐻1∈𝑁_𝑡 be an arbitrary admissible couple. Then 𝜌_𝐻1𝜒_(0,1)𝑅_{min(𝑘,𝑛)}𝑔≲𝜌_𝐸(𝐻)𝑔,𝑔∈𝐷₁. (66) It is enough to check that 𝜌_𝐻1𝜒_(0,1)𝑔≲𝜌_𝐻𝑔,𝑔∈𝑀_{min(𝑘,𝑛)}. (67) Let 𝑘<𝑛. (The case 𝑘≥𝑛 is easier.) We introduce a better function 𝑔₁(𝑡)=𝑔(𝑡^𝑛/𝑘). Then 𝑔₁ is quasiconcave; therefore 𝑔₁(𝑡)≈∫^𝑡₀ℎ₁(𝑢)𝑑𝑢 and ℎ₁∈𝐷₁. By changing the variables, we get 𝑔≈𝑅_𝑘ℎ with ℎ∈𝐷₁. Then 𝜌_𝐻1𝜒_(0,1)𝑔≲𝜌_𝐻1𝜒_(0,1)𝑅_𝑘ℎ≲𝜌_𝐸(𝐻)(ℎ)≈𝜌_𝐻𝑅_𝑘ℎ≈𝜌_𝐻𝑔. (68) Thus (67) is proved. To prove the equivalence (64), we use 𝑡^{min(𝑘,𝑛)/𝑛}𝑔(𝑡)≲𝑅_{min(𝑘,𝑛)}𝑔(𝑡),𝑔∈𝐷₁ and Minkowski’s inequality as follows: 𝜌^𝑝_𝐻𝑅_{min(𝑘,𝑛)}𝑔≲⁰_𝑗=−∞ℎ^𝑝_𝐻2^𝑗𝜌^𝑝_𝐻𝑡^{min(𝑘,𝑛)/𝑛}𝑔(𝑡),𝑔∈𝐷₁,𝛼_𝐻>0, (69) whence 𝜌_𝐸(𝐻)(𝑔)≲𝜌_𝐻(𝑡^{min(𝑘,𝑛)/𝑛}𝑔(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=𝐿¹_∗(𝑣), where 𝜌_𝐻(𝑔)=∫^∞₀𝑣(𝑡)𝑔(𝑡)𝑑𝑡/𝑡 and 𝜌_𝐻∈𝑁_𝑡. Using Theorem 15, we can construct an optimal domain 𝐸, where 𝜌_𝐸𝑔=𝜌_𝐻𝑅_{min(𝑘,𝑛)}𝑔=^∞₀𝑡^{min(𝑘,𝑛)/𝑛}𝑤(𝑡)𝑔(𝑡)𝑑𝑡𝑡 (70) and 𝑤(𝑡)=∫^∞_𝑡𝑣(𝑢)𝑑𝑢/𝑢. Hence 𝐸=Λ¹(𝑡^{min(𝑘,𝑛)/𝑛}𝑤), and this couple is optimal. Also 𝛼_𝐸=𝛽_𝐸=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 𝜌_𝐻(𝑔)=sup𝑣(𝑡)𝑔(𝑡) and 𝜌_𝐻∈𝑁_𝑡 and let 𝜌_𝐸𝑔=sup𝑣(𝑡)^𝑡₀𝑢^{min(𝑘,𝑛)/𝑛}𝑔(𝑢)𝑑𝑢𝑢. (71) Then by Theorem 15, the domain 𝐸 is optimal and the couple is optimal. In particular, the couple 𝐿^{𝑛/min(𝑘,𝑛),1},𝐶⁰ is optimal. If 𝑘=𝑛+𝑗, 𝑗≥0, this means that the embedding 𝑊₁^𝑛+𝑗↪𝐶^𝑗 is optimal.

