On the dependence of parameters in the equivalence theorem for the real method

vol. 55, no. 2 (2015), 79–87

On the dependence of parameters in the equivalence theorem for the real method

Fernando Cobos, Luz M. Fernández-Cabrera, Georgi E. Karadzhov, and Thomas Kühn

Summary. We determine the exact dependence on θ , q, p of the constants in the equivalence theorem for the real interpolation method (A

, A

)

with pairs of p-normed spaces.

Keywords real interpolation;

K -functional;

J-functional;

p-normed spaces

MSC 2010 46M35; 46B70 Received: 2015-12-15, Accepted: 2016-01-15

Dedicated to Professor Henryk Hudzik on the occasion of his 70th birthday.

1. Introduction

The real interpolation method (A 0 , A ₁ ) ^θ ,q is very useful in applications of interpolation theory to function spaces, PDEs, operator theory and approximation theory (see, for exam- ple, the books by Butzer and Berens [4], Bergh and Löfström [2], Triebel [15,16], König [12], Bennett and Sharpley [1] or Brudny˘ ı and Krugljak [3]). It can be realized either by means

Fernando Cobos , Departamento de Análisis Matemático, Facultad de Matemáticas, Universidad Complutense de Madrid, Plaza de Ciencias 3, 28040 Madrid, Spain (e-mail: cobos@mat.ucm.es)

Luz M. Fernández-Cabrera , Sección Departamental de Matemática Aplicada, Facultad de Estudios Estadísticos, Universidad Complutense de Madrid, 28040 Madrid, Spain (e-mail: luz_fernandez-c@mat.ucm.es)

Georgi E. Karadzhov , Institute of Mathematics and Informatics, Bulgarian Academy of Sciences, 1113 Sofia, Bulgaria and Abdus Salam School of Mathematical Scienses, GC University, Lahore, Pakistan (e-mail: geremika@yahoo.com) Thomas Kühn , Mathematisches Institut, Universität Leipzig, Augustusplatz 10, 04109 Leipzig, Germany

(e-mail: kuehn@math.uni-leipzig.de)

DOI 10.14708/cm.v55i2.833 © 2015 Polish Mathematical Society

of Peetre’s K -functional (A 0 , A ₁ ) ^θ ,q;K or J-functional (A 0 , A ₁ ) ^θ ,q;J , the dual functional to K .

In order to be able to combine interpolation techniques with those of extrapolation theory, it is important to work with normalized scales of real interpolation spaces (see, for example, the papers by Jawerth and Milman [9], Milman [13], Karadzhov and Mil- man [10], Fiorenza and Karadzhov [8] or Cobos and Kühn [7]). In the case of the K -scale (A 0 , A ₁ ) ^θ ,q;K generated by a Banach couple, this is achieved by multiplying the norm by the factor (θ(1 − θ)q) ^1/q . This modification ensures that the embeddings

A ₀ ∩ A 1 ↪ (A 0 , A ₁ ) ^θ ,q;K ↪ A 0 + A 1

and

(A 0 , A ₁ ) ^θ ,q;K ↪ (A 0 , A ₁ ) ^θ ,r;K (q < r) have uniformly bounded quasi-norms with respect to θ .

It is also of interest to determine the exact dependence on the parameters θ , q for the quasi-norms of the embeddings between K -spaces and J-spaces, and vice versa. Accordin- gly, we study such problems here for couples of p-normed spaces with 0 < p ⩽ 1.

2. Preliminaries

Let A be a vector space, and let ∥ ⋅ ∥ be a quasi-norm on A with a constant c ⩾ 1 in the quasi-triangle inequality. Let 0 < p ⩽ 1 be such that c = 2 ^1/p−1 . It is well kown that there is a p-norm ∥∣⋅∥∣ on Aequivalent to ∥⋅∥ (see [ 11, § 15.10] or [12, Proposition 1.c.5]). On the other hand, it is clear that any p-norm is a quasi-norm with constant 2 ^1/p in the quasi-triangle inequality.

