Weighted Hardy Operators in Complementary Morrey Spaces

Abstract

We study the weighted 𝑝→𝑞-boundedness of the multidimensional weighted Hardy-type operators 𝐻^𝛼_𝑤 and ℋ^𝛼_𝑤 with radial type weight 𝑤=𝑤(|𝑥|), in the generalized complementary Morrey spaces ^∁ℒ^𝑝,𝜓_{0}(ℝ^𝑛) defined by an almost increasing function 𝜓=𝜓(𝑟). We prove a theorem which provides conditions, in terms of some integral inequalities imposed on 𝜓 and 𝑤, for such a boundedness. These conditions are sufficient in the general case, but we prove that they are also necessary when the function 𝜓 and the weight 𝑤 are power functions. We also prove that the spaces ^∁ℒ^𝑝,𝜓_{0}(Ω) over bounded domains Ω are embedded between weighted Lebesgue space 𝐿^𝑝 with the weight 𝜓 and such a space with the weight 𝜓, perturbed by a logarithmic factor. Both the embeddings are sharp.

1. Introduction

Hardy operators and related Hardy inequalities are widely studied in various function spaces, and we refer to the books [1–4] and references therein. They continue to attract attention of researchers both as an interesting mathematical object and a useful tool for many purposes: see for instance the recent papers [5, 6]. Results on weighted estimations of Hardy operators in Lebesgue spaces may be found in the abovementioned books. In the papers [7–9] the weighted boundedness of the Hardy type operators was studied in Morrey spaces.

In this paper we study multi-dimensional weighted Hardy operators 𝐻^𝛼_𝑤𝑓(𝑥)∶=|𝑥|^𝛼−𝑛𝑤(|𝑥|)_{|𝑦|<|𝑥|}𝑓𝑦𝑑𝑦𝑤𝑦,ℋ^𝛼_𝑤𝑓(𝑥)∶=|𝑥|^𝛼𝑤(|𝑥|)_{|𝑦|>|𝑥|}𝑓𝑦𝑑𝑦𝑦^𝑛𝑤𝑦, (1.1) where 𝛼≥0, in the so called complementary Morrey spaces. The one-dimensional case will include the versions 𝐻^𝛼_𝑤𝑓(𝑥)=𝑥^𝛼−1𝑤(𝑥)^𝑥₀𝑓(𝑡)𝑑𝑡𝑤(𝑡),ℋ^𝛼_𝑤𝑓(𝑥)=𝑥^𝛼𝑤(𝑥)^∞_𝑥𝑓(𝑡)𝑑𝑡𝑡𝑤(𝑡),𝑥>0 (1.2) adjusted for the half-axis ℝ¹₊, so that in the sequel ℝ^𝑛 with 𝑛=1 may be read either as ℝ¹ or ℝ¹₊.

The classical Morrey spaces ℒ^𝑝,𝜆(Ω),Ω⊆ℝ^𝑛, defined by the norm 𝑓_𝑝,𝜆∶=sup_{𝑥∈Ω,𝑟>0}1𝑟^𝜆𝐵_(𝑥,𝑟)𝑓𝑦^𝑝𝑑𝑦^1/𝑝,1≤𝑝<∞,0≤𝜆≤𝑛, (1.3) where 𝐵(𝑥,𝑟)=Ω∩𝐵(𝑥,𝑟), are well known, in particular, because of their usage in the study of regularity properties of solutions to PDE; see for instance the books [10–12] and references therein. There are also known various generalizations of the classical Morrey spaces ℒ^𝑝,𝜆, and we refer for instance to the surveying paper [13]. One of the direct generalizations is obtained by replacing 𝑟^𝜆 in (1.3) by a function 𝜑(𝑟), usually satisfying some monotonicity type conditions. We also denote it as ℒ^𝑝,𝜑(Ω) without danger of confusion. Such spaces appeared in [14, 15] and were widely studied in [16, 17]. The spaces ℒ^𝑝,𝜑_loc(Ω), defined by the norm 𝑓_{𝑝,𝜑;loc}∶=sup_𝑟>01𝜑(𝑟)𝐵_(𝑥0,𝑟)𝑓𝑦^𝑝𝑑𝑦^1/𝑝, (1.4) where 𝑥₀∈Ω, are known as generalized local Morrey spaces

Hardy type operators (1.1) in the spaces ℒ^𝑝,𝜑(Ω),ℒ^𝑝,𝜑_loc(Ω) have been studied in [7–9].

The norm in Morrey spaces controls the smallness of the integral ∫_{𝐵(𝑥,𝑟)}|𝑓(𝑦)|^𝑝𝑑𝑦 over small balls 𝐵(𝑥,𝑟) (and also a possible growth of this integral for 𝑟→∞ in the case Ω is unbounded). There are also known spaces ^∁ℒ^𝑝,𝜓_{𝑥0}(Ω), called complementary Morrey spaces, with the norm controlling possible growth, as 𝑟→0, of the integral _{Ω⧵𝐵(𝑥0,𝑟)}𝑓𝑦^𝑝𝑑𝑦 (1.5) over exterior of balls. Such spaces have sense in the local setting only. It is introduced in [16, 17] that the space ^∁ℒ^𝑝,𝜓_{𝑥0}(Ω) is defined as the space of all functions 𝑓∈𝐿^𝑝_loc(Ω⧵{𝑥₀}) with the finite norm 𝑓_∁ℒ0}^𝑝,𝜓_{𝑥(Ω)=sup_𝑟>0𝜓^1/𝑝(𝑟)𝑓_{𝐿𝑝(Ω⧵𝐵(𝑥0,𝑟))}, (1.6) where admission of the multiplier 𝜓(𝑟) vanishing at 𝑟=0 controls the growth of the norm ‖𝑓‖_𝐿^𝑝_{(Ω⧵𝐵(𝑥0,𝑟))}. Such spaces were also studied in some later papers, and we refer for instance to [18]. We refer also to the paper [19] where variable exponent complementary spaces of such type were introduced.

During the last decades various classical operators, such as maximal, singular, and potential operators were widely investigated both in classical and generalized Morrey spaces, including the complementary Morrey spaces. The mapping properties of Hardy type operators in complementary Morrey spaces were not known. In this paper we obtain conditions for the 𝑝→𝑞-boundedness of Hardy type operators in complementary Morrey spaces. Thus we make certain contributions to the known theory of Hardy type inequalities; see, for example, the books [2–4] and references given there.

We also prove a new property for the generalized complementary Morrey spaces; see Theorem 3.1, by showing that the spaces ^∁ℒ^𝑝,𝜓_{0}(Ω) over bounded domains Ω are embedded between the weighted Lebesgue space 𝐿^𝑝 with the weight 𝜓, and such a space with the weight 𝜓, perturbed by a logarithmic factor. In the case where 𝜓 was a power function, this was proved in [19].

The paper is organized as follows. In Section 2 we give definitions and necessary preliminaries, including conditions for radial functions to belong to complementary Morrey spaces. In Section 3 we prove the abovementioned embedding of the generalized Morrey space between Lebesgue weighted spaces. In Section 4, which plays a crucial role in the preparation of the proofs of the main results, we prove pointwise estimate for the Hardy-type constructions via the norm defining the complementary Morrey space. In Section 5 we give the final theorems on the weighted 𝑝→𝑞-boundedness of Hardy operators in complementary Morrey spaces. In the appendix we collect various properties of weights from the Bary-Stechkin class which we need when we formulate some sufficient conditions for the boundedness in terms of the Matuszewska-Orlicz indices of the functions 𝜓 and 𝑤.

