Some New Iterated Hardy-Type Inequalities

Abstract

We characterize the validity of the Hardy-type inequality ‖‖∫^∞_sℎ(𝑧)𝑑𝑧‖_{𝑝,𝑢,(0,𝑡)}‖_{𝑞,𝑤,(0,∞)}≤𝑐‖ℎ‖_{𝜃,𝑣(0,∞)}, where 0<𝑝<∞, 0<𝑞≤∞, 1<𝜃≤∞, 𝑢, 𝑤, and 𝑣 are weight functions on (0,∞). Some fairly new discretizing and antidiscretizing techniques of independent interest are used.

1. Introduction

Everywhere in the paper, 𝑢, 𝑣, and 𝑤 are weights, that is, locally integrable nonnegative functions on (0,∞), and we denote 𝑈(𝑠)=^𝑠₀𝑢(𝑡)𝑉𝑑𝑡,_𝜃(𝑡)=∫^∞_𝑡𝑣(𝑠)^1−𝜃′∫𝑑𝑠for1<𝜃<∞,_∞𝑡𝑑𝑠𝑣(𝑠)for𝜃=∞, (1.1) where 𝜃^′𝜃∶=∞if𝜃=1,𝜃−1if1<𝜃<∞,1if𝜃=∞. (1.2) We assume that 𝑢 is such that 𝑈(𝑡)>0 for every 𝑡∈(0,∞).

For 0<𝑝<∞ and 𝑤, a weight function on (𝑎,𝑏)⊆(0,∞), let us denote by 𝐿_𝑝,𝑤(𝑎,𝑏) the weighted Lebesgue space defined as the set of all measurable functions 𝑢 on (𝑎,𝑏) for which the quantity ‖𝑢‖_{𝑝,𝑤,(𝑎,𝑏)}=∫^𝑏_𝑎|𝑢(𝑥)|^𝑝𝑤(𝑥)𝑑𝑥^1/𝑝for1≤𝑝<∞,esssup_{𝑎<𝑥<𝑏}|𝑢(𝑥)|𝑤(𝑥)for𝑝=∞ (1.3) is finite.

In this paper we characterize the validity of the inequality ^∞_𝑠ℎ(𝑧)𝑑𝑧_{𝑝,𝑢,(0,𝑡)𝑞,𝑤,(0,∞)}≤𝑐‖ℎ‖_{𝜃,𝑣,(0,∞)}, (1.4) where 0<𝑝<∞, 0<𝑞≤∞, 1<𝜃≤∞, 𝑢, 𝑤, and 𝑣 are weight functions on (0,∞). Note that inequality (1.4) has been considered in the case 𝑝=1 in [1] (see also [2]), where the result is presented without proof, in the case 𝑝=∞ in [3] and in the case 𝜃=1 in [4, 5], where weight functions 𝑣 of special type were considered. For general weight functions 𝑣, the characterization of the inequality (1.4) in the case 𝜃=1 does not follow directly by this method (there are some technical problems) and we are working on it.

It is worth to mention that, by Fubini’s theorem, ^𝑥₀𝑢(𝑡)^∞_𝑡𝑔(𝑠)𝑑𝑠𝑑𝑡≈𝑈(𝑥)^∞₀𝑔(𝑠)𝑈(𝑠)𝑑𝑠𝑈(𝑥)(𝑠).+𝑈 (1.5) Hence, we see that the inequality (1.4) (with 𝑝=1) is equivalent with the following inequality: ‖𝑆ℎ‖_𝑞,𝑈^𝑞_𝑤,(0,∞)≤𝑐‖ℎ‖_𝜃,𝑈^−𝜃_𝑣,(0,∞), (1.6) where the operator 𝑆 defined by (𝑆ℎ)(𝑥)=^∞₀ℎ(𝑡)𝑑𝑡𝑈(𝑥)+𝑈(𝑡) (1.7) for all nonnegative measurable functions ℎ on (0,∞). We call this operator the generalized Stieltjes transform; the usual Stieltjes transform is obtained on putting 𝑈(𝑥)≡𝑥.

In the case 𝑈(𝑥)≡𝑥^𝜆, 𝜆>0, the boundedness of the operator 𝑆 between weighted 𝐿^𝑝 and 𝐿^𝑞 spaces was investigated in [6] (when 1≤𝑝≤𝑞≤∞) and in [7, 8] (when 1≤𝑞<𝑝≤∞).

Our approach is based on discretization and antidiscretization methods developed in [4, 9, 10]. Some basic facts concerning these methods and other preliminaries are presented in Section 2. The main results (Theorems 3.1 and 3.2) are stated and proved in Section 3.

Throughout the paper, we always denote by 𝑐 or 𝐶 a positive constant which is independent of the main parameters, but it may vary from line to line. However a constant with subscript such as 𝑐₁ does not change in different occurrences. By 𝑎≲𝑏, (𝑏≳𝑎), we mean that 𝑎≤𝜆𝑏, where 𝜆>0 depends on inessential parameters. If 𝑎≲𝑏 and 𝑏≲𝑎, we write 𝑎≈𝑏 and say that 𝑎 and 𝑏 are equivalent. We put 1/∞=0, 0⋅∞=0, 0/0=0, and ∞/∞=0.

2. Preliminaries

Let us now recall some definitions and basic facts concerning discretization and antidiscretization which can be found in [4, 9, 10].

Definition 2.1. Let {𝑎_𝑘} be a sequence of positive real numbers. One says that {𝑎_𝑘} is strongly increasing or strongly decreasing and write 𝑎_𝑘↑↑ or 𝑎_𝑘↓↓ when inf_𝑘∈ℤ𝑎_𝑘+1𝑎_𝑘>1orsup_𝑘∈ℤ𝑎_𝑘+1𝑎_𝑘<1, (2.1) respectively.

Definition 2.2. Let 𝑈 be a continuous strictly increasing function on [0,∞) such that 𝑈(0)=0 and lim_𝑡→∞𝑈(𝑡)=∞. Then One says that 𝑈 is admissible.
Let 𝑈 be an admissible function. We say that a function 𝜑 is 𝑈-quasiconcave if 𝜑 is equivalent to an increasing function on (0,∞) and 𝜑/𝑈 is equivalent to a decreasing function on (0,∞). We say that a 𝑈-quasiconcave function 𝜑 is nondegenerate if lim_𝑡→0+𝜑(𝑡)=lim_𝑡→∞1𝜑(𝑡)=lim_𝑡→∞𝜑(𝑡)𝑈(𝑡)=lim_𝑡→0+𝑈(𝑡)𝜑(𝑡)=0. (2.2) The family of nondegenerate 𝑈-quasiconcave functions will be denoted by Ω_𝑈. We say that 𝜑 is quasiconcave when 𝜑∈Ω_𝑈 with 𝑈(𝑡)=𝑡. A quasiconcave function is equivalent to a concave function. Such functions are very important in various parts of analysis. Let us just mention that, for example, the Hardy operator 𝐻𝑓(𝑥)=∫^𝑥₀𝑓(𝑡)𝑑𝑡 of a decreasing function, the Peetre 𝐾-functional in interpolation theory, and the fundamental function ‖𝜒_𝐸‖_𝑋, 𝑋 is a rearrangement invariant space, all are quasiconcave.

Definition 2.3. Assume that 𝑈 is admissible and 𝜑∈Ω_𝑈. One says that {𝑥_𝑘}_𝑘∈ℤ is a discretizing sequence for 𝜑 with respect to 𝑈 if(i)𝑥₀=1 and 𝑈(𝑥_𝑘)↑↑;(ii)𝜑(𝑥_𝑘)↑↑ and 𝜑(𝑥_𝑘)/𝑈(𝑥_𝑘)↓↓;(iii) there is a decomposition ℤ=ℤ₁∪ℤ₂ such that ℤ₁∩ℤ₂=∅ and for every 𝑡∈[𝑥_𝑘,𝑥_𝑘+1]𝜑𝑥_𝑘≈𝜑(𝑡)if𝑘∈ℤ₁,𝜑𝑥_𝑘𝑈𝑥_𝑘≈𝜑(𝑡)𝑈(𝑡)if𝑘∈ℤ₂. (2.3)
Let us recall (see [9, Lemma 2.7]) that if 𝜑∈Ω_𝑈, then there always exists a discretizing sequence for 𝜑 with respect to 𝑈.

Definition 2.4. Let 𝑈 be an admissible function, and let 𝜈 be a nonnegative Borel measure on [0,∞). We say that the function 𝜑 defined by 𝜑(𝑡)=𝑈(𝑡)_[0,∞)𝑑𝜈(𝑠)𝑈(𝑠)(𝑡)(),+𝑈,𝑡∈0,∞ (2.4) is the fundamental function of the measure 𝜈 with respect to 𝑈. One will also say that 𝜈 is a representation measure of 𝜑 with respect to 𝑈.
We say that 𝜈 is nondegenerate if the following conditions are satisfied for every 𝑡∈(0,∞): 𝑑𝜈(_[0,∞)𝑠)𝑈(𝑠)+𝑈(𝑡)<∞,_[0,1]𝑑𝜈(𝑠)𝑈(𝑠)=_[1,∞)𝑑𝜈(𝑠)=∞. (2.5)
We recall from [9, Remark 2.10] that 𝜑(𝑡)≈_[0,𝑡]𝑑𝜈(𝑠)+𝑈(𝑡)_[𝑡,∞)𝑈(𝑠)⁻¹𝑑𝜈(𝑠),𝑡∈(0,∞). (2.6)

Corollary 2.5 (see [10, Lemma 1.5]). Let 𝑢, 𝑤 be weights, and let 𝜑 be defined by 𝜑(𝑡)=esssup_{()𝑠∈0,𝑡}𝑈(𝑠)^1/𝑝esssup_{()𝜏∈𝑠,∞}𝑤(𝜏)𝑈(𝜏)^1/𝑝,𝑡∈(0,∞). (2.7) Then 𝜑 is the least 𝑈^1/𝑝-quasiconcave majorant of 𝑤, and sup_{()𝑡∈0,∞}𝜑(𝑡)1𝑈(𝑡)^𝑡₀^∞_𝑠ℎ(𝑧)𝑑𝑧^𝑝𝑢(𝑠)𝑑𝑠^1/𝑝=esssup_{()𝑡∈0,∞}𝑤(𝑡)1𝑈(𝑡)^𝑡₀^∞_𝑠ℎ(𝑧)𝑑𝑧^𝑝𝑢(𝑠)𝑑𝑠^1/𝑝 (2.8) for any nonnegative measurable ℎ on (0,∞). Further, for 𝑡∈(0,∞), 𝜑(𝑡)=esssup_{()𝜏∈0,∞}𝑤(𝜏)𝑈(𝑡)min1,𝑈(𝜏)^1/𝑝=𝑈(𝑡)^1/𝑝esssup_{()𝑠∈𝑡,∞}1𝑈(𝑠)^1/𝑝esssup_{()𝜏∈0,𝑠}𝑤(𝜏),𝜑(𝑡)≈esssup_{()𝑠∈0,∞}𝑤(𝑠)𝑈(𝑡)𝑈(𝑠)+𝑈(𝑡)^1/𝑝. (2.9)

Theorem 2.6 (see [9, Theorem 2.11]). Let 𝑝,𝑞,𝑟∈(0,∞). Assume that 𝑈 is an admissible function, 𝜈 is a nonnegative nondegenerate Borel measure on [0,∞), and 𝜑 is the fundamental function of 𝜈 with respect to 𝑈^𝑞 and 𝜎∈Ω_𝑈^𝑝. If {𝑥_𝑘} is a discretizing sequence for 𝜑 with respect to 𝑈^𝑞, then _[0,∞)𝜑(𝑡)^{(𝑟/𝑞)−1}𝜎(𝑡)^𝑟/𝑝𝑑𝜈(𝑡)≈_𝑘∈ℤ𝜑𝑥_𝑘^𝑟/𝑞𝜎𝑥_𝑘^𝑟/𝑝. (2.10)

Lemma 2.7 (see [9, Corollary 2.13]). Let 𝑞∈(0,∞). Assume that 𝑈 is an admissible function, 𝑓∈Ω_𝑈, 𝜈 is a nonnegative nondegenerate Borel measure on [0,∞), and 𝜑 is the fundamental function of 𝜈 with respect to 𝑈^𝑞. If {𝑥_𝑘} is a discretizing sequence for 𝜑 with respect to 𝑈^𝑞, then _[0,∞)𝑓(𝑡)𝑈(𝑡)^𝑞𝑑𝜈(𝑡)^1/𝑞≈_𝑘∈ℤ𝑓𝑥_𝑘𝑈𝑥_𝑘^𝑞𝜑𝑥_𝑘^1/𝑞. (2.11)

