Generalized Dunkl-Lipschitz spaces

Kallel, Samir

doi:10.1007/s11868-016-0149-9

Cited by 4 publications

(20 citation statements)

References 23 publications

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“…Now, let us recall the following characterization of k ‐Poisson transform (see ), which will be needed later. Theorem (Characterization of k ‐Poisson transform) Let

p \in [1, \infty]

and let

U (x, t)

be k ‐harmonic on

I R_{+}^{2}

.…”

Section: The Relation Of Generalized Dunkl–lipschitz Spaces To Dunkl–mentioning

confidence: 99%

“…Before giving a central result of this section, we need to recall the following definitions and results (see ). Definition For any

α \geq 0

and

f \in L^{p} {true(I R, false| x false|}^{2 k} dx true)

when

1 \leq p \leq \infty

, the Dunkl–Bessel potential (or the k ‐ Bessel potential )

{scriptJ}_{α}^{k} f

of order α of f is given by

{scriptJ}_{α}^{k} false(f false) : = {scriptB}_{α}^{k} *_{k} f, if α > 0 and {scriptJ}_{0}^{k} false(f false) : = f,

where

{scriptF}_{k} ({scriptB}_{α}^{k}) false(x false) = {(1 + x^{2})}^{- \frac{α}{2}}

x \in I R

.…”

Section: The Relation Of Generalized Dunkl–lipschitz Spaces To Dunkl–mentioning

confidence: 99%

“…(iii)

{scriptB}_{α_{1} + α_{2}}^{k} = {scriptB}_{α_{1}}^{k} *_{k} {scriptB}_{α_{2}}^{k}

, if α 1 ,

α_{2} > 0

. Remark (see , Remark 6.9) We have If

β > 0

, we define

P_{t}^{k} ({scriptB}_{- β}^{k})

as follows

P_{t}^{k} ({scriptB}_{- β}^{k}) false(x false) = P_{t}^{k} ({scriptB}_{2 - β}^{k}) false(x false) + \partial_{t}^{2} P_{t}^{k} ({scriptB}_{2 - β}^{k}) false(x false), when 0 < β < 2,

and for arbitrary

β > 0

by the rule

P_{t}^{k} ({scriptB}_{- β}^{k}) false(x false) = P_{\frac{t}{2}}^{k} ({scriptB}_{- γ}^{k}) *_{k} P_{\frac{t}{2}}^{k} ({scriptB}_{- δ}^{k}) false(x false), whenever γ + δ = β .

β > 0

, we define the k ‐Bessel potential

{scriptJ}_{- β}^{k} f false(x false)...

…”

Section: The Relation Of Generalized Dunkl–lipschitz Spaces To Dunkl–mentioning

confidence: 99%

“…Remark We have: (see [, p. 553])

{∥ \partial_{t}^{\bar{α}} P_{t}^{k} ({scriptB}_{α}^{k}) ∥}_{k, 1} \leq B false(k, α false) t^{α - \bar{α}}, α > 0 .

For

α > 0

p, q \in [1, \infty]

, we set

\land_{α, p, q}^{k} false(I R false) : = \{f \in L^{p} true(I R, {false| x false|}^{2 k} dx true) : A_{p, q}^{k} true(t^{\bar{α} - α} \partial_{t}^{\bar{α}} P_{t}^{k} (f) true) < \infty\} .

The

\land_{α, p, q}^{k}

‐norms are defined by

{∥ f ∥}_{\land_{α, p, q}^{k}} : = {∥ f ∥}_{k, p} + {scriptA}_{p, q}^{k} (t^{\bar{α} - α} \partial_{t}^{\bar{α}} P_{t}^{k} false(f false)) .

…”

Section: The Relation Of Generalized Dunkl–lipschitz Spaces To Dunkl–mentioning

confidence: 99%

“…In , we have introduced and characterized the generalized Dunkl–Lipschitz spaces

\land_{α, p, q}^{k} false(I R false)

, when

α \in I R

and

1 \leq p, q \leq \infty

. Also, for any α,

β \in I R

, the Dunkl–Bessel potential

J_{β}^{k}

is proved to be a topological isomorphism from

\land_{α, p, q}^{k} false(I R false)

onto

\land_{α + β, p, q}^{k} false(I R false)

.…”

Section: Introductionmentioning

confidence: 99%

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Some results on generalized Dunkl–Lipschitz spaces

Kallel

2019

Mathematische Nachrichten

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In this paper we study certain relations between the Dunkl-Sobolev classes of fractional order  , ( ) and the generalized Dunkl-Lipschitz classes ∧ , , ( ). Next, we show a Dunkl-Hardy-Littlewood theorem for -temperatures and its dual. Also, we give some theorems on Dunkl convolutions of Dunkl-Lipschitz functions. Finally, we establish some results concerning the Dunkl transforms of functions and distributions belonging to generalized Dunkl-Lipschitz spaces. K E Y W O R D SDunkl operator, Dunkl transform, generalized Dunkl-Lipschitz space, maximal functions M S C ( 2 0 1 0 ) 26A16, 42A38, 42B25, 46E30 www.mn-journal.org

show abstract

“…Now, let us recall the following characterization of k ‐Poisson transform (see ), which will be needed later. Theorem (Characterization of k ‐Poisson transform) Let

p \in [1, \infty]

and let

U (x, t)

be k ‐harmonic on

I R_{+}^{2}

.…”