2. Complementary Morrey Spaces and Their Properties

2.1. Definitions

Let Ω be an open set in ℝ^𝑛, Ω⊆ℝ^𝑛 and ℓ=diamΩ, 0<ℓ≤∞, 𝐵(𝑥,𝑟)={𝑦∈ℝ^𝑛∶|𝑥−𝑦|<𝑟} and 𝐵(𝑥,𝑟)=𝐵(𝑥,𝑟)∩Ω. Let also 𝜓(𝑟) be a continuous function nonnegative on (0,ℓ) with 𝜓(0)∶=lim_𝑟→0𝜓(𝑟)=0 which may arbitrarily grow as 𝑟→∞. Let also 1≤𝑝<∞.

Definition 2.1. Let 1≤𝑝<∞. The complementary Morrey space ^∁ℒ^𝑝,𝜓_{𝑥0}(Ω), where 𝑥₀∈Ω, is the space of functions 𝑓∈𝐿^𝑝_loc(Ω⧵{𝑥₀}) such that 𝑓_∁ℒ0}^𝑝,𝜓_{𝑥(Ω)∶=sup_𝑟>0𝜓(𝑟)_Ω⧵𝐵_(𝑥0,𝑟)𝑓𝑦^𝑝𝑑𝑦^1/𝑝<∞. (2.1)

In the case of power factor 𝜓(𝑟)≡𝑟^𝜆, 𝜆>0, we also denote ^∁ℒ_𝑥^𝑝,𝜆₀(Ω)∶=^∁ℒ^𝑝,𝜓_{𝑥0}(Ω)|_{𝜓≡𝑟𝜆} (2.2) without danger of confusion.

The space ^∁ℒ^𝑝,𝜓_{𝑥0}(Ω) is nontrivial in the case of any locally bounded function 𝜓(𝑟) with an arbitrary behaviour at infinity, because bounded functions with compact support belong then to this space.

We will also use the following notation for the modular: ℳ_𝑝,𝜓𝑓;𝑥₀,𝑟∶=𝜓(𝑟)_Ω⧵𝐵_(𝑥0,𝑟)𝑓𝑦^𝑝𝑑𝑦. (2.3)

The function 𝜓, defining the complementary Morrey space, will be called definitive function.

Remark 2.2. The function 𝜑 in (1.4), defining the Morrey spaces, in some papers is called weight function. We prefer to call both 𝜑 and 𝜓 as definitive functions, keeping the word weight for its natural use in the theory of function spaces, that is, for the cases where the function 𝑓 itself is controlled by a weight.

2.2. On Belongness of Radial Functions to the Space ^∁ℒ^𝑝,𝜓_{𝑥0}(Ω)

Lemma 2.3. For a nonnegative radial type functions 𝑢=𝑢(|𝑥−𝑥₀|), 𝑥₀∈Ω, the condition sup_0<𝑟<ℓ𝜓(𝑟)^ℓ_𝑟𝑢(𝑡)^𝑝𝑡^𝑛−1𝑑𝑡<∞ (2.4) is sufficient to belong to ^∁ℒ^𝑝,𝜓_{𝑥0}(Ω). (It is also necessary, when Ω=ℝ^𝑛 or Ω is a ball centered at 𝑥₀.)

The proof of Lemma 2.3 is obvious. By means of this lemma we easily obtain the following corollary.

Corollary 2.4. Let Ω be bounded. A power function |𝑥−𝑥₀|^𝛾 with 𝛾≠−𝑛/𝑝 belongs to the space ^∁ℒ^𝑝,𝜓_{𝑥0}(Ω) if and only if sup_𝑟>0𝑟^{min{𝑛+𝛾𝑝,0}}𝜓(𝑟)<∞. (2.5) In the case 𝑛+𝛾𝑝=0, the same holds with 𝑟^{min{𝑛+𝛾𝑝,0}} replaced by |ln𝑟|. When Ω=ℝ^𝑛, the necessary and sufficient conditions are 𝑛+𝛾𝑝<0 and sup_𝑟>0𝑟^{𝑛+𝛾𝑝}𝜓(𝑟)<∞.

As is known, there may be given sufficient numerical inequalities for the validity of the integral condition in (2.4) in terms of Matuszewska-Orlicz indices 𝑚(𝑢) and 𝑀(𝑢) of the function 𝑢. We give below such sufficient inequalities, but in order not to interrupt the main body of the paper by notions connected with such indices and related Bary-Stechkin function classes ℤ_𝛼,𝛽, we put these notions in the appendix.

Let now 𝑢 be a nonnegative function in Bary-Stechkin class, namely, 𝑢∈ℤ_−𝑛/𝑝([0,ℓ])ifℓ<∞,𝑢∈ℤ_{−𝑛/𝑝,−𝑛/𝑝}ℝ¹₊ifℓ=∞. (2.6) Then ∫^ℓ_𝑟𝑢^𝑝(𝑡)𝑡^𝑛−1𝑑𝑡∼𝑟^𝑛𝑢^𝑝(𝑟), 0<𝑟<ℓ, so that the condition sup_0<𝑟<ℓ𝑟^𝑛𝜓(𝑟)𝑢^𝑝(𝑟)<∞ is sufficient for 𝑢=𝑢(|𝑥−𝑥₀|) to be in ^∁ℒ^𝑝,𝜓_{𝑥0}(Ω). Note also that (2.6) is equivalent to the inequalities 𝑝𝑀(𝑢)+𝑛<0, 𝑝𝑀_∞(𝑢)+𝑛<0 in terms of the indices, where the second inequality is to be used only in the case ℓ=∞; see (A.17).

3. Complementary Morrey Spaces Are Closely Embedded between Weighted Lebesgue Spaces

The complementary Morrey spaces are close to a certain weighted Lebesgue space as stated in the following theorem, which provides sharp embeddings of independent interest. The notation for the weighted space is taken in the form 𝐿^𝑝(Ω,𝜚)∶={𝑓∶∫_Ω𝜚(𝑥)|𝑓(𝑥)|^𝑝𝑑𝑥<∞}. The class 𝑊₀([0,ℓ]), used in this theorem, is defined in the appendix; see Definitions A.1-A.2 therein. In the case where 𝜓(𝑟) is a power function, Theorem 3.1 was proved in [19].