Lemma 2.8 (see [9, Lemma 3.5]). Let 𝑝,𝑞,𝑟∈(0,∞). Assume that 𝑈 is an admissible function, 𝜑∈Ω_𝑈^𝑞, and 𝑔∈Ω_𝑈^𝑝. If {𝑥_𝑘} is a discretizing sequence for 𝜑 with respect to 𝑈^𝑞 and {𝜆_𝑘} is a discretizing sequence of 𝑔 with respect to 𝑈^𝑝, then _𝑘∈ℤ𝜑𝑥_𝑘^𝑟/𝑞𝑔𝑥_𝑘^𝑟/𝑝≈_ℓ∈ℤ𝜑𝜆_ℓ^𝑟/𝑞𝑔𝜆_ℓ^𝑟/𝑝,sup_{()𝑡∈0,∞}𝜑(𝑡)^1/𝑞𝑔(𝑡)^1/𝑝≈sup_𝑘∈ℤ𝜑𝑥_𝑘^1/𝑞𝑔𝑥_𝑘^1/𝑝≈sup_ℓ∈ℤ𝜑𝜆_ℓ^1/𝑞𝑔𝜆_ℓ^1/𝑝. (2.12)

Lemma 2.9 (see [4, Lemma 2.5]). If 𝜏_𝑘↓↓, then _𝑘∈ℤ^𝑥𝑘₀ℎ^𝑞𝜏_𝑘≈_𝑘∈ℤ^𝑥𝑘_𝑥𝑘−1ℎ^𝑞𝜏_𝑘,sup_𝑘∈ℤ^𝑥𝑘₀ℎ^𝑞𝜏_𝑘≈sup_𝑘∈ℤ^𝑥𝑘_𝑥𝑘−1ℎ^𝑞𝜏_𝑘,_{𝑘∈ℤ∞𝑥𝑘}ℎ^𝑞𝜏_𝑘⁻¹≈_𝑘∈ℤ^𝑥𝑘+1_𝑥𝑘ℎ^𝑞𝜏_𝑘⁻¹,sup_𝑘∈ℤ^∞_𝑥𝑘ℎ^𝑞𝜏_𝑘⁻¹≈sup_𝑘∈ℤ^𝑥𝑘+1_𝑥𝑘ℎ^𝑞𝜏_𝑘⁻¹. (2.13)

Lemma 2.10 (see [9, Lemma 3.6]). Let 𝑞∈(0,∞). Assume that 𝑈 is an admissible function, 𝜈 is a nondegenerate nonnegative Borel measure on [0,∞), 𝜑 is the fundamental function of 𝜈 with respect to 𝑈^𝑞, and 𝑓 is a measurable function on [0,∞). If {𝑥_𝑘} is a discretizing sequence for 𝜑 with respect to 𝑈^𝑞, then ^∞₀^∞₀|𝑓(𝑡)|𝑑𝑡𝑈(𝑡)+𝑈(𝑥)^𝑞𝑑𝜈(𝑥)≈_𝑘∈ℤ^∞₀|𝑓(𝑡)|𝑑𝑡𝑈(𝑡)+𝑈(𝑥_𝑘)^𝑞𝜑𝑥_𝑘≈_𝑘∈ℤ𝑈⁻¹(𝑥_𝑘)^𝑥𝑘_𝑥𝑘−1|𝑓(𝑦)|𝑑𝑦+^𝑥𝑘+1_𝑥𝑘|𝑓(𝑦)|𝑈⁻¹(𝑦)𝑑𝑦^𝑞𝜑𝑥_𝑘≈_𝑘∈ℤ^𝑥𝑘+1_𝑥𝑘|𝑓(𝑦)|𝑈⁻¹(𝑦)𝜑(𝑦)^1/𝑞𝑑𝑦^𝑞. (2.14)

Lemma 2.11 (see [9, Lemma 3.7]). Let 𝑞∈(0,∞). Assume that 𝑈 is an admissible function, 𝜈 is a nondegenerate nonnegative Borel measure on [0,∞), 𝜑 is the fundamental function of 𝜈 with respect to 𝑈^𝑞, and 𝑓 is a measurable function on [0,∞). If {𝑥_𝑘} is a discretizing sequence for 𝜑 with respect to 𝑈^𝑞, then _[0,∞)esssup_{()𝑦∈0,∞}𝑓𝑦𝑈(𝑥)+𝑈𝑦^𝑞𝑑𝜈(𝑥)≈_𝑘∈ℤesssup_{()𝑦∈0,∞}𝑓𝑦𝑈𝑥_𝑘+𝑈𝑦^𝑞𝜑𝑥_𝑘≈_𝑘∈ℤ𝑈⁻¹(𝑥_𝑘)esssup_{𝑥𝑘−1≤𝑦<𝑥𝑘}𝑓𝑦+esssup_{𝑥𝑘≤𝑦<𝑥𝑘+1}𝑓𝑦𝑈⁻¹𝑦^𝑞𝜑𝑥_𝑘≈_𝑘∈ℤesssup_{𝑥𝑘≤𝑦<𝑥𝑘+1}𝑓𝑦^𝑞𝑈^−𝑞𝑦𝜑𝑦. (2.15)

Lemma 2.12 (see [9, Lemma 3.8]). Let 𝑞∈(0,∞). Assume that 𝑈 is an admissible function, 𝜑∈Ω_𝑈^𝑞, {𝑥_𝑘} is a discretizing sequence for 𝜑 with respect to 𝑈^𝑞, and 𝑓 is a measurable function on [0,∞). Then sup_{𝑥∈(0,∞)}^∞₀|𝑓(𝑡)|𝑑𝑡𝑈(𝑡)+𝑈(𝑥)^𝑞𝜑(𝑥)≈sup_𝑘∈ℤ^∞₀𝑓(𝑡)𝑑𝑡𝑈(𝑡)+𝑈𝑥_𝑘^𝑞𝜑𝑥_𝑘≈sup_𝑘∈ℤ𝑈⁻¹𝑥_𝑘^𝑥𝑘_𝑥𝑘−1𝑓𝑦𝑑𝑦+^𝑥𝑘+1_𝑥𝑘𝑓𝑦𝑈⁻¹𝑦𝑑𝑦^𝑞𝜑𝑥_𝑘≈sup_𝑘∈ℤ^𝑥𝑘+1_𝑥𝑘𝑓𝑦𝑈⁻¹𝑦𝜑𝑦^1/𝑞𝑑𝑦^𝑞. (2.16)

Lemma 2.13 (see [9, Lemma 3.9]). Let 𝑈 be an admissible function, 𝜑∈Ω_𝑈, {𝑥_𝑘} be a discretizing sequence for 𝜑 with respect to 𝑈, and 𝑓 be a measurable function on [0,∞). Then sup_{()𝑥∈0,∞}𝜑(𝑥)esssup_{()𝑦∈0,∞}𝑓𝑦𝑈(𝑥)+𝑈𝑦≈sup_𝑘∈ℤ𝜑𝑥_𝑘esssup_{()𝑦∈0,∞}𝑓𝑦𝑈𝑥_𝑘+𝑈𝑦≈sup_𝑘∈ℤ𝜑𝑥_𝑘𝑈𝑥_𝑘⁻¹esssup_{𝑥𝑘−1≤𝑦<𝑥𝑘}𝑓𝑦+sup_𝑘∈ℤ𝜑𝑥_𝑘esssup_{𝑥𝑘≤𝑦<𝑥𝑘+1}𝑓𝑦𝑈𝑦⁻¹≈sup_𝑘∈ℤesssup_{𝑥𝑘≤𝑦<𝑥𝑘+1}𝑓𝑦𝑈𝑦⁻¹𝜑𝑦. (2.17)

Proposition 2.14 (see [9, Proposition 4.1]). Let {𝜔_𝑘} and {𝜐_𝑘}, 𝑘∈ℤ, be two sequences of positive real numbers. Let 𝑝,𝑞∈(0,∞), and assume that the inequality _𝑘∈ℤ𝑎^𝑞_𝑘𝜐_𝑘^1/𝑞≤𝑐_𝑘∈ℤ𝑎^𝑝_𝑘𝜔_𝑘^1/𝑝 (2.18) is satisfied for every sequence {𝑎_𝑘} of positive real numbers.(i)If 𝑝≤𝑞, then sup_𝑘∈ℤ𝜔_𝑘^−𝑞/𝑝𝜐_𝑘<∞. (2.19)(ii)If 𝑝>𝑞 and 𝑟=𝑝𝑞/(𝑝−𝑞), then_𝑘∈ℤ𝜔_𝑘^−𝑟/𝑝𝜐_𝑘^{𝑟/𝑞1/𝑟}<∞. (2.20)

Lemma 2.15. One has the following Hardy-type inequalities.(a) Let 1<𝜃≤𝑝<∞. Then the inequality ^𝑥𝑘_𝑠ℎ(𝑧)𝑑𝑧_{𝑝,𝑢,(𝑥𝑘−1,𝑥𝑘)}≤𝑐‖ℎ‖_{𝜃,𝑣,(𝑥𝑘−1,𝑥𝑘)} (2.21) holds for all nonnegative measurable ℎ if and only if 𝐴∶=sup_{𝑥𝑘−1<𝑡<𝑥𝑘}^𝑡_𝑥𝑘−1𝑢(𝑠)𝑑𝑠^{1/𝑝𝑥𝑘}_𝑡𝑣(𝑠)^1−𝜃′𝑑𝑠^1/𝜃′<∞, (2.22) and the best constant in (2.21) satisfies 𝑐≈𝐴.(b) Let 1<𝜃<∞, 𝑝=∞. Then the inequality (2.21) holds if and only if 𝐵∶=sup_{𝑥𝑘−1<𝑡<𝑥𝑘}esssup_{𝑥𝑘−1<𝑠<𝑡}𝑢(𝑠)^𝑥𝑘_𝑡𝑣(𝑠)^1−𝜃′𝑑𝑠^1/𝜃′<∞, (2.23) and the best constant in (2.21) satisfies 𝑐≈𝐵.(c) Let 𝜃=𝑝=∞. Then the inequality (2.21) holds if and only if 𝐶∶=sup_{𝑥𝑘−1<𝑡<𝑥𝑘}esssup_{𝑥𝑘−1<𝑠<𝑡}𝑢(𝑠)^𝑥𝑘_𝑡𝑑𝑠𝑣(𝑠)<∞, (2.24) and the best constant in (2.21) satisfies 𝑐≈𝐶.(d) Let 1<𝜃<∞, 0<𝑝<𝜃, and 1/𝑟=1/𝑝−1/𝜃. Then the inequality (2.21) holds if and only if 𝐷∶=^𝑥𝑘_𝑥𝑘−1^𝑡_𝑥𝑘−1𝑢(𝑠)𝑑𝑠^{𝑟/𝑝𝑥𝑘}_𝑡𝑣(𝑠)^1−𝜃′𝑑𝑠^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡^1/𝑟<∞, (2.25) and the best constant in (2.21) satisfies 𝑐≈𝐷.(e) Let 𝜃=∞, 0<𝑝<∞. Then the inequality (2.21) holds if and only if 𝐸∶=^𝑥𝑘_𝑥𝑘−1^𝑡_𝑥𝑘−1𝑢(𝑠)𝑑𝑠^𝑥𝑘_𝑡𝑑𝑠𝑣(𝑠)^𝑝−1𝑑𝑡𝑣(𝑡)^1/𝑝<∞, (2.26) and the best constant in (2.21) satisfies 𝑐≈𝐸.

These results are just classical results of Maz’ja [11] and Sinnamon [12] (cf. also [13, 14]).

3. The Main Results

In this section we characterize the validity of the inequalities ^∞₀1𝑈(𝑡)^𝑡₀^∞_𝑠ℎ(𝑧)𝑑𝑧^𝑝𝑢(𝑠)𝑑𝑠^𝑞/𝑝𝑤(𝑡)𝑑𝑡^1/𝑞≤𝑐‖ℎ‖_{𝜃,𝑣,(0,∞)},esssup_{()𝑡∈0,∞}𝑤(𝑡1)𝑈(𝑡)^𝑡₀^∞_𝑠ℎ(𝑧)𝑑𝑧^𝑝𝑢(𝑠)𝑑𝑠^1/𝑝≤𝑐‖ℎ‖_{𝜃,𝑣,(0,∞)}. (3.1)(3.2)

Denote 𝒰()𝑈(𝑥)𝑥,𝑡∶=𝑈(𝑡)(𝑥).+𝑈 (3.3)

First we characterize (3.1) as follows.

