Non-doubling Measure Research Articles

Fractional maximal operator in hyperbolic spaces

In this article, we introduce the fractional maximal operator on the Hyperbolic space, a non-doubling measure space, and study its weighted boundedness. Motivated by the weighted boundedness of the Hardy-Littlewood maximal studied by Antezana and Ombrosi in [1], we give conditions for the weak type and strong type estimate for fractional maximal. Also, we present examples of weights for which the fractional maximal operator satisfies weak type (p,q) inequality but strong type (p,q) inequality fails.

Trudinger-type inequalities on Musielak-Orlicz-Morrey spaces of an integral form

Our aim in this paper is to give Trudinger-type inequalities for Riesz potentials of functions in Musielak-Orlicz-Morrey spaces of an integral form over non-doubling metric measure spaces. Our result are new even for the doubling metric measure setting. As a corollary, we give Trudinger-type inequalities on Musielak-Orlicz-Morrey spaces of an integral form in the framework of double phase functions with variable exponents.

Sobolev's inequality in central Herz-Morrey-Musielak-Orlicz spaces over metric measure spaces

ABSTRACT We give the boundedness of the Hardy-Littlewood maximal operator , , on central Herz-Morrey-Musielak-Orlicz spaces over bounded non-doubling metric measure spaces and to establish a generalization of Sobolev's inequality for Riesz potentials , , , of functions in such spaces. As an application and example, we obtain the boundedness of and for double phase functionals Φ such that . These results are new even for the doubling metric measure setting.

Weighted estimates for iterated commutators of multilinear Calderón-Zygmund operators on non-homogeneous metric measure spaces

Let ( $${\cal X}$$ , d,μ) be a non-homogeneous metric measure space satisfying the so-called upper doubling and geometrically doubling conditions, which includes the space of homogeneous type and the Euclidean space with the non-doubling measure as special cases. Let T be a multilinear Calderon-Zygmund operator and $$\vec b: = ({b_1}, \ldots ,{b_m})$$ be a finite family of $$\widetilde{{\rm{RBMO}}}(\mu)$$ functions. In this paper, some weak-type multiple weighted estimates for the iterated commutator $${{\rm{T}}_{\Pi \vec b}}$$ generated by T and $$\vec b$$ are obtained.

Generalized Riesz potentials of functions in Morrey spaces L (1,ϕ;κ)(G) over non-doubling measure spaces

Abstract This paper proves the boundedness of the generalized Riesz potentials I ρ , μ , τ ⁢ f {I_{\rho,\mu,\tau}f} of functions in the Morrey space L ( 1 , φ ; κ ) ⁢ ( G ) {L^{(1,\varphi;\kappa)}(G)} over a general measure space X, with G a bounded open set in X (or G is X ) {X)} , as an extension of earlier results. The modification parameter τ is introduced for the purpose of including the case where the underlying measure does not satisfy the doubling condition. What is new in the present paper is that ρ depends on x ∈ X {x\in X} . An example in the end of this article convincingly explains why the modification parameter τ must be introduced.

Boundedness of generalized Riesz potentials of functions in Morrey spaces L^(1,φ;κ)(G) over non-doubling measure spaces

Boundedness of generalized Riesz potentials of functions in Morrey spaces L^(1,φ;κ)(G) over non-doubling measure spaces

Littlewood-Paley $g^{\ast}_{\lambda,\mu}$-function and Its Commutator on Non-homogeneous Generalized Morrey Spaces

Let $\mu$ be a non-negative Radon measure on $\mathbb{R}^{d}$ which may be a non-doubling measure. In this paper, the authors prove that the Littlewood-Paley $g^{\ast}_{\lambda,\mu}$-function is bounded on the generalized Morrey space $\mathcal{L}^{p,\phi}(\mu)$, and also obtain that the commutator $g^{\ast}_{\lambda,\mu,b}$ generated by the Littlewood-Paley function $g^{\ast}_{\lambda,\mu}$ and the regular bounded mean oscillation space $(=$RBMO$)$, which is due to X. Tolsa, is bounded on $\mathcal{L}^{p,\phi}(\mu)$. As a corollary, the authors prove that the commutator $g^{\ast}_{\lambda,\mu,b}$ is bounded on the Morrey space $\mathcal{M}^{p}_{q}(\mu)$ defined by Sawano and Tanaka when we take $\phi(t)=t^{1-\frac{p}{q}}$ with $1<p<q<\infty$.

