Multi–dimensional Hardy type inequalities in Hölder spaces

Inequalities

Volume 12, Number 3 (2018), 719–729 doi:10.7153/jmi-2018-12-55

MULTI–DIMENSIONAL HARDY TYPE INEQUALITIES IN H ¨ OLDER SPACES

E VGENIYA B URTSEVA , S TAFFAN L UNDBERG , L ARS -E RIK P ERSSON AND

N ATASHA S AMKO

(Communicated by J. Peˇcari´c)

Abstract. Most Hardy type inequalities concern boundedness of the Hardy type operators in Lebesgue spaces. In this paper we prove some new multi-dimensional Hardy type inequalities in H ¨older spaces.

1. Introduction

The original Hardy inequality from 1925 (see [2])reads:

∞ Z

0



 1 x

x Z

0

f (y) dy





p

dx 6

 p

p − 1

 p ∞ Z 0

f ^p (x) dx, p > 1.

Since the constant 

p p−1

 p

is sharp this means that the Hardy operator H defined by H f (x) := ¹ _x

x R

0

f (y) dy maps L p into L p with the operator norm p ^′ := _p−1 ^p .

After this fundamental discovery by Hardy it was an almost unbelievable develop- ment of this area which today usually is referred to as Hardy type inequalities. A great number of papers and even books have been published on the subject and the research in this area is still very intensive. One important reason for that is that Hardy type in- equalities are especially useful for various types of applications within different parts of Mathematics but also in other Sciences, see e.g. the books [5], [6] and [7] and the references therein.

Most of the developments described above are devoted to study the boundedness of Hardy type operators between weighted Lebesgue spaces and most of the results are for the one-dimensional case. But for applications it is also often required to con- sider the boundedness between other function spaces. Unfortunately, there exist not so many results concerning the boundedness of Hardy type operators in other func- tion spaces. However, some results of this type can be found in Chapter 11 of the

Mathematics subject classification (2010): 46E15, 26D15, 47B38.

Keywords and phrases: Inequalities, Hardy type inequalities, H¨older spaces, Hardy type operators, boundedness, compactification.

c

, Zagreb

Paper JMI-12-55 719

book [6], where it is reported on Hardy type inequalities in Orlicz, Lorentz and rear- rangement invariant spaces and also on some really first not complete results in general Banach function spaces. Moreover, in [15] some corresponding Hardy type inequali- ties in weighted Morrey spaces were proved; in [13] the weighted estimates for multi- dimensional Hardy type operators were proved in generalized Morrey spaces; in [1]

was proved the weighted boundedness of some multi-dimensional Hardy type operators from generalized Morrey to Orlicz-Morrey spaces. For more information concerning Hardy type inequalities in Morrey type spaces and their applications we refer to [1], [9], [10], [12], [16] and references therein.

In this paper we continue this research by investigating Hardy type inequalities in H¨older spaces in the multi-dimensional case. H¨older spaces on unbounded sets can be defined with compactification at infinity (see Definition 3.1) or without.

We study multi-dimensional Hardy operators of order α ∈ [0, 1) as defined in (1.1). We refer to the paper [19] where a version of Hardy operators of the order α = 0 was studied within the frameworks of Triebel-Lizorkin spaces. This version may be regarded as a one-dimensional Hardy type operator in a given direction _|x| ^x of a function f of many variables. Multi-dimensional Hardy operators in our paper are of different nature.

By C ^λ (Ω), 0 < λ 6 1, where Ω is an open set in R ⁿ , Ω ⊆ R ⁿ , n > 1, we denote the class of bounded H¨older continuous functions, defined by the seminorm

[ f ] _λ := sup

x,x+h∈Ω

|h|<1

| f (x + h) − f (x)|

|h| ^λ < ∞.

Equipped with the norm

k f k _C λ = sup

x∈Ω

| f (x)| + [ f ] _λ

C ^λ (Ω) is a Banach space. We shall deal with the case Ω = B R , where B R = B(0, R) :=

{x ∈ R ⁿ : |x| < R} , 0 < R 6 ∞.