Theorem 3.1. Let 0<𝑞<∞, 0<𝑝<∞, 1<𝜃≤∞, and let 𝑢,𝑣,𝑤 be weights. Assume that 𝑢 is such that 𝑈^𝑞/𝑝 is admissible and the measure 𝑤(𝑡)𝑑𝑡 is nondenerate with respect to 𝑈^𝑞/𝑝. Then the inequality (3.1) holds for every measurable function 𝑓 on (0,∞) if and only if(i)1<𝜃≤𝑝<∞, 𝜃≤𝑞𝐴₁∶=sup_{()𝑥∈0,∞}^∞₀𝒰(𝑥,𝑡)^𝑞/𝑝𝑤(𝑡)𝑑𝑡^1/𝑞𝑈(𝑥)^−1/𝑝sup_{()𝑡∈0,𝑥}𝑈(𝑡)^1/𝑝𝑉_𝜃(𝑡)^1/𝜃′<∞. (3.4) Moreover, the best constant 𝑐 in (3.1) satisfies 𝑐≈𝐴₁.(ii)1<𝜃≤𝑝<∞, 0<𝑞<𝜃<∞,  𝑙=𝜃𝑞/(𝜃−𝑞)𝐴₂∶=^∞₀^∞₀𝒰(𝑥,𝑡)^𝑞/𝑝𝑤(𝑡)𝑑𝑡^{(𝑙−𝑞)/𝑞}𝑤(𝑥)𝑈(𝑥)^−𝑙/𝑝sup_{𝑡∈(0,𝑥)}𝑈(𝑡)^𝑙/𝑝𝑉_𝜃(𝑡)^𝑙/𝜃′𝑑𝑥^1/𝑙<∞. (3.5) Moreover, the best constant 𝑐 in (3.1) satisfies 𝑐≈𝐴₂.(iii)0<𝑝<𝜃<∞, 1<𝜃≤𝑞<∞, 𝑟=𝜃𝑝/(𝜃−𝑝)𝐴₃∶=sup_{()𝑥∈0,∞}^∞₀𝒰(𝑥,𝑡)^𝑞/𝑝𝑤(𝑡)𝑑𝑡^1/𝑞∞₀𝒰(𝑡,𝑥)^𝑟/𝑝𝑉_𝜃(𝑡)^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡^1/𝑟<∞. (3.6) Moreover, the best constant 𝑐 in (3.1) satisfies 𝑐≈𝐴₃.(iv)0<𝑝<𝜃<∞, 1<𝜃<∞, 𝑞<𝜃,  𝑟=𝜃𝑝/(𝜃−𝑝),  𝑙=𝜃𝑞/(𝜃−𝑞), 𝐴₄∶=^∞₀^∞₀𝒰(𝑥,𝑡)^𝑞/𝑝𝑤(𝑡)𝑑𝑡^{(𝑙−𝑞)/𝑞}𝑤(𝑥)^∞₀𝒰(𝑡,𝑥)^𝑟/𝑝𝑉_𝜃(𝑡)^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡^𝑙/𝑟𝑑𝑥^1/𝑙<∞. (3.7) Moreover, the best constant 𝑐 in (3.1) satisfies 𝑐≈𝐴₄.(v) Let 𝜃=∞, 0<𝑝<∞, 0<𝑞<∞, 𝐴₅∶=^∞₀𝒰(𝑡,𝑥)𝑉_𝜃(𝑡)^𝑝−1𝑑𝑡𝑣(𝑡)^𝑞/𝑝𝑤(𝑥)𝑑𝑥^1/𝑞<∞. (3.8) Moreover, the best constant 𝑐 in (3.1) satisfies 𝑐≈𝐴₅.