Predual spaces of generalized grand Morrey spaces over non-doubling measure spaces

Abstract The predual spaces of generalized grand Morrey spaces over non-doubling measure spaces are investigated. The case of the grand Lebesgue spaces is covered, which is also new. An example shows that the modification of Morrey spaces is essential.

The endpoint Fefferman-Stein inequality for the strong maximal function with respect to nondoubling measure

The endpoint Fefferman-Stein inequality for the strong maximal function with respect to nondoubling measure

Two-Weight Weak Type Inequality for Modified Fractional Maximal Function Defined with Respect to Non-Doubling Measure and Some Applications in Non-standard Lebesgue Spaces

Let $$ \mu $$ a nonnegative Radon measure on $$ {\mathbb{R}}^{d} $$ ; $$ p,q,\gamma ,k $$ real numbers; $$ M_{\mu ,k}^{\gamma } $$ a fractional maximal operator; $$ A_{p,q}^{\gamma ,k} \left( \mu \right) $$ a Muckenhoupt condition associated to $$ \mu $$ ; $$ L^{p( \cdot )} ({\mathbb{R}}^{d} , \mu ) $$ and $$ F(q, p,\alpha ,\mu )({\mathbb{R}}^{d} ) $$ two generalized Lebesgue spaces. The purpose of the present work is double:

Generalized fractional integral operators over non-doubling metric measure spaces

ABSTRACTWe shall deal with the boundedness and weak-type boundedness for the generalized fractional integral operators on generalized Orlicz-Morrey spaces of the second kind over non-doubling metric measure spaces in this paper. We also discuss a necessary condition for the boundedness of the generalized fractional integral operators.

Trudinger’s inequality and continuity for Riesz potential of functions in Orlicz spaces of two variable exponents over nondoubling measure spaces

In this article, we consider Trudinger’s inequality and continuity for Riesz potentials of functions in Orlicz spaces of two variable exponents near Sobolev’s exponent over nondoubling metric measure spaces.

EXPONENTIAL INTEGRABILITY FOR LOGARITHMIC POTENTIALS OF FUNCTIONS IN GENERALIZED LEBESGUE SPACES $L(\log L)^{q(\cdot)}$ OVER NON-DOUBLING MEASURE SPACES

In this paper, we are concerned with exponential integrability for logarithmic potentials of functions in generalized Lebesgue spaces $L(\log L)^{q(\cdot)}$ over non-doubling measure spaces. Here $q$ satisfies the loglog-Holder condition.

Hardy spaces H p over non-homogeneous metric measure spaces and their applications