We consider the Hardy type operators H ^α f (x) = |x| ^α−n

Z

|y|<|x|

f (y)dy and H ^α f (x) = |x| ^α Z

|y|>|x|

f (y)

|y| ⁿ dy, α > 0, (1.1) where x ∈ B R , 0 < R 6 ∞ for the operator H ^α , and R = ∞ for the operator H ^α . We write H = H ^α and H = H ^α in the case α = 0.

The operator H ^α , α = 0, may be considered in both with and without compact- ification settings, but a consideration of H requires the compactification due to the needed convergence of integrals at infinity. We provide details for the operator H ^α , α ^> 0, without compactification, and for both the operators H and H with compact- ification. We also show that in the setting of the spaces with compactification we may consider only the case α = 0.

In Sections 2 and 3 we present and prove our new results on the boundedness of

the Hardy type operator H ^α in H¨older spaces without compactification (Theorem 2.2),

and for the operators H and H in the case with compactification (Theorems 3.5 and 3.6).

2. Boundedness of the Hardy type operator H ^α in a H¨older type space Denote

C ₀ ^λ (B R ) = { f ∈ C ^λ (B R ) : f (0) = 0}.

For the Hardy operator H ^α defined by H ^α f (x) := |x| ^α−n

Z

|y|<|x|

f (y)dy, α ^> 0,

we show that it maps H¨older space into itself in the case α = 0 and we prove a bound- edness result of the type C ^λ → C ^{λ +α} in the case α > 0 provided that λ + α ⁶ 1, see Theorem 2.2.

In the case α > 0 we will need the following Lemma:

L EMMA 2.1. Let

g(r) = 1 r ⁿ

Z

|y|<r

f (y)dy, 0 < r < R,

where f ∈ C ^λ (B R ), 0 < λ ⁶ 1 , 0 < R 6 ∞. Then

|g ^′ (r)| 6 C _n,λ [ f ] λ

r ^1−λ , 0 < r < R, (2.1) where C _n,λ depends only on n and λ .

Proof. Passing to polar coordinates, we have

g(r) = 1 r ⁿ

r Z

0

t ⁿ⁻¹ Φ(t)dt, Φ(t) = Z

S n−1

f (t σ )d σ .

Hence,

g ^′ (r) = − n r ⁿ⁺¹

r Z

0

t ⁿ⁻¹ Φ(t)dt + Φ(r)

r = n

r ⁿ⁺¹

r Z

0

t ⁿ⁻¹ [Φ(r) − Φ(t)]dt.

Therefore,

|g ^′ (r)| 6 n r ⁿ⁺¹

r Z

0

t ⁿ⁻¹ |Φ(r) − Φ(t)|dt.

It is easily seen that

|Φ(r) − Φ(t)| 6 [ f ] _λ |S ⁿ⁻¹ |(r − t) ^λ .

Consequently,

|g ^′ (r)| 6 n|S ⁿ⁻¹ |[ f ] _λ r ⁿ⁺¹

r Z

0

t ⁿ⁻¹ (r − t) ^λ dt = n|S ⁿ⁻¹ |[ f ] _λ r ^1−λ

1 Z

0

s ⁿ⁻¹ (1 − s) ^λ ds,

and we arrive at (2.1). The proof is complete. 

In the following theorem we deal also with the space ˜ C ₀ ^λ (Ω) consisting of func- tions f for which [ f ] _λ < ∞ and f (0) = 0. This space contains functions which are unbounded in the case Ω is unbounded. Note that [ f ] _λ is a norm in this space.

Now we are in a position to prove the following theorem:

T HEOREM 2.2. Let α ^> 0 , λ > 0 and λ + α ⁶ 1. In the case α = 0 the Hardy operator H ^α is bounded in C ^λ (B R ) and [H ^α f | α=0 ] _λ 6 C[ f ] _λ . In the case α > 0 the operator H ^α is bounded from ˜ C ₀ ^λ (B R ) into ˜ C ^{λ +α} ₀ (B R ), 0 < R 6 ∞.

