On the boundedness of pseudo-differential operators on Triebel–Lizorkin and Besov spaces

Park, Bae Jun

doi:10.1016/j.jmaa.2017.12.065

Cited by 12 publications

(17 citation statements)

References 26 publications

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“…It was already proved in that for any

s, m \in R

and

0 < p, t ⩽ \infty

, we have

false∥ T_{[a^{(j)}]} {f ∥}_{F_{p}^{0, t}} ≲ {∥ f ∥}_{F_{p}^{s, t}}, j = 1, 2,

which clearly implies

T_{[a^{(j)}]} : F_{p}^{s_{1}, q} \to F_{p}^{s_{2}, t}, j = 1, 2

for all

0 < q, t ⩽ \infty

and

s_{1}, s_{2} \in R

. For the sake of completeness, a sketch of those bounds is provided in what follows.…”

Section: Proof Of Theoremmentioning

confidence: 80%

“…The proof is based on the paradifferential technique as in . Let

a_{j, k} false(x, ξ false) = \{\begin{matrix} left ϕ_{j} * a (\cdot, ξ) (x) \overset{ϕ_{k}}{true} (ξ) & j, k ⩾ 0 \\ left 0 & o t h e r w i s e . \end{matrix}

Write

\begin{matrix} true \\ right a false(x, ξ false) & left = \sum_{j = 3}^{\infty} \sum_{k = 0}^{j - 3} a_{j, k} (x, ξ) + \sum_{k = 0}^{\infty} \sum_{j = k - 2}^{k + 2} a_{j, k} (x, ξ) + \sum_{k = 3}^{\infty} \sum_{j = 0}^{k - 3} a_{j, k} (x, ξ) \\ left = : a^{false(1 false)} (x, ξ) + a^{false(2 false)} (x, ξ) + a^{false(3 false)} (x, ξ) . \end{matrix}

Note that

a^{(j)} \in {scriptS}_{0, 0}^{m}

for each

j = 1, 2, 3

.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…For

0 ⩽ δ ⩽ ρ < 1

, the boundedness of

T_{[a]} \in O p {scriptS}_{ρ, δ}^{m}

was studied, for example, by Calderón and Vaillancourt in , by Fefferman in , and by Päivärinta and Somersalo in . The operators are bounded on

h^{p}

for

0 < p < \infty

m ⩽ - d false(1 - ρ false) |1 / 2 - 1 / p| .

The author generalized this result to Triebel–Lizorkin and Besov spaces for

0 < ρ < 1

. Theorem Let

0 < ρ < 1

0 < p, q, t ⩽ \infty

, and

s_{1}, s_{2} \in R

.…”

Section: Introductionmentioning

confidence: 99%

“…For

(ρ, δ) \neq (1, 1)

, the operators form a class invariant under taking adjoints, and thus we have

T_{[a]} : S^{'} false(R^{d} false) \to S^{'} false(R^{d} false)

via duality. See for details.…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, it suffices to prove that

∥ T_{[a]} f {true∥}_{F_{p}^{s_{2}, p} false(R^{d} false)} ≲ {∥ f ∥}_{F_{p}^{s_{1}, \infty} false(R^{d} false)} for 0 < p ⩽ 1,

∥ T_{[a]} f {true∥}_{F_{p}^{s_{2}, t} false(R^{d} false)} ≲ {∥ f ∥}_{F_{p}^{s_{1}, p} false(R^{d} false)} for 2 < p ⩽ \infty, 0 < t < 1,

∥ T_{[a]} f {true∥}_{B_{p}^{s_{2}, q} false(R^{d} false)} ≲ {∥ f ∥}_{B_{p}^{s_{1}, q} false(R^{d} false)} for 0 < q ⩽ \infty

due to the embedding

F_{p}^{s, q} ↪ F_{p}^{s, t}

for

q ⩽ t ⩽ \infty

. We will not pursue the case

1 < p < 2

here because it clearly follows from the duality consideration in [, p. 556].…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Boundedness of pseudo‐differential operators of type (0,0) on Triebel–Lizorkin and Besov spaces

Park

2019

Bull. London Math. Soc.

