# Weighted Hardy Operators in Complementary Morrey Spaces

Authors: Dag Lukkassen; Lars-Erik Persson; Stefan Samko
Journal: Journal of Function Spaces and Applications
 (2012)

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## Abstract

We study the weightedp→q-boundedness of the multidimensional weighted Hardy-type operators Hwα and ℋwα with radial type weight w=w(|x|), in the generalized complementary Morrey spaces ℒ∁{0}p,ψ(ℝn) 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∁{0}p,ψ(Ω) over bounded domains Ω are embedded between weighted Lebesgue space Lp with the weight ψ and such a space with the weight ψ, perturbed by a logarithmic factor. Both the embeddings are sharp.


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## Body

## 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(1.1)Hwαf(x):=|x|α-nw(|x|)∫|y|<|x|f(y)dyw(|y|),ℋwαf(x):=|x|αw(|x|)∫|y|>|x|f(y)dy|y|nw(|y|),
where α≥0, in the so called complementary Morrey spaces. The one-dimensional case will include the versions
(1.2)Hwαf(x)=xα-1w(x)∫0x‍f(t)dtw(t),ℋwαf(x)=xαw(x)∫x∞‍f(t)dttw(t),x>0
adjusted for the half-axis ℝ+1, so that in the sequel ℝn with n=1 may be read either as ℝ1 or ℝ+1.The classical Morrey spacesℒp,λ(Ω),Ω⊆ℝn, defined by the norm
(1.3)∥f∥p,λ:=supx∈Ω,r>0(1rλ∫B~(x,r)‍|f(y)|pdy)1/p,1≤p<∞,0≤λ≤n,
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 ℒp,λ, and we refer for instance to the surveying paper [13]. One of the direct generalizations is obtained by replacing rλ in (1.3) by a function φ(r), usually satisfying some monotonicity type conditions. We also denote it as ℒp,φ(Ω) without danger of confusion. Such spaces appeared in [14, 15] and were widely studied in [16, 17]. The spaces ℒlocp,φ(Ω), defined by the norm
(1.4)∥f∥p,φ;loc:=supr>0(1φ(r)∫B~(x0,r)‍|f(y)|pdy)1/p,
where x0∈Ω, are known as generalized local Morrey spacesHardy type operators (1.1) in the spaces ℒp,φ(Ω),ℒlocp,φ(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∁{x0}p,ψ(Ω), called complementary Morrey spaces, with the norm controlling possible growth, as r→0, of the integral
(1.5)∫Ω∖B(x0,r)‍|f(y)|pdy
over exterior of balls. Such spaces have sense in the local setting only. It is introduced in [16, 17] that the space ℒ∁{x0}p,ψ(Ω) is defined as the space of all functions f∈Llocp(Ω∖{x0}) with the finite norm
(1.6)∥f∥ℒ∁{x0}p,ψ(Ω)=supr>0ψ1/p(r)∥f∥Lp(Ω∖B(x0,r)),
where admission of the multiplier ψ(r) vanishing at r=0 controls the growth of the norm ∥f∥Lp(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 thep→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 Theorem3.1, by showing that the spaces ℒ∁{0}p,ψ(Ω) over bounded domains Ω are embedded between the weighted Lebesgue space Lp 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 Section2  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

LetΩ be an open set in ℝn, Ω⊆ℝn and ℓ=diamΩ,  0<ℓ≤∞, B(x,r)={y∈ℝn:|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)∶=limr→0ψ(r)=0 which may arbitrarily grow as r→∞. Let also 1≤p<∞.Definition 2.1.
Let1≤p<∞. The complementary Morrey space ℒ∁{x0}p,ψ(Ω), where x0∈Ω, is the space of functions f∈Llocp(Ω∖{x0}) such that
(2.1)∥f∥ℒ∁{x0}p,ψ(Ω)∶=supr>0(ψ(r)∫Ω∖B~(x0,r)‍|f(y)|pdy)1/p<∞.In the case of power factorψ(r)≡rλ, λ>0, we also denote
(2.2)ℒ∁{x0}p,λ(Ω)∶=ℒ∁{x0}p,ψ(Ω)|ψ≡rλ
without danger of confusion.The spaceℒ∁{x0}p,ψ(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:(2.3)ℳp,ψ(f;x0,r)∶=ψ(r)∫Ω∖B~(x0,r)‍|f(y)|pdy.The functionψ, defining the complementary Morrey space, will be called definitive function.Remark 2.2.
The functionφ 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 Spaceℒ∁{x0}p,ψ(Ω)

Lemma 2.3.
For a nonnegative radial type functionsu=u(|x-x0|), x0∈Ω, the condition
(2.4)sup0<r<ℓψ(r)∫rℓ‍u(t)ptn-1dt<∞
is sufficient to belong to ℒ∁{x0}p,ψ(Ω). (It is also necessary, when Ω=ℝn or Ω is a ball centered at x0.)The proof of Lemma2.3 is obvious. By means of this lemma we easily obtain the following corollary.Corollary 2.4.
LetΩ be bounded. A power function |x-x0|γ with γ≠-n/p belongs to the space ℒ∁{x0}p,ψ(Ω) if and only if
(2.5)supr>0rmin{n+γp,0}ψ(r)<∞.
In the case n+γp=0, the same holds with rmin{n+γp,0} replaced by |lnr|. When Ω=ℝn, the necessary and sufficient conditions are n+γp<0 and supr>0rn+γpψ(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 ℤα,β, we put these notions in the appendix.Let nowu be a nonnegative function in Bary-Stechkin class, namely,
(2.6)u∈ℤ-n/p([0,ℓ])ifℓ<∞,u∈ℤ-n/p,-n/p(ℝ+1)ifℓ=∞.
Then ∫rℓ‍up(t)tn-1dt~rnup(r), 0<r<ℓ, so that the condition sup0<r<ℓrnψ(r)up(r)<∞ is sufficient for u=u(|x-x0|) to be in ℒ∁{x0}p,ψ(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.16).


