Weighted Bourgain–Morrey–Besov type and Triebel–Lizorkin type spaces associated with operators

Recently in [6]–[7] are studied Sobolev, Besov and Triebel Lizorkin type spaces and Hankel potentials on the basis of the Hankel transformation. In this paper we obtain new characterizations of Sobolev type spaces and Hankel potentials spaces. Moreover we characterize Lipschitz and Besov type spaces in terms of the Poisson integral given in [17].

Let $$(M, \rho ,\mu )$$ be a space of homogeneous type satisfying the reverse doubling condition and the non-collapsing condition. In this paper, the authors introduce Besov-type spaces $$B_{p,q}^{s,\tau }(M)$$ and Triebel–Lizorkin-type spaces $$F_{p,q}^{s,\tau }(M)$$ associated to a nonnegative self-adjoint operator $$L$$ whose heat kernel satisfies sub-Gaussian upper bound estimate, Hölder continuity, and stochastic completeness. The novelty in this article is that the indices $$p,q,s,\tau $$ here can be take full range of all possible values as in the Euclidean setting. Characterizations of these spaces via Peetre maximal functions and the heat semigroup are established for full range of possible indices. Also, frame characterizations of these spaces are given. When $$L$$ is the Laplacian operator on $$\mathbb R^n$$ , these spaces coincide with the Besov-type and Triebel–Lizorkin-type spaces on $$\mathbb R^n$$ studied in (Yuan et al. Lecture Notes in Mathematics, vol 2005, 2010). In the case $$\tau =0$$ and the smoothness index $$s$$ is around zero, comparisons of these spaces with the Besov and Triebel–Lizorkin spaces studied in (Han et al. Abstr Appl Anal 1–250, 2008, Art ID 893409) are also presented.

In this article, the authors construct some counterexamples to show that the generalized Carleson measure space and the Triebel–Lizorkin-type space are not equivalent for certain parameters, which was claimed to be true in Lin and Wang [C.-C. Lin and K.Wang, Equivalency between the generalized Carleson measure spaces and Triebel–Lizorkin-type spaces, Taiwanese J. Math. 15 (2011), pp. 919–926]. Moreover, the authors show that for some special parameters, the generalized Carleson measure space, the Triebel–Lizorkin-type space and the Besov-type space coincide with certain Triebel–Lizorkin space, which answers a question posed in Remark 6.11(i) of Yuan et al. [W. Yuan, W. Sickel and D. Yang, Morrey and Campanato Meet Besov, Lizorkin and Triebel, Lecture Notes in Mathematics 2005, Springer-Verlag, Berlin, 2010]. In conclusion, the Triebel–Lizorkin-type space and the Besov-type space become the classical Besov spaces, when the fourth parameter is sufficiently large.

Let $X$ be a space of homogeneous type and $L$ be a nonnegative self-adjoint operator on $L^{2}(X)$ satisfying Gaussian upper bounds on its heat kernels. In this paper, we develop the theory of weighted Besov spaces ${\dot{B}}_{p,q,w}^{\unicode[STIX]{x1D6FC},L}(X)$ and weighted Triebel–Lizorkin spaces ${\dot{F}}_{p,q,w}^{\unicode[STIX]{x1D6FC},L}(X)$ associated with the operator $L$ for the full range $0<p,q\leqslant \infty$ , $\unicode[STIX]{x1D6FC}\in \mathbb{R}$ and $w$ being in the Muckenhoupt weight class $A_{\infty }$ . Under rather weak assumptions on $L$ as stated above, we prove that our new spaces satisfy important features such as continuous characterizations in terms of square functions, atomic decompositions and the identifications with some well-known function spaces such as Hardy-type spaces and Sobolev-type spaces. One of the highlights of our result is the characterization of these spaces via noncompactly supported functional calculus. An important by-product of this characterization is the characterization via the heat kernel for the full range of indices. Moreover, with extra assumptions on the operator $L$ , we prove that the new function spaces associated with $L$ coincide with the classical function spaces. Finally we apply our results to prove the boundedness of the fractional power of $L$ , the spectral multiplier of $L$ in our new function spaces and the dispersive estimates of wave equations.

Let (X, d, mu ) be a space of homogeneous type. Let L be a nonnegative self-adjoint operator on L^2(X) satisfying certain conditions on the heat kernel estimates which are motivated from the heat kernel of the Schrödinger operator on mathbb {R}^n. The main aim of this paper is to prove a new atomic decomposition for the Besov space dot{B}^{0, L}_{1,1}(X) associated with the operator L. As a consequence, we prove the boundedness of the Riesz transform associated with L on the Besov space dot{B}^{0, L}_{1,1}(X).