Example 19. Let 𝐻 be as in the previous example. Since 𝜌_𝐸𝑔≤sup𝑡^{min(𝑘,𝑛)/𝑛}𝑤(𝑡)𝑔(𝑡),1𝑣(𝑡)=^𝑡₀1𝑤(𝑢)𝑑𝑢𝑢, (72) it follows that the couple 𝐸=Λ^∞(𝑡^{min(𝑘,𝑛)/𝑛}𝑤),𝐻=𝐿^∞_∗(𝑣) is admissible. In order to prove that 𝜌_𝐻 is optimal, take any 𝑔∈𝑀₀, and define ℎ from 𝑡^{min(𝑘,𝑛)/𝑛}𝑤(𝑡)ℎ(𝑡)=sup_𝑢≥𝑡𝑣(𝑢)𝑔(𝑢). Then ℎ∈𝐷₁ and 𝜌_𝐸1(ℎ)≲𝜌_𝐻(𝑔). On the other hand 𝑅_{min(𝑘,𝑛)}ℎ(𝑡)=^𝑡₀sup_𝑥≥𝑢𝑣(𝑥)𝑔(𝑥)1𝑤(𝑢)𝑑𝑢𝑢≥sup_𝑢≥𝑡𝑣(𝑢)𝑔(𝑢)1𝑣(𝑡)≥𝑔(𝑡). (73) Hence 𝜌_𝐻(𝐸)(𝑔)≤𝜌_𝐸1(ℎ)≲𝜌_𝐻(𝑔); therefore 𝜌_𝐻 is optimal.

Example 20 (case 𝑘≥𝑛). Let 𝐻=𝐿^𝑞_∗(𝑣), 0<𝑞≤∞, where 𝜌_𝐻(𝑔)=(∫^∞₀[𝑣(𝑡)𝑔(𝑡)]^𝑞𝑑𝑡/𝑡)^1/𝑞 and 𝜌_𝐻∈𝑁_𝑡 and let 𝑘≥𝑛. Using Remark 16, we can construct an optimal domain 𝐸=Γ^𝑞(𝑡𝑣) and this couple is optimal. Also 𝛼_𝐸=𝛽_𝐸=1 if 𝑣 is slowly varying.

4.2. Optimal Target Quasinorms

Definition 21 (construction of the optimal target quasi-norm). For a given domain quasinorm 𝜌_𝐸∈𝑁_𝑑, we set 𝜌_𝐻(𝐸)𝑔∶=inf𝜌_𝐸(ℎ)∶𝑔≤𝑅_{min(𝑘,𝑛)}ℎ,ℎ∈𝐷₁,𝑔∈𝑀_{min(𝑘,𝑛)}. (74) Note that 𝛼_𝐻(𝐸)=𝛽_𝐻(𝐸)=min(𝑘,𝑛)/𝑛−𝛼_𝐸.

Theorem 22. The target quasinorm 𝜌_𝐻(𝐸) belongs to 𝑁_𝑡, the couple 𝜌_𝐸,𝜌_𝐻(𝐸) is admissible, and the target quasinorm is optimal.

Proof. The property “𝜌_𝐻(𝐸)(𝑔)=0 implies 𝑔=0” follows from (30). Also, since 𝜌_𝐸∈𝑁_𝑑 it is easy to check that 𝜌_𝐻(𝐸)∈𝑁_𝑡. The couple is admissible since 𝜌_𝐻(𝐸)(𝑅_{min(𝑘,𝑛)}ℎ)≤𝜌_𝐸(ℎ), ℎ∈𝐷₁. Suppose that the couple 𝜌_𝐸, 𝜌_𝐻1∈𝑁_𝑡 is admissible. Then 𝜌_𝐻1(𝜒_(0,1)𝑅_{min(𝑘,𝑛)}ℎ)≲𝜌_𝐸(ℎ), ℎ∈𝐷₁. Therefore if 𝜒_(0,1)𝑔≤𝑅_{min(𝑘,𝑛)}ℎ, ℎ∈𝐷₁, then 𝜌_𝐻1(𝜒_(0,1)𝑔)≤𝜌_𝐻1(𝜒_(0,1)𝑅_{min(𝑘,𝑛)}ℎ)≲𝜌_𝐸(ℎ), whence 𝜌_𝐻1(𝜒_(0,1)𝑔)≲𝜌_𝐻(𝐸)(𝑔), 𝑔∈𝑀_{min(𝑘,𝑛)}. Hence 𝜌_𝐻(𝐸) is optimal.

Theorem 23 (supercritical case). If 𝛼_𝐸<min(𝑘,𝑛)/𝑛 and 𝜌_𝐸(𝜒_(1,∞)(𝑡)𝑡^{−min(𝑘,𝑛)/𝑛})<∞, then 𝜌_𝐻(𝐸)𝑔≈𝜌_𝐸𝑡^{−min(𝑘,𝑛)/𝑛}𝑔(𝑡),𝑔∈𝑀_{min(𝑘,𝑛)}. (75) Moreover, the couple 𝜌_𝐸,𝜌_𝐻(𝐸) is optimal.