## 2.1. Definitions

LetΩ be an open set in ℝn, Ω⊆ℝn and ℓ=diamΩ,  0<ℓ≤∞, B(x,r)={y∈ℝn:|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)∶=limr→0ψ(r)=0 which may arbitrarily grow as r→∞. Let also 1≤p<∞.Definition 2.1.
Let1≤p<∞. The complementary Morrey space ℒ∁{x0}p,ψ(Ω), where x0∈Ω, is the space of functions f∈Llocp(Ω∖{x0}) such that
(2.1)∥f∥ℒ∁{x0}p,ψ(Ω)∶=supr>0(ψ(r)∫Ω∖B~(x0,r)‍|f(y)|pdy)1/p<∞.In the case of power factorψ(r)≡rλ, λ>0, we also denote
(2.2)ℒ∁{x0}p,λ(Ω)∶=ℒ∁{x0}p,ψ(Ω)|ψ≡rλ
without danger of confusion.The spaceℒ∁{x0}p,ψ(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:(2.3)ℳp,ψ(f;x0,r)∶=ψ(r)∫Ω∖B~(x0,r)‍|f(y)|pdy.The functionψ, defining the complementary Morrey space, will be called definitive function.Remark 2.2.
The functionφ 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 Spaceℒ∁{x0}p,ψ(Ω)

Lemma 2.3.
For a nonnegative radial type functionsu=u(|x-x0|), x0∈Ω, the condition
(2.4)sup0<r<ℓψ(r)∫rℓ‍u(t)ptn-1dt<∞
is sufficient to belong to ℒ∁{x0}p,ψ(Ω). (It is also necessary, when Ω=ℝn or Ω is a ball centered at x0.)The proof of Lemma2.3 is obvious. By means of this lemma we easily obtain the following corollary.Corollary 2.4.
LetΩ be bounded. A power function |x-x0|γ with γ≠-n/p belongs to the space ℒ∁{x0}p,ψ(Ω) if and only if
(2.5)supr>0rmin{n+γp,0}ψ(r)<∞.
In the case n+γp=0, the same holds with rmin{n+γp,0} replaced by |lnr|. When Ω=ℝn, the necessary and sufficient conditions are n+γp<0 and supr>0rn+γpψ(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 ℤα,β, we put these notions in the appendix.Let nowu be a nonnegative function in Bary-Stechkin class, namely,
(2.6)u∈ℤ-n/p([0,ℓ])ifℓ<∞,u∈ℤ-n/p,-n/p(ℝ+1)ifℓ=∞.
Then ∫rℓ‍up(t)tn-1dt~rnup(r), 0<r<ℓ, so that the condition sup0<r<ℓrnψ(r)up(r)<∞ is sufficient for u=u(|x-x0|) to be in ℒ∁{x0}p,ψ(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.16).