Self Cite

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show abstract

“…It was already proved in that for any

s, m \in R

and

0 < p, t ⩽ \infty

, we have

false∥ T_{[a^{(j)}]} {f ∥}_{F_{p}^{0, t}} ≲ {∥ f ∥}_{F_{p}^{s, t}}, j = 1, 2,

which clearly implies

T_{[a^{(j)}]} : F_{p}^{s_{1}, q} \to F_{p}^{s_{2}, t}, j = 1, 2

for all

0 < q, t ⩽ \infty

and

s_{1}, s_{2} \in R

. For the sake of completeness, a sketch of those bounds is provided in what follows.…”

Section: Proof Of Theoremmentioning

confidence: 80%

“…The proof is based on the paradifferential technique as in . Let

a_{j, k} false(x, ξ false) = \{\begin{matrix} left ϕ_{j} * a (\cdot, ξ) (x) \overset{ϕ_{k}}{true} (ξ) & j, k ⩾ 0 \\ left 0 & o t h e r w i s e . \end{matrix}

Write

\begin{matrix} true \\ right a false(x, ξ false) & left = \sum_{j = 3}^{\infty} \sum_{k = 0}^{j - 3} a_{j, k} (x, ξ) + \sum_{k = 0}^{\infty} \sum_{j = k - 2}^{k + 2} a_{j, k} (x, ξ) + \sum_{k = 3}^{\infty} \sum_{j = 0}^{k - 3} a_{j, k} (x, ξ) \\ left = : a^{false(1 false)} (x, ξ) + a^{false(2 false)} (x, ξ) + a^{false(3 false)} (x, ξ) . \end{matrix}

Note that

a^{(j)} \in {scriptS}_{0, 0}^{m}

for each

j = 1, 2, 3

.…”

Section: Proof Of Theoremmentioning

confidence: 99%

“…For

0 ⩽ δ ⩽ ρ < 1

, the boundedness of

T_{[a]} \in O p {scriptS}_{ρ, δ}^{m}

was studied, for example, by Calderón and Vaillancourt in , by Fefferman in , and by Päivärinta and Somersalo in . The operators are bounded on

h^{p}

for

0 < p < \infty

m ⩽ - d false(1 - ρ false) |1 / 2 - 1 / p| .

The author generalized this result to Triebel–Lizorkin and Besov spaces for

0 < ρ < 1

. Theorem Let

0 < ρ < 1

0 < p, q, t ⩽ \infty

, and

s_{1}, s_{2} \in R

.…”

Section: Introductionmentioning

confidence: 99%

“…For

(ρ, δ) \neq (1, 1)

, the operators form a class invariant under taking adjoints, and thus we have

T_{[a]} : S^{'} false(R^{d} false) \to S^{'} false(R^{d} false)

via duality. See for details.…”

Section: Introductionmentioning

confidence: 99%

“…Furthermore, it suffices to prove that

∥ T_{[a]} f {true∥}_{F_{p}^{s_{2}, p} false(R^{d} false)} ≲ {∥ f ∥}_{F_{p}^{s_{1}, \infty} false(R^{d} false)} for 0 < p ⩽ 1,

∥ T_{[a]} f {true∥}_{F_{p}^{s_{2}, t} false(R^{d} false)} ≲ {∥ f ∥}_{F_{p}^{s_{1}, p} false(R^{d} false)} for 2 < p ⩽ \infty, 0 < t < 1,

∥ T_{[a]} f {true∥}_{B_{p}^{s_{2}, q} false(R^{d} false)} ≲ {∥ f ∥}_{B_{p}^{s_{1}, q} false(R^{d} false)} for 0 < q ⩽ \infty

due to the embedding

F_{p}^{s, q} ↪ F_{p}^{s, t}

for

q ⩽ t ⩽ \infty

. We will not pursue the case

1 < p < 2

here because it clearly follows from the duality consideration in [, p. 556].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Boundedness of pseudo‐differential operators of type (0,0) on Triebel–Lizorkin and Besov spaces

Park

2019

Bull. London Math. Soc.

Self Cite

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show abstract

Pseudo-multipliers and Smooth Molecules on Hermite Besov and Hermite Triebel–Lizorkin Spaces

Naibo

2021

J Fourier Anal Appl

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Fourier multiplier theorems for Triebel–Lizorkin spaces

Park

2018

Math. Z.

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On the boundedness of pseudo-differential operators on Triebel–Lizorkin and Besov spaces

Abstract: Abstract. In this work we show endpoint boundedness properties of pseudo-differential operators of type (ρ, ρ), 0 < ρ < 1, on Triebel-Lizorkin and Besov spaces. Our results are sharp and they also cover operators defined by compound symbols.

Cited by 12 publications

References 26 publications

Boundedness of pseudo‐differential operators of type (0,0) on Triebel–Lizorkin and Besov spaces

Boundedness of pseudo‐differential operators of type (0,0) on Triebel–Lizorkin and Besov spaces

Pseudo-multipliers and Smooth Molecules on Hermite Besov and Hermite Triebel–Lizorkin Spaces

Fourier multiplier theorems for Triebel–Lizorkin spaces

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