Relations between functional norms of a non-negative function and its square root on the positive cone of Besov and Triebel-Lizorkin spaces

Relations between functional norms of a non-negative function and its square root on the positive cone of

Besov and Triebel-Lizorkin spaces

Lubomir T. Dechevsky ¹ and Niklas Grip ²

Outline

We will derive direct estimates of functionals depending on f by respective functionals depending on √

f and inverse estimates, where the role of f and √

f is reversed.

Organization of the talk

1 Relations between the inner-product wavelet coefficients of f and √

f

Sobolev norms, variable metric index, fixed integer smoothness index

We will derive direct estimates of functionals depending on f by respective functionals depending on √

f and inverse estimates, where the role of f and √

f is reversed.

Organization of the talk

1 Relations between the inner-product wavelet coefficients of f and √ f

2 Sobolev norms, variable metric index, fixed integer smoothness index

3 Besov and Triebel-Lizorkin spaces, fixed metric index, variable (fractional) smoothness index Direct results

Inverse results

4 Interpolation results

Outline

We will derive direct estimates of functionals depending on f by respective functionals depending on √

f and inverse estimates, where the role of f and √

f is reversed.

Organization of the talk

1 Relations between the inner-product wavelet coefficients of f and √

f

Sobolev norms, variable metric index, fixed integer smoothness index

We will derive direct estimates of functionals depending on f by respective functionals depending on √

f and inverse estimates, where the role of f and √

f is reversed.

Organization of the talk

1 Relations between the inner-product wavelet coefficients of f and √ f

2 Sobolev norms, variable metric index, fixed integer smoothness index

3 Besov and Triebel-Lizorkin spaces, fixed metric index, variable (fractional) smoothness index Direct results

Inverse results

4 Interpolation results

Relations between inner-product wavelet coeffs of f and√ f

Relations between the inner-product wavelet coefficients of f and √

f

Background: Different statistical wavelet approximations [PD,PV] of a distribution density f (defined on R), that preserve the crucial properties

f (x) ≥ 0 and Z

R

f (x) dx = 1.

Relations between the inner-product wavelet coefficients of f and √

f

Background: Different statistical wavelet approximations [PD,PV] of a distribution density f (defined on R), that preserve the crucial properties

f (x) ≥ 0 and Z

R

f (x) dx = 1.

It was also proved in [PD] that this approximation achieves certain assymptotycally minimax rates. However, these rates were in terms of functional norms depending on the wavelet coefficients of √

f rather than the ones of f .

[PD] S. Penev, and L. Dechevsky, J. Nonparametr. Statist. 7, 365–394 (1997), ISSN 1048-5252.

[PV] A. Pinheiro, and B. Vidakovic, Comput. Statist. 25, 399–415 (1997), ISSN 0943-4062.

Relations between inner-product wavelet coeffs of f and√ f

Assumptions and notation:

ϕ is real-valued, compactly supported and bounded, ϕ j ,k (x) = 2

ϕ(2 ^j x − k), x ∈ R, j ∈ Z, k ∈ Z.

No assumptions about (bi)orthogonality of the wavelet basis, thus

allowing also for non-biorthogonal wavelet constructions, such as those

Assumptions and notation:

ϕ is real-valued, compactly supported and bounded, ϕ j ,k (x) = 2

ϕ(2 ^j x − k), x ∈ R, j ∈ Z, k ∈ Z.

No assumptions about (bi)orthogonality of the wavelet basis, thus allowing also for non-biorthogonal wavelet constructions, such as those considered, e.g. in [DPa,DPb].

[DPa] L. T. Dechevsky, and S. I. Penev, Stochastic Anal. Appl. 15, 187–215 (1997), ISSN 0736-2994.

[DPb] L. T. Dechevsky, and S. I. Penev, Stochastic Anal. Appl. 16, 423–462 (1998), ISSN 0736-2994.

.