Theorem 3.1. Let Ω be a bounded open set, 1≤𝑝<∞, 𝜓∈𝑊₀([0,ℓ]), ℓ=diamΩ, and let 𝜓 be absolutely continuous and 𝑃∶=sup_0<𝑡<ℓ𝑡𝜓^′(𝑡)𝜓(𝑡)<∞. (3.1) Then 𝐿^𝑝Ω,𝜓𝑦−𝑥₀↪^∁ℒ_𝑥^𝑝,𝜓₀(Ω)↪_𝜀>0𝐿^𝑝Ω,𝜓_𝜀𝑦−𝑥₀, (3.2) where 𝜓_𝜀(𝑡)=𝜓(𝑡)/(ln(𝐴/𝑡))^1+𝜀, 𝐴>ℓ. The left-hand side embedding in (3.2) is strict, when 𝜓 satisfies the condition ^ℓ_𝑟𝑑𝑡(𝑡)𝐶𝑡𝜓𝑑𝑡≤𝜓(𝑟) (3.3) (in particular, if 𝜓(𝑟)=𝑟^𝜆, 𝜆>0); that is, there exists a function 𝑓₀=𝑓₀(𝑥) such that 𝑓₀∈^∁ℒ_𝑥^𝑝,𝜓₀(Ω),but𝑓₀∉𝐿^𝑝Ω,𝜓𝑦−𝑥₀. (3.4) The right-hand side embedding in (3.2) is strict, when 𝜓 satisfies the doubling condition 𝜓(2𝑟)≤𝐶𝜓(𝑟), (in particular, if 𝜓(𝑟)=𝑟^𝜆, 𝜆≥0), there exists a function 𝑔₀=𝑔₀(𝑥) such that 𝑔₀∈_𝜀>0𝐿^𝑝Ω,𝜓_𝜀𝑥−𝑥₀,but𝑔₀∉^∁ℒ_𝑥^𝑝,𝜓₀(Ω). (3.5)

Proof. Without loss of generality, we may take 𝑥₀=0 for simplicity, supposing that 0∈Ω.
1⁰. The left-hand side embedding: since the function 𝜓 is almost increasing, we have _Ω𝜓𝑦𝑓𝑦|^𝑝𝑑𝑦^1/𝑝≥_Ω⧵𝐵_(0,𝑟)𝜓𝑦𝑓𝑦|^𝑝𝑑𝑦^1/𝑝≥𝐶𝜓(𝑟)_Ω⧵𝐵_(0,𝑟)𝑓𝑦^𝑝𝑑𝑦^1/𝑝 (3.6) so that 𝑓_{𝐿𝑝(Ω,𝜓(|𝑦|))}≥𝐶𝑓_∁ℒ^𝑝,𝜓_{0}(Ω). (3.7)
2⁰. The right-hand side embedding: we have 𝐵_(0,𝑟)𝑓𝑦^𝑝𝜓_𝜀𝑦𝐵𝑑𝑦=_(0,𝑟)𝑓𝑦^𝑝₀^|𝑦|𝑑𝜓𝑑𝑡^𝜀(𝑡)𝑑𝑡𝑑𝑦, (3.8) where 𝜓^′_𝜀(𝑡)=𝜓(𝑡)𝑡𝑡𝜓^′(𝑡)𝜓(𝑡)+1+𝜀()1ln𝐴/𝑡(ln(𝐴/𝑡))^1+𝜀, (3.9) so that 𝜓^′_𝜀(𝑡≤𝐶)^𝜀𝜓(𝑡)𝑡(ln(𝐴/𝑡))^1+𝜀,𝐶_𝜀=𝑃+1+𝜀().ln𝐴/ℓ (3.10) Therefore, 𝐵_(𝑥0,𝑟)𝑓𝑦^𝑝𝜓_𝜀𝑦𝑑𝑦≤^𝑡₀𝜓^′_𝜀(𝑡)_{{𝑦∈Ω∶𝑡<|𝑥0−𝑦|<𝑟}}𝑓𝑦^𝑝𝑑𝑦𝑑𝑡≤^ℓ₀𝜓^′_𝜀(𝑡)⋅𝑓^𝑝_{𝐿𝑝Ω⧵}𝐵_𝑥0,𝑡≤𝑓𝑑𝑠_{𝑝∁ℒ𝑥0𝑝,𝜓(Ω)ℓ0}𝜓^′_𝜀(𝑡)𝜓(𝑡)𝑑𝑡, (3.11) where the last integral converges when 𝜀>0 since |𝜓^′_𝜀(𝑡)|/𝜓(𝑡)≤𝐶_𝜀/(𝑡(ln(𝐴/𝑡))^1+𝜀).
3⁰. The strictness of the embeddings: the corresponding counterexamples are 𝑓₀(𝑥)=1𝜓^1/𝑝(|𝑥|)|𝑥|^𝑛/𝑝,𝑔₀(𝑥)=ln(ln(𝐵/|𝑥|))|𝑥|^𝑛/𝑝𝜓^1/𝑝(|𝑥|),𝐵>ℓ𝑒^𝑒. (3.12)
Calculations for the function 𝑓₀, which is obviously not in 𝐿^𝑝(Ω,𝜓(|𝑦|), are easy. Indeed, 𝑓₀^𝑝_{∁ℒ𝑝,𝜓{0}(Ω)}≤𝕊^𝑛−1𝜓(𝑟)^ℓ_𝑟𝑑𝑡(𝑡),𝑡𝜓 (3.13) where the right-hand side is bounded by (3.3).
In the case of the function 𝑔₀ we have 𝑔₀^𝑝_{𝐿𝑝Ω,𝜓𝜀}=_Ωln^𝑝(ln(𝐵/|𝑥|))|𝑥|^𝑛(ln(𝐴/|𝑥|))^1+𝜀𝑑𝑥≤𝕊^𝑛−1ℓ₀ln^𝑝(ln(𝐵/𝑡))𝑡(ln(𝐴/𝑡))^1+𝜀𝑑𝑡<∞ (3.14) for every 𝜀>0. However, for small 𝑟∈(0,𝛿/2), where 𝛿=dist(0,𝜕Ω), we obtain 𝜓(𝑟)_{𝑥∈Ω∶|𝑥|>𝑟}𝑔^𝑝₀(|𝑥|)𝑑𝑥≥𝜓(𝑟)_{𝑥∈Ω∶𝑟<|𝑥|<𝛿}ln^𝑝(ln(𝐵/|𝑥|))𝑑𝑥|𝑥|^𝑛𝜓𝑥=𝕊(||)_𝑛−1𝜓(𝑟)^𝛿_𝑟ln^𝑝(ln(𝐵/𝑡))𝑑𝑡(𝑡)≥𝕊𝑡𝜓_𝑛−1𝜓(𝑟)_𝑟^2𝑟ln^𝑝(ln(𝐵/𝑡))𝑑𝑡(𝑡).𝑡𝜓 (3.15)
Taking into account that 𝜓(𝑡) is almost increasing and satisfies the doubling condition, we get 𝜓(𝑟)_𝑟^2𝑟ln^𝑝(ln(𝐵/𝑡))𝑑𝑡(𝑡)𝜓(𝑟)𝑡𝜓≥𝐶𝜓(2𝑟)ln^𝑝𝐵ln2𝑟_𝑟^2𝑟𝑑𝑡𝑡≥𝐶ln^𝑝𝐵ln𝑟→∞as𝑟→0, (3.16) which completes the proof of the theorem.

Remark 3.2. Under the assumption (3.3), the upper embedding in (3.2) does not hold with 𝜀=0. The corresponding counterexample is 𝑓(𝑥)=1/(|𝑥−𝑥₀|^𝑛𝜓(|𝑥−𝑥₀|)) which is in ^∁ℒ^𝑝,𝜓_{𝑥0}(Ω), but 𝑓∉𝐿^𝑝(Ω,𝜓_𝜀(|𝑦−𝑥₀|))|_𝜀=0.

4. Weighted Estimates of Functions in Complementary Morrey Spaces

In the following lemmas we give a pointwise estimate of the “Hardy-type” constructions in terms of the modular 𝔐_𝑝,𝜓(𝑓;𝑥₀,𝑟) with 𝑥₀=0. These estimates are of independent interest and also crucial for our study of the Hardy operators in the complementary Morrey space.

