Lipschitz functions on spaces of homogeneous type

In this paper, using the discrete Calderon reproducing formula on spaces of homogeneous type obtained by the author, we obtain the Plancherel-Pôlya type inequalities on spaces of homogeneous type. These inequalities give new characterizations of the Besov spaces B ˙ p α , q \dot B_p^{\alpha ,q} and the Triebel-Lizorkin spaces F ˙ p α , q \dot F_p^{\alpha ,q} on spaces of homogeneous type introduced earlier by the author and E. T. Sawyer and also allow us to generalize these spaces to the case where p , q ≤ 1 p,q\le 1 . Moreover, using these inequalities, we can easily show that the Littlewood-Paley G G -function and S S -function are equivalent on spaces of homogeneous type, which gives a new characterization of the Hardy spaces on spaces of homogeneous type introduced by Macias and Segovia.

In this paper, we establish the two weight commutator theorem of Calderon–Zygmund operators in the sense of Coifman–Weiss on spaces of homogeneous type, by studying the weighted Hardy and BMO space for $$A_2$$ weights and by proving the sparse operator domination of commutators. The main tool here is the Haar basis, the adjacent dyadic systems on spaces of homogeneous type, and the construction of a suitable version of a sparse operator on spaces of homogeneous type. As applications, we provide a two weight commutator theorem (including the high order commutators) for the following Calderon–Zygmund operators: Cauchy integral operator on $${\mathbb {R}}$$ , Cauchy–Szego projection operator on Heisenberg groups, Szego projection operators on a family of unbounded weakly pseudoconvex domains, the Riesz transform associated with the sub-Laplacian on stratified Lie groups, as well as the Bessel Riesz transforms (in one and several dimensions).

Given a space of homogeneous type we give sufficient conditions on a variable exponent {p(.)} so that the fractional maximal operator {M_{\eta}} maps {L^{p(.)}(X)} to {L^{q(.)}(X)}, where {1/p(.) - 1/q(.) = {\eta}}. In the endpoint case we also prove the corresponding weak type inequality. As an application we prove norm inequalities for the fractional integral operator {I_{\eta}}. Our proof for the fractional maximal operator uses the theory of dyadic cubes on spaces of homogeneous type, and even in the Euclidean setting it is simpler than existing proofs. For the fractional integral operator we extend a pointwise inequality of Welland to spaces of homogeneous type. Our work generalizes results from the Euclidean case and extends recent work by Adamowicz, et al. on the Hardy-Littlewood maximal operator on spaces of homogeneous type.

WeightedLpforp∈(1,∞)and weak-type endpoint estimates with general weights are established for commutators of the Hardy-Littlewood maximal operator with BMO symbols on spaces of homogeneous type. As an application, a weighted weak-type endpoint estimate is proved for maximal operators associated with commutators of singular integral operators with BMO symbols on spaces of homogeneous type. All results with no weight on spaces of homogeneous type are also new.

In this article, the authors introduce the spaces of Lipschitz type on spaces of homogeneous type in the sense of Coifman and Weiss, and discuss their relations with Besov and Triebel–Lizorkin spaces. As an application, the authors establish the difference characterization of Besov and Triebel–Lizorkin spaces on spaces of homogeneous type. A major novelty of this article is that all results presented in this article get rid of the dependence on the reverse doubling assumption of the considered measure of the underlying space \({{\mathcal {X}}}\) via using the geometrical property of \({{\mathcal {X}}}\) expressed by its dyadic reference points, dyadic cubes, and the (local) lower bound. Moreover, some results when \(p\le 1\) but near to 1 are new even when \({{\mathcal {X}}}\) is an RD-space.

Assume that (X, d, μ) is a space of homogeneous type in the sense of Coifman and Weiss (1971, 1977). In this article, motivated by the breakthrough work of Auscher and Hytonen (2013) on orthonormal bases of regular wavelets on spaces of homogeneous type, we introduce a new kind of approximations of the identity with exponential decay (for short, exp-ATI). Via such an exp-ATI, motivated by another creative idea of Han et al. (2018) to merge the aforementioned orthonormal bases of regular wavelets into the frame of the existed distributional theory on spaces of homogeneous type, we establish the homogeneous continuous/discrete Calderon reproducing formulae on (X, d, μ), as well as their inhomogeneous counterparts. The novelty of this article exists in that d is only assumed to be a quasi-metric and the underlying measure μ a doubling measure, not necessary to satisfy the reverse doubling condition. It is well known that Calderon reproducing formulae are the cornerstone to develop analysis and, especially, harmonic analysis on spaces of homogeneous type.