## 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 formLp(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.
LetΩ be a bounded open set, 1≤p<∞, ψ∈W0([0,ℓ]), ℓ=diamΩ, and let ψ be absolutely continuous and
(3.1)P∶=sup0<t<ℓt|ψ′(t)|ψ(t)<∞.
Then
(3.2)Lp(Ω,ψ(|y-x0|))↪ℒ∁{x0}p,ψ(Ω)↪⋂ε>0‍Lp(Ω,ψε(|y-x0|)),
where ψε(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
(3.3)∫rℓ‍dttψ(t)dt≤Cψ(r)
(in particular, if ψ(r)=rλ, λ>0); that is, there exists a function f0=f0(x) such that
(3.4)f0∈ℒ∁{x0}p,ψ(Ω),butf0∉Lp(Ω,ψ(|y-x0|)).
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)=rλ, λ≥0), there exists a function g0=g0(x) such that
(3.5)g0∈⋂ε>0‍Lp(Ω,ψε(|x-x0|)),butg0∉ℒ∁{x0}p,ψ(Ω).Proof.
Without loss of generality, we may takex0=0 for simplicity, supposing that 0∈Ω.
10. The left-hand side embedding: since the function ψ is almost increasing, we have
(3.6)(∫Ω‍ψ(|y|)f(y)|pdy)1/p≥(∫Ω∖B~(0,r)‍ψ(|y|)f(y)|pdy)1/p≥C(ψ(r)∫Ω∖B~(0,r)‍|f(y)|pdy)1/p
so that
(3.7)∥f∥Lp(Ω,ψ(|y|))≥C∥f∥ℒ∁{0}p,ψ(Ω).
2
0. The right-hand side embedding: we have
(3.8)∫B~(0,r)‍|f(y)|pψε(|y|)dy=∫B~(0,r)‍|f(y)|p(∫0|y|‍ddtψε(t)dt)dy,
where
(3.9)ψε′(t)=ψ(t)t[tψ′(t)ψ(t)+1+εln(A/t)]1(ln(A/t))1+ε,
so that
(3.10)|ψε′(t)|≤Cεψ(t)t(ln(A/t))1+ε,Cε=P+1+εln(A/ℓ).
Therefore,
(3.11)∫B~(x0,r)‍|f(y)|pψε(|y|)dy≤∫0t‍|ψε′(t)|(∫{y∈Ω:t<|x0-y|<r}‍|f(y)|pdy)dt≤∫0ℓ‍|ψε′(t)|·∥f∥Lp(Ω∖B~(x0,t))pds≤∥f∥ℒ∁{x0}p,ψ(Ω)p∫0ℓ‍|ψε′(t)|ψ(t)dt,
where the last integral converges when ε>0 since |ψε′(t)|/ψ(t)≤Cε/(t(ln(A/t))1+ε).
30. The strictness of the embeddings: the corresponding counterexamples are
(3.12)f0(x)=1ψ1/p(|x|)|x|n/p,g0(x)=ln(ln(B/|x|))|x|n/pψ1/p(|x|),B>ℓee.
Calculations for the functionf0, which is obviously not in Lp(Ω,ψ(|y|), are easy. Indeed,
(3.13)∥f0∥ℒ∁{0}p,ψ(Ω)p≤|𝕊n-1|ψ(r)∫rℓ‍dttψ(t),
where the right-hand side is bounded by (3.3).
In the case of the functiong0 we have
(3.14)∥g0∥Lp(Ω,ψε)p=∫Ω‍lnp(ln(B/|x|))|x|n(ln(A/|x|))1+εdx≤|𝕊n-1|∫0ℓ‍lnp(ln(B/t))t(ln(A/t))1+εdt<∞
for every ε>0. However, for small r∈(0,δ/2), where δ=dist(0,∂Ω), we obtain
(3.15)ψ(r)∫x∈Ω:|x|>r‍g0p(|x|)dx≥ψ(r)∫x∈Ω:r<|x|<δ‍lnp(ln(B/|x|))dx|x|nψ(|x|)=|𝕊n-1|ψ(r)∫rδ‍lnp(ln(B/t))dttψ(t)≥|𝕊n-1|ψ(r)∫r2r‍lnp(ln(B/t))dttψ(t).
Taking into account thatψ(t) is almost increasing and satisfies the doubling condition, we get
(3.16)ψ(r)∫r2r‍lnp(ln(B/t))dttψ(t)≥Cψ(r)ψ(2r)lnp(lnB2r)∫r2r‍dtt≥Clnp(lnBr)→∞asr→0,
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∁{x0}p,ψ(Ω), but f∉Lp(Ω,ψε(|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 modular𝔐p,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.
Let1≤p<∞, 0<s≤p, let v,ψs/pv∈W¯([0,∞]), and v(2t)≤cv(t). Then
(4.1)∫|z|<r‍|f(z)|sv(|z|)dz≤C∫0r‍Vs(t)tdt·ℳp,ψs/p(f;0,r),r>0,
for f∈Llocp(Ω∖{0}), where C>0 does not depend on f and r∈(0,∞) and
(4.2)Vs(t)=tn(1-(s/p))v(t)ψs/p(t).Proof.
We use the dyadic decomposition as follows:(4.3)∫|z|<r‍|f(z)|sv(|z|)dz=∑k=0∞‍∫Bk(r)‍|f(z)|sv(|z|)dz,
where Bk(r)={z:2-k-1r<|z|<2-kr}. Since there exists a β such that tβv(t) is almost increasing, we observe that
(4.4)1v(|z|)≤Cv(2-k-1r)
on Bk(r). Applying this in (4.3) and making use of the Hölder inequality with the exponent p/s≥1, we obtain
(4.5)∫|z|<r‍|f(z)|sv(|z|)dz≤C∑k=0∞‍(2-k-1r)n(1-(s/p))v(2-k-1r)(∫Bk(r)‍|f(z)|pdz)s/p.
Hence
(4.6)∫|z|<r‍|f(z)|sv(|z|)dz≤C∑k=0∞‍(2-k-1r)n(1-(s/p))ℳp,ψs/p(f;0,2-k-1r)v(2-k-1r)ψs/p(2-k-1r).
On the other hand we have
(4.7)∫0r‍Vs(t)tℳp,ψs/p(f;0,t)dt=∑k=0∞‍∫2-k-1r2-kr‍tn(1-(s/p))-1ℳp,ψs/p(f;0,t)v(t)ψS/p(t)dt.
The function 1/(ψ(t))ℳp,ψ(f;0,t) is decreasing, and the function 1/(v(t)) is almost decreasing after multiplication by some power functions. Therefore,
(4.8)∫0rVs(t)tℳp,ψs/p(f;0,t)dt≥C∑k=0∞‍(2-kr)n(1-(s/p))ℳp,ψs/p(f;0,2-kr)v(2-kr)ψS/p(2-kr)≥C∑k=1∞‍(2-kr)n(1-(s/p))ℳp,ψs/p(f;0,2-kr)v(2-kr)ψs/p(2-kr)=C1∑k=0∞‍(2-k-1r)n(1-(s/p))ℳp,ψs/p(f;0,2-k-1r)v(2-k-1r)ψs/p(2-k-1r).
Then (4.1) follows from (4.6) and (4.8).Lemma 4.2.
Let1≤p<∞, 0≤s≤p, v∈W¯(ℝ+1) and v(2t)≤Cv(t), ψ(2t)≤Cψ(t). Let also
(4.9)Vs(t)∶=tn(1-(s/p))v(t)ψs/p(t).
Then
(4.10)∫|z|>r‍v(|z|)|f(z)|sdz≤C∫r∞‍Vs(t)tℳp,ψs/p(f;0,t)dt,
where C>0 does not depend on r>0 and f.Proof.
We use the corresponding dyadic decomposition:(4.11)∫|z|>r‍v(t)|f(z)|sdz=∑k=0∞‍∫Bk(r)‍v(z)|f(z)|sdz,
where Bk(r)={z:2kr<|z|<2k+1r}. Since there exists a β∈ℝ1 such that tβv(t) is almost increasing, we obtain
(4.12)∑k=0∞‍∫Bk(r)‍v(|z|)|f(z)|sdz≤C∑k=0∞‍v(2k+1r)∫Bk(r)‍|f(z)|sdz,
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
(4.13)∫|z|>r‍v(|z|)|f(z)|sdz≤C∑k=0∞‍v(2k+1r)(2kr)n(1-(s/p))(∫Bk(r)‍|f(z)|pdz)s/p≤C∑k=0∞‍v(2k+1r)(2kr)n(1-(s/p))ψ-s/p(2kr)ℳp,ψs/p(f;0,2kr).
On the other hand, the integral on the right-hand side of (4.10) can be estimated as follows:
(4.14)∫r∞‍Vs(t)tℳp,ψs/p(f;0,t)dt=∑k=0∞‍∫2kr2k+1r‍tn-1-(ns/p)[ℳp,ψ(f;0,t)ψ(t)]s/pv(t)dt≥C∑k=0∞‍v(2kr)[ℳp,ψ(f;0,2k+1r)ψ(2k+1r)]s/p(2kr)n(1-s/p)≥C∑k=0∞‍v(2k+1r)ψ-1/p(2kr)(2kr)n(1-s/p)ℳp,ψs/p(f;0,2k+1r),
which completes the proof.Corollary 4.3.
Let1≤p<∞, let w,wψ1/p∈W¯([0,ℓ]),w(2t)≤cw(t), 0<ℓ≤∞ and 0∈Ω. Let also
(4.15)V(r):=1wp(r)ψ(r)∈ℤ0.
Then
(4.16)ℳp,ψ(fw;0,r)≤Cwp(r)ℳp,ψ(f;0,r),0<r<ℓ.