Proof. If 𝑔≤𝑅_{min(𝑘,𝑛)}ℎ, ℎ∈𝐷₁, then, by Minkowski’s inequality and since min(𝑘,𝑛)/𝑛>𝛼_𝐸, it follows 𝜌_𝐸𝑡^{−min(𝑘,𝑛)/𝑛}𝑔(𝑡)≤𝜌_𝐸𝑡^{−min(𝑘,𝑛)/𝑛}𝑅_{min(𝑘,𝑛)}ℎ(𝑡)≲𝜌_𝐸(ℎ). (76) Hence, taking the infimum, we get 𝜌_𝐸(𝑡^{−min(𝑘,𝑛)/𝑛}𝑔(𝑡))≲𝜌_𝐻(𝐸)(𝑔).
On the other hand, for 𝑔∈𝑀_{min(𝑘,𝑛)}, we have 𝑔≲𝑅_{min(𝑘,𝑛)}ℎ, ℎ(𝑡)=𝑡^{−min(𝑘,𝑛)/𝑛}𝑔(𝑡). Since ℎ∈𝐷₁ it follows 𝜌_𝐻(𝐸)(𝑔)≲𝜌_𝐸(ℎ)≲𝜌_𝐸(𝑡^{−min𝑘,𝑛/𝑛}𝑔(𝑡)).
The domain quasinorm 𝜌_𝐸 is also optimal since for 𝑓∈𝐿¹(Ω), 𝜌_{𝐸(𝐻(𝐸))}𝑓^∗=𝜌_𝐻(𝐸)𝑅_{min(𝑘,𝑛)}𝑓^∗≈𝜌_𝐸𝑡^{−min(𝑘,𝑛)/𝑛}𝑅_{min(𝑘,𝑛)}𝑓^∗(𝑡)≳𝜌_𝐸𝑓^∗. (77)

Example 24. Consider the space 𝐸=Λ^𝑞(𝑤), 0<𝑞≤∞, min(𝑘,𝑛)/𝑛>𝛽_𝐸=𝛼_𝐸>0. Then by Theorem 23 the couple 𝐸,𝐻, 𝐻=𝐿^𝑞_∗(𝑡^{−min(𝑘,𝑛)/𝑛}𝑤) is optimal. In particular, using also Theorem 13, the embedding 𝑊^𝑘𝐿^𝑝,∞↪𝒞^{𝑘−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 Σ^𝑞(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=𝑙𝑔_𝑗, 𝑔_𝑗∈𝐴_𝑗, 𝐴_𝑗∶=(𝐴₀,𝐴₁)_{𝑎(𝑡)𝑡}^𝑗_−1/2,𝑞, with a quasinorm 𝑓_{Σ𝑞(𝑀(𝜎)(𝐴0,𝐴1)𝑎(𝑡)𝑡−𝜎,𝑞)}=inf^∞_𝑗=𝑙𝑀2^−𝑗𝑔_𝑗𝐴𝑗^𝑞1/𝑞, (78) where the infimum is taken with respect to all representations 𝑓=∑^∞_𝑗=𝑙𝑔_𝑗.

This space can be characterized as an interpolation space.

Theorem 25 (see [3]). Let 𝑎(𝑡)=𝑡^−𝜃𝑏(𝑡),  𝑏-slowly varying, 0<𝜃<1. Then Σ^𝑞𝑀(𝜎)𝐴₀,𝐴_{1𝑎(𝑡)𝑡−𝜎,𝑞}=𝐴₀,𝐴_1𝑤,𝑞, (79) where 1𝑤(𝑡)=1𝑎(𝑡)^𝜎0₀𝑡^𝜎𝑀(𝜎)^𝑟𝑑𝜎𝜎^1/𝑟 (80) 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=Λ^𝑞(𝑡^𝑘/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 𝜌_𝐸∈𝑁_𝑑 and 𝜌_𝐻∈𝑁_𝑡. Then this couple is admissible and the target quasinorm is optimal.

