Sobolev‐Jawerth embedding of Triebel‐Lizorkin‐Morrey‐Lorentz spaces and fractional integral operator on Hardy type spaces

Abstract

A Sobolev type embedding for Triebel‐Lizorkin‐Morrey‐Lorentz spaces is established in this paper. As an application of this result, the boundedness of the fractional integral operator on some generalizations of Hardy spaces such as Hardy‐Morrey spaces and Hardy‐Lorentz spaces are obtained.

Fourier integrals and Sobolev embedding on rearrangement invariant quasi-Banach function spaces

“…In particular, the above result recovers the mapping properties for the fractional integral operators on Hardy-Lorentz spaces, see [20,Theorem 5.5].…”

Partial Regularity for Fractional Harmonic Maps into Spheres

“…Proof of Theorem C.3. In [26], the author introduced a more refined scale of homogeneous Triebel-Lizorkin spaces of Morrey-Lorentz type, denoted by Ḟ s,u M p,q,λ (R n ). In the case u = p = q, those spaces coincide with the homogeneous Triebel-Lizorkin-Morrey spaces above, namely Ḟ s,p M p,p,λ (R n ) = Ės p,p, np λ (R n ) for every p ∈ (0, ∞), λ ∈ (0, n], and s ∈ R. More precisely, their defining semi-norms are equivalent (in one case the supremum is taken over all dyadic cubes, while in the other it is taken over balls).…”

“…In the case u = p = q, those spaces coincide with the homogeneous Triebel-Lizorkin-Morrey spaces above, namely Ḟ s,p M p,p,λ (R n ) = Ės p,p, np λ (R n ) for every p ∈ (0, ∞), λ ∈ (0, n], and s ∈ R. More precisely, their defining semi-norms are equivalent (in one case the supremum is taken over all dyadic cubes, while in the other it is taken over balls). By [26,Theorem 4.1], under condition (C.1) the space Ḟ α2,q2 M q 2 ,q 2 ,λ (R n ) embeds continuously into Ḟ α1,q1 M q 1 ,q 1 ,λ (R n ). In other words, Ėα2…”

“…A main difference with [17] lies in the fact that an additional "error term" appears when rewriting the s-harmonic map equation in the suitable form where compensation can be seen. To control this error term in arbitrary dimensions, we make use of a recent embedding result between Triebel-Lizorkin-Morrey type spaces [26] and various characterizations of these spaces [42,57].…”

Definability of singular integral operators on Morrey–Banach spaces

“…Recently, the studies of Morrey spaces is extending to Morrey space built on some non Lebesgue spaces such as Morrey-Lorentz spaces [3,18,31], Orlicz-Morrey spaces [11,29,28], Morrey spaces with variable exponents [1,13,15,19,22,24,25,26,33,32]. On the other hand, for instance, in [15,19], we are lack of a precise definition of the action of singular integral operators on the above mentioned Morrey type spaces.…”