Let $({\mathcal X},d,\mu)$ be a metric measure space satisfying both the geometrically doubling and the upper doubling conditions. Let $\rho\in (1,\infty)$, $0<p\le1\le q\le\infty$, $p\neq q$, $\gamma\in[1,\infty)$ and $\epsilon\in(0,\infty)$. In this article, the authors introduce the atomic Hardy space ${\widetilde H_{\mathrm{atb},\,\rho}^{p,\,q,\,\gamma}(\mu)}$ and the molecular Hardy space ${\widetilde H_{\rm{mb},\,\rho}^{p,\,q,\,\gamma,\,\epsilon}(\mu)}$ via the discrete coefficient $\widetilde{K}^{(\rho),\,p}_{B,\,S}$, and prove that the Calder\'on-Zygmund operator is bounded from ${\widetilde H_{\rm{mb},\,\rho}^{p,\,q,\,\gamma,\,\delta}(\mu)}$ (or ${\widetilde H_{\rm{atb},\,\rho}^{p,\,q,\,\gamma}(\mu)}$) into $L^p(\mu)$, and from ${\widetilde H_{\rm{atb},\,\rho(\rho+1)}^{p,\,q,\,\gamma+1}(\mu)}$ into ${\widetilde H_{\rm{mb},\,\rho}^{p,\,q,\,\gamma,\,\frac12(\delta -\frac{\nu}{p}+\nu)}(\mu)}$ whose genealized fractional versions are also obtained. The authors also introduce the $\rho$-weakly doubling condition, with $\rho\in (1,\infty)$, of the measure $\mu$ and construct a non-doubling measure $\mu$ satisfying this condition. If $\mu$ is $\rho$-weakly doubling, the authors further introduce the Campanato space ${\mathcal E}^{\alpha,\,q}_{\rho,\,\eta,\,\gamma}(\mu)$ and show that ${\mathcal E}^{\alpha,\,q}_{\rho,\,\eta,\,\gamma}(\mu)$ is independent of the choices of $\rho$, $\eta$, $\gamma$ and $q$; the authors then introduce the atomic Hardy space $\widehat H_{\rm{atb},\,\rho}^{p,\,q,\,\gamma}(\mu)$ and the molecular Hardy space $\widehat H_{\rm{mb},\,\rho}^{p,\,q,\,\gamma,\,\epsilon}(\mu)$, which coincide with each other; the authors finally prove that $\widehat{H}_{\rm{atb},\,\rho}^{p,\,q,\,\gamma}(\mu)$ is the predual of ${\mathcal E}^{1/p-1,\,1}_{\rho,\,\rho,\,1}(\mu)$. Moreover, relations of these Hardy spaces are also discussed.

Littlewood-Paley theory on metric spaces with non doubling measures and its applications

The purpose of this paper is to extend the Littlewood-Paley theory to a geometrically doubling metric space with a non-doubling measure satisfying a weak growth condition. Moreover, we prove that our setting mentioned above, is equivalent to the one introduced and studied by Hytonen (2010) in his remarkable framework, i.e., the geometrically doubling metric space with a non-doubling measure satisfying a so-called upper doubling condition. As an application, we obtain the T1 theorem in this more general setting. Moreover, the Gaussian measure is also discussed.

Sobolev embeddings for Riesz potentials of functions in Musielak–Orlicz–Morrey spaces over non-doubling measure spaces

In this paper we are concerned with Sobolev embeddings for Riesz potentials of functions in Musielak–Orlicz–Morrey spaces over non-doubling measure spaces. We deal with the boundedness of the generalized maximal operators, Sobolev's inequality, Trudinger's inequality and continuity for Riesz potentials of variable order. We prove the results in full generality: our function spaces will cover Morrey spaces, variable Lebesgue spaces as well as Lebesgue spaces with a Radon measure μ. Also, we can readily handle function spaces on metric measure spaces. Our results will cover the ones in our earlier paper [Sawano Y, Shimomura T. Sobolev embeddings for Riesz potentials of functions in non-doubling Morrey spaces of variable exponents. Collect. Math. 2013;64:313–350]. We shall show that the Trudinger exponential integrability is available in our generalized setting, whose counterpart on Lebesgue spaces was not considered in Mizuta et al. [Maximal functions, Riesz potentials and Sobolev embeddings on Musielak–Orlicz–Morrey spaces of variable exponent in Rn. Rev. Mat. Complut. 2012;25:413–434.]

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.

Sobolev embeddings for Riesz potentials of functions in grand Morrey spaces of variable exponents over non-doubling measure spaces

Our aim in this paper is to deal with the boundedness of the Hardy-Littlewood maximal operator on grand Morrey spaces of variable exponents over non-doubling measure spaces. As an application of the boundedness of the maximal operator, we establish Sobolev’s inequality for Riesz potentials of functions in grand Morrey spaces of variable exponents over non-doubling measure spaces. We are also concerned with Trudinger’s inequality and the continuity for Riesz potentials.

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.