## 5. Weighted Hardy Operators in Complementary Morrey Spaces

### 5.1. Pointwise Estimations

The proof of our main result of this Section given in Theorems5.3 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.
Let1≤p≤∞, let w,wψ1/p∈W¯, and w(2t)≤Cw(t). The condition
(5.1)∫0ε‍V(t)tdt<∞withε>0,whereV(t)=tn/p′w(t)ψ1/p(t)
is sufficient for the Hardy operator Hwα to be defined on the space ℒ∁{0}p,ψ(ℝn). Under this condition, the pointwise estimate
(5.2)|Hwαf(x)|≤C|x|α-nw(|x|)∫0|x|‍V(t)tdt∥f∥ℒ∁{0}p,ψ
holds.Proof.
Apply Lemma4.1 with s=1.Theorem 5.2.
Let1≤p≤∞ and 1/w∈W¯, or w∈W¯ and w(2t)≤Cw(t). The condition
(5.3)∫ε∞‍𝒱(t)tdt<∞,withε>0,where𝒱(t)=1w(t)ψ1/p(t)tn/p
is sufficient for the Hardy operator ℋwα to be defined on the space ℒ∁{x0}p,ψ(ℝn), and in this case the following pointwise estimate:
(5.4)|ℋwαf(x)|≤C|x|αw(|x|)∫|x|∞‍𝒱(t)tdt∥f∥ℒ∁{x0}p,ψ
holds.Proof.
Apply Lemma4.2 with s=1.