Section: The Relation Of Generalized Dunkl–lipschitz Spaces To Dunkl–mentioning

confidence: 99%

“…Before giving a central result of this section, we need to recall the following definitions and results (see ). Definition For any

α \geq 0

and

f \in L^{p} {true(I R, false| x false|}^{2 k} dx true)

when

1 \leq p \leq \infty

, the Dunkl–Bessel potential (or the k ‐ Bessel potential )

{scriptJ}_{α}^{k} f

of order α of f is given by

{scriptJ}_{α}^{k} false(f false) : = {scriptB}_{α}^{k} *_{k} f, if α > 0 and {scriptJ}_{0}^{k} false(f false) : = f,

where

{scriptF}_{k} ({scriptB}_{α}^{k}) false(x false) = {(1 + x^{2})}^{- \frac{α}{2}}

x \in I R

.…”

Section: The Relation Of Generalized Dunkl–lipschitz Spaces To Dunkl–mentioning

confidence: 99%

“…(iii)

{scriptB}_{α_{1} + α_{2}}^{k} = {scriptB}_{α_{1}}^{k} *_{k} {scriptB}_{α_{2}}^{k}

, if α 1 ,

α_{2} > 0

. Remark (see , Remark 6.9) We have If

β > 0

, we define

P_{t}^{k} ({scriptB}_{- β}^{k})

as follows

P_{t}^{k} ({scriptB}_{- β}^{k}) false(x false) = P_{t}^{k} ({scriptB}_{2 - β}^{k}) false(x false) + \partial_{t}^{2} P_{t}^{k} ({scriptB}_{2 - β}^{k}) false(x false), when 0 < β < 2,

and for arbitrary

β > 0

by the rule

P_{t}^{k} ({scriptB}_{- β}^{k}) false(x false) = P_{\frac{t}{2}}^{k} ({scriptB}_{- γ}^{k}) *_{k} P_{\frac{t}{2}}^{k} ({scriptB}_{- δ}^{k}) false(x false), whenever γ + δ = β .

β > 0

, we define the k ‐Bessel potential

{scriptJ}_{- β}^{k} f false(x false)...

…”

Section: The Relation Of Generalized Dunkl–lipschitz Spaces To Dunkl–mentioning

confidence: 99%

“…Remark We have: (see [, p. 553])

{∥ \partial_{t}^{\bar{α}} P_{t}^{k} ({scriptB}_{α}^{k}) ∥}_{k, 1} \leq B false(k, α false) t^{α - \bar{α}}, α > 0 .

For

α > 0

p, q \in [1, \infty]

, we set

\land_{α, p, q}^{k} false(I R false) : = \{f \in L^{p} true(I R, {false| x false|}^{2 k} dx true) : A_{p, q}^{k} true(t^{\bar{α} - α} \partial_{t}^{\bar{α}} P_{t}^{k} (f) true) < \infty\} .

The

\land_{α, p, q}^{k}

‐norms are defined by

{∥ f ∥}_{\land_{α, p, q}^{k}} : = {∥ f ∥}_{k, p} + {scriptA}_{p, q}^{k} (t^{\bar{α} - α} \partial_{t}^{\bar{α}} P_{t}^{k} false(f false)) .

…”

Section: The Relation Of Generalized Dunkl–lipschitz Spaces To Dunkl–mentioning

confidence: 99%

“…In , we have introduced and characterized the generalized Dunkl–Lipschitz spaces

\land_{α, p, q}^{k} false(I R false)

, when

α \in I R

and

1 \leq p, q \leq \infty

. Also, for any α,

β \in I R

, the Dunkl–Bessel potential

J_{β}^{k}

is proved to be a topological isomorphism from

\land_{α, p, q}^{k} false(I R false)

onto

\land_{α + β, p, q}^{k} false(I R false)

.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Some results on generalized Dunkl–Lipschitz spaces

Kallel

2019

Mathematische Nachrichten

Self Cite

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show abstract

Lipschitz and Triebel–Lizorkin spaces associated with the Dunkl operators on $$\mathbb {R}^d$$ R d

Kallel

2019

Positivity

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Duality of generalized Dunkl-Lipschitz spaces

Kallel

2017

Acta Mathematica Scientia

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Generalized Dunkl-Lipschitz spaces

Abstract: This paper deals with generalized Lipschitz spaces ∧

Cited by 4 publications

References 23 publications

Some results on generalized Dunkl–Lipschitz spaces

Some results on generalized Dunkl–Lipschitz spaces

Lipschitz and Triebel–Lizorkin spaces associated with the Dunkl operators on $$\mathbb {R}^d$$ R d

Duality of generalized Dunkl-Lipschitz spaces

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