Relations between inner-product wavelet coeffs of f and√ f

Assumptions and notation:

ϕ is real-valued, compactly supported and bounded, ϕ j ,k (x) = 2

ϕ(2 ^j x − k), x ∈ R, j ∈ Z, k ∈ Z.

No assumptions about (bi)orthogonality of the wavelet basis, thus

allowing also for non-biorthogonal wavelet constructions, such as those

Assumptions and notation:

ϕ is real-valued, compactly supported and bounded, ϕ j ,k (x) = 2

ϕ(2 ^j x − k), x ∈ R, j ∈ Z, k ∈ Z.

No assumptions about (bi)orthogonality of the wavelet basis, thus allowing also for non-biorthogonal wavelet constructions, such as those considered, e.g. in [DPa,DPb].

[DPa] L. T. Dechevsky, and S. I. Penev, Stochastic Anal. Appl. 15, 187–215 (1997), ISSN 0736-2994.

[DPb] L. T. Dechevsky, and S. I. Penev, Stochastic Anal. Appl. 16, 423–462 (1998), ISSN 0736-2994.

.

Relations between inner-product wavelet coeffs of f and√ f

Proposition (Direct results) Assume that

f ∈ L 1 (R) ∩ L ∞ ((R))

|hf , ϕ ^j ,k i| ≥ d or |hf , ϕ ^j ,k i| = 0 (i.e. hard thresholded coeffs) (1) Then |hf , ϕ ^j ,k i| ≤ c 5

D √

f , ϕ _j,k E 

, where 0 < c 5 < ∞, c 5 depends on

kf k _∞ and d.

Proposition (Direct results) Assume that

f ∈ L 1 (R) ∩ L ∞ ((R))

|hf , ϕ ^j ,k i| ≥ d or |hf , ϕ ^j ,k i| = 0 (i.e. hard thresholded coeffs) (1) Then |hf , ϕ ^j ,k i| ≤ c 5

D √

f , ϕ _j,k E 

, where 0 < c 5 < ∞, c 5 depends on kf k _∞ and d.

Proposition (Inverse results) Under the assumptions (1), 

D √ f , ϕ j ,k

E 

 ≤ c ⁶ 2 ⁻

|hf , ϕ ^j ,k i|

where 0 < c 6 < ∞ and c ⁶ depends on kf k _∞ , d, kϕk _∞ and on |supp ϕ|.

If, additionally, it is also known that f | supp ϕ

≥ c ^f > 0 (c f = c f (j, k)), then 

D √ f , ϕ j ,k

E 

 ≤ c 7 ≤ |hf , ϕ ^j ,k i| holds, where 0 < c 7 < ∞ and c 7

depends on c _f .

Relations between inner-product wavelet coeffs of f and√ f

Proposition (Direct results) Assume that

f ∈ L 1 (R) ∩ L ∞ ((R))

|hf , ϕ ^j ,k i| ≥ d or |hf , ϕ ^j ,k i| = 0 (i.e. hard thresholded coeffs) (1) Then |hf , ϕ ^j ,k i| ≤ c 5

D √

f , ϕ _j,k E 

, where 0 < c ₅ < ∞, c 5 depends on

kf k _∞ and d.

Proposition (Direct results) Assume that

f ∈ L 1 (R) ∩ L ∞ ((R))

|hf , ϕ ^j ,k i| ≥ d or |hf , ϕ ^j ,k i| = 0 (i.e. hard thresholded coeffs) (1) Then |hf , ϕ ^j ,k i| ≤ c 5

D √

f , ϕ _j,k E 

, where 0 < c 5 < ∞, c 5 depends on kf k _∞ and d.

Proposition (Inverse results) Under the assumptions (1), 

D √ f , ϕ j ,k

E 

 ≤ c ⁶ 2 ⁻

|hf , ϕ ^j ,k i|

where 0 < c 6 < ∞ and c ⁶ depends on kf k _∞ , d, kϕk _∞ and on |supp ϕ|.