### 5.2. Weightedp→q Boundedness of Hardy Operators in Complementary Morrey Spaces

#### 5.2.1. The Case of the OperatorHwα

TheLp→Lq-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.
Let1≤p≤∞,1≤q≤∞ and let
(5.5)w,ψ1/pw∈W¯(ℝ+1),w(2t)≤cw(t).
The operator Hwα is bounded from ℒ∁{0}p,ψ(ℝn) to ℒ∁{0}q,ψ(ℝn) if supr>0𝕎(r)<∞, where
(5.6)𝕎(r):=ψ(r)∫r∞‍ϱn-1(ϱ(α-(n/p))ψ1/p(ϱ)V(ϱ)∫0ϱ‍V(t)tdt)qdϱ,
and V(t) is the same as in (5.1). Under this condition,
(5.7)∥Hwαf∥ℒ∁{0}q,ψ≤Csupr>0𝕎1/q(r)∥f∥ℒ∁{0}p,ψ.Proof.
From the estimate (5.2) of Theorem 5.1 we have
(5.8)|Hwαf(x)|≤C|x|α-(n/p)ψq/p(|x|)(1V(|x|)∫0|x|‍V(t)tdt)∥f∥ℒ∁{0}q,ψ.
Hence, calculating ∥Hwαf∥ℒ∁{0}p,ψ, passing to polar coordinates, we obtain (5.6).Remark 5.4.
Note that(5.9)1V(ϱ)∫0ϱ‍V(t)tdt≃1
in (5.6) if we suppose that V∈ℤ0.The following corollary gives sufficient conditions for the boundedness of the operatorHwα 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.
Let1≤p≤q<∞ (with q=p admitted in the case α=0) and the conditions (5.5) be satisfied. Suppose also that
(5.10)ψ(r)≥Cr(αpq/q-p)-n
in the case α≠0. The operator Hwα is bounded from ℒ∁{0}p,ψ(ℝn) to ℒ∁{0}q,ψ(ℝn) if
(5.11)min{m(V),m∞(V)}>0,min{m(ψ),m∞(ψ)}>0.
The conditionmin{m(V),m∞(V)}>0, is guaranteed by the inequalities
(5.12)M(w)<np′-M(ψ),M∞(w)<np′-M∞(ψ).
In the power case ψ(r)=rλ, λ>0, and w(r)=rμ, the conditions (5.10)-(5.11) reduce to
(5.13)1≤p<n+λα,1q=1p-αn+λ,μ<np′-λp;
conditions (5.13) are also necessary for the operator Hwα to be bounded from ℒ∁{0}p,ψ(ℝn) to ℒ∁{0}q,ψ(ℝn).Proof.
We have to check that the conditionsupr>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
(5.14)𝕎(r)≤Cψ(r)∫r∞‍ϱn-1ϱq(α-(n/p))ψq/p(ϱ)dϱ.
We representψq/p(ϱ) as ψq/p(ϱ)=ψ(q/p)-1(ϱ)ψ(ϱ) and by (5.10) obtain
(5.15)𝕎(r)≤Cψ(r)∫r∞‍dϱϱψ(ϱ),
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∥Hwα∥ℒ∁{0}q,ψ(ℝn)≤C∥f∥ℒ∁{0}p,ψ(ℝn) with ψ(r)=rλ,λ>0, and w(r)=rμ, by standard homogeneity arguments with the use of the dilation operator Πδf(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
(5.16)∥Πδf∥ℒ∁{0}p,ψ(ℝn)=δ-(n+λ)/p∥f∥ℒ∁{0}p,ψ(ℝn),HwαΠδf=δ-αΠδHwαf.
(We take into account that we excluded q=∞ in this theorem.)
The necessity of the remaining conditionμ<(n/p′)-(λ/p) in (5.13) follows from the fact that |y|-(n+λ)/p∈ℒ∁{0}p,ψ(ℝn) by Corollary 2.4, so that this condition is necessary for the operator Hwα with w=rμ to be defined on the space ℒ∁{0}p,ψ(ℝn).


#### 5.2.2. The Case of the Operatorℋwα

Let(5.17)𝒲(r)∶=ψ(r)∫r∞‍wq(ϱ)ϱqα+n-1(∫ϱ∞‍𝒱(t)tdt)qdϱ,
where 𝒱 is the same as in (5.3).Theorem 5.6.
Let1≤p<∞, 1≤q<∞ and
(5.18)wψ1/p∈W_(ℝ+1)orw∈W¯(ℝ+1),w(2t)≤Cw(t).
The operator ℋwα is bounded from ℒ∁{0}p,ψ(ℝn) to ℒ∁{0}q,ψ(ℝn) if
(5.19)supr>0𝒲(r)<∞,
and then
(5.20)∥ℋwαf∥ℒ∁{0}q,ψ≤Csupr>0𝒲1/q(r)∥f∥ℒ∁{0}p,ψ.Proof.
From the estimate (5.4) we obtain
(5.21)ℳq,ψ(ℋwαf;0,r)≤Cψ(r)∫r∞‍ϱn-1+q(α-(n/p))ψ-q/p(ϱ)(1𝒱(ϱ)∫ϱ∞‍𝒱(t)tdt)qdϱ∥f∥ℒ∁{0}p,ψq,
from which (5.20) follows.As above, we provide also sufficient conditions for the boundedness of the operatorℋwβ in terms of the Matuszewska-Orlicz indices.Corollary 5.7.
Let1≤p≤q<∞ (with q=p admitted in the case α=0 and w and ψ satisfy the conditions (5.18). The operator ℋwα is bounded from ℒ∁{0}p,ψ(ℝn) to ℒ∁{0}q,ψ(ℝn) if
(5.22)max{M(𝒱),M∞(𝒱)}<0,min{m(ψ),m∞(ψ)}>0,
and the condition (5.10) holds; the assumption max{M(𝒱),M∞(𝒱)}<0 is guaranteed by the conditions
(5.23)m(w)>-m(ψ)+np,m∞(w)>-m∞(ψ)+np.
In the case ψ(r)=rλ,λ>0, and w(r)=rμ, the conditions (5.22) and (5.10) reduce to
(5.24)1≤p<n+λα,1q=1p-αn+λ,μ>n+λp;
conditions (5.24) are also necessary for the operator ℋwα to be bounded from ℒ∁{0}p,ψ(ℝn) to ℒ∁{0}q,ψ(ℝn).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
(5.25)𝒲(r)≤Cψ(r)∫r∞‍ϱn-1+q(α-(n/p))ψ-q/p(ϱ)(1𝒱(ϱ)∫ϱ∞‍𝒱(t)tdt)qdρ≤Cψ(r)∫r∞‍ϱn-1+q(α-(n/p))ψ-q/p(ϱ)dρ
by the assumption max{M(𝒱),M∞(𝒱)}<0. With ψq/p(ρ)=ψ(q/p)-1(ρ)ψ(ρ) and the condition (5.10) we arrive at 𝒲(r)≤Cψ(r)∫r∞‍dϱ/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 Corollary5.5.