Lemma 4.1. Let 1≤𝑝<∞, 0<𝑠≤𝑝, let 𝑣,𝜓^𝑠/𝑝𝑣∈𝑊([0,∞]), and 𝑣(2𝑡)≤𝑐𝑣(𝑡). Then _|𝑧|<𝑟𝑓(𝑧)^𝑠𝑣(|𝑧|)𝑑𝑧≤𝐶^𝑟₀𝑉_𝑠(𝑡)𝑡𝑑𝑡⋅ℳ^𝑠/𝑝_𝑝,𝜓𝑓;0,𝑟,𝑟>0, (4.1) for 𝑓∈𝐿^𝑝_loc(Ω⧵{0}), where 𝐶>0 does not depend on 𝑓 and 𝑟∈(0,∞) and 𝑉_𝑠(𝑡)=𝑡^{𝑛(1−(𝑠/𝑝))}𝑣(𝑡)𝜓^𝑠/𝑝(𝑡). (4.2)

Proof. We use the dyadic decomposition as follows: _|𝑧|<𝑟𝑓(𝑧)^𝑠𝑣(|𝑧|)𝑑𝑧=^∞_{𝑘=0𝐵𝑘(𝑟)}𝑓(𝑧)^𝑠𝑣(|𝑧|)𝑑𝑧, (4.3) where 𝐵_𝑘(𝑟)={𝑧∶2^−𝑘−1𝑟<|𝑧|<2^−𝑘𝑟}. Since there exists a 𝛽 such that 𝑡^𝛽𝑣(𝑡) is almost increasing, we observe that 1𝑣𝑧≤𝐶(||)𝑣2^−𝑘−1𝑟 (4.4) on 𝐵_𝑘(𝑟). Applying this in (4.3) and making use of the Hölder inequality with the exponent 𝑝/𝑠≥1, we obtain _|𝑧|<𝑟𝑓(𝑧)^𝑠𝑣(|𝑧|)𝑑𝑧≤𝐶^∞_𝑘=02^−𝑘−1𝑟^{𝑛(1−(𝑠/𝑝))}𝑣2^−𝑘−1𝑟^𝐵_𝑘^(𝑟)𝑓(𝑧)^𝑝𝑑𝑧^𝑠/𝑝. (4.5) Hence _|𝑧|<𝑟𝑓(𝑧)^𝑠𝑣(|𝑧|)𝑑𝑧≤𝐶^∞_𝑘=02^−𝑘−1𝑟^{𝑛(1−(𝑠/𝑝))}ℳ^𝑠/𝑝_𝑝,𝜓𝑓;0,2^−𝑘−1𝑟𝑣2^−𝑘−1𝑟𝜓^𝑠/𝑝2^−𝑘−1𝑟. (4.6) On the other hand we have ^𝑟₀𝑉_𝑠(𝑡)𝑡ℳ^𝑠/𝑝_𝑝,𝜓𝑓;0,𝑡𝑑𝑡=^∞_𝑘=0^2−𝑘𝑟_{2−𝑘−1𝑟}𝑡^{𝑛(1−(𝑠/𝑝))−1}ℳ^𝑠/𝑝_𝑝,𝜓𝑓;0,𝑡𝑣(𝑡)𝜓^𝑆/𝑝(𝑡)𝑑𝑡. (4.7) The function 1/(𝜓(𝑡))ℳ_𝑝,𝜓(𝑓;0,𝑡) is decreasing, and the function 1/(𝑣(𝑡)) is almost decreasing after multiplication by some power functions. Therefore, ^𝑟₀𝑉_𝑠(𝑡)𝑡ℳ^𝑠/𝑝_𝑝,𝜓𝑓;0,𝑡𝑑𝑡≥𝐶^∞_𝑘=02^−𝑘𝑟^{𝑛(1−(𝑠/𝑝))}ℳ^𝑠/𝑝_𝑝,𝜓𝑓;0,2^−𝑘𝑟𝑣2^−𝑘𝑟𝜓^𝑆/𝑝2^−𝑘𝑟≥𝐶^∞_𝑘=12^−𝑘𝑟^{𝑛(1−(𝑠/𝑝))}ℳ^𝑠/𝑝_𝑝,𝜓𝑓;0,2^−𝑘𝑟𝑣2^−𝑘𝑟𝜓^𝑠/𝑝2^−𝑘𝑟=𝐶₁^∞_𝑘=02^−𝑘−1𝑟^{𝑛(1−(𝑠/𝑝))}ℳ^𝑠/𝑝_𝑝,𝜓𝑓;0,2^−𝑘−1𝑟𝑣2^−𝑘−1𝑟𝜓^𝑠/𝑝2^−𝑘−1𝑟. (4.8) Then (4.1) follows from (4.6) and (4.8).

Lemma 4.2. Let 1≤𝑝<∞, 0≤𝑠≤𝑝, 𝑣∈𝑊(ℝ¹₊) and 𝑣(2𝑡)≤𝐶𝑣(𝑡), 𝜓(2𝑡)≤𝐶𝜓(𝑡). Let also 𝑉^𝑠(𝑡)𝑡∶=^{𝑛(1−(𝑠/𝑝))}𝑣(𝑡)𝜓^𝑠/𝑝(𝑡). (4.9) Then _|𝑧|>𝑟𝑣(|𝑧|)𝑓(𝑧)^𝑠𝑑𝑧≤𝐶^∞_𝑟𝑉^𝑠(𝑡)𝑡ℳ^𝑠/𝑝_𝑝,𝜓𝑓;0,𝑡𝑑𝑡, (4.10) where 𝐶>0 does not depend on 𝑟>0 and 𝑓.

Proof. We use the corresponding dyadic decomposition: _|𝑧|>𝑟𝑣(𝑡)𝑓(𝑧)^𝑠𝑑𝑧=^∞_{𝑘=0𝐵𝑘(𝑟)}𝑣(𝑧)𝑓(𝑧)^𝑠𝑑𝑧, (4.11) where 𝐵^𝑘(𝑟)={𝑧∶2^𝑘𝑟<|𝑧|<2^𝑘+1𝑟}. Since there exists a 𝛽∈ℝ¹ such that 𝑡^𝛽𝑣(𝑡) is almost increasing, we obtain ^∞_{𝑘=0𝐵𝑘(𝑟)}𝑣(|𝑧|)𝑓(𝑧)^𝑠𝑑𝑧≤𝐶^∞_𝑘=0𝑣2^𝑘+1𝑟_{𝐵𝑘(𝑟)}𝑓(𝑧)^𝑠𝑑𝑧, (4.12) where 𝐶 may depend on 𝛽 but does not depend on 𝑟 and 𝑓. Applying the Hölder inequality with the exponent 𝑝/𝑠, we get _|𝑧|>𝑟𝑣(|𝑧|)𝑓(𝑧)^𝑠𝑑𝑧≤𝐶^∞_𝑘=0𝑣2^𝑘+1𝑟2^𝑘𝑟^{𝑛(1−(𝑠/𝑝))}_{𝐵𝑘(𝑟)}𝑓(𝑧)^𝑝𝑑𝑧^𝑠/𝑝≤𝐶^∞_𝑘=0𝑣2^𝑘+1𝑟2^𝑘𝑟^{𝑛(1−(𝑠/𝑝))}𝜓^−𝑠/𝑝2^𝑘𝑟ℳ^𝑠/𝑝_𝑝,𝜓𝑓;0,2^𝑘𝑟. (4.13)
On the other hand, the integral on the right-hand side of (4.10) can be estimated as follows: ^∞_𝑟𝑉^𝑠(𝑡)𝑡ℳ^𝑠/𝑝_𝑝,𝜓𝑓;0,𝑡𝑑𝑡=^∞_𝑘=0^2𝑘+1𝑟_2𝑘𝑟𝑡^{𝑛−1−(𝑛𝑠/𝑝)}ℳ_𝑝,𝜓𝑓;0,𝑡𝜓(𝑡)^𝑠/𝑝𝑣(𝑡)𝑑𝑡≥𝐶^∞_𝑘=0𝑣2^𝑘𝑟ℳ_𝑝,𝜓𝑓;0,2^𝑘+1𝑟𝜓2^𝑘+1𝑟^𝑠/𝑝2^𝑘𝑟^{𝑛(1−𝑠/𝑝)}≥𝐶^∞_𝑘=0𝑣2^𝑘+1𝑟𝜓^−1/𝑝2^𝑘𝑟2^𝑘𝑟^{𝑛(1−𝑠/𝑝)}ℳ^𝑠/𝑝_𝑝,𝜓𝑓;0,2^𝑘+1𝑟, (4.14) which completes the proof.

