Non-doubling Measures Research Articles

A Necessary Condition on a Singular Kernel for the Continuity of an Integral Operator in Hölder Spaces

We prove that a condition of boundedness of the maximal function of a singular integral operator, that is known to be sufficient for the continuity of a corresponding integral operator in Hölder spaces, is actually also necessary in case the action of the integral operator does not decrease the regularity of a function. We do so in the frame of metric measured spaces with a measure satisfying certain growth conditions that include nondoubling measures. Then we present an application to the case of an integral operator defined on a compact differentiable manifold.

Integral operators in Hölder spaces on upper Ahlfors regular sets

Volume and layer potentials are integrals on a subset Y of the Euclidean space {\mathbb{R}}^n that depend on a variable in a subset X of {\mathbb{R}}^n . Here we present a unified approach to some results by assuming that X and Y are subsets of a metric space M and that Y is equipped with a measure \nu that satisfies upper Ahlfors growth conditions that include non-doubling measures. We prove continuity statements in the frame of (generalized) Hölder spaces upon variation both of the density functions on Y and of the off-diagonal potential kernel and T1 theorems that generalize corresponding results of J. García-Cuerva and A. E. Gatto in case X=Y for kernels that include the standard ones.

Spectral multipliers in group algebras and noncommutative Calderón-Zygmund theory

In this paper we solve three problems in noncommutative harmonic analysis which are related to endpoint inequalities for singular integrals. In first place, we prove that an L2-form of Hörmander's kernel condition suffices for the weak type (1,1) of Calderón-Zygmund operators acting on matrix-valued functions. To that end, we introduce an improved CZ decomposition for martingale filtrations in von Neumann algebras, and apply a very simple unconventional argument which notably avoids pseudolocalization. In second place, we establish as well the weak L1 endpoint for matrix-valued CZ operators over nondoubling measures of polynomial growth, in the line of the work of Tolsa and Nazarov/Treil/Volberg. The above results are valid for other von Neumann algebras and solve in the positive two open problems formulated in 2009. An even more interesting problem is the lack of L1 endpoint inequalities for singular Fourier and Schur multipliers over nonabelian groups. Given a locally compact group G equipped with a conditionally negative length ψ:G→R+, we prove that Herz-Schur multipliers with symbol m∘ψ satisfying a Mikhlin condition in terms of the ψ-cocycle dimension are of weak type (1,1). Our result extends to Fourier multipliers for amenable groups and imposes sharp regularity conditions on the symbol. The proof crucially combines our new CZ methods with novel forms of recent transference techniques. This L1 endpoint gives a very much expected inequality which complements the L∞→BMO estimates proved in 2014 by Junge, Mei and Parcet.

Poincaré inequalities and compact embeddings from Sobolev type spaces into weighted Lq spaces on metric spaces

We study compactness and boundedness of embeddings from Sobolev type spaces on metric spaces into Lq spaces with respect to another measure. The considered Sobolev spaces can be of fractional order and some statements allow also nondoubling measures. Our results are formulated in a general form, using sequences of covering families and local Poincaré type inequalities. We show how to construct such suitable coverings and Poincaré inequalities. For locally doubling measures, we prove a self-improvement property for two-weighted Poincaré inequalities, which applies also to lower-dimensional measures.We simultaneously treat various Sobolev spaces, such as the Newtonian, fractional Hajłasz and Poincaré type spaces, for rather general measures and sets, including fractals and domains with fractal boundaries. By considering lower-dimensional measures on the boundaries of such domains, we obtain trace embeddings for the above spaces. In the case of Newtonian spaces we exactly characterize when embeddings into Lq spaces with respect to another measure are compact. Our tools are illustrated by concrete examples. For measures satisfying suitable dimension conditions, we recover several classical embedding theorems on domains and fractal sets in Rn.