New applications of Besov-type and Triebel–Lizorkin-type spaces

Let $$({\mathcal {X}},\rho , \mu )$$ be a space of homogeneous type. Suppose that $$p(\cdot ),\ q(\cdot ):\ {\mathcal {X}}\rightarrow (0,\infty ]$$ are such that both $$1/p(\cdot )$$ and $$1/q(\cdot )$$ satisfy the globally log-Holder continuous condition, and $$s(\cdot ):\ {\mathcal {X}}\rightarrow \mathbb R$$ is a bounded function satisfying the locally log-Holder continuous condition. In this article, the authors introduce the variable Besov space $$B_{p(\cdot ),q(\cdot )}^{s(\cdot ),L}({\mathcal {X}})$$ , associated with a nonnegative self-adjoint operator L whose heat kernels satisfy small time Gaussian upper bound estimates, the Holder continuity, and the Markov property, which is new even on the sphere and the ball of $$\mathbb R^d$$ . Equivalent characterizations of this space, in terms of Peetre maximal functions and the heat semigroup, are established. Moreover, under the additional assumptions that $$\mu $$ satisfies the reverse doubling condition and the non-collapsing condition, its frame characterization is obtained. When L is the Laplacian operator on $$\mathbb R^d$$ , this space coincides with the existing variable Besov space.

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.

Let $X$ be a space of homogeneous type and let $L$ be a nonnegative self-adjoint operator on $L^2(X)$ that satisfies a Gaussian estimate on its heat kernel. In this paper we prove a Hörmander-type spectral multiplier theorem for $L$ on the Besov and Triebel–Lizorkin spaces associated to $L$. Our work not only recovers the boundedness of the spectral multipliers on $L^p$ spaces and Hardy spaces associated to $L$ but also is the 1st one that proves the boundedness of a general spectral multiplier theorem on Besov and Triebel–Lizorkin spaces.

Let $X$ be a metric measure space with a doubling measure and $L$ be a nonnegative self-adjoint operator acting on $L^2(X)$. Assume that $L$ generates an analytic semigroup $e^{-tL}$ whose kernels $p_t(x,y)$ satisfy Gaussian upper bounds but without any assumptions on the regularity of space variables $x$ and $y$. In this article we continue a study in \cite{SY} to give an atomic decomposition for the Hardy spaces $ H^p_{L,max}(X)$ in terms of the nontangential maximal function associated with the heat semigroup of $L$, and hence we establish characterizations of Hardy spaces associated to an operator $L$, via an atomic decomposition or the nontangential maximal function. We also obtain an equivalence of $ H^p_{L, max}(X)$ in terms of the radial maximal function.

Let θ ∈ (0, 1), s0, s1 ∈ ℝ, τ0, τ1 ∈ [0, ∞), p0, p1 ∈ (0, ∞), q0, q1 ∈ (0, ∞], s = s0(1 - θ) + s1θ, τ = τ0(1-θ) + τ1θ, [Formula: see text] and [Formula: see text]. In this paper, under the restriction [Formula: see text], the authors establish the complex interpolation, on Triebel–Lizorkin-type spaces, that [Formula: see text], where [Formula: see text] denotes the closure of the Schwartz functions in [Formula: see text]. Similar results on Besov-type spaces and Besov–Morrey spaces are also presented. As a corollary, the authors obtain the complex interpolation for Morrey spaces that, for all 1 < p0 ≤ u0 < ∞, 1 < p1 ≤ u1 < ∞ and 1 < p ≤ u < ∞ such that [Formula: see text], [Formula: see text] and p0u1 = p1u0, [Formula: see text], where [Formula: see text] denotes the closure of the Schwartz space in [Formula: see text]. It is known that, if p0u1 ≠ p1u0, these conclusions on Morrey spaces may not be true.

We extend the Hairer reconstruction theorem for distributions due to Caravenna and Zambotti [EMS Surv. Math. Sci. 7.2 (2020)] to general function spaces satisfying a translation and scaling condition. This includes Besov type spaces with exponents below 1 and Triebel–Lizorkin type spaces.

Let (X,d,μ) be a space of homogenous type and L be a non-negative self-adjoint operator on L2(X) with heat kernels satisfying Gaussian upper bounds. In this paper, we introduce the variable Besov–Morrey space associated with the operator L and prove that this space can be characterized via the Peetre maximal functions. Then, we establish its atomic decomposition.

Spectral multipliers on spaces of distributions associated with non-negative self-adjoint operators

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