Proof. Define 𝜑(𝑥)=^∞₀𝒰(𝑥,𝑠)^𝑞/𝑝𝑤(𝑠)𝑑𝑠. (3.9) Then 𝜑∈Ω_𝑈^𝑞/𝑝, and therefore there exists a discretizing sequence for 𝜑 with respect to 𝑈^𝑞/𝑝. Let {𝑥_𝑘} be one such sequence. Then 𝜑(𝑥_𝑘)↑↑ and 𝜑(𝑥_𝑘)𝑈^−𝑞/𝑝↓↓. Furthermore, there is a decomposition ℤ=ℤ₁∪ℤ₂, ℤ₁∩ℤ₂=∅ such that for every 𝑘∈ℤ₁ and 𝑡∈[𝑥_𝑘,𝑥_𝑘+1], 𝜑(𝑥_𝑘)≈𝜑(𝑡) and for every 𝑘∈ℤ₂ and 𝑡∈[𝑥_𝑘,𝑥_𝑘+1], 𝜑(𝑥_𝑘)𝑈(𝑥_𝑘)^−𝑞/𝑝≈𝜑(𝑡)𝑈(𝑡)^−𝑞/𝑝.
For the left-hand side of (3.1), by using Lemma 2.7 with 𝑑𝜈(𝑡)=𝑤(𝑡)𝑑𝑡,𝑓(𝑡)=^𝑡₀^∞_𝑠ℎ(𝑧)𝑑𝑧^𝑝𝑢(𝑠)𝑑𝑠, (3.10) we get that 𝐽∶=^∞₀1𝑈(𝑡)^𝑡₀^∞_𝑠ℎ(𝑧)𝑑𝑧^𝑝𝑢(𝑠)𝑑𝑠^𝑞/𝑝𝑤(𝑡)𝑑𝑡^1/𝑞≈_𝑘∈ℤ^𝑥𝑘₀^∞_𝑠ℎ(𝑧)𝑑𝑧^𝑝𝑢(𝑠)𝑑𝑠^𝑞/𝑝𝜑𝑥_𝑘𝑈^𝑞/𝑝𝑥_𝑘^1/𝑞. (3.11) Moreover, by using Lemma 2.9, we get that 𝐽≈_𝑘∈ℤ^𝑥𝑘_𝑥𝑘−1^∞_𝑠ℎ(𝑧)𝑑𝑧^𝑝𝑢(𝑠)𝑑𝑠^𝑞/𝑝𝜑𝑥_𝑘𝑈^𝑞/𝑝𝑥_𝑘^1/𝑞=_𝑘∈ℤ^𝑥𝑘_𝑥𝑘−1^𝑥𝑘_𝑠ℎ(𝑧)𝑑𝑧+^∞_𝑥𝑘ℎ(𝑧)𝑑𝑧^𝑝𝑢(𝑠)𝑑𝑠^𝑞/𝑝𝜑𝑥_𝑘𝑈^𝑞/𝑝𝑥_𝑘^1/𝑞≈_𝑘∈ℤ^𝑥𝑘_𝑥𝑘−1^𝑥𝑘_𝑠ℎ(𝑧)𝑑𝑧^𝑝𝑢(𝑠)𝑑𝑠^𝑞/𝑝𝜑𝑥_𝑘𝑈^𝑞/𝑝𝑥_𝑘^1/𝑞+_𝑘∈ℤ^𝑥𝑘_𝑥𝑘−1^∞_𝑥𝑘ℎ(𝑧)𝑑𝑧^𝑝𝑢(𝑠)𝑑𝑠^𝑞/𝑝𝜑(𝑥_𝑘)𝑈^𝑞/𝑝𝑥_𝑘^1/𝑞≈_𝑘∈ℤ^𝑥𝑘_𝑥𝑘−1^𝑥𝑘_𝑠ℎ(𝑧)𝑑𝑧^𝑝𝑢(𝑠)𝑑𝑠^𝑞/𝑝𝜑𝑥_𝑘𝑈^𝑞/𝑝𝑥_𝑘^1/𝑞+_𝑘∈ℤ^∞_𝑥𝑘ℎ(𝑧)𝑑𝑧^𝑞𝑥𝑘_𝑥𝑘−1𝑢(𝑠)𝑑𝑠^𝑞/𝑝𝜑𝑥_𝑘𝑈^𝑞/𝑝𝑥_𝑘^1/𝑞. (3.12)
By now using the fact that ∫^𝑥𝑘_𝑥𝑘−1𝑢(𝑠)𝑑𝑠=𝑈(𝑥_𝑘)−𝑈(𝑥_𝑘−1)≈𝑈(𝑥_𝑘), we find that 𝐽≈_𝑘∈ℤ^𝑥𝑘_𝑥𝑘−1^𝑥𝑘_𝑠ℎ(𝑧)𝑑𝑧^𝑝𝑢(𝑠)𝑑𝑠^𝑞/𝑝𝜑𝑥_𝑘𝑈^𝑞/𝑝𝑥_𝑘^1/𝑞+_𝑘∈ℤ^∞_𝑥𝑘ℎ(𝑧)𝑑𝑧^𝑞𝜑𝑥_𝑘^1/𝑞, (3.13) is, by using Lemma 2.9 on the second term, 𝐽≈_𝑘∈ℤ^𝑥𝑘_𝑥𝑘−1^𝑥𝑘_𝑠ℎ(𝑧)𝑑𝑧^𝑝𝑢(𝑠)𝑑𝑠^𝑞/𝑝𝜑𝑥_𝑘𝑈^𝑞/𝑝𝑥_𝑘^1/𝑞+_𝑘∈ℤ^𝑥𝑘+1_𝑥𝑘ℎ(𝑧)𝑑𝑧^𝑞𝜑𝑥_𝑘^1/𝑞∶=𝐼+𝐼𝐼. (3.14) Now we will distinguish several cases. We start with the case 1<𝜃≤𝑝<∞. Then, by using Lemma 2.15, we get that 𝐼=_𝑘∈ℤ^𝑥𝑘_𝑥𝑘−1^𝑥𝑘_𝑠ℎ(𝑧)𝑑𝑧^𝑝𝑢(𝑠)𝑑𝑠^𝑞/𝑝𝜑𝑥_𝑘𝑈^𝑞/𝑝𝑥_𝑘^1/𝑞≤_𝑘∈ℤ𝜑𝑥_𝑘𝑈^𝑞/𝑝𝑥_𝑘sup_{𝑥𝑘−1<𝑡<𝑥𝑘}𝑈(𝑡)^𝑞/𝑝𝑉_𝜃(𝑡)^{𝑞/𝜃′𝑥𝑘}_𝑥𝑘−1ℎ(𝑧)^𝜃𝑣(𝑧)𝑑𝑧^{𝑞/𝜃1/𝑞}. (3.15) Moreover, by applying Hölder’s inequality for 𝐼𝐼, we find that 𝐼𝐼=_𝑘∈ℤ𝜑𝑥_𝑘^𝑥𝑘+1_𝑥𝑘ℎ(𝑧)𝑑𝑧^𝑞1/𝑞≤_𝑘∈ℤ𝜑𝑥_𝑘^𝑥𝑘+1_𝑥𝑘𝑣(𝑧)^1−𝜃′𝑑𝑧^{𝑞/𝜃′𝑥𝑘+1}_𝑥𝑘ℎ(𝑧)^𝜃𝑣(𝑧)𝑑𝑧^{𝑞/𝜃1/𝑞}≤_𝑘∈ℤ𝜑𝑥_𝑘𝑉_𝜃𝑥_𝑘^{𝑞/𝜃′𝑥𝑘+1}_𝑥𝑘ℎ(𝑧)^𝜃𝑣(𝑧)𝑑𝑧^{𝑞/𝜃1/𝑞}. (3.16)(i) In the case 𝑞/𝜃≥1, according to (3.15), we have that 𝐼≤sup_𝑘∈ℤ𝜑𝑥_𝑘^1/𝑞𝑈^1/𝑝𝑥_𝑘sup_{𝑥𝑘−1<𝑡<𝑥𝑘}𝑈(𝑡)^1/𝑝𝑉_𝜃(𝑡)^1/𝜃′‖ℎ‖_{𝜃,𝑣,(0,∞)}. (3.17) Similarly, if  𝑞/𝜃≥1, then, according to (3.16), we obtain that 𝐼𝐼≤sup_𝑘∈ℤ𝜑𝑥_𝑘^1/𝑞𝑉_𝜃𝑥_𝑘^1/𝜃′‖ℎ‖_{𝜃,𝑣,(0,∞)}, (3.18) and, finally, by using (3.9), Lemma 2.13, and (3.14), we get that 𝐽≲sup_𝑘∈ℤ𝜑𝑥_𝑘^1/𝑞𝑈^−1/𝑝𝑥_𝑘sup_{𝑥𝑘−1<𝑡<𝑥𝑘}𝑈(𝑡)^1/𝑝𝑉_𝜃(𝑡)^1/𝜃′+𝑉_𝜃𝑥_𝑘^1/𝜃′‖ℎ‖_{𝜃,𝑣,(0,∞)}≈sup_{()𝑥∈0,∞}𝜑(𝑥)^1/𝑞sup_{()𝑡∈0,∞}𝒰(𝑡,𝑥)^1/𝑝𝑉_𝜃(𝑡)^1/𝜃′‖ℎ‖_{𝜃,𝑣,(0,∞)}=sup_{()𝑥∈0,∞}^∞₀𝒰(𝑥,𝑡)^𝑞/𝑝𝑤(𝑡)𝑑𝑡^1/𝑞sup_{()𝑡∈0,∞}𝒰(𝑡,𝑥)^1/𝑝𝑉_𝜃(𝑡)^1/𝜃′‖ℎ‖_{𝜃,𝑣,(0,∞)}≈sup_{()𝑥∈0,∞}^∞₀𝒰(𝑥,𝑡)^𝑞/𝑝𝑤(𝑡)𝑑𝑡^1/𝑞𝑈(𝑥)^−1/𝑝sup_{()𝑡∈0,𝑥}𝑈(𝑡)^1/𝑝𝑉_𝜃(𝑡)^1/𝜃′‖ℎ‖_{𝜃,𝑣,(0,∞)}. (3.19)(ii) For the case 0<𝑞<𝜃<∞, 𝑙=𝜃𝑞/(𝜃−𝑞), by applying Hölder’s inequality for sums to the right-hand side of (3.15) and (3.16) with exponents 𝜃/𝑞 and 𝑙/𝑞, we find that 𝐼≤_𝑘∈ℤ𝜑𝑥_𝑘^𝑙/𝑞𝑈^𝑙/𝑝𝑥_𝑘sup_{𝑥𝑘−1<𝑡<𝑥𝑘}𝑈(𝑡)^𝑙/𝑝𝑉_𝜃(𝑡)^{𝑙/𝜃′1/𝑙}‖ℎ‖_{𝜃,𝑣,(0,∞)},𝐼𝐼≤_𝑘∈ℤ𝜑𝑥_𝑘^𝑙/𝑞𝑉_𝜃𝑥_𝑘^{𝑙/𝜃′1/𝑙}‖ℎ‖_{𝜃,𝑣,(0,∞)}. (3.20) Therefore, we get that 𝐼+𝐼𝐼≤_𝑘∈ℤ𝜑𝑥_𝑘^𝑙/𝑞𝑈𝑥_𝑘^−𝑙/𝑝sup_{𝑥𝑘−1<𝑡<𝑥𝑘}𝑈(𝑡)^𝑙/𝑝𝑉_𝜃(𝑡)^𝑙/𝜃′+𝑉_𝜃𝑥_𝑘^{𝑙/𝜃′1/𝑙}‖ℎ‖_{𝜃,𝑣,(0,∞)}, (3.21) so that, in view of Lemma 2.11, Theorem 2.6, and (3.14), 𝐽≲_𝑘∈ℤ𝜑𝑥_𝑘^𝑙/𝑞sup_{()𝑡∈0,∞}𝒰𝑡,𝑥_𝑘^𝑙/𝑝𝑉_𝜃(𝑡)^{𝑙/𝜃′1/𝑙}‖ℎ‖_{𝜃,𝑣,(0,∞)}≈^∞₀𝜑(𝑥)^{(𝑙/𝑞)−1}sup_{()𝑡∈0,∞}𝒰(𝑡,𝑥)^𝑙/𝑝𝑉_𝜃(𝑡)^𝑙/𝜃′𝑤(𝑥)𝑑𝑥^1/𝑙‖ℎ‖_{𝜃,𝑣,(0,∞)}≈^∞₀^∞₀𝒰(𝑥,𝑡)^𝑞/𝑝𝑤(𝑡)𝑑𝑡^{(𝑙−𝑞)/𝑞}𝑤(𝑥)𝑈(𝑥)^−𝑙/𝑝sup_{()𝑡∈0,𝑥}𝑈(𝑡)^𝑙/𝑝𝑉_𝜃(𝑡)^𝑙/𝜃′𝑑𝑥^1/𝑙‖ℎ‖_{𝜃,𝑣,(0,∞)}. (3.22) Now let us assume that 0<𝑝<𝜃<∞, 1<𝜃<∞, 1/𝑟=1/𝑝−1/𝜃. By Lemma 2.15, we have that 𝐼≤_𝑘∈ℤ𝜑𝑥_𝑘𝑈^𝑞/𝑝𝑥_𝑘^𝑥𝑘_𝑥𝑘−1^𝑡_𝑥𝑘−1𝑢(𝑠)𝑑𝑠^{𝑟/𝑝𝑥𝑘}_𝑡𝑣(𝑠)^1−𝜃′𝑑𝑠^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡^𝑞/𝑟×^𝑥𝑘_𝑥𝑘−1ℎ(𝑧)^𝜃𝑣(𝑧)𝑑𝑧^{𝑞/𝜃1/𝑞}. (3.23) Moreover, by applying Hölder’s inequality for 𝐼𝐼, we find that 𝐼𝐼≤_𝑘∈ℤ𝜑𝑥_𝑘𝑉_𝜃𝑥_𝑘^{𝑞/𝜃′𝑥𝑘+1}_𝑥𝑘ℎ(𝑧)^𝜃𝑣(𝑧)𝑑𝑧^{𝑞/𝜃1/𝑞}. (3.24)(iii) Now, we assume that 𝑞/𝜃≥1. Then, according to (3.7) and (3.24), we obtain that 𝐼≲sup_𝑘∈ℤ𝜑𝑥_𝑘^1/𝑞𝑈^1/𝑝𝑥_𝑘^𝑥𝑘_𝑥𝑘−1𝑈(𝑡)^𝑟/𝑝𝑉_𝜃(𝑡)^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡^1/𝑟‖ℎ‖_{𝜃,𝑣,(0,∞)},𝐼𝐼≤sup_𝑘∈ℤ𝜑𝑥_𝑘^1/𝑞𝑉_𝜃𝑥_𝑘^1/𝜃′‖ℎ‖_{𝜃,𝑣,(0,∞)}. (3.25) Hence, using Lemmas 2.9 and 2.12, and (3.14), we get that 𝐽≲sup_𝑘∈ℤ𝜑𝑥_𝑘^1/𝑞𝑈^−𝑟/𝑝𝑥_𝑘^𝑥𝑘_𝑥𝑘−1𝑈(𝑡)^𝑟/𝑝𝑉_𝜃(𝑡)^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡+^∞_𝑥𝑘𝑉_𝜃(𝑡)^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡^1/𝑟×‖ℎ‖_{𝜃,𝑣,(0,∞)}≈sup_𝑘∈ℤ𝜑𝑥_𝑘^1/𝑞𝑈^−𝑟/𝑝𝑥_𝑘^𝑥𝑘_𝑥𝑘−1𝑈(𝑡)^𝑟/𝑝𝑉_𝜃(𝑡)^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡+^𝑥𝑘+1_𝑥𝑘𝑉_𝜃(𝑡)^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡^1/𝑟×‖ℎ‖_{𝜃,𝑣,(0,∞)}≈sup_𝑘∈ℤ𝜑𝑥_𝑘^1/𝑞∞₀𝒰𝑡,𝑥_𝑘^𝑟/𝑝𝑉_𝜃(𝑡)^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡^1/𝑟‖ℎ‖_{𝜃,𝑣,(0,∞)}≈sup_{()𝑥∈0,∞}𝜑(𝑥)^1/𝑞∞₀𝒰(𝑡,𝑥)^𝑟/𝑝𝑉_𝜃(𝑡)^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡^1/𝑟‖ℎ‖_{𝜃,𝑣,(0,∞)}=sup_{()𝑥∈0,∞}^∞₀𝒰(𝑥,𝑡)^𝑞/𝑝𝑤(𝑡)𝑑𝑡^1/𝑞∞₀𝒰(𝑡,𝑥)^𝑟/𝑝𝑉_𝜃(𝑡)^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡^1/𝑟×‖ℎ‖_{𝜃,𝑣,(0,∞)}. (3.26)(iv) Next, we