If, additionally, it is also known that f | supp ϕ

≥ c ^f > 0 (c f = c f (j, k)), then 

D √ f , ϕ j ,k

E 

 ≤ c 7 ≤ |hf , ϕ ^j ,k i| holds, where 0 < c 7 < ∞ and c 7

depends on c _f .

Relations between inner-product wavelet coeffs of f and√ f

Proposition (Direct results) Assume that

f ∈ L 1 (R) ∩ L ∞ ((R))

|hf , ϕ ^j ,k i| ≥ d or |hf , ϕ ^j ,k i| = 0 (i.e. hard thresholded coeffs) (1) Then |hf , ϕ ^j ,k i| ≤ c 5

D √

f , ϕ _j,k E 

, where 0 < c 5 < ∞, c 5 depends on

kf k _∞ and d.

Proposition (Direct results) Assume that

f ∈ L 1 (R) ∩ L ∞ ((R))

|hf , ϕ ^j ,k i| ≥ d or |hf , ϕ ^j ,k i| = 0 (i.e. hard thresholded coeffs) (1) Then |hf , ϕ ^j ,k i| ≤ c 5

D √

f , ϕ _j,k E 

, where 0 < c 5 < ∞, c 5 depends on kf k _∞ and d.

Proposition (Inverse results) Under the assumptions (1), 

D √ f , ϕ j ,k

E 

 ≤ c ⁶ 2 ⁻

|hf , ϕ ^j ,k i|

where 0 < c 6 < ∞ and c ⁶ depends on kf k _∞ , d, kϕk _∞ and on |supp ϕ|.

If, additionally, it is also known that f | supp ϕ

≥ c ^f > 0 (c f = c f (j, k)), then 

D √ f , ϕ j ,k

E 

 ≤ c 7 ≤ |hf , ϕ ^j ,k i| holds, where 0 < c 7 < ∞ and c 7

depends on c _f .

Relations between inner-product wavelet coeffs of f and√ f

Proposition (Direct results) Assume that

f ∈ L 1 (R) ∩ L ∞ ((R))

|hf , ϕ ^j ,k i| ≥ d or |hf , ϕ ^j ,k i| = 0 (i.e. hard thresholded coeffs) (1) Then |hf , ϕ ^j ,k i| ≤ c 5

D √

f , ϕ _j,k E 

, where 0 < c 5 < ∞, c 5 depends on

kf k _∞ and d.

Proposition (Direct results) Assume that

f ∈ L 1 (R) ∩ L ∞ ((R))

|hf , ϕ ^j ,k i| ≥ d or |hf , ϕ ^j ,k i| = 0 (i.e. hard thresholded coeffs) (1) Then |hf , ϕ ^j ,k i| ≤ c 5

D √

f , ϕ _j,k E 

, where 0 < c 5 < ∞, c 5 depends on kf k _∞ and d.

Proposition (Inverse results) Under the assumptions (1), 

D √ f , ϕ j ,k

E 

 ≤ c ⁶ 2 ⁻

|hf , ϕ ^j ,k i|

where 0 < c 6 < ∞ and c ⁶ depends on kf k _∞ , d, kϕk _∞ and on |supp ϕ|.

If, additionally, it is also known that f | supp ϕ

≥ c ^f > 0 (c f = c f (j, k)), then 

D √ f , ϕ j ,k

E 

 ≤ c 7 ≤ |hf , ϕ ^j ,k i| holds, where 0 < c ₇ < ∞ and c 7

depends on c _f .

Relations between inner-product wavelet coeffs of f and√ f

Proposition (Direct results) Assume that

f ∈ L 1 (R) ∩ L ∞ ((R))

|hf , ϕ ^j ,k i| ≥ d or |hf , ϕ ^j ,k i| = 0 (i.e. hard thresholded coeffs) (1) Then |hf , ϕ ^j ,k i| ≤ c 5

D √

f , ϕ _j,k E 

, where 0 < c 5 < ∞, c 5 depends on

kf k _∞ and d.