Subsequently we work with pairs of p-normed quasi-Banach spaces (A 0 , A ₁ ). By this we mean two p-normed quasi-Banach spaces A j which are continuously embedded in the same Hausdorff topological vector space. For t > 0, Peetre’s K- and J-functionals are defined by

K (t, a) = K(t, a; A 0 , A ₁ ) = inf{∥a 0 ∥ ^A

+ t∥a 1 ∥ ^A

∶ a = a 0 + a 1 , a j ∈ A ^j } where a ∈ A 0 + A 1 and

J (t, a) = J(t, a; A 0 , A ₁ ) = max{∥a∥ ^A

, t ∥a∥ ^A

}, a ∈ A 0 ∩ A 1 .

Note that K (1, ⋅) is the quasi-norm of A 0 + A 1 and J (1, ⋅) the quasi-norm of A 0 ∩ A 1 . Moreover, J (t, ⋅) is a p-norm on A 0 ∩ A 1 . It will be also useful to work with the functional

K _p (t, a) = inf{(∥a 0 ∥ A ^p

+ t ^p ∥a 1 ∥ ^p A

) ^1/p ∶ a = a 0 + a 1 , a _j ∈ A ^j }

which is a p-norm on A ₀ + A 1 and satisfies that

K (t, a) ⩽ K ^p (t, a) ⩽ 2 ^1/p K (t, a), a ∈ A 0 + A 1 . (1) For 0 < θ < 1 and 0 < q ⩽ ∞, the real interpolation space realized by means of the K -functional (A 0 , A ₁ ) ^θ ,q;K consists of all a ∈ A 0 + A 1 having a finite quasi-norm

∥a∥ ^(A

,A

)

= ( ∑ ^∞

m=−∞

(2 ^{−θ m} K (2 ^m , a )) ^q ) ^1/q

(as usual, the sum should be replaced by the supremum if q = ∞). The corresponding space defined in terms of the J-functional (A 0 , A ₁ ) ^θ ,q;J is the collection of all sums a =

∑ ^{∞ m=−∞} u m (convergence in A ₀ + A 1 ), where (u ^m ) ⊆ A 0 ∩ A 1 and ( ∑ ^∞

m=−∞

(2 ^{−θ m} J (2 ^m , u _m )) ^q ) ^1/q < ∞.

The quasi-norm on (A 0 , A ₁ ) ^θ ,q;J is given by

∥a∥ ^(A

,A

)

= inf{( ∑ ^∞

m=−∞

(2 ^{−θ m} J (2 ^m , u m )) ^q ) ^1/q < ∞ ∶ a = ∑ ^∞

m=−∞

u m }.

The equivalence theorem shows that (A 0 , A ₁ ) ^θ ,q;K = (A 0 , A ₁ ) ^θ ,q;J with equivalent quasi- -norms (see [2, 14, 15] or [3]). Next we describe the dependence of the constants involved

in the equivalence of quasi-norms on the parameters θ , q, p.

3. The results

We start with the embeddings from K -spaces into J-spaces. We write [x] for the largest integer which is less than or equal to x . In the proof we use similar decompositions as in [5, 6].

3.1. Theorem. Let (A 0 , A ₁ ) be a pair of p-normed quasi-Banach spaces, let 0 < θ < 1 and 0 < q ⩽ ∞. Then

∥a∥ ^(A

,A

)

⩽ 2 ^4+1/p θ (1 − θ)∥a∥ ^(A

,A

)

. Proof. Take any a ∈ (A 0 , A ₁ ) ^θ ,q;K . Since

min (1, t ^{− 1} )K(t, a) = min(t ^θ , t ^{θ − 1} ) K (t, a)

t ^θ ⩽ c min(t ^θ , t ^{θ − 1} )∥a∥ ^(A

,A

)

, we have that min (1, t ^{− 1} )K(t, a) → 0 as t → 0 or t → ∞. Given ν ∈ Z, let µ ^ν = 2 ^{ν[ 1/θ]} . Take any ε > 0. We can decompose a = a 0,ν + a 1,ν in such a way that a _{j ,ν} ∈ A ^j and

∥a 0,ν ∥ ^p A

+ µ ^p ν+ 1 ∥a 1,ν ∥ ^p A

⩽ (1 + ε)K ^p (µ ^ν+ 1 , a ) ^p .