consider the case 0<𝑞<𝜃, 1/𝑙=1/𝑞−1/𝜃. By using Hölder’s inequality for sums to the right-hand side of (3.7) and (3.24) with exponents 𝜃/𝑞 and 𝑙/𝑞, we get that 𝐼≤_𝑘∈ℤ𝜑𝑥_𝑘^𝑙/𝑞𝑈^𝑙/𝑝𝑥_𝑘^𝑥𝑘𝑥_𝑘−1𝑈(𝑡)^𝑟/𝑝𝑉_𝜃(𝑡)^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡^{𝑙/𝑟1/𝑙}‖ℎ‖_{𝜃,𝑣,(0,∞)},𝐼𝐼≤_𝑘∈ℤ𝜑𝑥_𝑘^𝑙/𝑞𝑉_𝜃𝑥_𝑘^{𝑙/𝜃′1/𝑙}‖ℎ‖_{𝜃,𝑣,(0,∞)}. (3.27) Therefore, using Lemmas 2.9 and 2.10, Theorem 2.6, and (3.14), we find that 𝐽≲_𝑘∈ℤ𝜑𝑥_𝑘^𝑙/𝑞𝑈^−𝑙/𝑝𝑥_𝑘^𝑥𝑘_𝑥𝑘−1𝑈(𝑡)^𝑟/𝑝𝑉_𝜃(𝑡)^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡^𝑙/𝑟+𝑉𝑥_𝑘^{𝑙/𝜃′1/𝑙}×‖ℎ‖_{𝜃,𝑣,(0,∞)}≈_𝑘∈ℤ𝜑𝑥_𝑘^𝑙/𝑞𝑈^−𝑟/𝑝𝑥_𝑘^𝑥𝑘_𝑥𝑘−1𝑈(𝑡)^𝑟/𝑝𝑉_𝜃(𝑡)^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡+^∞_𝑥𝑘𝑉_𝜃(𝑡)^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡^{𝑙/𝑟1/𝑙}×‖ℎ‖_{𝜃,𝑣,(0,∞)}≈_𝑘∈ℤ𝜑𝑥_𝑘^𝑙/𝑞𝑈^−𝑟/𝑝𝑥_𝑘^𝑥𝑘_𝑥𝑘−1𝑈(𝑡)^𝑟/𝑝𝑉_𝜃(𝑡)^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡+^𝑥𝑘+1_𝑥𝑘𝑉_𝜃(𝑡)^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡^{𝑙/𝑟1/𝑙}×‖ℎ‖_{𝜃,𝑣,(0,∞)}≈_𝑘∈ℤ𝜑𝑥_𝑘^𝑙/𝑞∞₀𝒰𝑡,𝑥_𝑘^𝑟/𝑝𝑉_𝜃(𝑡)^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡^{𝑙/𝑟𝑙/𝑙}‖ℎ‖_{𝜃,𝑣,(0,∞)}≈^∞₀𝜑(𝑡)^{𝑙/𝑞−1∞}₀𝒰(𝑡,𝑥)^𝑟/𝑝𝑉_𝜃(𝑡)^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡^𝑙/𝑟𝑤(𝑥)𝑑𝑥^𝑙/𝑙‖ℎ‖_{𝜃,𝑣,(0,∞)}=^∞₀^∞₀𝒰(𝑥,𝑡)^𝑞/𝑝𝑤(𝑡)𝑑𝑡^{(𝑙−𝑞)/𝑞}𝑤(𝑥)^∞₀𝒰(𝑡,𝑥)^𝑟/𝑝𝑉_𝜃(𝑡)^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡^𝑙/𝑟𝑑𝑥^1/𝑙×‖ℎ‖_{𝜃,𝑣,(0,∞)}. (3.28)(v) Let 𝜃=∞, 0<𝑝<∞, 0<𝑞<∞. According to Lemma 2.15, we have that 𝐼≤_𝑘∈ℤ𝜑𝑥_𝑘𝑈^𝑞/𝑝𝑥_𝑘^𝑥𝑘_𝑥𝑘−1^𝑡_𝑥𝑘−1𝑢(𝑠)𝑑𝑠^𝑥𝑘_𝑡𝑑𝑧𝑣(𝑧)^𝑝−1𝑑𝑡𝑣(𝑡)^{𝑞/𝑝1/𝑞}‖ℎ‖_{𝜃,𝑣,(0,∞)}. (3.29) Moreover, it yields that 𝐼𝐼≤_𝑘∈ℤ𝜑𝑥_𝑘^𝑥𝑘+1_𝑥𝑘𝑑𝑧𝑣(𝑧)^𝑞1/𝑞‖ℎ‖_{𝜃,𝑣,(0,∞)}. (3.30) Hence, by integrating by parts, using Lemmas 2.9 and 2.10, and (3.14), we get that 𝐽≲_𝑘∈ℤ𝜑𝑥_𝑘𝑈𝑥_𝑘^−1𝑥𝑘_𝑥𝑘−1𝑈(𝑠)^∞_𝑠𝑑𝑧𝑣(𝑧)^𝑝−1𝑑𝑠𝑣(𝑠)+^∞_𝑥𝑘^∞_𝑠𝑑𝑧𝑣(𝑧)^𝑝−1𝑑𝑠𝑣(𝑠)^{𝑞/𝑝1/𝑞}×‖ℎ‖_{∞,𝑣,(0,∞)}≈_𝑘∈ℤ𝜑𝑥_𝑘𝑈𝑥_𝑘^−1𝑥𝑘_𝑥𝑘−1𝑈(𝑠)^∞_𝑠𝑑𝑧𝑣(𝑧)^𝑝−1𝑑𝑠𝑣(𝑠)+^𝑥𝑘+1_𝑥𝑘^∞_𝑠𝑑𝑧𝑣(𝑧)^𝑝−1𝑑𝑠𝑣(𝑠)^{𝑞/𝑝1/𝑞}×‖ℎ‖_{∞,𝑣,(0,∞)}≈^∞₀𝒰(𝑡,𝑥)^∞_𝑡𝑑𝑧𝑣(𝑧)^𝑝−1𝑑𝑡𝑣(𝑡)^𝑞/𝑝𝜔(𝑥)𝑑𝑥^1/𝑞‖ℎ‖_{∞,𝑣,(0,∞)}. (3.31)
Now we prove the lower bounds (necessity). Let 0<𝑞<∞ and {𝑥_𝑘} be a discretizing sequence for 𝜑 from (3.9). Then, by (3.14), we find that _𝑘∈ℤ^𝑥𝑘_𝑥𝑘−1^𝑥𝑘_𝑠ℎ(𝑧)𝑑𝑧^𝑝𝑢(𝑠)𝑑𝑠^𝑞/𝑝𝜑𝑥_𝑘𝑈^𝑞/𝑝(𝑥_𝑘)^1/𝑞+_𝑘∈ℤ^𝑥𝑘+1_𝑥𝑘ℎ(𝑧)𝑑𝑧^𝑞𝜑𝑥_𝑘^1/𝑞≲_𝑘∈ℤ^𝑥𝑘_𝑥𝑘−1ℎ(𝑧)^𝜃𝑣(𝑧)𝑑𝑧^1/𝜃. (3.32) Let 1<𝜃≤𝑝<∞. For 𝑘∈ℤ, let ℎ_𝑘 be functions that saturate the Hardy inequality (2.21) and Hölder’s inequality, that is, functions ℎ_𝑘 satisfying suppℎ_𝑘⊂𝑥_𝑘−1,𝑥_𝑘,_{𝑥𝑘𝑥𝑘−1}ℎ_𝑘(𝑡)^𝜃𝑣(𝑡)𝑑𝑡=1,_𝑥𝑘𝑠ℎ_𝑘(𝑧)𝑑𝑧_{𝑝,𝑢,(𝑥𝑘−1,𝑥𝑘)}≳sup_{𝑥𝑘−1<𝑡<𝑥𝑘}^𝑡_𝑥𝑘−1𝑢(𝑠)𝑑𝑠^{1/𝑝𝑥𝑘}_𝑡𝑣(𝑠)^1−𝜃′𝑑𝑠^1/𝜃′,_{𝑥𝑘𝑥𝑘−1}ℎ_𝑘(𝑡)𝑑𝑡≳^𝑥𝑘_𝑥𝑘−1𝑣(𝑡)^1−𝜃′𝑑𝑡^1/𝜃′. (3.33) Now we define the test function ℎ(𝑡)=_𝑘∈ℤ𝑎_𝑘ℎ_𝑘(𝑡), (3.34) where {𝑎_𝑘} is a sequence of positive real numbers. Thus, using test function (3.34) in (3.32), we get that _𝑘∈ℤ𝑎^𝑞_𝑘𝜑𝑥_𝑘𝑈^𝑞/𝑝𝑥_𝑘sup_{𝑥𝑘−1<𝑡<𝑥𝑘}^𝑡_𝑥𝑘−1𝑢(𝑠)𝑑𝑠^{𝑞/𝑝𝑥𝑘}_𝑡𝑣(𝑠)^1−𝜃′𝑑𝑠^{𝑞/𝜃′1/𝑞}+_𝑘∈ℤ𝑎^𝑞_𝑘𝜑𝑥_𝑘^𝑥𝑘+1_𝑥𝑘𝑣(𝑡)^1−𝜃′𝑑𝑡^{𝑞/𝜃′1/𝑞}≲_𝑘∈ℤ𝑎^𝜃_𝑘^1/𝜃. (3.35) Now using Proposition 2.14 for the case 𝜃≤𝑞, we obtain that sup_𝑘∈ℤ𝜑𝑥_𝑘^{1/𝑞𝑥𝑘+1}_𝑥𝑘𝑣(𝑡)^1−𝜃′𝑑𝑡^1/𝜃′+sup_𝑘∈ℤ𝜑𝑥_𝑘^1/𝑞𝑈^1/𝑝𝑥_𝑘sup_{𝑥𝑘−1<𝑡<𝑥𝑘}^𝑡_𝑥𝑘−1𝑢(𝑠)𝑑𝑠^{1/𝑝𝑥𝑘}_𝑡𝑣(𝑠)^1−𝜃′𝑑𝑠^1/𝜃′<∞. (3.36) On the other hand, using Lemma 2.9, we get that 𝐴₁≲sup_𝑘∈ℤ𝜑𝑥_𝑘^1/𝑞𝑈^1/𝑝𝑥_𝑘sup_{𝑥𝑘−1<𝑡<𝑥𝑘}𝑈(𝑡)^1/𝑝𝑉_𝜃(𝑡)^1/𝜃′+sup_𝑘∈ℤ𝜑𝑥_𝑘^1/𝑞𝑉_𝜃𝑥_𝑘^1/𝜃′≲sup_𝑘∈ℤ𝜑𝑥_𝑘^1/𝑞𝑈^1/𝑝𝑥_𝑘sup_{𝑥𝑘−1<𝑡<𝑥𝑘}𝑈(𝑡)^{1/𝑝𝑥𝑘}_𝑡𝑣(𝑠)^1−𝜃′𝑑𝑠^1/𝜃′+sup_𝑘∈ℤ𝜑𝑥_𝑘^1/𝑞𝑉_𝜃𝑥_𝑘^1/𝜃′≲sup_𝑘∈ℤ𝜑𝑥_𝑘^1/𝑞𝑈^1/𝑝𝑥_𝑘sup_{𝑥𝑘−1<𝑡<𝑥𝑘}^𝑡_𝑥𝑘−1𝑢(𝑠)𝑑𝑠^{1/𝑝𝑥𝑘}_𝑡𝑣(𝑠)^1−𝜃′𝑑𝑠^1/𝜃′+sup_𝑘∈ℤ𝜑𝑥_𝑘^1/𝑞𝑉_𝜃𝑥_𝑘^1/𝜃′≲sup_𝑘∈ℤ𝜑𝑥_𝑘^1/𝑞𝑈^1/𝑝𝑥_𝑘sup_{𝑥𝑘−1<𝑡<𝑥𝑘}^𝑡_𝑥𝑘−1𝑢(𝑠)𝑑𝑠^{1/𝑝𝑥𝑘}_𝑡𝑣(𝑠)^1−𝜃′𝑑𝑠^1/𝜃′+sup_𝑘∈ℤ𝜑𝑥_𝑘^{1/𝑞𝑥𝑘+1}_𝑥𝑘𝑣(𝑠)^1−𝜃′𝑑𝑠^1/𝜃′<∞. (3.37) Let 0<𝑞<𝜃<∞. From (3.35) and Proposition 2.14, we obtain that _𝑘∈ℤ𝜑𝑥_𝑘^{𝑙/𝑞𝑥𝑘+1}_𝑥𝑘𝑣(𝑡)^1−𝜃′𝑑𝑡^{𝑙/𝜃′1/𝑙}+_𝑘∈ℤ𝜑𝑥_𝑘^𝑙/𝑞𝑈^𝑙/𝑝𝑥_𝑘sup_{𝑥𝑘−1<𝑡<𝑥𝑘}^𝑡_𝑥𝑘−1𝑢(𝑠)𝑑𝑠^{𝑙/𝑝𝑥𝑘}_𝑡𝑣(𝑠)^1−𝜃′𝑑𝑠^{𝑙/𝜃′1/𝑙}<∞. (3.38) Since 𝐴₂≈_𝑘∈ℤ𝜑𝑥_𝑘^𝑙/𝑞𝑉_𝜃𝑥_𝑘^{𝑙/𝜃′1/𝑙}+_𝑘∈ℤ𝜑𝑥_𝑘^𝑙/𝑞𝑈𝑥_𝑘^𝑙/𝑝sup_{𝑥𝑘−1<𝑡<𝑥𝑘}𝑈(𝑡)^𝑙/𝑝𝑉_𝜃(𝑡)^{𝑙/𝜃′1/𝑙}≲_𝑘∈ℤ𝜑𝑥_𝑘^𝑙/𝑞𝑉_𝜃𝑥_𝑘^{𝑙/𝜃′1/𝑙}+_𝑘∈ℤ𝜑𝑥_𝑘^𝑙/𝑞𝑈𝑥_𝑘^𝑙/𝑝sup_{𝑥𝑘−1<𝑡<𝑥𝑘}𝑈(𝑡)^{𝑙/𝑝𝑥𝑘}_𝑡𝑣(𝑠)^1−𝜃′𝑑𝑠^{𝑙/𝜃′1/𝑙}≲_𝑘∈ℤ𝜑𝑥_𝑘^𝑙/𝑞𝑉_𝜃𝑥_𝑘^{𝑙/𝜃′1/𝑙}+_𝑘∈ℤ𝜑𝑥_𝑘^𝑙/𝑞𝑈𝑥_𝑘^𝑙/𝑝sup_{𝑥𝑘−1<𝑡<𝑥𝑘}^𝑡_𝑥𝑘−1𝑢(𝑠)𝑑𝑠^{𝑙/𝑝𝑥𝑘}_𝑡𝑣(𝑠)^1−𝜃′𝑑𝑠^{𝑙/𝜃′1/𝑙}, (3.39) by Lemma 2.9, we arrive at 