In this work, we establish a Plancherel–Polya inequality for functions in Besov spaces on spaces of homogeneous type as defined in Han and Yang (Diss Math 403:1–102, 2002) in the spirit of their recent counterpart for $${\mathbb {R}}^d$$ established by Jaming and Malinnikova (J Fourier Anal Appl 22:768–786, 2016. The main tool is the wavelet decomposition presented by Deng and Han (Harmonic Analysis on Spaces of Homogeneous Type, Springer, New York, 2009).

We consider a version of M. Riesz fractional integral operator on a space of homogeneous type and show an analogue of the well-known Hardy-Littlewood-Sobolev theorem in this context. In our main result, we investigate the dependence of the operator norm on weighted spaces on the weight constant, and find the relationship between these two quantities. It it shown that the estimate obtained is sharp in any given space of homogeneous type with infinitely many points. Our result generalizes the recent Euclidean result by Lacey, Moen, Pérez and Torres [21].

In this paper, we shall study the Hardy-boundedness for the multilinear commuta- tors related to the singular integral operators on the space of homogeneous type. By using the Holder's inequalities and the L q (1 < q < ∞) boundedness for the singular integral operators on the space of homogeneous type, we obtain the (H p �b ,L p ) and (H ˙ K α,p q,�b , ˙ K α,p q ) type boundedness for the multilinear commutators on the space of homogeneous type.

The authors introduce a class of generalized Riesz potentials with kernels having weak regularity on spaces of homogeneous type in the sense of Coifman and Weiss and establish their boundedness on Lebesgue spaces and Hardy spaces. As applications, the authors obtain the boundedness on Lebesgue spaces and Hardy spaces of commutators generated by Lipschitz functions and generalized Riesz potentials or Calderon-Zygmund operators with kernels having weak regularity on spaces of homogeneous type. Mathematics subject classification (2010): 31C15, 42B20, 47B47, 42B30, 43A99.

It was well known that geometric considerations enter in a decisive way in many questions. The embedding theorem arises in several problems from partial differential equations, analysis, and geometry. The purpose of this paper is to provide a deep understanding of analysis and geometry with a particular focus on embedding theorems for spaces of homogeneous type in the sense of Coifman and Weiss, where the quasi-metric d may have no regularity and the measure $$\mu $$ satisfies the doubling property only. We prove that embedding theorems hold on spaces of homogeneous type if and only if geometric conditions, namely the measures of all balls have lower bounds, hold. We make no additional geometric assumptions on the quasi-metric or the doubling measure, and thus, the results of this paper extend to the full generality of all related previous ones, in which the extra geometric assumptions were made on both the quasi-metric d and the measure $$\mu .$$ As applications, our results provide new and sharp previous related embedding theorems for the Sobolev, Besov, and Triebel–Lizorkin spaces. The crucial tool used in this paper is the remarkable orthonormal wavelet basis constructed recently by Auscher–Hytonen on spaces of homogeneous type in the sense of Coifman and Weiss.

BMO from dyadic BMO via expectations on product spaces of homogeneous type

The purpose of this note is to give an adequate Calderon-Zygmund type lemma in order to extend to the general setting of spaces of homogeneous type the Ap weighted Lp boundedness for the Hardy-Littlewood maximal operator given by M. Christ and R. Fefferman. Recently Michael Christ and Robert Fefferman gave in (1) a remarkable proof of the weighted norm inequality for the Hardy-Littlewood maximal function operator in R, \\Mf\\Lp(w) 1. In (2), A. P. Calderon proved this boundedness property for spaces such that the measure of balls is continuous as a function of the radius. In (3), R. Macias and C. Segovia extended this result to general spaces of homogeneous type (defined below) constructing an adequate quasi-distance. In both cases, the reverse Holder inequality must be extended to this general setting, while the proof given in (1) does not make use of this property and only depends on an adequate Calderon-Zygmund type lemma, the proof of which for cubes in R is very simple. The purpose of this note is to obtain a decomposition lemma which allows us to extend the proof of

Some equivalent characterizations for boundedness of maximal singular integral operators on spaces of homogeneous type are given via certain norm inequalities on John-Stromberg sharp maximal functions and without resorting the boundedness of these operators themselves. As a corollary, the results of Grafakos on Euclidean spaces are generalized to spaces of homogeneous type. Moreover, applications to maximal Monge-Ampere singular integral operators and maximal Nagel-Stein singular integral operators on certain specific smooth manifolds are also presented.

Orthonormal bases of regular wavelets in spaces of homogeneous type