## 5.1. Pointwise Estimations

The proof of our main result of this Section given in Theorems5.3 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.
Let1≤p≤∞, let w,wψ1/p∈W¯, and w(2t)≤Cw(t). The condition
(5.1)∫0ε‍V(t)tdt<∞withε>0,whereV(t)=tn/p′w(t)ψ1/p(t)
is sufficient for the Hardy operator Hwα to be defined on the space ℒ∁{0}p,ψ(ℝn). Under this condition, the pointwise estimate
(5.2)|Hwαf(x)|≤C|x|α-nw(|x|)∫0|x|‍V(t)tdt∥f∥ℒ∁{0}p,ψ
holds.Proof.
Apply Lemma4.1 with s=1.Theorem 5.2.
Let1≤p≤∞ and 1/w∈W¯, or w∈W¯ and w(2t)≤Cw(t). The condition
(5.3)∫ε∞‍𝒱(t)tdt<∞,withε>0,where𝒱(t)=1w(t)ψ1/p(t)tn/p
is sufficient for the Hardy operator ℋwα to be defined on the space ℒ∁{x0}p,ψ(ℝn), and in this case the following pointwise estimate:
(5.4)|ℋwαf(x)|≤C|x|αw(|x|)∫|x|∞‍𝒱(t)tdt∥f∥ℒ∁{x0}p,ψ
holds.Proof.
Apply Lemma4.2 with s=1.


## 5.2. Weightedp→q Boundedness of Hardy Operators in Complementary Morrey Spaces

### 5.2.1. The Case of the OperatorHwα

TheLp→Lq-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.
Let1≤p≤∞,1≤q≤∞ and let
(5.5)w,ψ1/pw∈W¯(ℝ+1),w(2t)≤cw(t).
The operator Hwα is bounded from ℒ∁{0}p,ψ(ℝn) to ℒ∁{0}q,ψ(ℝn) if supr>0𝕎(r)<∞, where
(5.6)𝕎(r):=ψ(r)∫r∞‍ϱn-1(ϱ(α-(n/p))ψ1/p(ϱ)V(ϱ)∫0ϱ‍V(t)tdt)qdϱ,
and V(t) is the same as in (5.1). Under this condition,
(5.7)∥Hwαf∥ℒ∁{0}q,ψ≤Csupr>0𝕎1/q(r)∥f∥ℒ∁{0}p,ψ.Proof.
From the estimate (5.2) of Theorem 5.1 we have
(5.8)|Hwαf(x)|≤C|x|α-(n/p)ψq/p(|x|)(1V(|x|)∫0|x|‍V(t)tdt)∥f∥ℒ∁{0}q,ψ.
Hence, calculating ∥Hwαf∥ℒ∁{0}p,ψ, passing to polar coordinates, we obtain (5.6).Remark 5.4.
Note that(5.9)1V(ϱ)∫0ϱ‍V(t)tdt≃1
in (5.6) if we suppose that V∈ℤ0.The following corollary gives sufficient conditions for the boundedness of the operatorHwα 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.
Let1≤p≤q<∞ (with q=p admitted in the case α=0) and the conditions (5.5) be satisfied. Suppose also that
(5.10)ψ(r)≥Cr(αpq/q-p)-n
in the case α≠0. The operator Hwα is bounded from ℒ∁{0}p,ψ(ℝn) to ℒ∁{0}q,ψ(ℝn) if
(5.11)min{m(V),m∞(V)}>0,min{m(ψ),m∞(ψ)}>0.
The conditionmin{m(V),m∞(V)}>0, is guaranteed by the inequalities
(5.12)M(w)<np′-M(ψ),M∞(w)<np′-M∞(ψ).
In the power case ψ(r)=rλ, λ>0, and w(r)=rμ, the conditions (5.10)-(5.11) reduce to
(5.13)1≤p<n+λα,1q=1p-αn+λ,μ<np′-λp;
conditions (5.13) are also necessary for the operator Hwα to be bounded from ℒ∁{0}p,ψ(ℝn) to ℒ∁{0}q,ψ(ℝn).Proof.
We have to check that the conditionsupr>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
(5.14)𝕎(r)≤Cψ(r)∫r∞‍ϱn-1ϱq(α-(n/p))ψq/p(ϱ)dϱ.
We representψq/p(ϱ) as ψq/p(ϱ)=ψ(q/p)-1(ϱ)ψ(ϱ) and by (5.10) obtain
(5.15)𝕎(r)≤Cψ(r)∫r∞‍dϱϱψ(ϱ),
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∥Hwα∥ℒ∁{0}q,ψ(ℝn)≤C∥f∥ℒ∁{0}p,ψ(ℝn) with ψ(r)=rλ,λ>0, and w(r)=rμ, by standard homogeneity arguments with the use of the dilation operator Πδf(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
(5.16)∥Πδf∥ℒ∁{0}p,ψ(ℝn)=δ-(n+λ)/p∥f∥ℒ∁{0}p,ψ(ℝn),HwαΠδf=δ-αΠδHwαf.
(We take into account that we excluded q=∞ in this theorem.)
The necessity of the remaining conditionμ<(n/p′)-(λ/p) in (5.13) follows from the fact that |y|-(n+λ)/p∈ℒ∁{0}p,ψ(ℝn) by Corollary 2.4, so that this condition is necessary for the operator Hwα with w=rμ to be defined on the space ℒ∁{0}p,ψ(ℝn).