𝐴₂≲_𝑘∈ℤ𝜑𝑥_𝑘^{𝑙/𝑞𝑥𝑘}_𝑡𝑣(𝑠)^1−𝜃′𝑑𝑠^{𝑙/𝜃′1/𝑙}+𝜑𝑥_𝑘^𝑙/𝑞𝑈𝑥_𝑘^𝑙/𝑝sup_{𝑥𝑘−1<𝑡<𝑥𝑘}^𝑡_𝑥𝑘−1𝑢(𝑠)𝑑𝑠^{𝑙/𝑝𝑥𝑘}_𝑡𝑣(𝑠)^1−𝜃′𝑑𝑠^{𝑙/𝜃′1/𝑙}<∞. (3.40) Let 1<𝜃<∞, 0<𝑝<𝜃. For 𝑘∈ℤ, let ℎ_𝑘 be functions that saturate the Hardy inequality (2.21) and Hölder’s inequality, that is, functions ℎ_𝑘 satisfying suppℎ_𝑘⊂𝑥_𝑘−1,𝑥_𝑘,_{𝑥𝑘𝑥𝑘−1}ℎ_𝑘(𝑡)^𝜃𝑣(𝑡)𝑑𝑡=1,_𝑥𝑘𝑠ℎ_𝑘(𝑧)𝑑𝑧_{𝑝,𝑢,(𝑥𝑘−1,𝑥𝑘)}≳^𝑥𝑘_𝑥𝑘−1^𝑡_𝑥𝑘−1𝑢(𝑠)𝑑𝑠^{𝑟/𝑝𝑥𝑘}_𝑡𝑣(𝑠)^1−𝜃′𝑑𝑠^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡^1/𝑟,_{𝑥𝑘𝑥𝑘−1}ℎ_𝑘(𝑡)𝑑𝑡≳^𝑥𝑘_𝑥𝑘−1𝑣(𝑡)^1−𝜃′𝑑𝑡^1/𝜃′. (3.41) Now we define the test function ℎ(𝑡)=_𝑘∈ℤ𝑎_𝑘ℎ_𝑘(𝑡), (3.42) where {𝑎_𝑘} is a sequence of positive real numbers. Thus, using test function (3.42) in (3.32), we get that _𝑘∑_∈ℤ𝑎^𝑞_𝑘𝜑𝑥_𝑘𝑈^𝑞/𝑝𝑥_𝑘∫^𝑥𝑘_𝑥𝑘−1∫^𝑡_𝑥𝑘−1𝑢(𝑠)𝑑𝑠^𝑟/𝑝∫^𝑥𝑘_𝑡𝑣(𝑠)^1−𝜃′𝑑𝑠^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡^{𝑞/𝑟1/𝑞}+_𝑘∈ℤ𝑎^𝑞_𝑘𝜑𝑥_𝑘^𝑥𝑘+1_𝑥𝑘𝑣(𝑡)^1−𝜃′𝑑𝑡^{𝑞/𝜃′1/𝑞}≲_𝑘∈ℤ𝑎^𝜃_𝑘^1/𝜃. (3.43) Now using Proposition 2.14 for the case 𝜃≤𝑞, we obtain that sup_𝑘∈ℤ𝜑𝑥_𝑘^1/𝑞𝑈^1/𝑝𝑥_𝑘^𝑥𝑘_𝑥𝑘−1^𝑡_𝑥𝑘−1𝑢(𝑠)𝑑𝑠^{𝑟/𝑝𝑥𝑘}_𝑡𝑣(𝑠)^1−𝜃′𝑑𝑠^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡^1/𝑟+sup_𝑘∈ℤ𝜑𝑥_𝑘^{1/𝑞𝑥𝑘+1}_𝑥𝑘𝑣(𝑡)^1−𝜃′𝑑𝑡^1/𝜃′<∞. (3.44) Since 𝐴₃≈sup_𝑘∈ℤ𝜑𝑥_𝑘^1/𝑞𝑈^−𝑟/𝑝𝑥_𝑘^𝑥𝑘_𝑥𝑘−1𝑈(𝑡)^𝑟/𝑝𝑉_𝜃(𝑡)^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡+^∞_𝑥𝑘𝑉_𝜃(𝑡)^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡^1/𝑟≲sup_𝑘∈ℤ𝜑𝑥_𝑘^1/𝑞𝑈^1/𝑝𝑥_𝑘^𝑥𝑘_𝑥𝑘−1𝑈(𝑡)^𝑟/𝑝𝑉_𝜃(𝑡)^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡^1/𝑟+sup_𝑘∈ℤ𝜑𝑥_𝑘^1/𝑞𝑉_𝜃𝑥_𝑘^1/𝜃′≲sup_𝑘∈ℤ𝜑𝑥_𝑘^1/𝑞𝑈^1/𝑝𝑥_𝑘^𝑥𝑘_𝑥𝑘−1^𝑡_𝑥𝑘−1𝑢(𝑠)𝑑𝑠^𝑟/𝑝𝑉_𝜃(𝑡)^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡^1/𝑟+sup_𝑘∈ℤ𝜑𝑥_𝑘^1/𝑞𝑉_𝜃𝑥_𝑘^1/𝜃′, (3.45) by integrating by parts, we get that 𝐴₃≲sup_𝑘∈ℤ𝜑𝑥_𝑘^1/𝑞𝑈^1/𝑝𝑥_𝑘^𝑥𝑘_𝑥𝑘−1^∞_𝑡𝑣(𝑠)^1−𝜃′𝑑𝑠^{(𝑟/𝑝′)+1}𝑢(𝑡)^𝑡_𝑥𝑘−1𝑢(𝑠)𝑑𝑠^{(𝑟/𝑝)−1}𝑑𝑡^1/𝑟+sup_𝑘∈ℤ𝜑𝑥_𝑘^1/𝑞𝑉_𝜃𝑥_𝑘^1/𝜃′≲sup_𝑘∈ℤ𝜑𝑥_𝑘^1/𝑞𝑈^1/𝑝𝑥_𝑘^𝑥𝑘_𝑥𝑘−1^𝑥𝑘_𝑡𝑣(𝑠)^1−𝜃′𝑑𝑠^{(𝑟/𝑝′)+1}𝑢(𝑡)^𝑡_𝑥𝑘−1𝑢(𝑠)𝑑𝑠^{(𝑟/𝑝)−1}𝑑𝑡^1/𝑟+sup_𝑘∈ℤ𝜑𝑥_𝑘^1/𝑞𝑉_𝜃𝑥_𝑘^1/𝜃′. (3.46) Again integrating by parts, we arrive at 𝐴₃≲sup_𝑘∈ℤ𝜑𝑥_𝑘^1/𝑞𝑈^1/𝑝𝑥_𝑘^𝑥𝑘_𝑥𝑘−1^𝑡_𝑥𝑘−1𝑢(𝑠)𝑑𝑠^{𝑟/𝑝𝑥𝑘}_𝑡𝑣(𝑠)^1−𝜃′𝑑𝑠^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡^1/𝑟+sup_𝑘∈ℤ𝜑𝑥_𝑘^1/𝑞𝑉_𝜃𝑥_𝑘^1/𝜃′<∞. (3.47) Now let 1<𝜃<∞, 0<𝑝<𝜃, 𝑞<𝜃. By using (3.43) and Proposition 2.14, we obtain _𝑘∈ℤ𝜑𝑥_𝑘^𝑙/𝑝𝑈^1/𝑝𝑥_𝑘^𝑥𝑘_𝑥𝑘−1^𝑡_𝑥𝑘−1𝑢(𝑠)𝑑𝑠^{𝑟/𝑝𝑥𝑘}_𝑡𝑣(𝑠)^1−𝜃′𝑑𝑠^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡^{𝑙/𝑟1/𝑙}+_𝑘∈ℤ𝜑𝑥_𝑘^{𝑙/𝑝𝑥𝑘+1}_𝑥𝑘𝑣(𝑡)^1−𝜃′𝑑𝑡^{𝑙/𝜃′1/𝑙}<∞. (3.48) Since 𝐴₄≈_𝑘∈ℤ𝜑𝑥_𝑘^𝑙/𝑞𝑈^−𝑙/𝑝(𝑥_𝑘)^𝑥𝑘_𝑥𝑘−1𝑈(𝑡)^𝑟/𝑝𝑉_𝜃(𝑡)^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡^𝑙/𝑟+𝑉_𝜃𝑥_𝑘^{𝑙/𝜃′1/𝑙}≲_𝑘∈ℤ𝜑𝑥_𝑘^𝑙/𝑞𝑈^𝑙/𝑝𝑥_𝑘^𝑥𝑘𝑥_𝑘−1𝑈(𝑡)^𝑟/𝑝𝑉_𝜃(𝑡)^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡^{𝑙/𝑟1/𝑙}+_𝑘∈ℤ𝜑𝑥_𝑘^𝑙/𝑞𝑉_𝜃𝑥_𝑘^{𝑙/𝜃′1/𝑙}≲_𝑘∈ℤ𝜑𝑥_𝑘^𝑙/𝑞𝑈^𝑙/𝑝𝑥_𝑘^𝑥𝑘𝑥_𝑘−1^𝑡𝑥_𝑘−1𝑢(𝑠)𝑑𝑠^𝑟/𝑝𝑉_𝜃(𝑡)^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡^{𝑙/𝑟1/𝑙}+_𝑘∈ℤ𝜑𝑥_𝑘^𝑙/𝑞𝑉_𝜃𝑥_𝑘^{𝑙/𝜃′1/𝑙}, (3.49) integrating by parts, we find that 𝐴₄≲_𝑘∈ℤ𝜑𝑥_𝑘^𝑙/𝑞𝑈^𝑙/𝑝𝑥_𝑘^𝑥𝑘𝑥_𝑘−1^∞𝑡𝑣(𝑠)^1−𝜃′𝑑𝑠^{(𝑟/𝑝′)+1}𝑢(𝑡)^𝑡_𝑥𝑘−1𝑢(𝑠)𝑑𝑠^{(𝑟/𝑝)−1}𝑑𝑡^{𝑙/𝑟1/𝑙}+_𝑘∈ℤ𝜑𝑥_𝑘^𝑙/𝑞𝑉_𝜃𝑥_𝑘^{𝑙/𝜃′1/𝑙}≲_𝑘∈ℤ𝜑𝑥_𝑘^𝑙/𝑞𝑈^𝑙/𝑝𝑥_𝑘^𝑥𝑘𝑥_𝑘−1^𝑥𝑘𝑡𝑣(𝑠)^1−𝜃′𝑑𝑠^{(𝑟/𝑝′)+1}𝑢(𝑡)^𝑡_𝑥𝑘−1𝑢(𝑠)𝑑𝑠^{(𝑟/𝑝)−1}𝑑𝑡^{𝑙/𝑟1/𝑙}+_𝑘∈ℤ𝜑𝑥_𝑘^𝑙/𝑞𝑉_𝜃𝑥_𝑘^{𝑙/𝜃′1/𝑙}. (3.50) Again integrating by parts, we arrive at 𝐴₄≲_𝑘∈ℤ𝜑𝑥_𝑘^𝑙/𝑞𝑈^𝑙/𝑝𝑥_𝑘^𝑥𝑘𝑥_𝑘−1^𝑡𝑥_𝑘−1𝑢(𝑠)𝑑𝑠^{𝑟/𝑝𝑥𝑘}_𝑡𝑣(𝑠)^1−𝜃′𝑑𝑠^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡^{𝑙/𝑟1/𝑙}+_𝑘∈ℤ𝜑𝑥_𝑘^𝑙/𝑞𝑉_𝜃𝑥_𝑘^{𝑙/𝜃′1/𝑙}<∞. (3.51)
Now let 0<𝑝<∞, 𝜃=∞, 0<𝑞<∞. For 𝑘∈ℤ, let ℎ_𝑘 be functions that saturate the Hardy inequality (2.21) and Hölder’s inequality for 𝜃=∞, that is, functions ℎ_𝑘 satisfying suppℎ_𝑘⊂𝑥_𝑘−1,𝑥_𝑘,ℎ_{𝑘∞,𝑣,(0,∞)}=1,_𝑥𝑘𝑠ℎ_𝑘(𝑧)𝑑𝑧_{𝑝,𝑢,(𝑥𝑘−1,𝑥𝑘)}≳^𝑥𝑘_𝑥𝑘−1^𝑥𝑘_𝑠𝑑𝑧𝑣(𝑧)^𝑝𝑢(𝑠)𝑑𝑠^1/𝑝,_{𝑥𝑘𝑥𝑘−1}ℎ_𝑘(𝑡)𝑑𝑡≳^𝑥𝑘+1_𝑥𝑘𝑑𝑧𝑣(𝑧). (3.52) Now we define the test function ℎ(𝑡)=_𝑘∈ℤ𝑎_𝑘ℎ_𝑘(𝑡), (3.53) where {𝑎_𝑘} is a sequence of positive real numbers. Thus, using test function (3.53) in (3.32), we get _𝑘∈ℤ𝑎^𝑞_𝑘𝜑𝑥_𝑘𝑈𝑥_𝑘^{𝑞/𝑝𝑥𝑘𝑥}_𝑘−1^𝑥𝑘𝑠𝑑𝑧𝑣(𝑧)^𝑝𝑢(𝑠)𝑑𝑠^{𝑞/𝑝1/𝑞}+_𝑘∈ℤ𝑎^𝑞_𝑘𝜑𝑥_𝑘^𝑥𝑘+1_𝑥𝑘𝑑𝑧𝑣(𝑧)^𝑞1/𝑞≲sup_𝑘∈ℤ𝑎_𝑘. (3.54) Hence, by Proposition 2.14, we have that _𝑘∈ℤ𝜑𝑥_𝑘𝑈𝑥_𝑘^{𝑞/𝑝𝑥𝑘𝑥}_𝑘−1^𝑥𝑘𝑠𝑑𝑧𝑣(𝑧)^𝑝𝑢(𝑠)𝑑𝑠^{𝑞/𝑝1/𝑞}+_𝑘∈ℤ𝜑𝑥_𝑘^𝑥𝑘+1_𝑥𝑘𝑑𝑧𝑣(𝑧)^𝑞1/𝑞<∞. (3.55) On the other hand, 𝐴₅≈_𝑘∈ℤ𝜑𝑥_𝑘𝑈𝑥_𝑘^−1𝑥𝑘_𝑥𝑘−1𝑈(𝑠)^∞_𝑠𝑑𝑧𝑣(𝑧)^𝑝−1𝑑𝑠𝑣(𝑠)+^∞_𝑥𝑘^∞_𝑠𝑑𝑧𝑣(𝑧)^𝑝−1𝑑𝑠𝑣(𝑠)^{𝑞/𝑝1/𝑞}≈_𝑘∈ℤ𝜑𝑥_𝑘𝑈𝑥_𝑘^{𝑞/𝑝𝑥𝑘𝑥}_𝑘−1𝑈(𝑠)^∞_𝑠𝑑𝑧𝑣(𝑧)^𝑝−1𝑑𝑠𝑣(𝑠)^{𝑞/𝑝1/𝑞}+_𝑘∈ℤ𝜑𝑥_𝑘^∞_𝑥𝑘𝑑𝑧𝑣(𝑧)^𝑞1/𝑞. (3.56) Integrating by part and using Lemma 2.9, we get that 𝐴₅≲_𝑘∈ℤ𝜑𝑥_𝑘𝑈𝑥_𝑘^{𝑞/𝑝𝑥𝑘𝑥}_𝑘−1^∞𝑠𝑑𝑧𝑣(𝑧)^𝑝𝑢(𝑠)𝑑𝑠^{𝑞/𝑝1/𝑞}+_𝑘∈ℤ𝜑𝑥_𝑘^∞_𝑥𝑘𝑑𝑧𝑣(𝑧)^𝑞1/𝑞≲_𝑘∈ℤ𝜑𝑥_𝑘𝑈𝑥_𝑘^{𝑞/𝑝𝑥𝑘𝑥}_𝑘−1^𝑥𝑘𝑠𝑑𝑧𝑣(𝑧)^𝑝𝑢(𝑠)𝑑𝑠^{𝑞/𝑝1/𝑞}+_𝑘∈ℤ𝜑𝑥_𝑘^𝑥𝑘+1_𝑥𝑘𝑑𝑧𝑣(𝑧)^𝑞1/𝑞<∞. (3.57) The proof is complete.