Corollary 4.3. Let 1≤𝑝<∞, let 𝑤,𝑤𝜓^1/𝑝∈𝑊([0,ℓ]),𝑤(2𝑡)≤𝑐𝑤(𝑡), 0<ℓ≤∞ and 0∈Ω. Let also 𝑉(𝑟)1∶=𝑤^𝑝(𝑟)𝜓(𝑟)∈ℤ₀. (4.15) Then ℳ_𝑝,𝜓𝑓𝑤≤𝐶;0,𝑟𝑤^𝑝(𝑟)ℳ_𝑝,𝜓𝑓;0,𝑟,0<𝑟<ℓ. (4.16)

5. Weighted Hardy Operators in Complementary Morrey Spaces

5.1. Pointwise Estimations

The proof of our main result of this Section given in Theorems 5.3 and 5.6 is prepared by the following Theorems 5.1 and 5.2 on the pointwise estimates of the Hardy-type operators.

Theorem 5.1. Let 1≤𝑝≤∞, let 𝑤,𝑤𝜓^1/𝑝∈𝑊, and 𝑤(2𝑡)≤𝐶𝑤(𝑡). The condition ^𝜀₀𝑉(𝑡)𝑡(𝑡)=𝑡𝑑𝑡<∞𝑤𝑖𝑡ℎ𝜀>0,𝑤ℎ𝑒𝑟𝑒𝑉^𝑛/𝑝′𝑤(𝑡)𝜓^1/𝑝(𝑡) (5.1) is sufficient for the Hardy operator 𝐻^𝛼_𝑤 to be defined on the space ^∁ℒ^𝑝,𝜓_{0}(ℝ^𝑛). Under this condition, the pointwise estimate 𝐻^𝛼_𝑤𝑓(𝑥)≤𝐶|𝑥|^𝛼−𝑛𝑤(|𝑥|)₀^|𝑥|𝑉(𝑡)𝑡𝑑𝑡𝑓_∁ℒ^𝑝,𝜓_{0} (5.2) holds.

Proof. Apply Lemma 4.1 with 𝑠=1.

Theorem 5.2. Let 1≤𝑝≤∞ and 1/𝑤∈𝑊, or 𝑤∈𝑊 and 𝑤(2𝑡)≤𝐶𝑤(𝑡). The condition ^∞_𝜀𝒱(𝑡)𝑡(𝑡)=1𝑑𝑡<∞,𝑤𝑖𝑡ℎ𝜀>0,𝑤ℎ𝑒𝑟𝑒𝒱𝑤(𝑡)𝜓^1/𝑝(𝑡)𝑡^𝑛/𝑝 (5.3) is sufficient for the Hardy operator ℋ^𝛼_𝑤 to be defined on the space ^∁ℒ^𝑝,𝜓_{𝑥0}(ℝ^𝑛), and in this case the following pointwise estimate: ℋ^𝛼_𝑤𝑓(𝑥)≤𝐶|𝑥|^𝛼𝑤(|𝑥|)^∞_|𝑥|𝒱(𝑡)𝑡𝑑𝑡𝑓_∁ℒ0}^𝑝,𝜓_{𝑥 (5.4) holds.

Proof. Apply Lemma 4.2 with 𝑠=1.

5.2. Weighted 𝑝→𝑞 Boundedness of Hardy Operators in Complementary Morrey Spaces

5.2.1. The Case of the Operator 𝐻^𝛼_𝑤

The 𝐿^𝑝→𝐿^𝑞-boundedness of the multidimensional Hardy operators within the frameworks of Lebesgue spaces (the case 𝜑≡1 in (1.4)) with 1<𝑝<∞ and 0<𝑞<∞ is well known: see for instance, [4, page 54]. For Morrey spaces, both local and global, the boundedness of Hardy operators was studied in [7, 8]. We call attention of the reader to the fact that, in contrast to the case of Lebesgue spaces, Hardy-type inequalities in both usual and complementary Morrey spaces different from Lebesgue spaces (i.e., in the case 𝜑(0)=0 or 𝜓(0)=0) admit the value 𝑝=1.

Theorem 5.3. Let 1≤𝑝≤∞,1≤𝑞≤∞ and let 𝑤,𝜓^1/𝑝𝑤∈𝑊ℝ¹₊,𝑤(2𝑡)≤𝑐𝑤(𝑡). (5.5) The operator 𝐻^𝛼_𝑤 is bounded from ^∁ℒ^𝑝,𝜓_{0}(ℝ^𝑛) to ^∁ℒ^𝑞,𝜓_{0}(ℝ^𝑛) if sup_𝑟>0𝕎(𝑟)<∞, where 𝕎(𝑟)∶=𝜓(𝑟)^∞_𝑟𝜚^𝑛−1𝜚^{(𝛼−(𝑛/𝑝))}𝜓^1/𝑝(𝜚)𝑉(𝜚)^𝜚₀𝑉(𝑡)𝑡𝑑𝑡^𝑞𝑑𝜚, (5.6) and 𝑉(𝑡) is the same as in (5.1). Under this condition, 𝐻^𝛼_𝑤𝑓_∁ℒ^𝑞,𝜓_{0}≤𝐶sup_𝑟>0𝕎^1/𝑞(𝑟)𝑓_∁ℒ^𝑝,𝜓_{0}. (5.7)

Proof. From the estimate (5.2) of Theorem 5.1 we have 𝐻^𝛼_𝑤𝑓(𝑥|𝑥|)≤𝐶^{𝛼−(𝑛/𝑝)}𝜓^𝑞/𝑝𝑥1(||)𝑉(|𝑥|)₀^|𝑥|𝑉(𝑡)𝑡𝑑𝑡𝑓_∁ℒ^𝑞,𝜓_{0}. (5.8) Hence, calculating ‖𝐻^𝛼_𝑤𝑓‖_∁ℒ^𝑝,𝜓_{0}, passing to polar coordinates, we obtain (5.6).

Remark 5.4. Note that 1𝑉𝜚^𝜚₀𝑉(𝑡)𝑡𝑑𝑡≃1 (5.9) in (5.6) if we suppose that 𝑉∈ℤ⁰.