Sobolev norms with variable metric indices and fixed constant integer smoothness index

Definition

(Inhomogeneous) Sobolev spaces with metric index 0 < p ≤ ∞ and smoothness index m = 0, 1, 2, . . . can be defined via their (quasi)norms

kf k W

def = kf k L

, 0 < p ≤ ∞,

and

kf k W

def = kf k L

+ f

(m) _L

p

, 0 < p ≤ ∞, m = 1, 2, . . .

Variable metric index, fixed integer smoothness index

Sobolev norms with variable metric indices and fixed constant integer smoothness index

Definition

(Inhomogeneous) Sobolev spaces with metric index 0 < p ≤ ∞ and smoothness index m = 0, 1, 2, . . . can be defined via their (quasi)norms

def

Theorem (Direct results)

For 0 < p ≤ ∞ and m = 0, 1, 2, . . . kf k W

≤ c 2 (p, m)

√ f

2 W

. Theorem (Inverse results)

Let 0 < p ≤ ∞. If m = 0, then the above result is also inverse:

√ f p =

√ f

1/2 p/2 . If m = 1, 2, . . ., assume additionally that √ ¹

f ∈ W q ^{m −1} for p ≤ q ≤ ∞.

Then

√ f

W

≤ c(p, m)

√ 1 f

w

· kf k W

with 1 r = 1

p − 1

q .

Variable metric index, fixed integer smoothness index

Theorem (Direct results)

For 0 < p ≤ ∞ and m = 0, 1, 2, . . . kf k W

≤ c 2 (p, m)

√ f

2 W

. Theorem (Inverse results)

Let 0 < p ≤ ∞. If m = 0, then the above result is also inverse:

Theorem (Direct results)

For 0 < p ≤ ∞ and m = 0, 1, 2, . . . kf k W

≤ c 2 (p, m)

√ f

2 W

. Theorem (Inverse results)

Let 0 < p ≤ ∞. If m = 0, then the above result is also inverse:

√ f p =

√ f

1/2 p/2 . If m = 1, 2, . . ., assume additionally that √ ¹

f ∈ W q ^{m −1} for p ≤ q ≤ ∞.

Then

√ f

W

≤ c(p, m)

√ 1 f

w

· kf k W

with 1 r = 1

p − 1

q .

Variable metric index, fixed integer smoothness index

Theorem (Direct results)

For 0 < p ≤ ∞ and m = 0, 1, 2, . . . kf k W

≤ c 2 (p, m)

√ f

2 W

. Theorem (Inverse results)

Let 0 < p ≤ ∞. If m = 0, then the above result is also inverse:

Theorem (Direct results)

For 0 < p ≤ ∞ and m = 0, 1, 2, . . . kf k W

≤ c 2 (p, m)

√ f

2 W

. Theorem (Inverse results)

Let 0 < p ≤ ∞. If m = 0, then the above result is also inverse:

√ f p =

√ f

1/2 p/2 . If m = 1, 2, . . ., assume additionally that √ ¹

f ∈ W q ^{m −1} for p ≤ q ≤ ∞.

Then

√ f

W

≤ c(p, m)

√ 1 f

w

· kf k W

with 1 r = 1

p − 1

q .

Variable metric index, fixed integer smoothness index

Theorem (Direct results)

For 0 < p ≤ ∞ and m = 0, 1, 2, . . . kf k W

≤ c 2 (p, m)

√ f

2 W

. Theorem (Inverse results)

Let 0 < p ≤ ∞. If m = 0, then the above result is also inverse:

Besov and Triebel-Lizorkin spaces with fixed metric index and variable, possibly fractional smoothness index

Direct results

Using pointwise multipliers to preserve the metric, we have derived four theorems of the kind

“  √

f ∈ B p,q ^s (or F _p,q ^s ) + additional assumptions 

⇒ f ∈ B p,q ^s (or F _p,q ^s )”.