We now state the announced characterization of (3.2).

Theorem 3.2. Let 0<𝑝<∞, 1<𝜃≤∞, and let 𝑢,𝑣,𝑤 be weights. Assume that 𝑢 is such that 𝑈^1/𝑝 is admissible and the measure 𝑤(𝑡)𝑑𝑡 is nondenerate with respect to 𝑈^1/𝑝. Then the inequality (3.2) holds for every measurable function 𝑓 on (0,∞) if and only if(i)1<𝜃≤𝑝<∞ and 𝐵₁∶=sup_{()𝑥∈0,∞}esssup_{()𝑠∈0,∞}𝑤(𝑠)𝒰(𝑥,𝑠)^1/𝑝sup_{()𝑡∈0,∞}𝑉_𝜃(𝑡)^1/𝜃′𝒰(𝑡,𝑥)^1/𝑝<∞. (3.58) Moreover, the best constant 𝑐 in (3.2) satisfies 𝑐≈𝐵₁.(ii)0<𝑝<𝜃<∞, 1<𝜃<∞,  𝑟=𝜃𝑝/(𝜃−𝑝) and 𝐵₂∶=sup_{()𝑥∈0,∞}esssup_{()𝑠∈0,∞}𝑤(𝑠)𝒰(𝑥,𝑠)^1/𝑝∞₀𝒰(𝑡,𝑥)^𝑟/𝑝𝑉_𝜃(𝑡)^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡^1/𝑟<∞. (3.59) Moreover, the best constant 𝑐 in (3.2) satisfies 𝑐≈𝐵₂.(iii)  0<𝑝<∞, 𝜃=∞ and 𝐵₃∶=sup_{𝑥∈(0,∞)}^∞₀𝒰(𝑠,𝑡)^∞_𝑠𝑑𝑧𝑣(𝑧)^𝑝−1𝑑𝑠𝑣(𝑠)^1/𝑝esssup_{𝑠∈(0,∞)}𝑤(𝑠)𝒰(𝑥,𝑠)^1/𝑝<∞. (3.60) Moreover, the best constant 𝑐 in (3.2) satisfies 𝑐≈𝐵₃.