The following corollary gives sufficient conditions for the boundedness of the operator 𝐻^𝛼_𝑤 in terms of the Matuszewska-Orlicz indices of the function 𝜓 and the weight 𝑤 (we refer to the appendix for these notions).

Corollary 5.5. Let 1≤𝑝≤𝑞<∞ (with 𝑞=𝑝 admitted in the case 𝛼=0) and the conditions (5.5) be satisfied. Suppose also that 𝜓(𝑟)≥𝐶𝑟^{(𝛼𝑝𝑞/𝑞−𝑝)−𝑛} (5.10) in the case 𝛼≠0. The operator 𝐻^𝛼_𝑤 is bounded from ^∁ℒ^𝑝,𝜓_{0}(ℝ^𝑛) to ^∁ℒ^𝑞,𝜓_{0}(ℝ^𝑛) if min𝑚(𝑉),𝑚_∞(𝑉)>0,min𝑚𝜓,𝑚_∞𝜓>0. (5.11)
The condition min{𝑚(𝑉),𝑚_∞(𝑉)}>0, is guaranteed by the inequalities 𝑀(𝑤)<𝑛𝑝^′−𝑀𝜓,𝑀_∞(𝑤)<𝑛𝑝^′−𝑀_∞𝜓. (5.12) In the power case 𝜓(𝑟)=𝑟^𝜆, 𝜆>0, and 𝑤(𝑟)=𝑟^𝜇, the conditions (5.10)-(5.11) reduce to 1≤𝑝<𝑛+𝜆𝛼,1𝑞=1𝑝−𝛼𝑛𝑛+𝜆,𝜇<𝑝^′−𝜆𝑝; (5.13) conditions (5.13) are also necessary for the operator 𝐻^𝛼_𝑤 to be bounded from ^∁ℒ^𝑝,𝜓_{0}(ℝ^𝑛) to ^∁ℒ^𝑞,𝜓_{0}(ℝ^𝑛).

Proof. We have to check that the condition sup_𝑟>0𝕎(𝑟)<∞ of Theorem 5.3 holds under the assumptions (5.10)-(5.11). From the inequality min{𝑚(𝑉),𝑚_∞(𝑉)}>0 it follows that 𝕎(𝑟)≤𝐶𝜓(𝑟)^∞_𝑟𝜚^𝑛−1𝜚^{𝑞(𝛼−(𝑛/𝑝))}𝜓^𝑞/𝑝𝜚𝑑𝜚. (5.14)
We represent 𝜓^𝑞/𝑝(𝜚) as 𝜓^𝑞/𝑝(𝜚)=𝜓^{(𝑞/𝑝)−1}(𝜚)𝜓(𝜚) and by (5.10) obtain 𝕎(𝑟)≤𝐶𝜓(𝑟)^∞_𝑟𝑑𝜚𝜚,𝜚𝜓 (5.15) which is bounded in view of the second assumption in (5.11), by the property (A.4).
The sufficiency of the conditions (5.13) in the case of power functions is then obvious since 𝑚(𝜓)=𝑚_∞(𝜓)=𝜆 and 𝑚(𝑤)=𝑚_∞(𝑤)=𝜇 in this case.
Let us show the necessity of these conditions. From the boundedness ‖𝐻^𝛼_𝑤‖_∁ℒ^𝑞,𝜓_{0}(ℝ𝑛)≤𝐶‖𝑓‖_∁ℒ^𝑝,𝜓_{0}(ℝ𝑛) with 𝜓(𝑟)=𝑟^𝜆,𝜆>0, and 𝑤(𝑟)=𝑟^𝜇, by standard homogeneity arguments with the use of the dilation operator Π_𝛿𝑓(𝑥)∶=𝑓(𝛿𝑥) it is easily derived that the conditions 1≤𝑝<(𝑛+𝜆)/𝛼, 1/𝑞=(1/𝑝)−(𝛼/(𝑛+𝜆)) necessarily hold, via the relation Π_𝛿𝑓_∁ℒ^𝑝,𝜓_{0}(ℝ𝑛)=𝛿^{−(𝑛+𝜆)/𝑝}𝑓_∁ℒ^𝑝,𝜓_{0}(ℝ𝑛),𝐻^𝛼_𝑤Π_𝛿𝑓=𝛿^−𝛼Π_𝛿𝐻^𝛼_𝑤𝑓. (5.16) (We take into account that we excluded 𝑞=∞ in this theorem.)
The necessity of the remaining condition 𝜇<(𝑛/𝑝^′)−(𝜆/𝑝) in (5.13) follows from the fact that |𝑦|^{−(𝑛+𝜆)/𝑝}∈^∁ℒ^𝑝,𝜓_{0}(ℝ^𝑛) by Corollary 2.4, so that this condition is necessary for the operator 𝐻^𝛼_𝑤 with 𝑤=𝑟^𝜇 to be defined on the space ^∁ℒ^𝑝,𝜓_{0}(ℝ^𝑛).

5.2.2. The Case of the Operator ℋ^𝛼_𝑤

Let 𝒲(𝑟)∶=𝜓(𝑟)^∞_𝑟𝑤^𝑞𝜚𝜚^{𝑞𝛼+𝑛−1∞}_𝜚𝒱(𝑡)𝑡𝑑𝑡^𝑞𝑑𝜚, (5.17) where 𝒱 is the same as in (5.3).

Theorem 5.6. Let 1≤𝑝<∞, 1≤𝑞<∞ and 𝑤𝜓^1/𝑝∈𝑊ℝ¹₊𝑜𝑟𝑤∈𝑊ℝ¹₊,𝑤(2𝑡)≤𝐶𝑤(𝑡). (5.18) The operator ℋ^𝛼_𝑤 is bounded from ^∁ℒ^𝑝,𝜓_{0}(ℝ^𝑛) to ^∁ℒ^𝑞,𝜓_{0}(ℝ^𝑛) if sup_𝑟>0𝒲(𝑟)<∞, (5.19) and then ℋ^𝛼_𝑤𝑓_∁ℒ^𝑞,𝜓_{0}≤𝐶sup_𝑟>0𝒲^1/𝑞(𝑟)𝑓_∁ℒ^𝑝,𝜓_{0}. (5.20)

Proof. From the estimate (5.4) we obtain ℳ_𝑞,𝜓ℋ^𝛼_𝑤𝑓;0,𝑟≤𝐶𝜓(𝑟)^∞_𝑟𝜚^{𝑛−1+𝑞(𝛼−(𝑛/𝑝))}𝜓^−𝑞/𝑝𝜚1𝒱𝜚^∞_𝜚𝒱(𝑡)𝑡𝑑𝑡^𝑞𝑑𝜚𝑓^𝑞_{∁ℒ𝑝,𝜓{0}}, (5.21) from which (5.20) follows.

As above, we provide also sufficient conditions for the boundedness of the operator ℋ^𝛽_𝑤 in terms of the Matuszewska-Orlicz indices.

