Fourier Multipliers on a Vector-Valued Function Space

Park, Bae Jun

doi:10.1007/s00365-021-09526-5

Cited by 5 publications

(5 citation statements)

References 20 publications

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“…Then m and 1 m are Fourier multipliers, see [22,Section 1.5.2] (B-scale), [22,Theorem,p. 75] (F -scale, p < ∞) and [15] (F -scale, p = ∞), and it follows that both (φ k ) k∈N 0 and the resolution of unity ( 1 m φ k ) k∈N 0 define equivalent (quasi-)norms.…”

Section: Function Spacesmentioning

confidence: 99%

See 1 more Smart Citation

Convolution inequalities for Besov and Triebel--Lizorkin spaces, and applications to convolution semigroups

Kühn¹,

Schilling²

2021

Preprint

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We establish convolution inequalities for Besov spaces B s p,q and Triebel-Lizorkin spaces F s p,q . As an application, we study the mapping properties of convolution semigroups, considered as operators on the function spaces A s p,q , A ∈ {B, F }. Our results apply to a wide class of convolution semigroups including the Gauß-Weierstraß semigroup, stable semigroups and heat kernels for higher-order powers of the Laplacian (−∆) m , and we can derive various caloric smoothing estimates.The convolution f * g of two functions f, g ∶ R n → R is defined byprovided that the integral exists. It is well known that the convolution f * g inherits the regularity of both f and g, in the sense that if, say,In this article, we study the regularizing properties of the convolution in the scales of Besov spaces B s p,q and Triebel-Lizorkin spaces F s p,q . Having in mind that the parameters s and p describe the regularity, resp., integrability properties of f ∈ A s p,q , A ∈ {B, F }, it is natural to expect that the implication f ∈ A s p1,q1 , g ∈ A u

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Section: Function Spacesmentioning

confidence: 99%

“…Since f and g play symmetric roles, it is enough to consider the case p 1 = ∞. Then, by (15), p = ∞ and p 2 = 1, i.e. we need to prove that…”

Section: Theorem Letmentioning

confidence: 99%

Convolution inequalities for Besov and Triebel--Lizorkin spaces, and applications to convolution semigroups

Kühn¹,

Schilling²

2021

Preprint

View full text Add to dashboard Cite

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“…Since one of p j 's is less or equal to 1, we benefit from the atomic decomposition for the Hardy space. Moreover, for other indices greater than 2, we employ the techniques of (variant) ϕ-transform, introduced by Frazier and Jawerth [8,9,10] and Park [26], which will be presented in Section 2. Then T σ (f 1 , f 2 , f 3 ) can be decomposed in the form…”

Section: Then the Two Estimatesmentioning

confidence: 99%

“…It is also known in [26] that Γ j f has a representation analogous to (2.7) with an equivalence similar to (2.8), while f = j∈Z Γ j f generally. Let θ := 2 n θ(2•) and θ j := 2 jn θ(…”

Section: 3mentioning

confidence: 99%

Trilinear Fourier multipliers on Hardy spaces

Lee¹,

Park²

2021

Preprint

Self Cite

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In this paper, we obtain the H p 1 × H p 2 × H p 3 → H p boundedness for trilinear Fourier multiplier operators, which is a trilinear analogue of the multiplier theorem of Calderón and Torchinsky [4]. Our result improves the trilinear estimate in [22] by additionally assuming an appropriate vanishing moment condition, which is natural in the boundedness into the Hardy space H p for 0 < p ≤ 1.

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“…[18,Exercise 13.21]); this allows us to let θ → ∞ on the left-hand side of the previous inequality. For the right-hand side we use once more the dominated convergence theorem, which is applicable because of (15). Thus, we arrive at…”

mentioning

confidence: 99%

Convolution inequalities for Besov and Triebel–Lizorkin spaces, and applications to convolution semigroups

Kühn

Schilling

2022

Studia Math.

View full text Add to dashboard Cite

We establish convolution inequalities for Besov spaces B s p,q and Triebel-Lizorkin spaces F s p,q . As an application, we study the mapping properties of convolution semigroups, considered as operators on the function spaces A s p,q , A ∈ {B, F }. Our results apply to a wide class of convolution semigroups including the Gauß-Weierstraß semigroup, stable semigroups and heat kernels for higher-order powers of the Laplacian (−∆) m , and we can derive various caloric smoothing estimates.Introduction. The convolution f * g of two functions f, g : R n → R is defined byprovided that the integral exists. It is well known that f * g inherits the regularity of both f and g, in the sense that if, say, f ∈In this article, we study the regularizing properties of the convolution in the scales of Besov spaces B s p,q and Triebel-Lizorkin spaces F s p,q . Having in mind that the parameters s and p describe the regularity, resp., integrability properties of f ∈ A s p,q , A ∈ {B, F }, it is natural to expect that the implicationWe establish inequalities of the form

show abstract

Fourier Multipliers on a Vector-Valued Function Space

Cited by 5 publications

References 20 publications

Convolution inequalities for Besov and Triebel--Lizorkin spaces, and applications to convolution semigroups

Convolution inequalities for Besov and Triebel--Lizorkin spaces, and applications to convolution semigroups

Trilinear Fourier multipliers on Hardy spaces

Convolution inequalities for Besov and Triebel–Lizorkin spaces, and applications to convolution semigroups

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