Proof. Let first α = 0. For H f = H ^α f | α=0 we have H f (x) = |x| ⁻ⁿ

Z

|y|<|x|

f (y)dy = Z

B(0,1)

f (|x|y)dy

so that

|H f (x + h) − H f (x)| 6 Z

B(0,1)

| f (|x + h|y) − f (|x|y)|dy

6 [ f ] λ Z

B(0,1)

||x + h| − |x|| ^λ |y| ^λ dy =: A.

Since, by triangle inequality ||x + h| − |x|| ^λ 6 |h| ^λ , λ > 0 , for all x, x + h ∈ R ⁿ , we obtain that

A 6 [ f ] _λ Z

B(0,1)

|h| ^λ |y| ^λ dy 6 [ f ] _λ |h| ^λ Z

B(0,1)

|y| ^λ dy = C|h| ^λ [ f ] _λ .

Thus, |H f (x + h) − H f (x)| 6 C|h| ^λ [ f ] _λ and therefore [H f ] _λ 6 C[ f ] _λ , with C not depending on x and h.

Since the inequality sup

x∈Ω

|H f (x)| 6 c sup

x∈Ω

| f (x)| is obvious, the proof is complete for α = 0.

Let now α > 0 and f ∈ ˜ C ₀ ^λ (B R ). We have H ^α f (x) = |x| ^α g(|x|), g(r) = 1

r ⁿ Z

B(0,r)

f (y)dy = Z

B(0,1)

f (ry)dy. (2.2)

Hence, by the triangle inequality,

|H ^α f (x + h) − H ^α f (x)| 6 ||x + h| ^α − |x| ^α | |g(|x + h|)| + |g(|x + h|) − g(|x|)||x| ^α 6 C[ f ] _λ ||x + h| ^α − |x| ^α | |x + h| ^λ + |g(|x + h|) − g(|x|)||x| ^α

=: ∆ 1 + ∆ 2 ,

where we used the fact that f (0) = 0 and consequently

|g(|x + h|)| = |H f (|x + h|)| 6 C|x + h| ^λ [ f ] _λ (2.3) according to the case α = 0 in the last passage.

We consider separately the cases |x + h| 6 2|h| and |x + h| > 2|h|.

The case |x + h| 6 2|h|.

In this case we also have |x| 6 3|h|.

Thus, by (2.3),

∆ 1 6 C[ f ] _λ |h| ^α |x + h| ^λ 6 C 1 [ f ] _λ |h| ^{λ +α} and

∆ 2 6 C[g] _λ |h| ^λ |x| ^α 6 C 1 [ f ] _λ |h| ^{λ +α} . The case |x + h| > 2|h|.

We have

∆ 1 6 C[ f ] _λ |x + h| ^{λ +α}    

1 −

 |x|

|x + h|

 α     . Since, |1 − t ^α | 6 |1 − t| for all 0 < t 6 1 , 0 < α ⁶ 1, we obtain

∆ 1 6 C[ f ] λ ||x + h| − |x||

|x + h| ^{1−λ −α} 6 C[ f ] λ |h| ^{λ +α} . For ∆ 2 we use the mean value theorem and find that

∆ 2 6 C  g ^′ ( ξ ) 

 ||x + h| − |x|||x| ^α 6 C|g ^′ ( ξ )||h||x| ^α with ξ between |x| and |x + h|.

If |x| 6 |x + h|, then, by Lemma 2.1, we get

∆ 2 6 C [ f ] _λ

| ξ | ^1−λ |x| ^α |h| 6 C [ f ] _λ

|x| ^{1−λ −α} |h| 6 C[ f ] _λ |h| ^{λ +α} because |x| > |x + h| − |h| > |h|. Finally, when |x| > |x + h|, we have

∆ 2 6 C [ f ] _λ

| ξ | ^1−λ |x| ^α |h| 6 C [ f ] _λ

|x + h| ^1−λ |x| ^α |h| = C [ f ] _λ

|x + h| ^{1−λ −α}

 |x|

|x + h|

 α

|h|,

where _|x+h| ^|x| 6 _|x+h| ^|h| + ^|x+h| _|x+h| 6 ³

2 . Therefore,

∆ 2 6 C[ f ] _λ |h| ^{λ +α} .