Corollary 5.7. Let 1≤𝑝≤𝑞<∞ (with 𝑞=𝑝 admitted in the case 𝛼=0 and 𝑤 and 𝜓 satisfy the conditions (5.18). The operator ℋ^𝛼_𝑤 is bounded from ^∁ℒ^𝑝,𝜓_{0}(ℝ^𝑛) to ^∁ℒ^𝑞,𝜓_{0}(ℝ^𝑛) if max𝑀(𝒱),𝑀_∞(𝒱)<0,min𝑚𝜓,𝑚_∞𝜓>0, (5.22) and the condition (5.10) holds; the assumption max{𝑀(𝒱),𝑀_∞(𝒱)}<0 is guaranteed by the conditions 𝑚(𝑤)𝑚𝜓>−+𝑛𝑝,𝑚_∞(𝑤)𝑚>−^∞𝜓+𝑛𝑝. (5.23) In the case 𝜓(𝑟)=𝑟^𝜆,𝜆>0, and 𝑤(𝑟)=𝑟^𝜇, the conditions (5.22) and (5.10) reduce to 1≤𝑝<𝑛+𝜆𝛼,1𝑞=1𝑝−𝛼𝑛+𝜆,𝜇>𝑛+𝜆𝑝; (5.24) conditions (5.24) are also necessary for the operator ℋ^𝛼_𝑤 to be bounded from ^∁ℒ^𝑝,𝜓_{0}(ℝ^𝑛) to ^∁ℒ^𝑞,𝜓_{0}(ℝ^𝑛).

Proof. We have to find, in terms of the Matuszewska-Orlicz indices, conditions sufficient for the validity of (5.19). For the latter we have 𝒲(𝑟)≤𝐶𝜓(𝑟)^∞_𝑟𝜚^{𝑛−1+𝑞(𝛼−(𝑛/𝑝))}𝜓^−𝑞/𝑝𝜚1𝒱𝜚^∞_𝜚𝒱(𝑡)𝑡𝑑𝑡^𝑞(𝑟)𝑑𝜌≤𝐶𝜓^∞_𝑟𝜚^{𝑛−1+𝑞(𝛼−(𝑛/𝑝))}𝜓^−𝑞/𝑝𝜚𝑑𝜌 (5.25) by the assumption max{𝑀(𝒱),𝑀_∞(𝒱)}<0. With 𝜓^𝑞/𝑝(𝜌)=𝜓^{(𝑞/𝑝)−1}(𝜌)𝜓(𝜌) and the condition (5.10) we arrive at 𝒲(𝑟)≤𝐶𝜓(𝑟)∫^∞_𝑟𝑑𝜚/𝜚𝜓(𝜚), where the boundedness of the right-hand side is guaranteed by the condition min{𝑚(𝜓),𝑚_∞(𝜓)}>0.
The proof for the case of power functions is similar to that in Corollary 5.5.

Appendix

A. Zygmund-Bary-Stechkin (ZBS) Classes and Matuszewska-Orlicz (MO) Type Indices

In the sequel, a nonnegative function 𝑓 on [0,ℓ], 0<ℓ≤∞, is called almost increasing (almost decreasing) if there exists a constant 𝐶 (≥1) such that 𝑓(𝑥)≤𝐶𝑓(𝑦) for all 𝑥≤𝑦 (𝑥≥𝑦, resp.). Equivalently, a function 𝑓 is almost increasing (almost decreasing) if it is equivalent to an increasing (decreasing, resp.) function 𝑔, that is, 𝑐₁𝑓(𝑥)≤𝑔(𝑥)≤𝑐₂𝑓(𝑥), 𝑐₁>0, 𝑐₂>0.

Definition A.1. Let 0<ℓ<∞:(1)by 𝑊=𝑊([0,ℓ]) we denote the class of continuous and positive functions 𝜑 on (0,ℓ] such that there exists finite or infinite limit lim_𝑥→0𝜑(𝑥);(2)by 𝑊₀=𝑊₀([0,ℓ]) we denote the class of almost increasing functions 𝜑∈𝑊 on (0,ℓ);(3)by 𝑊=𝑊([0,ℓ]) we denote the class of functions 𝜑∈𝑊 such that 𝑥^𝑎𝜑(𝑥)∈𝑊₀ for some 𝑎=𝑎(𝜑)∈ℝ¹;(4)by 𝑊=𝑊([0,ℓ]) we denote the class of functions 𝜑∈𝑊 such that 𝜑(𝑡)/𝑡^𝑏 is almost decreasing for some 𝑏∈ℝ¹.

Definition A.2. Let 0<ℓ<∞:(1)by 𝑊_∞=𝑊_∞([ℓ,∞]) we denote the class of functions 𝜑 which are continuous and positive and almost increasing on [ℓ,∞) and which have the finite or infinite limit lim_𝑥→∞𝜑(𝑥),(2)by 𝑊_∞=𝑊_∞([ℓ,∞)) we denote the class of functions 𝜑∈𝑊_∞ such 𝑥^𝑎𝜑(𝑥)∈𝑊_∞ for some 𝑎=𝑎(𝜑)∈ℝ¹.

By 𝑊(ℝ¹₊) we denote the set of functions on ℝ¹₊ whose restrictions onto (0,1) are in 𝑊([0,1]) and restrictions onto [1,∞) are in 𝑊_∞([1,∞)). Similarly, the set 𝑊(ℝ¹₊) is defined.

A.1. ZBS Classes and MO Indices of Weights at the Origin

In this subsection we assume that ℓ<∞.

We say that a function 𝜑 belongs to a Zygmund class ℤ^𝛽, 𝛽∈ℝ¹, if 𝜑∈𝑊([0,ℓ]) and ∫^𝑥₀(𝜑(𝑡)/𝑡^1+𝛽)𝑑𝑡≤𝑐(𝜑(𝑥)/𝑥^𝛽), 𝑥∈(0,ℓ), and to a Zygmund class ℤ_𝛾, 𝛾∈ℝ¹, if 𝜑∈𝑊([0,ℓ]) and ∫^ℓ_𝑥(𝜑(𝑡)/𝑡^1+𝛾)𝑑𝑡≤𝑐(𝜑(𝑥)/𝑥^𝛾), 𝑥∈(0,ℓ). We also denote Φ^𝛽_𝛾∶=ℤ^𝛽ℤ_𝛾, (A.1) the latter class being also known as Bary-Stechkin-Zygmund class [20].

For a function 𝜑∈𝑊, the numbers 𝑚𝜑=sup_0<𝑥<1lnlimsup_ℎ→0𝜑(ℎ𝑥)/𝜑(ℎ)ln𝑥=lim_𝑥→0lnlimsup_ℎ→0𝜑(ℎ𝑥)/𝜑(ℎ),𝑀𝜑ln𝑥=sup_𝑥>1lnlimsup_ℎ→0𝜑(ℎ𝑥)/𝜑(ℎ)ln𝑥=lim_𝑥→∞lnlimsup_ℎ→0𝜑(ℎ𝑥)/𝜑(ℎ)ln𝑥 (A.2) are known as the Matuszewska-Orlicz type lower and upper indices of the function 𝜑(𝑟). The property of functions to be almost increasing or almost decreasing after the multiplication (division) by a power function is closely connected with these indices. We refer to [22–28] for such a property and these indices.