Theorem

Let −∞ < s < ∞, 0 < p ≤ ∞, 0 < q ≤ ∞ and σ > max n

s, _min{p,1} ¹ − 1 − s o . If √

f ∈ B _∞,∞ ^σ ∩ B p,q ^s (which is possible only for a density with bounded support), then

f ∈ B p,q ^s and kf k B

≤ c

√ f B

∞,∞

√ f

B

.

Fixed metric index, variable (fractional) smoothness index Direct results

Besov and Triebel-Lizorkin spaces with fixed metric index and variable, possibly fractional smoothness index

Direct results

Using pointwise multipliers to preserve the metric, we have derived four theorems of the kind

“  √

f ∈ B p,q ^s (or F _p,q ^s ) + additional assumptions 

⇒ f ∈ B p,q ^s (or F _p,q ^s )”.

Theorem

Let −∞ < s < ∞, 0 < p < ∞, 0 < q < ∞ and σ > max



s, ¹

min { ^p,q,1 } − 1 − s

 . If √

f ∈ B _∞,∞ ^σ ∩ F p,q ^s , , then

f ∈ F p,q ^s and kf k F

≤ c

√ f B

√ f

F

.

Underlining denotes what is different from the previous theorem.

Fixed metric index, variable (fractional) smoothness index Direct results

Theorem

Let −∞ < s < ∞, 0 < p < ∞, 0 < q < ∞ and σ > max



s, ¹

min { ^p,q,1 } − 1 − s

 . If √

f ∈ B _∞,∞ ^σ ∩ F p,q ^s , , then

√ √

Theorem

Let −∞ < s < ∞, 0 < p < ∞, 0 < q < ∞ and σ > max



s, ¹

min { ^p,q,1 } − 1 − s

 . If √

f ∈ B _∞,∞ ^σ ∩ F p,q ^s , , then

f ∈ F p,q ^s and kf k F

≤ c

√ f B

√ f

F

.

Underlining denotes what is different from the previous theorem.

Fixed metric index, variable (fractional) smoothness index Direct results

Theorem

Let −∞ < s < ∞, 0 < p < ∞, 0 < q < ∞ and σ > max



s, ¹

min { ^p,q,1 } − 1 − s

 . If √

f ∈ B _∞,∞ ^σ ∩ F p,q ^s , , then

√ √

The next direct result is related to the continuity of f and √

f in view of the range of s and p for which Sobolev-type embeddings hold true.

Theorem

Let 0 < p ≤ ∞ and either 1 < q ≤ ∞, s > 1/p or 0 < q ≤ 1, s ≥ 1/p. If

√ f ∈ B p,q ^s , then f ∈ B p,q ^s and kf k B

≤ c

√ f

2 B

. Theorem

Let either 0 < p < q ≤ ∞, s > ¹ p or 0 < p ≤ q < ∞, s > ¹ ₂ 

1 p + ¹ _q 

. If √

f ∈ F p,q ^s , then f ∈ F p,q ^s and kf k F

≤ c

√ f

2 F

.

Here, again, we point out with underlining what is different from the

previous theorem.

Fixed metric index, variable (fractional) smoothness index Direct results

The next direct result is related to the continuity of f and √

f in view of the range of s and p for which Sobolev-type embeddings hold true.

Theorem

Let 0 < p ≤ ∞ and either 1 < q ≤ ∞, s > 1/p or 0 < q ≤ 1, s ≥ 1/p. If

√ f ∈ B p,q ^s , then f ∈ B p,q ^s and kf k B

≤ c

√ f

2

B

.

The next direct result is related to the continuity of f and √

f in view of the range of s and p for which Sobolev-type embeddings hold true.

