Pseudodifferential operators on mixed‐norm Besov and Triebel–Lizorkin spaces

Abstract

We prove boundedness of pseudodifferential operators on anisotropic mixed‐norm Besov and Triebel–Lizorkin spaces. Our proof relies only on general maximal function estimates and provides a new perspective even in the case of spaces without mixed norms. Moreover, we cover the case of Fourier multipliers on the above mentioned spaces. As application we establish boundedness of pseudodifferential operators and Fourier multipliers on anisotropic mixed‐norm Sobolev spaces.

Wavelet transforms for homogeneous mixed-norm Triebel–Lizorkin spaces

“…[20][21][22]. Continuity of pseudo-differential operators in this set-up has been treated by Georgiadis and Nielsen [15].…”

“…Corollary C.4. When the seminorm | ·, ψ | in Proposition C.2 is given by a ψ ∈ S having vanishing moments of order M ≥ 0, then R m , ψ = O(t n+max(M,m)+1−d ) (C. 15) and each term in P m has ∂ α t n Φ(t·) * f , ψ = 0 for |α| ≤ M and else is O(t n+|α|−d ).…”

Atomic and Littlewood–Paley Characterizations of Anisotropic Mixed-Norm Hardy Spaces and Their Applications

“…Later, in 1970, Lizorkin [57] further studied both the theory of multipliers of Fourier integrals and estimates of convolutions in the mixed-norm Lebesgue spaces. Moreover, very recently, there appears a renewed increasing interest in the theory of mixed-norm function spaces, including mixed-norm Lebesgue spaces, mixed-norm Hardy spaces, mixed-norm Besov spaces and mixed-norm Triebel-Lizorkin spaces; see, for example, [19,20,30,38,39,40,41]. For more developments of mixed-norm function spaces, we refer the reader to [16,18,27,29,42,43].…”

Time-Periodic Solutions to the Equations of Magnetohydrodynamics with Background Magnetic Field

“…In this section we are going to define an anisotropic distance function. The idea goes back to Fabes and Rivière [32], was extended by Yamazaki [85], and has become one of the standard tools to define anisotropic functions spaces, see for example Yamazaki [85,86], Johnsen and Sickel [55,56] or Georgiadis and Nielsen [39]. Because the proofs of properties of the anisotropic distance function are a bit sparse in the literature but often quite direct, we will state and prove the needed ones.…”