Compactness for iterated commutators of general bilinear fractional integral operators on Morrey spaces with non-doubling measures

<abstract><p>In the present article, we obtain the compactness of iterated commutators generated by general bilinear fractional operator with RVMO functions on Morrey spaces with non-doubling measures.</p></abstract>

Analysis on Trees with Nondoubling Flow Measures

We consider trees with root at infinity endowed with flow measures, which are nondoubling measures of at least exponential growth and which do not satisfy the isoperimetric inequality. In this setting, we develop a Calderón–Zygmund theory and we define BMO and Hardy spaces, proving a number of desired results extending the corresponding theory as known in more classical settings.

Boundedness of Integral Operators in Generalized Weighted Grand Lebesgue Spaces with Non-doubling Measures

The goal of this paper is to prove the boundedness of fundamental integral operators of Harmonic Analysis in generalized weighted grand Lebesgue spaces $$L^{p),\varphi }_w$$ defined on domains in $${\mathbb {R}}^n$$ without assuming that the underlying measure $$\mu $$ is doubling. Operators under consideration involve maximal and Calderon–Zygmund operators, also commutators of singular integrals with BMO functions. Essential part of this paper is devoted to the weighted Sobolev-type inequalities in these spaces for fractional integrals with non-doubling measures satisfying the growth condition. We discuss the case when a weight function defines the absolute continuous measure of integration, or plays the role of multiplier in the definition of the norm. All these results are new even for the classical weighted grand Lebesgue spaces $$L^{p),\theta }_w$$ defined with respect to non-doubling measures. The results are obtained under the Muckenhoupt condition on weights. The results are obtained for weight functions satisfying Muckenhoupt $$A_p$$ conditions defined with respect to non-homogeneous spaces.

Iterated Commutators for Multilinear Singular Integral Operators on Morrey Space with Non-Doubling Measures

Let μ be a non-negative Radon measure on Rd which only satisfies the following growth condition that there exists a positive constant C such that μ(B(x,r)) ≤ Crn for all x∈ Rd, r > 0 and some fixed n ∈ (0,d]. This paper is interested in the properties of the iterated commutators of multilinear singular integral operators on Morrey spaces .Precisely speaking, we show that the iterated commutators generated by multilinear singular integrals operators are bounded from to where (Regular Bounded Mean Oscillation space) and 1 qj ≤ pj ∞ with 1/p = 1/p1 + ... + 1/pm and 1/q = 1/q1+ ... + 1/qm.

Nondoubling Calderón–Zygmund theory: a dyadic approach

Given a measure $$\mu $$ of polynomial growth, we refine a deep result by David and Mattila to construct an atomic martingale filtration of $$\mathrm {supp}(\mu )$$ which provides the right framework for a dyadic form of nondoubling harmonic analysis. Despite this filtration being highly irregular, its atoms are comparable to balls in the given metric—which in turn are all doubling—and satisfy a weaker but crucial form of regularity. Our dyadic formulation is effective to address three basic questions: Our martingale RBMO space preserves the crucial properties of Tolsa’s original definition and reveals its interpolation behavior with the $$L_p$$ scale in the category of Banach spaces, unknown so far. On the other hand, due to some known obstructions for Haar shifts and related concepts over nondoubling measures, our pointwise domination theorem via sparsity naturally deviates from its doubling analogue. In a different direction, matrix-valued harmonic analysis over noncommutative $$L_p$$ spaces has recently produced profound applications. Our analogue for nondoubling measures was expected for quite some time. Finally, we also find a dyadic form of the Calderon–Zygmund decomposition which unifies those by Tolsa and Lopez-Sanchez/Martell/Parcet.

Pointwise Multipliers on BMO Spaces with Non-doubling Measures

Let $\mu$ be a non-negative Radon measure satisfying the polynomial growth condition. In this paper, the authors characterize the set of pointwise multipliers on a BMO type space $\operatorname{RBMO}(\mu)$ introduced by Tolsa.