Proof. Using Corollary 2.5, Lemmas 2.8 and 2.9, we obtain for the left-hand side 𝐽₀ of (3.2) that (𝜑 is defined by (2.7)) 𝐽₀≈sup_{𝑡∈(0,∞)}𝜑(𝑡)𝑈(𝑡)^{1/𝑝𝑡0∞𝑠}ℎ(𝑧)𝑑𝑧^𝑝𝑢(𝑠)𝑑𝑠^1/𝑝≈sup_𝑘∈ℤ𝜑𝑥_𝑘𝑈𝑥_𝑘^{1/𝑝𝑥𝑘0∞𝑠}ℎ(𝑧)𝑑𝑧^𝑝𝑢(𝑠)𝑑𝑠^1/𝑝≈sup_𝑘∈ℤ𝜑𝑥_𝑘𝑈𝑥_𝑘^{1/𝑝𝑥𝑘𝑥}_𝑘−1^𝑥𝑘𝑠ℎ(𝑧)𝑑𝑧^𝑝𝑢(𝑠)𝑑𝑠^1/𝑝+sup_𝑘∈ℤ𝜑𝑥_𝑘^𝑥𝑘+1_𝑥𝑘ℎ(𝑧)𝑑𝑧∶=𝐼𝐼𝐼+𝐼𝑉. (3.61)(i) For the case 1<𝜃≤𝑝<∞, by using Lemma 2.15 for 𝐼𝐼𝐼 and applying Hölder’s inequality for 𝐼𝑉, we arrive at 𝐼𝐼𝐼≲sup_𝑘∈ℤ𝜑𝑥_𝑘𝑈𝑥_𝑘^1/𝑝sup_{𝑥𝑘−1<𝑡<𝑥𝑘}𝑈(𝑡)^1/𝑝𝑉_𝜃(𝑡)^{1/𝜃′𝑥𝑘}_𝑥𝑘−1ℎ(𝑧)^𝜃𝑣(𝑧)𝑑𝑧^1/𝜃,𝐼𝑉≲sup_𝑘∈ℤ𝜑𝑥_𝑘𝑉_𝜃𝑥_𝑘^{1/𝜃′𝑥𝑘}_𝑥𝑘−1ℎ(𝑧)^𝜃𝑣(𝑧)𝑑𝑧^1/𝜃, (3.62) so that, by Lemma 2.13 and (3.61), we obtain that 𝐽₀≲sup_𝑘∈ℤ𝜑𝑥_𝑘𝑈𝑥_𝑘^1/𝑝sup_{𝑥𝑘−1<𝑡<𝑥𝑘}𝑈(𝑡)^1/𝑝𝑉_𝜃(𝑡)^1/𝜃′+sup_𝑘∈ℤ𝜑𝑥_𝑘𝑉_𝜃𝑥_𝑘^1/𝜃′‖ℎ‖_{𝜃,𝑣,(0,∞)}≈sup_{()𝑥∈0,∞}𝜑(𝑥)sup_0<𝑡<∞𝒰(𝑡,𝑥)^1/𝑝𝑉_𝜃(𝑡)^1/𝜃′‖ℎ‖_{𝜃,𝑣,(0,∞)}=sup_{()𝑥∈0,∞}esssup_{()𝑠∈0,∞}𝑤(𝑠)𝒰(𝑥,𝑠)^1/𝑝sup_0<𝑡<∞𝑉_𝜃(𝑡)^1/𝜃′𝒰(𝑡,𝑥)^1/𝑝‖ℎ‖_{𝜃,𝑣,(0,∞)}. (3.63)(ii)Let now 0<𝑝<𝜃<∞, 1<𝜃<∞, 𝑟=𝜃𝑝/(𝜃−𝑝). By using Lemma 2.15 for 𝐼𝐼𝐼 and applying Hölder’s inequality for 𝐼𝑉, we find that 𝐼𝐼𝐼≲sup_𝑘∈ℤ𝜑𝑥_𝑘𝑈𝑥_𝑘^{1/𝑝𝑥𝑘𝑥}_𝑘−1^𝑡𝑥_𝑘−1𝑢(𝑠)𝑑𝑠^{𝑟/𝑝𝑥𝑘}_𝑡𝑣(𝑠)^1−𝜃′𝑑𝑠^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡^1/𝑟×^𝑥𝑘_𝑥𝑘−1ℎ(𝑧)^𝜃𝑣(𝑧)𝑑𝑧^1/𝜃,𝐼𝑉≲sup_𝑘∈ℤ𝜑𝑥_𝑘𝑉_𝜃𝑥_𝑘^{1/𝜃′𝑥𝑘}_𝑥𝑘−1ℎ(𝑧)^𝜃𝑣(𝑧)𝑑𝑧^1/𝜃 (3.64) and, by Lemmas 2.9 and 2.12, and (3.61), we get that 𝐽₀≲sup_𝑘∈ℤ𝜑𝑥_𝑘𝑈𝑥_𝑘^{−𝑟/𝑝𝑥𝑘}_𝑥𝑘−1𝑈(𝑡)^𝑟/𝑝𝑉_𝜃(𝑡)^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡+^∞_𝑥𝑘𝑉_𝜃(𝑡)^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡^1/𝑟×‖ℎ‖_{𝜃,𝑣,(0,∞)}≈sup_𝑘∈ℤ𝜑𝑥_𝑘𝑈𝑥_𝑘^{−𝑟/𝑝𝑥𝑘}_𝑥𝑘−1𝑈(𝑡)^𝑟/𝑝𝑉_𝜃(𝑡)^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡+^𝑥𝑘+1_𝑥𝑘𝑉_𝜃(𝑡)^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡^1/𝑟×‖ℎ‖_{𝜃,𝑣,(0,∞)}≈sup_{()𝑥∈0,∞}𝜑(𝑥)^∞₀𝒰(𝑡,𝑥)^𝑟/𝑝𝑉_𝜃(𝑡)^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡^1/𝑟‖ℎ‖_{𝜃,𝑣,(0,∞)}=sup_{𝑥∈(0,∞)}esssup_{()𝑠∈0,∞}𝑤(𝑠)𝒰(𝑥,𝑠)^1/𝑝∞₀𝒰(𝑡,𝑥)^𝑟/𝑝𝑉_𝜃(𝑡)^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡^1/𝑟‖ℎ‖_{𝜃,𝑣,(0,∞)}. (3.65)(iii) Now let 0<𝑝<∞, 𝜃=∞. By using Lemma 2.15 for 𝐼𝐼𝐼, we deduce that 𝐼𝐼𝐼≲sup_𝑘∈ℤ𝜑𝑥_𝑘𝑈𝑥_𝑘^{1/𝑝𝑥𝑘𝑥}_𝑘−1^𝑥𝑘𝑠𝑑𝑧𝑣(𝑧)^𝑝𝑢(𝑠)𝑑𝑠^1/𝑝‖ℎ‖_{∞,𝑣,(0,∞)}. (3.66) Moreover, for 𝐼𝑉, it yields that 𝐼𝑉≲sup_𝑘∈ℤ𝜑𝑥_𝑘^𝑥𝑘+1_𝑥𝑘𝑑𝑧𝑣(𝑧)‖ℎ‖_{∞,𝑣,(0,∞)}. (3.67) Therefore, by using integration by parts, Lemma 2.12, and (3.61), we get that 𝐽₀≲sup_𝑘∈ℤ𝜑𝑥_𝑘𝑈𝑥_𝑘^−1𝑥𝑘_𝑥𝑘−1𝑈(𝑠)^∞_𝑠𝑑𝑧𝑣(𝑧)^𝑝−1𝑑𝑠𝑣(𝑠)+^∞_𝑥𝑘^∞_𝑠𝑑𝑧𝑣(𝑧)^𝑝−1𝑑𝑠𝑣(𝑠)^1/𝑝×‖ℎ‖_{∞,𝑣,(0,∞)}≈sup_{()𝑥∈0,∞}^∞₀𝒰(𝑠,𝑡)^∞_𝑠𝑑𝑧𝑣(𝑧)^𝑝−1𝑑𝑠𝑣(𝑠)^1/𝑝𝜑(𝑥)‖ℎ‖_{∞,𝑣,(0,∞)}=sup_{𝑥∈(0,∞)}^∞₀𝒰(𝑠,𝑡)^∞_𝑠𝑑𝑧𝑣(𝑧)^𝑝−1𝑑𝑠𝑣(𝑠)^1/𝑝esssup_{()𝑠∈0,∞}𝑤(𝑠)𝒰(𝑥,𝑠)^1/𝑝‖ℎ‖_{∞,𝑣,(0,∞)}. (3.68)
Now we prove the lower bounds (necessity). Let {𝑥_𝑘} be a discretizing sequence for 𝜑 defined by (2.7). Then, by (3.61), we have sup_𝑘∈ℤ𝜑𝑥_𝑘𝑈𝑥_𝑘^{1/𝑝𝑥𝑘𝑥}_𝑘−1^𝑥𝑘𝑠ℎ(𝑧)𝑑𝑧^𝑝𝑢(𝑠)𝑑𝑠^1/𝑝+sup_𝑘∈ℤ𝜑𝑥_𝑘^𝑥𝑘+1_𝑥𝑘ℎ(𝑧)≲𝑑𝑧_{𝑘∈ℤ𝑥𝑘𝑥𝑘−1}ℎ(𝑧)^𝜃𝑣(𝑧)𝑑𝑧^1/𝜃. (3.69) Let 1<𝜃≤𝑝<∞. If we use in (3.69) the test function defined by (3.34), we obtain that sup_𝑘∈ℤ𝑎_𝑘𝜑𝑥_𝑘𝑈𝑥_𝑘^1/𝑝sup_{𝑥𝑘−1<𝑡<𝑥𝑘}^𝑡_𝑥𝑘−1𝑢(𝑠)𝑑𝑠^{1/𝑝𝑥𝑘}_𝑡𝑣(𝑠)^1−𝜃′𝑑𝑠^1/𝜃′+sup_𝑘∈ℤ𝑎_𝑘𝜑𝑥_𝑘^𝑥𝑘+1_𝑥𝑘𝑣(𝑡)^1−𝜃′𝑑𝑡^1/𝜃′≲_𝑘∈ℤ𝑎^𝜃_𝑘^1/𝜃. (3.70) Therefore, by Proposition 2.14, we have that sup_𝑘∈ℤ𝜑𝑥_𝑘𝑈𝑥_𝑘^1/𝑝sup_{𝑥𝑘−1<𝑡<𝑥𝑘}^𝑡_𝑥𝑘−1𝑢(𝑠)𝑑𝑠^{1/𝑝𝑥𝑘}_𝑡𝑣(𝑠)^1−𝜃′𝑑𝑠^1/𝜃′+sup_𝑘∈ℤ𝜑𝑥_𝑘^𝑥𝑘+1_𝑥𝑘𝑣(𝑡)^1−𝜃′𝑑𝑡^1/𝜃′<∞. (3.71) Since 𝐵₁≈sup_𝑘∈ℤ𝜑𝑥_𝑘𝑈𝑥_𝑘^1/𝑝sup_{𝑥𝑘−1<𝑡<𝑥𝑘}𝑈(𝑡)^1/𝑝𝑉_𝜃(𝑡)^1/𝜃′+sup_𝑘∈ℤ𝜑𝑥_𝑘𝑉_𝜃𝑥_𝑘^1/𝜃′≲sup_𝑘∈ℤ𝜑𝑥_𝑘𝑈𝑥_𝑘^1/𝑝sup_{𝑥𝑘−1<𝑡<𝑥𝑘}^𝑡_𝑥𝑘−1𝑢(𝑠)𝑑𝑠^{1/𝑝𝑥𝑘}_𝑡𝑣(𝑠)^1−𝜃′𝑑𝑠^1/𝜃′+sup_𝑘∈ℤ𝜑𝑥_𝑘𝑉_𝜃𝑥_𝑘^1/𝜃′, (3.72) by Lemma 2.9, we get that 𝐵₁≲sup_𝑘∈ℤ𝜑𝑥_𝑘𝑈𝑥_𝑘^1/𝑝sup_{𝑥𝑘−1<𝑡<𝑥𝑘}^𝑡_𝑥𝑘−1𝑢(𝑠)𝑑𝑠^{1/𝑝𝑥𝑘}_𝑡𝑣(𝑠)^1−𝜃′𝑑𝑠^1/𝜃′+sup_𝑘∈ℤ𝜑𝑥_𝑘^𝑥𝑘+1_𝑥𝑘𝑣(𝑡)^1−𝜃′𝑑𝑡^1/𝜃′<∞. (3.73) Now let 0<𝑝<𝜃<∞, 1<𝜃<∞, 𝑟=𝜃𝑝/(𝜃−𝑝). By using in (3.69) the test function defined by (3.42), we obtain that sup_𝑘∈ℤ𝑎_𝑘𝜑𝑥_𝑘𝑈𝑥_𝑘^{1/𝑝𝑥𝑘𝑥}_𝑘−1^𝑡𝑥_𝑘−1𝑢(𝑠)𝑑𝑠^{𝑟/𝑝𝑥𝑘}_𝑡𝑣(𝑠)^1−𝜃′𝑑𝑠^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡^1/𝑟+sup_𝑘∈ℤ𝑎_𝑘𝜑𝑥_𝑘^𝑥𝑘+1_𝑥𝑘𝑣(𝑡)^1−𝜃′𝑑𝑡^1/𝜃′≲_𝑘∈ℤ𝑎^𝜃_𝑘^1/𝜃. (3.74) Then, by Proposition 2.14, we get that sup_𝑘∈ℤ𝜑𝑥_𝑘𝑈𝑥_𝑘^{1/𝑝𝑥𝑘𝑥}_𝑘−1^𝑡𝑥_𝑘−1𝑢(𝑠)𝑑𝑠^{𝑟/𝑝𝑥𝑘}_𝑡𝑣(𝑠)^1−𝜃′𝑑𝑠^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡^1/𝑟+sup_𝑘∈ℤ𝜑𝑥_𝑘^𝑥𝑘+1_𝑥𝑘𝑣(𝑡)^1−𝜃′𝑑𝑡^1/𝜃′<∞. (3.75) Since 𝐵₂≈sup_𝑘∈ℤ𝜑𝑥_𝑘𝑈𝑥_𝑘^{−𝑟/𝑝𝑥𝑘}_𝑥𝑘−1𝑈(𝑡)^𝑟/𝑝𝑉_𝜃(𝑡)^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡+^∞_𝑥𝑘𝑉_𝜃(𝑡)^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡^1/𝜃′≲sup_𝑘∈ℤ𝜑𝑥_𝑘𝑈𝑥_𝑘^{1/𝑝𝑥𝑘𝑥}_𝑘−1𝑈(𝑡)^𝑟/𝑝𝑉_𝜃(𝑡)^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡^1/𝑟+sup_𝑘∈ℤ𝜑𝑥_𝑘𝑉_𝜃𝑥_𝑘^1/𝜃′≲sup_𝑘∈ℤ𝜑𝑥_𝑘𝑈𝑥_𝑘^{1/𝑝𝑥𝑘𝑥}_𝑘−1^𝑡𝑥_𝑘−1𝑢(𝑠)𝑑𝑠^𝑟/𝑝𝑉_𝜃(𝑡)^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡^1/𝑟+sup_𝑘∈ℤ𝜑𝑥_𝑘𝑉_𝜃𝑥_𝑘^1/𝜃′, (3.76) by integrating by parts, we find that 𝐵₂≲sup_𝑘∈ℤ𝜑𝑥_𝑘𝑈𝑥_𝑘^{1/𝑝𝑥𝑘𝑥}_𝑘−1^∞𝑡𝑣(𝑠)^1−𝜃′𝑑𝑠^{(𝑟/𝑝′)+1}𝑢(𝑡)^𝑡_𝑥𝑘−1𝑢(𝑠)𝑑𝑠^{(𝑟/𝑝)−1}𝑑𝑡^1/𝑟+sup_𝑘∈ℤ𝜑𝑥_𝑘𝑉_𝜃𝑥_𝑘^1/𝜃′≲sup_𝑘∈ℤ𝜑𝑥_𝑘𝑈𝑥_𝑘^{1/𝑝𝑥𝑘𝑥}_𝑘−1^𝑥𝑘𝑡𝑣(𝑠)^1−𝜃′𝑑𝑠^{(𝑟/𝑝′)+1}𝑢(𝑡)^𝑡_𝑥𝑘−1𝑢(𝑠)𝑑𝑠^{(𝑟/𝑝)−1}𝑑𝑡^1/𝑟+sup_𝑘∈ℤ𝜑𝑥_𝑘𝑉_𝜃𝑥_𝑘^1/𝜃′. (3.77) Moreover, by again integrating by parts, we arrive at 𝐵₂≲sup_𝑘∈ℤ𝜑𝑥_𝑘𝑈𝑥_𝑘^{1/𝑝𝑥𝑘𝑥}_𝑘−1^𝑡𝑥_𝑘−1𝑢(𝑠)𝑑𝑠^{𝑟/𝑝𝑥𝑘}_𝑡𝑣(𝑠)^1−𝜃′𝑑𝑠^𝑟/𝑝′𝑣(𝑡)^1−𝜃′𝑑𝑡^1/𝑟+sup_𝑘∈ℤ𝜑𝑥_𝑘𝑉_𝜃𝑥_𝑘^1/𝜃′<∞. (3.78)
Finally, let 0<𝑝<∞, 𝜃=∞. By using the test function defined by (3.53) in (3.69), we get that sup_𝑘∈ℤ𝑎_𝑘𝜑𝑥_𝑘𝑈𝑥_𝑘^{1/𝑝𝑥𝑘𝑥}_𝑘−1^𝑥𝑘𝑠𝑑𝑧𝑣(𝑧)^𝑝𝑢(𝑠)𝑑𝑠^1/𝑝+sup_𝑘∈ℤ𝑎_𝑘𝜑𝑥_𝑘^𝑥𝑘+1_𝑥𝑘𝑑𝑧𝑣(𝑧)≲sup_𝑘∈ℤ𝑎_𝑘. (3.79) Hence, by Proposition 2.14, we have that sup_𝑘∈ℤ𝜑𝑥_𝑘𝑈𝑥_𝑘^{1/𝑝𝑥𝑘𝑥}_𝑘−1^𝑥𝑘𝑠𝑑𝑧𝑣(𝑧)^𝑝𝑢(𝑠)𝑑𝑠^1/𝑝+sup_𝑘∈ℤ𝜑𝑥_𝑘^𝑥𝑘+1_𝑥𝑘𝑑𝑧𝑣(𝑧)<∞. (3.80) Since 𝐵₃≈sup_𝑘∈ℤ𝜑𝑥_𝑘𝑈𝑥_𝑘^−1𝑥𝑘_𝑥𝑘−1𝑈(𝑠)^∞_𝑠𝑑𝑧𝑣(𝑧)^𝑝−1𝑑𝑠𝑣(𝑠)+^∞_𝑥𝑘^∞_𝑠𝑑𝑧𝑣(𝑧)^𝑝−1𝑑𝑠𝑣(𝑠)^1/𝑝≲sup_𝑘∈ℤ𝜑𝑥_𝑘𝑈𝑥_𝑘^{1/𝑝𝑥𝑘𝑥}_𝑘−1𝑈(𝑠)^∞_𝑠𝑑𝑧𝑣(𝑧)^𝑝−1𝑑𝑠𝑣(𝑠)^1/𝑝+sup_𝑘∈ℤ𝜑𝑥_𝑘^∞_𝑥𝑘𝑑𝑧𝑣(𝑧)≲sup_𝑘∈ℤ𝜑𝑥_𝑘𝑈𝑥_𝑘^{1/𝑝𝑥𝑘𝑥}_𝑘−1𝑈(𝑠)^𝑥𝑘_𝑠𝑑𝑧𝑣(𝑧)^𝑝−1𝑑𝑠𝑣(𝑠)^1/𝑝+sup_𝑘∈ℤ𝜑𝑥_𝑘^∞_𝑥𝑘𝑑𝑧𝑣(𝑧), (3.81) by integrating by parts and using Lemma 2.9, we obtain that 𝐵₃≲sup_𝑘∈ℤ𝜑𝑥_𝑘𝑈𝑥_𝑘^{1/𝑝𝑥𝑘𝑥}_𝑘−1^𝑥𝑘𝑠𝑑𝑧𝑣(𝑧)^𝑝𝑢(𝑠)𝑑𝑠^1/𝑝+sup_𝑘∈ℤ𝜑𝑥_𝑘^𝑥𝑘+1_𝑥𝑘𝑑𝑧𝑣(𝑧)<∞. (3.82) The proof is complete.

Acknowledgments

The authors thank the anonymous referee for his/her helpful remarks, which have improved the final version of this paper. The research of the first author was partly supported by the Grant 201/08/0383 of the Grant Agency of the Czech Republic and RVO: 67985840. The research of the second author was supported by the Science Development Foundation under the President of the Republic of Azerbaijan Project no. EIF-2010-1(1)-40/06-1.