Put u _ν = a 0,ν − a 0,ν−1 = a 1,ν−1 − a 1,ν . Then

∥a − ∑ ^M

ν=−N

u ν ∥

A

+A

= ∥a − a 0,M + a 0,−N−1 ∥ ^A

^+A

⩽ ∥a 0,−N−1 ∥ ^A

+ ∥a 1,M ∥ ^A

→ 0

as M → ∞ and N → ∞. Hence a = ∑ ^∞ ν=−∞ u ν in A ₀ + A 1 . Moreover, using that (µ ^ν ) is an increasing sequence and that J (t, ⋅) is a p-norm, we obtain

J (µ ^ν , u _ν ) ^p ⩽ ∥a 0,ν ∥ ^p A

+ ∥a 0,ν−1 ∥ A ^p

+ µ ^{ν p} ∥a 1,ν−1 ∥ ^p A

+ µ ^{ν p} ∥a 1,ν ∥ ^p A

⩽ (1 + ε)K ^p (µ ^ν+ 1 , a ) ^p + (1 + ε)K ^p (µ ^ν , a ) ^p

⩽ 2(1 + ε)K ^p (µ ^ν+ 1 , a ) ^p .

Let I _ν = [µ ^ν− 1 , µ _ν ). The number of integer numbers m such that 2 ^m ∈ I ^ν is

♯{m ∈ Z ∶ 2 ^{(ν− 1)[1/θ]} ⩽ 2 ^m < 2 ^{ν[ 1/θ]} } = [1/θ].

Put w m = [1/θ] ^{− 1} u _ν if 2 ^m ∈ I ^ν , ν ∈ Z. This sequence is also contained in A 0 ∩ A 1 and, by (1),

K _p (1, a − ∑ ^M

m=−N

w _m ) ^p ⩽ K ^p (1, a −

Q

∑

m=−P

u _m ) ^p + d 1 K _p (1, u ^Q ) + d 2 K _p (1, u ^−P )

⩽ 2∥a − ∑ ^Q

m=−P

u m ∥ ^p

A

+A

+ d 1 ∥u ^Q ∥ ^p A

+ d 2 ∥u ^−P ∥ ^p A

for some 0 ⩽ d 1 , d ₂ < 1. This yields that a = ∑ ^∞ m=−∞ w m is in A ₀ + A 1 . We obtain that

∥a∥ ^(A

,A

)

⩽ ( ∑ ^∞

m=−∞

(2 ^{−θ m} J (2 ^m , w m )) ^q ) ^1/q

= ( ∑ ^∞

ν=−∞

2 ∑

∈I

((2 ^{−θ m} J (2 ^m , w m )) ^p ) ^{q/ p} ) ^1/q

⩽ ( ∑ ^∞

ν=−∞

2 ∑

∈I

(2 ^{−θ m p} [1/θ] ^{− p} J (µ ^ν , u _ν ) ^p ) ^{q/ p} ) ^1/q

⩽ ( ∑ ^∞

ν=−∞

2 ∑

∈I

2 ^{−θ m q} [1/θ] ^−q (2(1 + ε)) ^{q/ p} K _p (µ ^ν+ 1 , a ) ^q ) ^1/q .

Note that 2 ^m ∈ I ^ν if and only if 2 ^{m+ 2[1/θ]} ∈ I ^ν+ 2 . So

2 ∑

∈I

2 ^{−θ m q} = 2 ^2θ[1/θ]q ∑

2

∈I

2 ^{−θ n q} ⩽ 2 ^2q ∑

2

∈I

2 ^{−θ n q} .

Therefore

∥a∥ ^(A

,A

)

⩽ ( ∑ ^∞

ν=−∞

[1/θ] ^−q (2(1 + ε)) ^{q/ p} K p (µ ^ν+ 1 , a ) ^q 2 ^2q ∑

2

∈I

2 ^{−θ n q} ) ^1/q

⩽ 2 ^2+1/p (1 + ε) ^1/p [1/θ] ^{− 1} ( ∑ ^∞

ν=−∞

2

∑ ∈I

2 ^{−θ n q} K _p (2 ⁿ , a ) ^q ) ^1/q . Since [1/θ] ^{− 1} ⩽ 2θ, using ( 1) and passing to the limit as ε → 0, we get

∥a∥ ^(A

,A

)

⩽ 2 ^3+1/p θ ∥a∥ ^(A

,A

)

. (2)

On the other hand,

2 ^{−θ m} K (2 ^m , a; A ₀ , A ₁ ) = 2 ^{( 1−θ)m} K (2 ^−m , a; A ₁ , A ₀ ), a ∈ A 0 + A 1 .