### 5.2.2. The Case of the Operatorℋwα

Let(5.17)𝒲(r)∶=ψ(r)∫r∞‍wq(ϱ)ϱqα+n-1(∫ϱ∞‍𝒱(t)tdt)qdϱ,
where 𝒱 is the same as in (5.3).Theorem 5.6.
Let1≤p<∞, 1≤q<∞ and
(5.18)wψ1/p∈W_(ℝ+1)orw∈W¯(ℝ+1),w(2t)≤Cw(t).
The operator ℋwα is bounded from ℒ∁{0}p,ψ(ℝn) to ℒ∁{0}q,ψ(ℝn) if
(5.19)supr>0𝒲(r)<∞,
and then
(5.20)∥ℋwαf∥ℒ∁{0}q,ψ≤Csupr>0𝒲1/q(r)∥f∥ℒ∁{0}p,ψ.Proof.
From the estimate (5.4) we obtain
(5.21)ℳq,ψ(ℋwαf;0,r)≤Cψ(r)∫r∞‍ϱn-1+q(α-(n/p))ψ-q/p(ϱ)(1𝒱(ϱ)∫ϱ∞‍𝒱(t)tdt)qdϱ∥f∥ℒ∁{0}p,ψq,
from which (5.20) follows.As above, we provide also sufficient conditions for the boundedness of the operatorℋwβ in terms of the Matuszewska-Orlicz indices.Corollary 5.7.
Let1≤p≤q<∞ (with q=p admitted in the case α=0 and w and ψ satisfy the conditions (5.18). The operator ℋwα is bounded from ℒ∁{0}p,ψ(ℝn) to ℒ∁{0}q,ψ(ℝn) if
(5.22)max{M(𝒱),M∞(𝒱)}<0,min{m(ψ),m∞(ψ)}>0,
and the condition (5.10) holds; the assumption max{M(𝒱),M∞(𝒱)}<0 is guaranteed by the conditions
(5.23)m(w)>-m(ψ)+np,m∞(w)>-m∞(ψ)+np.
In the case ψ(r)=rλ,λ>0, and w(r)=rμ, the conditions (5.22) and (5.10) reduce to
(5.24)1≤p<n+λα,1q=1p-αn+λ,μ>n+λp;
conditions (5.24) are also necessary for the operator ℋwα to be bounded from ℒ∁{0}p,ψ(ℝn) to ℒ∁{0}q,ψ(ℝn).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
(5.25)𝒲(r)≤Cψ(r)∫r∞‍ϱn-1+q(α-(n/p))ψ-q/p(ϱ)(1𝒱(ϱ)∫ϱ∞‍𝒱(t)tdt)qdρ≤Cψ(r)∫r∞‍ϱn-1+q(α-(n/p))ψ-q/p(ϱ)dρ
by the assumption max{M(𝒱),M∞(𝒱)}<0. With ψq/p(ρ)=ψ(q/p)-1(ρ)ψ(ρ) and the condition (5.10) we arrive at 𝒲(r)≤Cψ(r)∫r∞‍dϱ/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 Corollary5.5.