Theorem

Let 0 < p ≤ ∞ and either 1 < q ≤ ∞, s > 1/p or 0 < q ≤ 1, s ≥ 1/p. If

√ f ∈ B p,q ^s , then f ∈ B p,q ^s and kf k B

≤ c

√ f

2 B

. Theorem

Let either 0 < p < q ≤ ∞, s > ¹ p or 0 < p ≤ q < ∞, s > ¹ ₂ 

1 p + ¹ _q 

. If √

f ∈ F p,q ^s , then f ∈ F p,q ^s and kf k F

≤ c

√ f

2 F

.

Here, again, we point out with underlining what is different from the

previous theorem.

Fixed metric index, variable (fractional) smoothness index Direct results

The next direct result is related to the continuity of f and √

f in view of the range of s and p for which Sobolev-type embeddings hold true.

Theorem

Let 0 < p ≤ ∞ and either 1 < q ≤ ∞, s > 1/p or 0 < q ≤ 1, s ≥ 1/p. If

√ f ∈ B p,q ^s , then f ∈ B p,q ^s and kf k B

≤ c

√ f

2

B

.

Besov and Triebel-Lizorkin spaces with fixed metric index and variable, possibly fractional smoothness index

Inverse results

Using modifications of the ideas behind the proofs of the direct results, we obtained the following four corresponding inverse results of the kind

“ f ∈ B p,q ^s (or F _p,q ^s ) + additional assumptions
⇒ √

f ∈ B p,q ^s (or F _p,q ^s )”.

Note, however, that the inverse estimates depend essentially on the behaviour of √ ¹

f : namely, in which Besov and Triebel-Lizorkin spaces it is a pointwise mutliplier. Thus we also get conditions on derivatives of the function 1/pf (x), which here is understood to vanish when f (x) = 0.

Theorem

Let 1 ≤ p ≤ ∞, 0 < q ≤ ∞, m ∈ N 0

def = N ∪ {0}, s > m. If

 √ 1 f

 (µ)

∈ B _∞,q ^{s −m} , µ = 0, 1, . . . , m and if f ∈ B p,q ^s , then ( √

f ) ^(m) ∈ B p,q ^{s −m}

and (

√ f ) ^(m) B

≤ c m P m µ=0

 √ 1 f

 (µ) _B

∞,∞

!

· kf k B

Fixed metric index, variable (fractional) smoothness index Inverse results

Besov and Triebel-Lizorkin spaces with fixed metric index and variable, possibly fractional smoothness index

Inverse results

Using modifications of the ideas behind the proofs of the direct results, we obtained the following four corresponding inverse results of the kind

“ f ∈ B p,q ^s (or F _p,q ^s ) + additional assumptions
⇒ √

f ∈ B p,q ^s (or F _p,q ^s )”.

Note, however, that the inverse estimates depend essentially on the behaviour of √ ¹

f : namely, in which Besov and Triebel-Lizorkin spaces it is

Besov and Triebel-Lizorkin spaces with fixed metric index and variable, possibly fractional smoothness index

Inverse results

Using modifications of the ideas behind the proofs of the direct results, we obtained the following four corresponding inverse results of the kind

“ f ∈ B p,q ^s (or F _p,q ^s ) + additional assumptions
⇒ √

f ∈ B p,q ^s (or F _p,q ^s )”.

Note, however, that the inverse estimates depend essentially on the behaviour of √ ¹

f : namely, in which Besov and Triebel-Lizorkin spaces it is a pointwise mutliplier. Thus we also get conditions on derivatives of the function 1/pf (x), which here is understood to vanish when f (x) = 0.

Theorem

Let 1 ≤ p ≤ ∞, 0 < q ≤ ∞, m ∈ N 0

def = N ∪ {0}, s > m. If

 √ 1 f

 (µ)

∈ B _∞,q ^{s −m} , µ = 0, 1, . . . , m and if f ∈ B p,q ^s , then ( √

f ) ^(m) ∈ B p,q ^{s −m}

and (

√ f ) ^(m) B

≤ c m P m µ=0

 √ 1 f

 (µ) _B

∞,∞

!