Proof. Step 1 (admissibility). Since 𝛼_𝐸=𝛽_𝐸=𝑘/𝑛<1, it will be enough to check that 𝜌_𝐻𝑅_𝑘𝑔^∗∗≲𝜌_𝐸𝑔^∗∗, (81) where 𝜌_𝐸𝑔=¹₀𝑡^𝑘/𝑛𝑐(𝑡)(1−ln𝑡)𝑔(𝑡)^𝑞𝑑𝑡𝑡^1/𝑞,𝜌_𝐻𝑔=¹₀𝑐(𝑡)𝑔(𝑡)^𝑞𝑑𝑡𝑡^1/𝑞. (82)
Applying Minkowski’s inequality we obtain for 0<𝜎<𝜎₀<𝑘/𝑛,  𝑏-slowly varying weight, 𝜎𝑅_𝑘𝑔^∗∗_𝐿^𝑞_{∗(𝑏(𝑡)𝑡−𝜎)}≲𝑔_{Γ𝑞(𝑡𝑘/𝑛−𝜎𝑏(𝑡))}. (83) In order to extrapolate these inequalities, we write Γ^𝑞𝑡^{𝑘/𝑛−𝜎}𝑏(𝑡)=𝐿¹,𝐿^∞_{𝑏(𝑡)𝑡𝑘/𝑛−1−𝜎,𝑞},𝐿^𝑞_∗𝑡^−𝜎𝑏(𝑡)=𝐿^𝑞_∗𝑡^1/2𝑏(𝑡),𝐿^𝑞_∗𝑡^−1/2𝑏(𝑡)_{1/2+𝜎,𝑞}𝜎₀<12. (84) This is true since 𝐾𝑡,𝑔;𝐿^𝑞_∗𝑤₀,𝐿^𝑞_∗𝑤₁≈¹₀𝑔(𝑢)min𝑤₀(𝑢),𝑡𝑤₁(𝑢)^𝑞𝑑𝑢𝑢^1/𝑞,0<𝑡<1. (85) Let 𝜎=2^−𝑗 and 𝑔=∑𝑔_𝑗 (convergence in 𝐿¹), where 𝑔_𝑗∈𝐿^∞. Then 𝑔^∗∗≤∑𝑔_𝑗^∗∗, whence 𝑅_𝑘(𝑔^∗∗)≤∑𝑅_𝑘(𝑔_𝑗^∗∗) and for 𝑀(𝜎)=𝜎⁻²,𝑝=min(𝑞,1), we have 𝐾^𝑝𝑡,𝑅_𝑘𝑔^∗∗;𝐵₀,𝐵₁≤𝐶_𝜈∶=_𝑗≥𝑙𝐾^𝑝𝑡,𝑅_𝑘𝑔_𝑗^∗∗;𝐵₀,𝐵₁, (86) where 𝐵₀=𝐿^𝑞_∗(𝑡^1/2𝑏(𝑡)),𝐵₁=𝐿^𝑞_∗(𝑡^−1/2𝑏(𝑡)). We can write 𝐶_𝜈=_𝑗≥𝑙𝑡^{−1/2−2−𝑗}2^−𝑗𝑀2^−𝑗𝐾𝑡,𝑅_𝑘𝑔_𝑗^∗∗;𝐵₀,𝐵₁^𝑝×𝑡^1/2+2−𝑗2^−𝑗𝑀2^−𝑗𝑝 (87) and using also Hölder’s inequality if 𝑞>1, we get [𝑣(𝑡)]^𝑝𝐶_𝜈≤_𝑗≥𝑙𝑡^{−1/2−2−𝑗}2^−𝑗𝑀2^−𝑗𝐾𝑡,𝑅_𝑘𝑔_𝑗^∗∗;𝐵₀,𝐵₁^𝑝, (88) where 1𝑣(𝑡)=_𝑗≥𝑙𝑡^1/2+2−𝑗2^−𝑗𝑀2^{−𝑗𝑟1/𝑟}. (89) Hence 𝑅_𝑘𝑔^∗∗_{(𝐵0,𝐵1)𝑣,𝑞}≲_𝑗≥𝑙2^−𝑗𝑀2^−𝑗𝑅_𝑘𝑔_𝑗^∗∗_{(𝐵0,𝐵1)1/2+2−𝑗,𝑞}^𝑞1/𝑞. (90) Since 2^−𝑗𝑅_𝑘𝑔_𝑗^∗∗_{(𝐵0,𝐵1)1/2+2−𝑗,𝑞}≲𝑔_{𝑗Γ𝑞(𝑡𝑘/𝑛−2−𝑗𝑏(𝑡))}, (91) we get 𝑅_𝑘𝑔^∗∗_{(𝐵0,𝐵1)𝑣,𝑞}≲𝑔_{Σ𝑞(𝑀(𝜎)(𝐿1,𝐿∞)𝑏(𝑡)𝑡𝑘/𝑛−1−𝜎,𝑞)}, (92) whence 𝑅_𝑘∶𝐿¹,𝐿^∞_𝑤,𝑞↦𝐿^𝑞_∗𝑡^1/2𝑏(𝑡),𝐿^𝑞_∗𝑡^−1/2𝑏(𝑡)_𝑣,𝑞, (93) where 𝑤 is given by (80) with 𝑎(𝑡)=𝑏(𝑡)𝑡^{𝑘/𝑛−1} and 𝑀(𝜎)=𝜎⁻². It is easy to calculate these weights, see [3]. We have 𝑤(𝑡)≈𝑏(𝑡)𝑡^{𝑘/𝑛−1}(1−ln𝑡)²,𝑣(𝑡)≈𝑡^−1/2(1−ln𝑡),0<𝑡<1. (94) Then for 𝑏(𝑡)=𝑐(𝑡)(1−ln𝑡)⁻¹ we get Γ^𝑞𝑡^𝑘/𝑛𝑐(𝑡)(1−ln𝑡)↪𝐿¹,𝐿^∞_𝑤,𝑞,𝐿^𝑞_∗𝑡^1/2𝑏(𝑡),𝐿^𝑞_∗𝑡^−1/2𝑏(𝑡)_𝑣,𝑞↪𝐿^𝑞_∗(𝑐). (95) Hence (81) is proved.