Boundedness of strong maximal functions with respect to non-doubling measures

The main purpose of this paper is to establish a boundedness result for strong maximal functions with respect to certain non-doubling measures in $\mathbb{R}^{n}$ . More precisely, let $d\mu(x_{1}, \ldots, x_{n})=d\mu_{1}(x_{1})\cdots d\mu_{n}(x_{n})$ be a product measure which is not necessarily doubling in $\mathbb{R}^{n}$ (only assuming $d\mu_{i}$ is doubling on $\mathbb{R}$ for $i=2, \ldots, n$ ), and let ω be a nonnegative and locally integral function such that $\omega _{i}(\cdot)=\omega(x_{1}, \ldots, x_{i-1}, \cdot, x_{i+1}, \ldots, x_{n})$ is in $A_{\infty}^{1}(d\mu_{i})$ uniformly in $x_{1}, \ldots, x_{i-1}, x_{i+1}, \ldots, x_{n}$ for each $i=1, \ldots, n-1$ , let $d\nu=\omega \,d\mu$ , $\nu(E)=\int_{E} \omega(y)\,d\mu(y)$ , and $M_{\omega \,d\mu}^{n}$ be the strong maximal function defined by $$M_{\omega \,d\mu}^{n} f(x)=\sup_{x\in R\in\mathcal{R}} \frac{1}{\nu (R)} \int_{R} \bigl\vert f(y) \bigr\vert \omega(y)\,d\mu(y), $$ where $\mathcal{R}$ is the collection of rectangles with sides parallel to the coordinate axes in $\mathbb{R}^{n}$ . Then we show that $M_{\omega \,d\mu}^{n} $ is bounded on $L^{p}_{\omega \,d\mu}(\mathbb{R}^{n})$ for $1< p<\infty$ . This extends an earlier result of Fefferman (Am. J. Math. 103:33-40, 1981) who established the $L^{p}$ boundedness when $d\mu=dx$ is the Lebesgue measure on $\mathbb{R}^{n}$ and $d\nu=\omega \,d\mu$ is doubling with respect to rectangles in $\mathbb{R}^{n}$ , ω satisfies a uniform $A^{1}_{\infty}$ condition in each of the variables except one. Moreover, we also establish some boundedness result for the Cordoba maximal functions (Cordoba A. in Harmonic Analysis in Euclidean Spaces, pp. 29-50, 1978) associated with the Cordoba-Zygmund dilation in $\mathbb{R}^{3}$ with respect to some non-doubling measures. This generalizes the result of Fefferman-Pipher (Am. J. Math. 119:337-369, 1997).

Boundedness of Commutators of Fractional Marcinkiewicz Integral Under Non-Doubling Measures

In this paper, the authors discuss the boundedness of fractional type Marcinkiewicz commutator $\mathcal {M}_{b}$ generated by the fractional type Marcinkiewicz integral $\mathcal {M}$ with Lipschitz function b. Under the assumption that the kernel of $\mathcal {M}$ satisfies certain slightly stronger Hormander-type condition, the authors prove that $\mathcal {M}_{b}$ is bounded from the Lebesgue space L p (μ) to the Lebesgue space L q (μ) $\left (1<p< \frac {n}{\beta }\right )$ , from the Lebesgue space L p (μ) to the Lipschitz space $\text {Lip}_{(\beta -\frac {n}{p})}(\mu )$ , and from the Lebesgue space $L^{\frac {n}{\beta }}(\mu )$ to the space RBMO(μ). In addition, the authors also discuss the boundedness of $\mathcal {M}_{b}$ on Morrey spaces and Hardy spaces with non-doubling measures.

Boundedness of θ-type Calderón-Zygmund operators on Hardy spaces with non-doubling measures

Let μ be a non-negative Radon measure on $R^{d}$ which only satisfies some growth condition. In this paper, we obtain the boundedness of θ-type Calderon-Zygmund operators on the Hardy space $H^{1}(\mu)$ .

FATOU PROPERTY OF PREDUAL MORREY SPACES WITH NON-DOUBLING MEASURES

The aim of this paper is to establish that the predual Morrey spaces are closed under taking increasing limit. As an application, the bound- edness property of the fractional integral operators on predual Morrey spaces is obtained. This result answers the (open) question from our earlier paper Predual spaces of Morrey spaces with non-doubling measures. This property will be used to supplement that paper, where the boundedness property of the fractional integral operators is missing.