This yields that (A 0 , A ₁ ) ^θ ,q∶K = (A 1 , A ₀ ) 1−θ ,q;K with equality of quasi-norms. Similarly, 2 ^{−θ m} J (2 ^m , u m ; A ₀ , A ₁ ) = 2 ^{−( 1−θ)(−m)} J (2 ^−m , u m ; A ₁ , A ₀ ), a ∈ A 0 ∩ A 1 .

So, if a = ∑ ^{∞ m=−∞} u _m is any J-representation of a ∈ (A 0 , A ₁ ) ^θ ,q;J , working with v m = u ^−m , which also satisfies a = ∑ ^{∞ m=−∞} v _m , we obtain that (A 0 , A ₁ ) ^θ ,q∶J = (A 1 , A ₀ ) 1−θ ,q;J with equality of quasi-norms. Consequently, by (2), for any a ∈ (A 0 , A ₁ ) ^θ ,q∶K we derive

∥a∥ ^(A

,A

)

= ∥a∥ ^(A

,A

)

⩽ 2 ^3+1/p (1 − θ)∥a∥ ^(A

,A

)

= 2 ^3+1/p (1 − θ)∥a∥ ^(A

,A

)

. This and (2) yield that

∥a∥ ^(A

,A

)

⩽ 2 ^3+1/p min {θ, 1 − θ}∥a∥ ^(A

,A

)

.

Finally, since min {θ, 1 − θ} ⩽ 2θ(1 − θ), the desired estimate follows.

Before passing to the embeddings of J-spaces into K -spaces we establish an auxiliary result.

3.2. Lemma. Let 0 < θ < 1, 0 < q < ∞ and put J = ( ∑ ^∞ m=−∞ 2 ^{−θ m q} min (1, 2 ^m ) ^q ) ^1/q . Then 2 ^{− 1/2} (θ(1 − θ)q log 2) ^{− 1/q} ⩽ J ⩽ 2 ^1/4 (θ(1 − θ)q log 2) ^{− 1/q} .

Proof. We have

J ^q = 1 + ∑ ^∞

m= 1

2 ^{−θ m q} + ∑ ^∞

m= 1

2 ^{−( 1−θ)mq}

= 1 + 2 ^{−θ q}

1 − 2 ^{−θ q} + 2 ^{−( 1−θ)q} 1 − 2 ^{−( 1−θ)q}

= 1 + 1

2 ^{θ q} − 1 + 1

2 ^{( 1−θ)q} − 1 .

For all x ⩾ 0 it holds that e ^x − 1 ⩾ x, whence (2 ^x − 1) ^{− 1} ⩽ (x log 2) ^{− 1} . This implies J ^q ⩽ 1 + 1

θ q log 2 + 1 (1 − θ)q log 2

= 1 + 1

θ (1 − θ)q log 2 .

Now using 1 + x ¹ = ^1+x x ⩽ ^e x

for x > 0 and θ(1 − θ) ⩽ 1/4 for 0 < θ < 1, we obtain the upper estimate

J ^q ⩽ e ^{θ (} 1−θ)q log 2

θ (1 − θ)q log 2 ⩽ 2 ^{q/ 4} θ (1 − θ)q log 2 .