It remains to gather the estimates for ∆ 1 and ∆ 2 .

In view of (2.2), the equality H ^α f (0) = 0 is obvious, so the proof is complete.  We define the generalized H¨older space C ^ω(·) (Ω) as the set of functions continu- ous in Ω having the finite norm

k f k _C ω (·) = sup

x∈Ω

| f (x)| + [ f ] _ω(·) with the seminorm

[ f ] _ω(·) = sup

x,x+h∈Ω

|h|<1

| f (x + h) − f (x)|

ω (|h|) ,

where ω : [0, 1] → R + is a non-negative increasing function in C([0, 1]) such that ω (0) = 0 and ω (t) > 0 for 0 < t 6 1. Such spaces are known in the literature, see for instance [8], [14], [17, Section 13.6], [18].

Let also C ₀ ^ω(·) (B R ) := { f ∈ C ^ω(·) (B R ) : f (0) = 0}.

As usual, by saying that a function ϕ is almost decreasing, we mean that ϕ (t) 6 C ϕ (s) for some C > 1 and for all t > s.

Following the same lines as in proof of Theorem 2.2 one can prove the following generalization of Theorem 2.2:

T HEOREM 2.3. Let ω ∈ C([0, 1]) be positive on (0, 1], increasing and such that ω (0) = 0 and _t ^ω(t) _{1− α} is almost decreasing. In the case α = 0 the operator H ^α | α=0 is bounded in C ^ω(·) (B R ). When α > 0, it is bounded from ˜ C ₀ ^ω(·) (B R ) into ˜ C ₀ ^{ω α (·)} (B R ), where ω α (t) = t ^α ω (t).

3. Boundedness of Hardy type operators in H¨older type spaces with compactification

Let ˙ R ⁿ denote the compactification of R ⁿ by a single infinite point.

D EFINITION 3.1. Let 0 6 λ < 1. We say that f belongs to C ^λ ( ˙ R ⁿ ), for all x, y ∈ R ⁿ , if

| f (x) − f (y)| 6 C |x − y| ^λ (1 + |x|) ^λ (1 + |y|) ^λ . The set C ^λ ( ˙ R ⁿ ) is a Banach space with respect to the norm

k f k _C λ ( ˙ R ⁿ ) = k f k _{C( ˙} R ⁿ ) + sup

x,y∈R ⁿ

| f (x) − f (y)|  (1 + |x|)(1 + |y|)

|x − y|

 λ

.

It may be shown that C ^λ ( ˙ R ⁿ ) is a subspace of C ^λ (R ⁿ ), which is invariant with respect to the inversion change of variables x _∗ = _|x| ^x 2 , i.e.

C ^λ ( ˙ R ⁿ ) = { f : f ∈ C ^λ (R ⁿ ) and f ∗ ∈ C ^λ (R ⁿ )},

where f ∗ = f (x ∗ ).

In the setting of the spaces C ^λ ( ˙ R ⁿ ) we consider only the case α = 0, see Remark 3.4 below.

3.1. Hardy operator H

Our main result in this case reads:

T HEOREM 3.2. Let 0 6 λ < 1. Then the operator H is bounded in C ^λ ( ˙ R ⁿ ).

Proof. We note that

H f (x) − H f (y) = Z

B(0,1)

[ f (|x|z) − f (|y|z)] dz.