· kf k B

Fixed metric index, variable (fractional) smoothness index Inverse results

Besov and Triebel-Lizorkin spaces with fixed metric index and variable, possibly fractional smoothness index

Inverse results

Using modifications of the ideas behind the proofs of the direct results, we obtained the following four corresponding inverse results of the kind

“ f ∈ B p,q ^s (or F _p,q ^s ) + additional assumptions
⇒ √

f ∈ B p,q ^s (or F _p,q ^s )”.

Note, however, that the inverse estimates depend essentially on the behaviour of √ ¹

f : namely, in which Besov and Triebel-Lizorkin spaces it is

Theorem

Let 0 < p ≤ ∞, 0 < q ≤ ∞, m ∈ N 0 and σ > s > m − a + _min{p,1} ¹ , s > m. If 

√ 1 f

 (µ)

∈ B _∞,∞ ^σ−m , µ = 0, 1, . . . , m and if f ∈ B p,q ^s , then ( √

f ) ^(m) ∈ B p,q ^{s −m} and

(

√ f ) ^(m)

B

≤ c m,p P m µ=0

 √ 1 f

 (µ) B

!

· kf k B

Fixed metric index, variable (fractional) smoothness index Inverse results

Theorem

Let 0 < p ≤ ∞, 0 < q ≤ ∞, m ∈ N 0 and σ > s > m + max n

1

min{p,q} , _min{p,q,1} ¹ − 1 o . If 

√ 1 f

 (µ)

∈ B _∞,∞ ^{s −m} , µ = 0, 1, . . . , m and if f ∈ F ^s , then ( √

f ) ^(m) ∈ F ^{s −m} and

Theorem

Let 0 < p ≤ ∞, m ∈ N 0 and

either 1 < q ≤ ∞, s > m + 1/p or 0 < q ≤ 1, s ≥ m + 1/p. If

 √ 1 f

 (µ)

∈ B p,q ^{s −m} , µ = 0, 1, . . . , m and if f ∈ B p,q ^s , then ( √

f ) ^(m) ∈ B p,q ^{s −m}

and (

√ f ) ^(m) _B

p,q

≤ c m,p P m µ=0

 √ 1 f

 (µ) _B

!

· kf k B

.

Fixed metric index, variable (fractional) smoothness index Inverse results

Theorem

Let m ∈ N 0 , 0 < p ≤ ∞, 0 < q ≤ ∞, s > _min{p,q} ¹ . If 

√ 1 f

 (µ)

∈ F p,q ^{s −m} , µ = 0, 1, . . . , m and if f ∈ F p,q ^s , then ( √

f ) ^(m) ∈ F p,q ^{s −m} and

Interpolation results

from real and complex interpolation of the so far presented estimates

Our next aim is to generalize the direct results in the first theorem of this talk for the Besov and Triebel–Lizorkin space scales for a continual (fractional) range of the smoothness parameter s. This results in . . . Theorem (Direct results)

Under the assumptions of Theorem 1, suppose that 1 ≤ p ≤ ∞. Then for m ∈ N, 1 ≤ p ≤ ∞, 0 < q ≤ ∞ and 0 < s < m,

it holds that

kf k B

≤ c(p, q, s, m) ·

√ f

2 B

.

If, moreover, p belongs to the more constrained range 1 < p < ∞, then kf k F

≤ c(p) ^1−θ · c(p, m) ^θ ·

√ f

2

F

.

Interpolation results

Interpolation results

from real and complex interpolation of the so far presented estimates

Our next aim is to generalize the direct results in the first theorem of this talk for the Besov and Triebel–Lizorkin space scales for a continual (fractional) range of the smoothness parameter s. This results in . . . Theorem (Direct results)

Under the assumptions of Theorem 1, suppose that 1 ≤ p ≤ ∞. Then for

Similar to Theorems 2 and 13, the results of Theorem 3 can also be upgraded using the real and complex interpolation methods.

Theorem (Inverse results)

Research in progress. . .