For the proof of the lower estimate we use the inequality e ^x − 1 ⩽ xe ^x , which gives (2 ^x − 1) ^{− 1} ⩾ (2 ^x x log 2 ) ^{− 1} for all x > 0. Therefore, we obtain

J ^q = 1 + 1

2 ^{θ q} − 1 + 1 2 ^{( 1−θ)q} − 1

⩾ 1

2 ^{θ q} θ q log 2 + 1

2 ^{( 1−θ)q} (1 − θ)q log 2

= ( 1 − θ)2 ^{( 1−θ)q} + θ2 ^{θ q} 2 ^q (1 − θ)θq log 2 . From the arithmetic-geometric mean inequality

(1 − θ)a + θb ⩾ a ^1−θ b ^θ for 0 < θ < 1 and a, b > 0, we get

(1 − θ)2 ^{( 1−θ)q} + θ2 ^{θ q} ⩾ 2 ^{( 1−θ)}

^q+θ

^q ⩾ 2 ^{q/ 2} ,

where we have also used that (1 − θ) ² + θ ² = 2(θ − ₂ ¹ ) ² + ¹ ₂ ⩾ ₂ ¹ . This implies the lower estimate

J ^q ⩾ 2 ^{−q/ 2} (θ(1 − θ)q log 2) ^{− 1} and completes the proof.

Note that if q = ∞, then sup

m∈Z

{2 ^{−θ m} min (1, 2 ^m )} = 1.

3.3. Theorem. Let (A 0 , A ₁ ) be a pair of p-normed quasi-Banach spaces, let 0 < θ < 1, 0 <

q ⩽ ∞ and put r = min(p, q). Then

∥a∥ ^(A

,A

)

⩽ 2 ^1/4 (θ(1 − θ)r log 2) ^{− 1/r} ∥a∥ ^(A

,A

)

.

Proof. First note that if u ∈ A 0 ∩ A 1 and n, m ∈ Z, then K _p (2 ⁿ , u ) ⩽ min(1, 2 ^n−m )J(2 ^m , u ).

Besides, since r ⩽ p, the functional K ^p (t, ⋅) is also an r-norm.

Given any a ∈ (A 0 , A ₁ ) ^θ ,q;J and any J-representation a = ∑ ^{∞ m=−∞} u _m of a, we have K p (2 ⁿ , a ) ^r ⩽ ∑ ^∞

m=−∞

K p (2 ⁿ , u m ) ^r ⩽ ∑ ^∞

m=−∞

min (1, 2 ^n−m ) ^r J (2 ^m , u m ) ^r

= ∑ ^∞

m=−∞

min (1, 2 ^−m ) ^r J (2 ^m+n , u _m+n ) ^r . Now, if r = q ⩽ p, using ( 1) and Lemma 3.2, we derive

∥a∥ ^(A

,A

)

⩽ ( ∑ ^∞

n=−∞

(2 ^{−θ n} K _p (2 ⁿ , a )) ^q ) ^1/q

⩽ ( ∑ ^∞

n=−∞

2 ^{−θ n q}

∞

∑

m=−∞

min (1, 2 ^−m ) ^q J (2 ^m+n , u _m+n ) ^q ) ^1/q

= ( ∑ ^∞

m=−∞

2 ^{θ m q} min (1, 2 ^−m ) ^q ∑ ^∞

n=−∞

2 ^{−θ (m+n)q} J (2 ^m+n , u _m+n ) ^q ) ^1/q

= ( ∑ ^∞

m=−∞

2 ^{θ m q} min (1, 2 ^−m ) ^q ) ^1/q ( ∑ ^∞

ν=−∞

2 ^{−θ ν q} J (2 ^ν , u ν ) ^q ) ^1/q

⩽ 2 ^1/4 (θ(1 − θ)q log 2) ^{− 1/q} ( ∑ ^∞

ν=−∞

2 ^{−θ ν q} J (2 ^ν , u ν ) ^q ) ^1/q .

If r = p < q, so that 1 < q/p, we derive with the help of the triangle inequality in ℓ q/ p that

∥a∥ ^(A

,A

)

⩽ ( ∑ ^∞

n=−∞

(2 ^{−θ n p} K p (2 ⁿ , a ) ^p ) ^{q/ p} ) ^1/q

⩽ ( ∑ ^∞

n=−∞

( ∑ ^∞

m=−∞

2 ^{θ m p} min (1, 2 ^−m ) ^p 2 ^{−θ (m+n) p} J (2 ^m+n , u _m+n ) ^p ) ^{q/ p} ) ^1/q

⩽ ( ∑ ^∞

m=−∞

2 ^{θ m p} min (1, 2 ^−m ) ^p ( ∑ ^∞

n=−∞

2 ^{−θ (m+n)q} J (2 ^m+n , u _m+n ) ^q ) ^p/q ) ^1/p

⩽ 2 ^1/4 (θ(1 − θ)p log 2) ^{− 1/p} ( ∑ ^∞

ν=−∞

2 ^{−θ ν q} J (2 ^ν , u ν ) ^q ) ^1/q . Consequently,

∥a∥ ^(A

,A

)

⩽ 2 ^1/4 (θ(1 − θ)r log 2) ^{− 1/r} ∥a∥ ^(A

,A

)

.