Hence,

|H f (x) − H f (y)| 6 c Z

B(0,1)

||x| − |y|| ^λ |z| ^λ (1 + |x||z|) ^λ (1 + |y||z|) ^λ dz

6 c ||x| − |y|| ^λ Z

B(0,1)

|z| ^λ

(1 + |x||z|) ^λ (1 + |y||z|) ^λ dz =: A (3.1) Let |x| > 1 , |y| > 1. Then

A 6 c ||x| − |y|| ^λ Z

B(0,1)

|z| ^λ

(|x||z|) ^λ (|y||z|) ^λ dz = c ||x| − |y|| ^λ

|x| ^λ |y| ^λ Z

B(0,1)

dz

|z| ^λ

6 C

 ||x| − |y||

(1 + |x|)(1 + |y|)

 λ Z B(0,1)

dz

|z| ^λ 6 C 1

|x − y| ^λ

(1 + |x|) ^λ (1 + |y|) ^λ , (3.2)

since _|x| ¹ < _1+|x| ² .

Let |x| < 1 , |y| < 1. Then A 6 c|x − y| ^λ

Z

B(0,1)

|z| ^λ dz = c 1 |x − y| ^λ

6 C |x − y| ^λ

(1 + |x|) ^λ (1 + |y|) ^λ , (3.3)

since 1 < _1+|x| ² .

Let |x| < 1 , |y| > 1. Then A 6 c|x − y| ^λ

Z

B(0,1)

|z| ^λ

(|y||z|) ^λ dz 6 C 1

|x − y| ^λ

(1 + |x|) ^λ (1 + |y|) ^λ .

Let |x| > 1 , |y| < 1. Then

A 6 c|x − y| ^λ Z

B(0,1)

|z| ^λ

(|x||z|) ^λ dz 6 C 1

|x − y| ^λ (1 + |x|) ^λ (1 + |y|) ^λ .

Since the inequality kH f k _{C( ˙} R ⁿ ) 6 ck f k _{C( ˙} R ⁿ ) is obvious, the proof is complete.  3.2. Hardy operator H

To formulate the corresponding result for the operator H we need to consider the following subspaces:

C ₀ ^λ ( ˙ R ⁿ ) = { f ∈ C ^λ ( ˙ R ⁿ ) : f (0) = 0}, C _∞ ^λ ( ˙ R ⁿ ) = { f ∈ C ^λ ( ˙ R ⁿ ) : f (∞) = 0}

and

C _∞,0 ^λ = C ^λ _∞ ∩C ^λ ₀ .

T HEOREM 3.3. Let 0 < λ < 1. Then the operator H is bounded from C ^λ _∞,0 ( ˙ R ⁿ ) to C _∞ ^λ ( ˙ R ⁿ )

Proof. Let f ∈ C _∞,0 ^λ ( ˙ R ⁿ ) and denote g(x) = H f (x). Clearly, g(∞) = 0, and

|g(x) − g(y)| =       

Z

|z|>1

[ f (|x|z) − f (|y|z)] dz

|z| ⁿ       

(3.4)

6 C|x − y| ^λ Z

|z|>1

|z| ^{λ −n} dz

(1 + |x||z|) ^λ (1 + |y||z|) ^λ =: ∆.

Let |x| > 1 , |y| > 1. Then

∆ 6 C |x − y| ^λ

|x| ^λ |y| ^λ Z

|z|>1

dz

|z| ^n+λ 6 C 1

|x − y| ^λ (1 + |x|) ^λ (1 + |y|) ^λ .

Hence g(x) ∈ C _∞,0 ^λ ( ˙ R ⁿ ).

Let |y| < |x| < 2.

Since f (0) = 0, we have | f (z)| 6 C|z| ^λ and then

|g(x) − g(y)| =       

|x|

Z

|y|

f (z)

|z| ⁿ dz       

6 C

|x|

Z

|y|

|z| ^{λ −n} dz = C 1

 |x| ^λ − |y| ^λ  6 C 2

|x − y| ^λ (1 + |x|) ^λ (1 + |y|) ^λ ,

since a ^λ − b ^λ 6 (a − b) ^λ , a > b > 0 , 0 6 λ ⁶ 1.

