Generalization of some regularity criteria for 3D Boussinesq equations in homogeneous Besov and Triebel–Lizorkin spaces

We deal with homogeneous Besov and Triebel–Lizorkin spaces in the setting of a doubling metric measure space in the presence of a non-negative self-adjoint operator whose heat kernel has Gaussian localization and the Markov property. The class of almost diagonal operators on the associated sequence spaces is developed and it is shown that this class is an algebra. The boundedness of almost diagonal operators is utilized for establishing smooth molecular and atomic decompositions for the above homogeneous Besov and Triebel–Lizorkin spaces. Spectral multipliers for these spaces are established as well.

Homogeneous Besov and Triebel–Lizorkin spaces associated to non-negative self-adjoint operators

We consider the composition operators acting on the real‐valued homogeneous Besov or Triebel–Lizorkin spaces, realized as dilation invariant subspaces of , denoted as . If and , then any function acting by composition on is necessarily linear. The above conditions are optimal: (i) in case , (Besov space), (Triebel–Lizorkin space), is a quasi‐Banach algebra for the pointwise product, (ii) in case , , , any function such that is a finite measure, and , acts by composition on .

The Littlewood–Paley theory of homogeneous product Besov and Triebel–Lizorkin spaces is developed in the spirit of the $$\varphi $$ -transform of Frazier and Jawerth. This includes the frame characterization of the product Besov and Triebel–Lizorkin spaces and the development of almost diagonal operators on these spaces. The almost diagonal operators are used to obtain product wavelet decomposition of the product Besov and Triebel–Lizorkin spaces. The main application of this theory is to nonlinear m-term approximation from product wavelets in $$L^p$$ and Hardy spaces. Sharp Jackson and Bernstein estimates are obtained in terms of product Besov spaces.

Decompositions of Besov–Hausdorff and Triebel–Lizorkin–Hausdorff spaces and their applications

Consider the Hermite operator $$H=-\Delta +|x|^2$$ on the Euclidean space $$\mathbb {R}^n$$ . The main aim of this article is to develop a theory of homogeneous and inhomogeneous Besov and Triebel–Lizorkin spaces associated to the Hermite operator. Our inhomogeneous Besov and Triebel–Lizorkin spaces are different from those introduced by Petrushev and Xu (J Fourier Anal Appl 14, 372–414 2008). As applications, we show the boundedness of negative powers and spectral multipliers of the Hermite operators on some appropriate Besov and Triebel–Lizorkin spaces.

By using the discrete Calderon reproducing formulae, the author first establishes the boundedness of the Riesz-potential-type operator in homogeneous Besov and Triebel–Lizorkin spaces over spaces of homogeneous type. Then, by use of the T1 theorems for these spaces, the author proves that this operator of Riesz potential type can be used as the lifting operator of these spaces.

Wavelet characterization of Besov and Triebel–Lizorkin spaces on spaces of homogeneous type and its applications

Using the realizations, we study some convolution inequalities in the realized homogeneous Besov spaces ??Bs p,q(Rn) and the realized homogeneous Triebel-Lizorkin spaces ??Fs p,q(Rn). We also deduce for the homogeneous Sobolev spaces ?Wmp(Rn) in certain sense.

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.

Molecular decomposition of anisotropic homogeneous mixed-norm spaces with applications to the boundedness of operators

Characterizations of Besov and Triebel–Lizorkin spaces via averages on balls

Consider the space X = ( 0 , ∞ ) X=(0,\infty ) equipped with the Euclidean distance and the measure d μ α ( x ) = x α d x d\mu _\alpha (x)=x^{\alpha }dx where α ∈ ( − 1 , ∞ ) \alpha \in (-1,\infty ) is a fixed constant and d x dx is the Lebesgue measure. Consider the Laguerre operator L = − d 2 d x 2 − α x d d x + x 2 \displaystyle L=-\frac {d^2}{dx^2} -\frac {\alpha }{x}\frac {d}{dx}+x^2 on X X . The aim of this article is threefold. Firstly, we establish a Calderón reproducing formula using a suitable distribution of the Laguerre operator. Secondly, we study certain properties of the Laguerre operator such as a Harnack type inequality on the solutions and subsolutions of Laplace equations associated to Laguerre operators. Thirdly, we establish the theory of the weighted homogeneous Besov and Triebel-Lizorkin spaces associated to the Laguerre operator. We define the weighted homogeneous Besov and Triebel-Lizorkin spaces by the square functions of the Laguerre operator, then show that these spaces have an atomic decomposition. We then study the fractional powers L − γ , γ > 0 L^{-\gamma }, \gamma >0 , and show that these operators map boundedly from one weighted Besov space (or one weighted Triebel-Lizorkin space) to another suitable weighted Besov space (or weighted Triebel-Lizorkin space). We also show that in particular cases of the indices, our new weighted Besov and Triebel-Lizorkin spaces coincide with the (expected) weighted Hardy spaces, the weighted L p L^p spaces or the weighted Sobolev spaces in Laguerre settings.

An RD-space $\mathcal X$ is a space of homogeneous type in the sense of Coifman and Weiss with the additional property that a reverse doubling property holds in $\mathcal X$. In this paper, the authors first give several equivalent characterizations of RD-spaces and show that the definitions of spaces of test functions on $\mathcal X$ are independent of the choice of the regularity $\epsilon\in (0,1)$; as a result of this, the Besov and Triebel-Lizorkin spaces on $\mathcal X$ are also independent of the choice of the underlying distribution space. Then the authors characterize the norms of inhomogeneous Besov and Triebel-Lizorkin spaces by the norms of homogeneous Besov and Triebel-Lizorkin spaces together with the norm of local Hardy spaces in the sense of Goldberg. Also, the authors obtain the sharp locally integrability of elements in Besov and Triebel-Lizorkin spaces.

Atomic decomposition plays an important role in establishing the boundedness of operators on function spaces. Let 0 < p , q < ∞ ${0<p, q<\infty }$ and α = ( α 1 , α 2 ) ∈ ℝ 2 ${\alpha =(\alpha _1,\alpha _2)\in \mathbb {R}^2}$ . In this paper, we introduce multi-parameter Triebel–Lizorkin spaces F ˙ p α , q ( ℝ m ) ${\dot{F}^{\hspace*{0.85358pt}\alpha ,\,q}_{p}(\mathbb {R}^{m})}$ associated with different homogeneities arising from the composition of two singular integral operators whose weak (1,1) boundedness was first studied by Phong and Stein [Amer. J. Math. 104 (1982), 141–172]. We then establish its atomic decomposition which is substantially different from that for the classical one-parameter Triebel–Lizorkin spaces. As an application of our atomic decomposition, we obtain the necessary and sufficient conditions for the boundedness of an operator T on the multi-parameter Triebel–Lizorkin type spaces. In the special case of α 1 = α 2 = 0 ${\alpha _1=\alpha _2=0}$ , q = 2 and 0 < p ≤ 1 ${0<p\le 1}$ , our spaces F ˙ p α , q ( ℝ m ) ${\dot{F}^{\hspace*{0.85358pt}\alpha ,\,q}_{p}(\mathbb {R}^{m})}$ coincide with the Hardy spaces H com p ${H^p_{\mathrm {com}}}$ associated with the composition of two different singular integrals (see []). Therefore, our results also give an atomic decomposition of H com p ${H^p_{\mathrm {com}}}$ . Our work appears to be the first result of atomic decomposition in the Triebel–Lizorkin spaces in the multi-parameter setting.