Estimates for fractional type Marcinkiewicz integrals with non-doubling measures

Under the assumption that μ is a non-doubling measure on satisfying the growth condition, the authors prove that the fractional type Marcinkiewicz integral ℳ is bounded from the Hardy space to the Lebesgue space for with kernel satisfying a certain Hormander-type condition. In addition, the authors show that for , ℳ is bounded from the Morrey space to the space and from the Lebesgue space to the space . MSC:46A20, 42B25, 42B35.

Endpoint estimates for multilinear fractional integrals with non-doubling measures

Under the assumption that µ is a non-doubling measure on ℝd which only satisfies the polynomial growth condition, the authors obtain the boundedness of the multilinear fractional integrals on Morrey spaces, weak-Morrey spaces and Lipschitz spaces associated with µ, which, in the case when µ is the d-dimensional Lebesgue measure, also improve the known results.

Equivalent boundedness of Marcinkiewicz integrals on non-homogeneous metric measure spaces

Let $({\mathcal X},\,d,\,\mu)$ be a metric measure space satisfying the upper doubling condition and the geometrically doubling condition in the sense of T. Hyt\"onen. In this paper, the authors prove that the $L^p(\mu)$ boundedness with $p\in(1,\,\infty)$ of the Marcinkiewicz integral is equivalent to either of its boundedness from $L^1(\mu)$ into $L^{1,\infty}(\mu)$ or from the atomic Hardy space $H^1(\mu)$ into $L^1(\mu)$. Moreover, the authors show that, if the Marcinkiewicz integral is bounded from $H^1(\mu)$ into $L^1(\mu)$, then it is also bounded from $L^\infty(\mu)$ into the space ${\mathop\mathrm{RBLO}}(\mu)$ (the regularized {\rm BLO}), which is a proper subset of ${\rm RBMO}(\mu)$ (the regularized {\rm BMO}) and, conversely, if the Marcinkiewicz integral is bounded from $L_b^\infty(\mu)$ (the set of all $L^\infty(\mu)$ functions with bounded support) into the space ${\rm RBMO}(\mu)$, then it is also bounded from the finite atomic Hardy space $H_{\rm fin}^{1,\,\infty}(\mu)$ into $L^1(\mu)$. These results essentially improve the known results even for non-doubling measures.

Linear and sublinear operators on generalized Morrey spaces with non-doubling measures

Linear and sublinear operators on generalized Morrey spaces with non-doubling measures

Estimates for Marcinkiewicz commutators with Lipschitz functions under nondoubling measures

Under the assumption that μ is a nondoubling measure on R d satisfying the growth condition, the author proves that the commutator M b generated by the Marcinkiewicz integral operator and the Lipschitz function is bounded from the Hardy space H fin 1 , ∞ , 0 (μ) into L q (μ) for 1/q=1−β/n with the kernel satisfying a certain Hörmander-type condition. Moreover, the author shows that for p=n/β, M b is bounded from the Morrey space M q p (μ) into RBMO(μ), from L n / β (μ) into RBMO(μ) and from M q p (μ) into Lip ( β − n p ) (μ), respectively.MSC:42B25, 47B47, 42B20, 47A30.

Boundedness of Calderón-Zygmund operators with finite non-doubling measures

Let µ be a nonnegative Radon measure on ℝ d which satisfies the polynomial growth condition that there exist positive constants C 0 and n ∈ (0, d] such that, for all x ∈ ℝ d and r > 0, µ(B(x, r)) ⩽ C 0 r n , where B(x, r) denotes the open ball centered at x and having radius r. In this paper, we show that, if µ(ℝ d ) < ∞, then the boundedness of a Calderon-Zygmund operator T on L 2(µ) is equivalent to that of T from the localized atomic Hardy space h 1(µ) to L 1,∞(µ) or from h 1(µ) to L 1(µ).