Note that in this definition 𝜑(𝑥) need not to be an 𝑁-function: only its behaviour at the origin is of importance. Observe that 0≤𝑚(𝜑)≤𝑀(𝜑)≤∞ for 𝜑∈𝑊₀, and −∞<𝑚(𝜑)≤𝑀(𝜑)≤∞for𝜑∈𝑊, and the following formulas are valid: 𝑚𝑥^𝑎𝜑(𝑥)=𝑎+𝑚𝜑,𝑀𝑥^𝑎𝜑(𝑥)=𝑎+𝑀𝜑,𝑎∈ℝ¹,𝑚𝜑(𝑥)^𝑎=𝑎𝑚𝜑,𝑀𝜑(𝑥)^𝑎𝜑𝑚1=𝑎𝑀,𝑎≥0,𝜑𝜑1=−𝑀,𝑀𝜑𝜑,𝑚()(𝑢)(𝑣)()(𝑢)(𝑣)=−𝑚𝑢𝑣≥𝑚+𝑚,𝑀𝑢𝑣≤𝑀+𝑀 (A.3)(A.4)(A.5)(A.6) for 𝜑,𝑢,𝑣∈𝑊.

The proof of the following statement may be found in [21], Theorems 3.1, 3.2, and 3.5. (In the formulation of Theorems in [21] it was supposed that 𝛽≥0, 𝛾>0 and 𝜑∈𝑊₀. It is evidently true also for 𝜑∈𝑊 and all 𝛽,𝛾∈ℝ¹, in view of formulas (A.3)).

Theorem A.3. Let 𝜑∈𝑊 and 𝛽,𝛾∈ℝ¹. Then 𝜑∈ℤ^𝛽⇔𝑚(𝜑)>𝛽 and 𝜑∈ℤ_𝛾⇔𝑀(𝜑)<𝛾. Besides this, 𝑚(𝜑)=sup{𝜇>0∶𝜑(𝑥)/𝑥^𝜇isalmostincreasing}, and 𝑀(𝜑)=inf{𝜈>0∶𝜑(𝑥)/𝑥^𝜈isalmostdecreasing}, and for 𝜑∈Φ^𝛽_𝛾 the inequalities 𝑐₁𝑥^{𝑀(𝜑)+𝜀}≤𝜑(𝑥)≤𝑐₂𝑥^{𝑚(𝜑)−𝜀} (A.7) hold with an arbitrarily small 𝜀>0 and 𝑐₁=𝑐₁(𝜀), 𝑐₂=𝑐₂(𝜀).

We define the following subclass in 𝑊₀: 𝑊_0,𝑏=𝜑∈𝑊₀∶𝜑(𝑡)𝑡^𝑏,isalmostincreasing,𝑏∈ℝ¹. (A.8)

A.2. ZBS Classes and MO Indices of Weights at Infinity

Following Section 2.2 in [29], we use the following notation.

Let −∞<𝛼<𝛽<∞. We put Ψ^𝛽_𝛼∶=ℤ^𝛽∩ℤ_𝛼, where ℤ^𝛽 is the class of functions 𝜑∈𝑊_∞ satisfying the condition ∫^∞_𝑥(𝑥/𝑡)^𝛽(𝜑(𝑡)𝑑𝑡/𝑡)≤𝑐𝜑(𝑥),𝑥∈(ℓ,∞), and ℤ_𝛼 is the class of functions 𝜑∈𝑊([ℓ,∞)) satisfying the condition ∫^𝑥_ℓ(𝑥/𝑡)^𝛼(𝜑(𝑡)𝑑𝑡/𝑡)≤𝑐𝜑(𝑥),𝑥∈(ℓ,∞), where 𝑐=𝑐(𝜑)>0 does not depend on 𝑥∈[ℓ,∞).

The indices 𝑚_∞(𝜑) and 𝑀_∞(𝜑) responsible for the behavior of functions 𝜑∈Ψ^𝛽_𝛼([ℓ,∞)) at infinity are introduced in the way similar to (A.2): 𝑚_∞𝜑=sup_𝑥>1lnliminf_ℎ→∞𝜑(𝑥ℎ)/𝜑(ℎ),𝑀ln𝑥_∞𝜑=inf_𝑥>1lnlimsup_ℎ→∞𝜑(𝑥ℎ)/𝜑(ℎ).ln𝑥 (A.9)

Properties of functions in the class Ψ^𝛽_𝛼([ℓ,∞)) are easily derived from those of functions in Φ^𝛼_𝛽([0,ℓ]) because of the following equivalence 𝜑∈Ψ^𝛽_𝛼([ℓ,∞))⟺𝜑_∗∈Φ^−𝛽_−𝛼0,ℓ^∗, (A.10) where 𝜑_∗(𝑡)=𝜑(1/𝑡) and ℓ_∗=1/ℓ. Direct calculation shows that 𝑚_∞𝜑=−𝑀𝜑_∗,𝑀_∞𝜑=−𝑚𝜑_∗,𝜑_∗(𝑡)1∶=𝜑𝑡. (A.11)

By (A.10) and (A.11), one can easily reformulate properties of functions of the class Φ^𝛽_𝛾 near the origin, given in Theorem A.3 for the case of the corresponding behavior at infinity of functions of the class Ψ^𝛽_𝛼 and obtain that 𝑐₁𝑡^{𝑚∞(𝜑)−𝜀}≤𝜑(𝑡)≤𝑐₂𝑡^{𝑀∞(𝜑)+𝜀},𝑡≥ℓ,𝜑∈𝑊_∞,𝑚_∞𝜑=sup𝜇∈ℝ¹∶𝑡^−𝜇𝜑(𝑡)[,𝑀isalmostincreasingonℓ,∞)_∞𝜑=inf𝜈∈ℝ¹∶𝑡^−𝜈𝜑(𝑡)isalmostdecreasingon[ℓ,∞). (A.12)

We say that a continuous function 𝜑 in (0,∞) is in the class 𝑊_0,∞(ℝ¹₊) if its restriction to (0,1) belongs to 𝑊([0,1]) and its restriction to (1,∞) belongs to 𝑊_∞([1,∞]). For functions in 𝑊_0,∞(ℝ¹₊) the notation ℤ^{𝛽0,𝛽∞}ℝ¹₊=ℤ^𝛽0([0,1])∩ℤ^𝛽∞([1,∞)),ℤ_{𝛾0,𝛾∞}ℝ¹₊=ℤ_𝛾0([0,1])∩ℤ_𝛾∞([1,∞)) (A.13) has an obvious meaning (note that in (A.13) we use ℤ^𝛽∞([1,∞)) and ℤ_𝛾∞([1,∞)), not ℤ^𝛽∞([1,∞)) and ℤ_𝛾∞([1,∞))). In the case where the indices coincide, that is, 𝛽₀=𝛽_∞∶=𝛽, we will simply write ℤ^𝛽(ℝ¹₊) and similarly for ℤ_𝛾(ℝ¹₊). We also denote Φ^𝛽_𝛾ℝ¹₊∶=ℤ^𝛽ℝ¹₊∩ℤ_𝛾ℝ¹₊. (A.14)

Making use of Theorem A.3 for Φ^𝛼_𝛽([0,1]) and relations (A.11), one easily arrives at the following statement.

Lemma A.4. Let 𝜑∈𝑊(ℝ¹₊). Then 𝜑∈ℤ^{𝛽0,𝛽∞}ℝ¹₊⟺𝑚𝜑>𝛽₀,𝑚_∞𝜑>𝛽_∞,𝜑∈ℤ_{𝛾0,𝛾∞}ℝ¹₊⟺𝑀𝜑<𝛾₀,𝑀_∞𝜑<𝛾_∞. (A.15)(A.16)

Acknowledgment

The third author thanks Luleå University of Technology for financial support for research visits in June of 2012.