## 5.2.1. The Case of the OperatorHwα

TheLp→Lq-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.
Let1≤p≤∞,1≤q≤∞ and let
(5.5)w,ψ1/pw∈W¯(ℝ+1),w(2t)≤cw(t).
The operator Hwα is bounded from ℒ∁{0}p,ψ(ℝn) to ℒ∁{0}q,ψ(ℝn) if supr>0𝕎(r)<∞, where
(5.6)𝕎(r):=ψ(r)∫r∞‍ϱn-1(ϱ(α-(n/p))ψ1/p(ϱ)V(ϱ)∫0ϱ‍V(t)tdt)qdϱ,
and V(t) is the same as in (5.1). Under this condition,
(5.7)∥Hwαf∥ℒ∁{0}q,ψ≤Csupr>0𝕎1/q(r)∥f∥ℒ∁{0}p,ψ.Proof.
From the estimate (5.2) of Theorem 5.1 we have
(5.8)|Hwαf(x)|≤C|x|α-(n/p)ψq/p(|x|)(1V(|x|)∫0|x|‍V(t)tdt)∥f∥ℒ∁{0}q,ψ.
Hence, calculating ∥Hwαf∥ℒ∁{0}p,ψ, passing to polar coordinates, we obtain (5.6).Remark 5.4.
Note that(5.9)1V(ϱ)∫0ϱ‍V(t)tdt≃1
in (5.6) if we suppose that V∈ℤ0.The following corollary gives sufficient conditions for the boundedness of the operatorHwα 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.
Let1≤p≤q<∞ (with q=p admitted in the case α=0) and the conditions (5.5) be satisfied. Suppose also that
(5.10)ψ(r)≥Cr(αpq/q-p)-n
in the case α≠0. The operator Hwα is bounded from ℒ∁{0}p,ψ(ℝn) to ℒ∁{0}q,ψ(ℝn) if
(5.11)min{m(V),m∞(V)}>0,min{m(ψ),m∞(ψ)}>0.
The conditionmin{m(V),m∞(V)}>0, is guaranteed by the inequalities
(5.12)M(w)<np′-M(ψ),M∞(w)<np′-M∞(ψ).
In the power case ψ(r)=rλ, λ>0, and w(r)=rμ, the conditions (5.10)-(5.11) reduce to
(5.13)1≤p<n+λα,1q=1p-αn+λ,μ<np′-λp;
conditions (5.13) are also necessary for the operator Hwα to be bounded from ℒ∁{0}p,ψ(ℝn) to ℒ∁{0}q,ψ(ℝn).Proof.
We have to check that the conditionsupr>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
(5.14)𝕎(r)≤Cψ(r)∫r∞‍ϱn-1ϱq(α-(n/p))ψq/p(ϱ)dϱ.
We representψq/p(ϱ) as ψq/p(ϱ)=ψ(q/p)-1(ϱ)ψ(ϱ) and by (5.10) obtain
(5.15)𝕎(r)≤Cψ(r)∫r∞‍dϱϱψ(ϱ),
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∥Hwα∥ℒ∁{0}q,ψ(ℝn)≤C∥f∥ℒ∁{0}p,ψ(ℝn) with ψ(r)=rλ,λ>0, and w(r)=rμ, by standard homogeneity arguments with the use of the dilation operator Πδf(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
(5.16)∥Πδf∥ℒ∁{0}p,ψ(ℝn)=δ-(n+λ)/p∥f∥ℒ∁{0}p,ψ(ℝn),HwαΠδf=δ-αΠδHwαf.
(We take into account that we excluded q=∞ in this theorem.)
The necessity of the remaining conditionμ<(n/p′)-(λ/p) in (5.13) follows from the fact that |y|-(n+λ)/p∈ℒ∁{0}p,ψ(ℝn) by Corollary 2.4, so that this condition is necessary for the operator Hwα with w=rμ to be defined on the space ℒ∁{0}p,ψ(ℝn).


## 5.2.2. The Case of the Operatorℋwα

Let(5.17)𝒲(r)∶=ψ(r)∫r∞‍wq(ϱ)ϱqα+n-1(∫ϱ∞‍𝒱(t)tdt)qdϱ,
where 𝒱 is the same as in (5.3).Theorem 5.6.
Let1≤p<∞, 1≤q<∞ and
(5.18)wψ1/p∈W_(ℝ+1)orw∈W¯(ℝ+1),w(2t)≤Cw(t).
The operator ℋwα is bounded from ℒ∁{0}p,ψ(ℝn) to ℒ∁{0}q,ψ(ℝn) if
(5.19)supr>0𝒲(r)<∞,
and then
(5.20)∥ℋwαf∥ℒ∁{0}q,ψ≤Csupr>0𝒲1/q(r)∥f∥ℒ∁{0}p,ψ.Proof.
From the estimate (5.4) we obtain
(5.21)ℳq,ψ(ℋwαf;0,r)≤Cψ(r)∫r∞‍ϱn-1+q(α-(n/p))ψ-q/p(ϱ)(1𝒱(ϱ)∫ϱ∞‍𝒱(t)tdt)qdϱ∥f∥ℒ∁{0}p,ψq,
from which (5.20) follows.As above, we provide also sufficient conditions for the boundedness of the operatorℋwβ in terms of the Matuszewska-Orlicz indices.Corollary 5.7.
Let1≤p≤q<∞ (with q=p admitted in the case α=0 and w and ψ satisfy the conditions (5.18). The operator ℋwα is bounded from ℒ∁{0}p,ψ(ℝn) to ℒ∁{0}q,ψ(ℝn) if
(5.22)max{M(𝒱),M∞(𝒱)}<0,min{m(ψ),m∞(ψ)}>0,
and the condition (5.10) holds; the assumption max{M(𝒱),M∞(𝒱)}<0 is guaranteed by the conditions
(5.23)m(w)>-m(ψ)+np,m∞(w)>-m∞(ψ)+np.
In the case ψ(r)=rλ,λ>0, and w(r)=rμ, the conditions (5.22) and (5.10) reduce to
(5.24)1≤p<n+λα,1q=1p-αn+λ,μ>n+λp;
conditions (5.24) are also necessary for the operator ℋwα to be bounded from ℒ∁{0}p,ψ(ℝn) to ℒ∁{0}q,ψ(ℝn).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
(5.25)𝒲(r)≤Cψ(r)∫r∞‍ϱn-1+q(α-(n/p))ψ-q/p(ϱ)(1𝒱(ϱ)∫ϱ∞‍𝒱(t)tdt)qdρ≤Cψ(r)∫r∞‍ϱn-1+q(α-(n/p))ψ-q/p(ϱ)dρ
by the assumption max{M(𝒱),M∞(𝒱)}<0. With ψq/p(ρ)=ψ(q/p)-1(ρ)ψ(ρ) and the condition (5.10) we arrive at 𝒲(r)≤Cψ(r)∫r∞‍dϱ/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 Corollary5.5.



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*Source: 283285-2012-11-11.xml*