Bessel and Flett Potentials associated with Dunkl Operators on $\Bbb R^d$

Salem, Néjib Ben; Garna, A. El; Kallel, Samir

doi:10.4310/maa.2008.v15.n4.a5

Cited by 9 publications

(8 citation statements)

References 13 publications

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“…By Minkowski's inequality,

K false(f_{1} + f_{2} false) false(t false) \leq K f_{1} false(t false) + K f_{2} false(t false)

for all

f_{1}, f_{2} \in L^{p} {true(I R, false| x false|}^{2 k} dx true)

and all

t > 0

, so that K is quasi‐linear. Further, by the following relation

{∥ G_{t}^{k} false(f false) ∥}_{k, r} \leq B false(k, p, r false) {[ω false(t false)]}^{- \frac{δ}{2 k + 1}} {∥ f ∥}_{k, p}

(see Theorem 2.4(ii) of ), for

1 \leq p \leq r

we have

K f false(t false) \leq {[ω false(t false)]}^{- \frac{1}{p}} {∥ f ∥}_{k, p} .

It follows that

\{t : K f false(t false) > s\} \subset \{t : {false[ω (t) false]}^{- \frac{1}{p}} {false∥ f false∥}_{k, p} > s\} = false] 0, b false[,

where

b = [s^{...}

…”

Section: Dunkl–hardy–littlewood Theorem For K‐temperatures and Its Dualmentioning

confidence: 92%

“…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%

See 1 more Smart Citation

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

“…By Minkowski's inequality,

K false(f_{1} + f_{2} false) false(t false) \leq K f_{1} false(t false) + K f_{2} false(t false)

for all

f_{1}, f_{2} \in L^{p} {true(I R, false| x false|}^{2 k} dx true)

and all

t > 0

, so that K is quasi‐linear. Further, by the following relation

{∥ G_{t}^{k} false(f false) ∥}_{k, r} \leq B false(k, p, r false) {[ω false(t false)]}^{- \frac{δ}{2 k + 1}} {∥ f ∥}_{k, p}

(see Theorem 2.4(ii) of ), for

1 \leq p \leq r

we have

K f false(t false) \leq {[ω false(t false)]}^{- \frac{1}{p}} {∥ f ∥}_{k, p} .

It follows that

\{t : K f false(t false) > s\} \subset \{t : {false[ω (t) false]}^{- \frac{1}{p}} {false∥ f false∥}_{k, p} > s\} = false] 0, b false[,

where

b = [s^{...}

…”

Section: Dunkl–hardy–littlewood Theorem For K‐temperatures and Its Dualmentioning

confidence: 92%

“…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%

Some results on generalized Dunkl–Lipschitz spaces

Kallel

2019

Mathematische Nachrichten

Self Cite

View full text Add to dashboard Cite

show abstract

“…Proof From Theorem 3.12 of [6] and Theorem 5.7, the result is proved. Now, we want to extend the Theorem 6.11 for all real α and β.…”

mentioning

confidence: 88%

“…We recall some properties of the k-heat transforms of a measurable function f and we refer for more details to the survey [6] and the references therein.…”

Section: Notationsmentioning

confidence: 99%