In the case of a pair of Banach spaces with p = 1, K(t, ⋅) is a norm and we can establish Theorems 3.1 and 3.3 working directly with K (t, ⋅).

We obtain the following result.

3.4. Corollary. Let (A 0 , A ₁ ) be a pair of Banach spaces, let 0 < θ < 1 and 1 ⩽ q ⩽ ∞. Then 1

2 ⁴ ∥a∥ ^(A

,A

)

⩽ θ(1 − θ)∥a∥ ^(A

,A

)

⩽ 2 ^1/4

log 2 ∥a∥ ^(A

,A

)

.

Comparing the estimates given in Theorems 3.1 and 3.3 and focusing on the exponent of the term in θ , we observe that if q < 1, then (θ(1 − θ)) ^1/q decreases faster than θ (1 − θ) as θ → 0 or θ → 1. We finish the paper by showing that the exponent of θ(1 − θ) in Theorem 3.1 can be improved if the p-normed couple (A 0 , A ₁ ) is a mutually closed pair, that is

A j = {a ∈ A 0 + A 1 ∶ sup

m∈ Z

{2 ^{−m j} K (2 ^m , a )} < ∞} for j = 0, 1.

The proof is based on the strong fundamental lemma. As usual, we write (A 0 + A 1 ) ^○ for the closure of A ₀ ∩ A 1 in A ₀ + A 1 .

3.5. Theorem. Let (A 0 , A ₁ ) be a mutually closed pair of p-normed quasi-Banach spaces.

Let 0 < θ < 1 and 0 < q ⩽ p. Then there is a constant c ^p ,q depending only on p and q such that

∥a∥ ^(A

,A

)

⩽ c ^p ,q (θ(1 − θ)) ^1/q ∥a∥ ^(A

,A

)

.

Proof. According to [14, Theorem 3.2], there is a constant d _{p ,q} depending only on p and q, such that for any a ∈ (A 0 + A 1 ) ^○ there is (u ^ν ) ⊆ A 0 ∩ A 1 such that a = ∑ ^{∞ ν=−∞} u _ν and

( ∑ ^∞

ν=−∞

( min(1, t/2 ^ν )J(2 ^ν , u ν )) ^q ) ^1/q ⩽ d ^p ,q K (t, a), t > 0.

Take any a ∈ (A 0 , A ₁ ) ^θ ,q;K . By the above representation and Lemma 3.2, we derive that

∥a∥ ^(A

,A

)

⩽ ( ∑ ^∞

ν=−∞

(2 ^{−θ ν} J (2 ^ν , u ν )) ^q ) ^1/q

⩽ √

2 (θ(1 − θ)q log 2) ^1/q ( ∑ ^∞

ν=−∞

∞

∑

m=−∞

2 ^{−θ (m−ν)q} min (1, 2 ^m−ν ) ^q 2 ^{−θ ν q} J (2 ^ν , u ν ) ^q ) ^1/q

= √

2 (θ(1 − θ)q log 2) ^1/q ( ∑ ^∞

m=−∞

2 ^{−θ m q}

∞

∑

ν=−∞

min (1, 2 ^m−ν ) ^q J (2 ^ν , u ν ) ^q ) ^1/q

⩽ √

2 (θ(1 − θ)q log 2) ^1/q d _{p ,q} ( ∑ ^∞

m=−∞

2 ^{−θ m q} K (2 ^m , a ) ^q ) ^1/q

= c ^p ,q (θ(1 − θ)) ^1/q ∥a∥ ^(A

,A

)

.

Acknowledgements F. Cobos, L. M. Fernández-Cabrera and T. Kühn have been suppor-

ted in part by the Spanish Ministerio de Economía y Competitividad (MTM2013-42220-P).

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