Step 2 (optimality of the target quasi-norm). We want to prove that 𝜌_𝐻 is an optimal target quasi-norm. It is sufficient to see that 𝜌_𝐻(𝐸)𝜒_(0,1)𝑔≲𝜌_𝐻𝑔,𝑔∈𝑀_𝑘, (96) where 𝜌_𝐻(𝐸) is defined by (74). To this end for any such 𝑔 we construct an ℎ∈𝐷₁ such that 𝜒_(0,1)𝑔≲𝑅_𝑘ℎ and 𝜌_𝐸(ℎ)≲𝜌_𝐻(𝑔). Let 0<𝑞<∞. (The case 𝑞=∞ is analogous, see Example 19.) Let ℎ^𝑞(𝑡)=𝜒_(0,2)(𝑡)∫²_𝑡(1−ln𝑢)^−𝑞𝑢^−𝑞𝑔^𝑞(√𝑢𝑒)𝑑𝑢/𝑢. Then ℎ∈𝐷₁ and 𝜌_𝐸(ℎ)≲𝜌_𝐻(𝑔). On the other hand, for 0<𝑡<1, 𝑅_𝑘ℎ(𝑡)≥^𝑡_𝑡2/𝑒𝑔√𝑒𝑢1−ln𝑢𝑑𝑢𝑢≳𝜒_(0,1)(𝑡)𝑔(𝑡), (97) since ∫^𝑡_𝑡2/𝑒(1−ln𝑢)⁻¹𝑑𝑢/𝑢=ln2. Then by the definition of 𝜌_𝐻(𝐸) we get 𝜌_𝐻(𝐸)𝜒_(0,1)𝑔≲𝜌_𝐸(ℎ)≲𝜌_𝐻𝑔. (98)

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, ¹_𝑡𝑐^𝑞(𝑢)𝑑𝑢𝑢^1/𝑞𝑡₀[𝑏(𝑢)]^−𝑟𝑑𝑢𝑢^1/𝑟1≲1,𝑞+1𝑟=1,0<𝑡<1. (99) Then the operator 𝑅_𝑛 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=Λ^𝑞(𝑡𝑐(𝑡)(1−ln𝑡)), 𝑐-decreasing slowly varying weight, 𝑐(+0)=∞, 𝑐(𝑡²)≈𝑐(𝑡), 1≤𝑞≤∞, 𝐻=𝐿^𝑞_∗(𝑐). We suppose that 𝜌_𝐸∈𝑁_𝑑 and 𝜌_𝐻∈𝑁_𝑡. 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.