Homogeneous Triebel-Lizorkin Spaces Research Articles

Bochner–Riesz Means and K-Functional on Triebel–Lizorkin Spaces

Based on n-dimensional Euclidean spaces and n-dimensional torus as underlying spaces, we investigate the rate for the norm convergence of the generalized Bochner–Riesz means on homogeneous Triebel–Lizorkin spaces, and establish the equivalence between the rate and the K-functional. Particularly, we show that such equivalence is closely related to the Bochner–Riesz conjecture on homogeneous Triebel–Lizorkin spaces. Thus, we obtain natural extensions of some well-known theorems by Fefferman, Tomas and Stein and by Carleson, Söjlin and Hörmander.

Triebel–Lizorkin space estimates for evolution equations with structure dissipation

In this paper, we are concerned with the generalized wave equations in the homogeneous Triebel–Lizorkin spaces. The long time decay estimates are obtained by the decomposition of the unit, duality property and the multiplier theorems, extending the known results for the generalized wave equations with structure dissipation in the real Hardy spaces.

Characterizations of realized homogeneous Besov and Triebel-Lizorkin spaces via differences

Based on the role of the polynomial functions on the homogeneous Besov spaces, on the homogeneous Triebel-Lizorkin spaces and on their realized versions, we study and obtain characterizations of these spaces via difference operators in a certain sense.

Realizations of homogeneous Besov and Triebel–Lizorkin spaces and an application to pointwise multipliers

We study the dilation commuting realizations of the homogeneous Besov spaces [Formula: see text] or the homogeneous Triebel–Lizorkin spaces [Formula: see text] in the case p, q > 0, and either s - (n/p) ∉ ℕ0or s - (n/p) ∈ ℕ0and 0 < q ≤ 1 (0 < p ≤ 1 in the [Formula: see text]-case). We present an application to pointwise multiplication if s ≤ n/p.

Fractional type Marcinkiewicz integral operators associated to surfaces

In this paper, we discuss the boundedness of the fractional type Marcinkiewicz integral operators associated to surfaces, and we extend a result given by Chen et al. (J. Math. Anal. Appl. 276:691-708, 2002). They showed that under certain conditions the fractional type Marcinkiewicz integral operators are bounded from the Triebel-Lizorkin spaces F ˙ p q α ( R n ) to L p ( R n ). Recently the second author, together with Xue and Yan, greatly weakened their assumptions. In this paper, we extend their results to the case where the operators are associated to the surfaces of the form {x=ϕ(|y|)y/|y|}⊂ R n ×( R n ∖{0}). To prove our result, we discuss a characterization of the homogeneous Triebel-Lizorkin spaces in terms of lacunary sequences.MSC:42B20, 42B25, 47G10.

Homogeneous Triebel-Lizorkin Spaces on Stratified Lie Groups

Homogeneous Triebel-Lizorkin spaces with full range of parameters are introduced on stratified Lie groups in terms of Littlewood-Paley-type decomposition. It is shown that the scale of these spaces is independent of the choice of Littlewood-Paley-type decomposition and the sub-Laplacian used for the construction of the decomposition. Some basic properties of these spaces are given. As the main result of this paper, boundedness of a class of singular integral operators on these function spaces is obtained.

Generalized Carleson Measure Spaces and Their Applications

We introduce the generalized Carleson measure spaces CM that extend BMO. Using Frazier and Jawerth′s φ‐transform and sequence spaces, we show that, for α ∈ ℝ and 0 < p ≤ 1, the duals of homogeneous Triebel‐Lizorkin spaces for 1 < q < ∞ and 0 < q ≤ 1 are CM and CM (for any r ∈ ℝ), respectively. As applications, we give the necessary and sufficient conditions for the boundedness of wavelet multipliers and paraproduct operators acting on homogeneous Triebel‐Lizorkin spaces.

CONTINUOUS CHARACTERIZATION OF THE TRIEBEL-LIZORKIN SPACES AND FOURIER MULTIPLIERS

We give a set of continuous characterizations for the homogeneous Triebel-Lizorkin spaces and use them to study boundedness properties of Fourier multiplier operators whose symbols satisfy a generalization of Hormander’s condition. As an application, we give new direct proofs of the imbedding theorems of the Sobolev type.

New Proof of the 𝑇(1) Theorem for Triebel–Lizorkin Spaces

Abstract We give a new proof of the 𝑇(1) theorem for reflexive homogeneous Triebel–Lizorkin spaces , which uses neither maximal functions nor atomic decompositions. The substitute tool is an estimate on translations in the 𝐿𝑝(ℓ𝑞) spaces.