Let now |y| < 1, |x| > 2. H f is bounded. Indeed,

|g(x) − g(y)| 6 Z

R ⁿ

f (z)

|z| ⁿ dz.

As already shown, for each function f ∈ C _∞,0 ^λ we have that | f (z)| 6 c|z| ^λ , 0 < |z| < 1 and | f (z)| 6 _|z| ^c _λ , |z| > 1. Therefore

|g(x)| 6 c 1 1 Z

0

1

|z| ^n−λ dz + c 2

∞ Z

1

1

|z| ^n+λ dz = C < ∞, for 0 6 λ < 1, and then

|g(x) − g(y)| 6 C.

It is easily checked that

1 6 6 |x − y|

(1 + |x|)(1 + |y|) , when |y| < 1, |x| > 2. (3.5) Consequently,

|g(x) − g(y)| 6 C 6 C 2

|x − y| ^λ (1 + |x|) ^λ (1 + |y|) ^λ , which proves that g(x) ∈ C ^λ _∞,0 ( ˙ R ⁿ ) also in this case.

The case |x| < 1 , |y| > 2 can be similarly treated.

Similarly as in Theorem 3.2 we note that the boundedness of the operator H in C( ˙ R ⁿ ) is obvious, so the proof is complete. 

R EMARK 3.4. When α > 0. Theorems 3.2 and 3.3 may not be extended to the setting C ^λ ( ˙ R ⁿ ) −→ C ^{λ +α} ( ˙ R ⁿ ), in which we require the H¨older behavior of func- tions also at the infinite point, in contrast to Theorem 2.2. In fact, the function f 0 =

1

(1 + x) ^λ ∈ C _∞ ^λ ( ˙ R ₊ ) provides a corresponding counterexample for both the operators H ^α and H ^α . For example, for the operator H ^α we have

H ^α f ₀ (x) = x ^α−1 1 − λ ^{[(1 + x)}

1−λ − 1].

Hence, when x → ∞ we obtain that H ^α f 0 (x) ∼ cx ^α−λ , while the inclusion H ^α f 0 (x)

∈ C ^{λ +α} _∞ ( ˙ R ₊ ) requires the behavior |H ^α f 0 (x)| 6 c(1 + x) ^−α−λ .

Corresponding generalizations of Theorems 3.2 and 3.3 may be also formulated

in terms of the generalized H¨older spaces C ^ω ( ˙ R ⁿ ), C _∞ ^ω ( ˙ R ⁿ ), C ₀ ^ω ( ˙ R ⁿ ) and C ^ω _∞,0 ( ˙ R ⁿ )

defined below.

D EFINITION 3.5. Let ω = ω (h) be an increasing function. The generalized H¨older space C ^ω ( ˙ R ⁿ ) is defined as consisting of all functions satisfying the condition

| f (x) − f (y)| 6 C ω

 |x − y|

(1 + |x|)(1 + |y|)



, x, y ∈ R ⁿ .

The subspaces C ^ω _∞ ( ˙ R ⁿ ), C ₀ ^ω ( ˙ R ⁿ ) and C _∞,0 ^ω ( ˙ R ⁿ ) of the space C ^ω ( ˙ R ⁿ ) are defined by the conditions f(∞) = 0 , f (0) = 0 and f (0) = f (∞) = 0, respectively.

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(Received January 15, 2017) Evgeniya Burtseva

Lule˚a University of Technology SE 971 87 Lule˚a, Sweden e-mail: Evgeniya.Burtseva@ltu.se Staffan Lundberg Lule˚a University of Technology SE 971 87 Lule˚a, Sweden e-mail: Staffan.Lundberg@ltu.se Natasha Samko Lule˚a University of Technology SE 971 87 Lule˚a, Sweden e-mail: Natasha.Samko@ltu.se Lars-Erik Persson Lule˚a University of Technology SE 971 87 Lule˚a, Sweden and UiT, The Arctic University of Norway NO 8505, Narvik, Norway e-mail: Lars-Erik.Persson@ltu.se

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