The main purpose of this paper is to investigate the behaviour of fractional integral operators associated to a measure on a metric space satisfying just a mild growth condition, namely that the measure of each ball is controlled by a fixed power of its radius. This allows, in particular, non--doubling measures. It turns out that this condition is enough to build up a theory that contains the classical results based upon the Lebesgue measure on euclidean space and their known extensions for doubling measures. We start by analyzing the images of the Lebesgue spaces associated to the measure. The Lipschitz spaces, defined in terms of the metric, play a basic role too. For a euclidean space equipped with one of these measures, we also consider the so-called `` regular\rq\rq $\bmo$ space introduced by X. Tolsa. We show that it contains the image of a Lebesgue space in the appropriate limit case and also that the image of the space `` regular\rq\rq $\bmo$ is contained in the adequate Lipschitz space.
We give conditions for boundedness of Hausdorff operators on real Hardy spaces $H^1$ over homogeneous spaces of locally compact groups with local doubling property. The special case of the hyperbolic plane is considered.
In the context of a finite measure metric space whose measure satisfies a growth condition, we prove "T1" type necessary and sufficient conditions for the boundedness of fractional integrals, singular integrals, and hypersingular integrals on inhomogeneous Lipschitz spaces. We also indicate how the results can be extended to the case of infinite measure. Finally we show applications to Real and Complex Analysis.
Let ${\mathcal X}$ be a doubling metric measure space. If ${\mathcal X}$ has the $δ$-annular decay property for some $δ\in (0,\,1]$, the authors then establish the boundedness of the Lusin-area function, which is defined via kernels modeled on the semigroup generated by the Schrödinger operator, from localized spaces ${\rm BMO}_ρ({\mathcal X})$ to ${\rm BLO}_ρ({\mathcal X})$ without invoking any regularity of considered kernels. The same is true for the $g^\ast_\labda$ function and unlike the Lusin-area function, in this case, ${\mathcal X}$ is not necessary to have the $δ$-annular decay prope…
Let $M$ be a manifold with ends constructed in \cite{GS} and $Δ$ be the Laplace-Beltrami operator on $M$. In this note, we show the weak type $(1,1)$ and $L^p$ boundedness of the Hardy-Littlewood maximal function and of the maximal function associated with the heat semigroup $\M_Δf(x)=\sup_{t> 0} |\exp (-tΔ)f(x)| $ on $L^p(M)$ for $1 < p \le \infty$. The significance of these results comes from the fact that $M$ does not satisfies the doubling condition.
Let $μ$ be a non-negative Radon measure on ${\mathbb R}^d$ which only satisfies the polynomial growth condition. Let ${\mathcal Y}$ be a Banach space and $H^1(μ)$ the Hardy space of Tolsa. In this paper, the authors prove that a linear operator $T$ is bounded from $H^1(μ)$ to ${\mathcal Y}$ if and only if $T$ maps all $(p, γ)$-atomic blocks into uniformly bounded elements of ${\mathcal Y}$; moreover, the authors prove that for a sublinear operator $T$ bounded from $L^1(μ)$ to $L^{1, \infty}(μ)$, if $T$ maps all $(p, γ)$-atomic blocks with $p\in(1, \infty)$ and $γ\in{\mathbb N}$ into uniformly …
Let $({\mathcal X},\,d,\,μ)$ be a metric measure space satisfying the upper doubling condition and the geometrically doubling condition in the sense of T. Hytönen. In this paper, the authors prove that the $L^p(μ)$ boundedness with $p\in(1,\,\infty)$ of the Marcinkiewicz integral is equivalent to either of its boundedness from $L^1(μ)$ into $L^{1,\infty}(μ)$ or from the atomic Hardy space $H^1(μ)$ into $L^1(μ)$. Moreover, the authors show that, if the Marcinkiewicz integral is bounded from $H^1(μ)$ into $L^1(μ)$, then it is also bounded from $L^\infty(μ)$ into the space ${\mathop\mathrm{RBLO}}…
Let $(\cx,\,d,\,μ)$ be a metric measure space and satisfy the so-called upper doubling condition and the geometrically doubling condition. In this paper, the authors show that for the maximal Calderón-Zygmund operator associated with a singular integral whose kernel satisfies the standard size condition and the Hörmander condition, its $L^p(μ)$ boundedness with $p\in(1,\infty)$ is equivalent to its boundedness from $L^1(μ)$ into $L^{1,\infty}(μ)$. Moreover, applying this, together with a new Cotlar type inequality, the authors show that if the Calderón-Zygmund operator $T$ is bounded on $L^2(μ…
Let $(\mathcal{X},d,μ)$ be a metric measure space satisfying the so-called upper doubling condition and the geometrically doubling condition. Let $T$ be a Calderón-Zygmund operator with kernel satisfying only the size condition and some Hörmander-type condition, and $b\in\rm{\widetilde{RBMO}(μ)}$ (the regularized BMO space with the discrete coefficient). In this paper, the authors establish the boundedness of the commutator $T_b:=bT-Tb$ generated by $T$ and $b$ from the atomic Hardy space $\widetilde{H}^1(μ)$ with the discrete coefficient into the weak Lebesgue space $L^{1,\,\infty}(μ)$. The b…
In this paper, the author studies the boundedness for a large class of sublinear operator $T_α, α\in[0,n)$ generated by Calder{ó}n-Zygmund operators ($α=0$) and generated by fractional integral operator ($α>0$) on generalized mixed Morrey spaces $M^φ_{\vec{q}}(\Bbb R^n)$. Moreover, the boundeness for the commutators of $T_α, α\in[0,n)$ on generalized mixed Morrey spaces $M^φ_{\vec{q}}(\Bbb R^n)$ is also studied. As applications, we obtain the boundedness for Hardy-Littlewood maximal operator, Calderón-Zygmund singular integral operators, fractional integral operator, fractional maximal operato…
In this paper, the fractional integral operator on non-homogeneous metric measure spaces is introduced, which contains the classic fractional integral operator, fractional integral with non-doubling measures and fractional integral with fractional kernel of order $α$ and regularity $ε$ introduced by García-Cuerva and Gatto as special cases. And the $(L^{p}(μ),L^{q}(μ))$-boundedness for fractional integral operators on non-homogeneous metric measure spaces is established. From this, the $(L^{p}(μ),L^{q}(μ))$-boundedness for commutators and multilinear commutators generated by fractional integra…
We obtain (essentially sharp) boundedness results for certain generalized local maximal operators between fractional weighted Sobolev spaces and their modifications. Concrete boundedness results between well known fractional Sobolev spaces are derived as consequences of our main result. We also apply our boundedness results by studying both generalized neighbourhood capacities and the Lebesgue differentiation of fractional weighted Sobolev functions.
In this article, we establish some conditions for the boundedness of fractional integral operators on the vanishing generalized weighted Morrey spaces. We also investigate corresponding commutators generated by BMO functions.
In the present paper, we shall give a characterization for weak/strong Adams-type boundedness of the fractional maximal operator on generalized Orlicz-Morrey spaces.
In this paper, some boundedness for commutators of fractional integrals are obtained on Herz-Morrey spaces with variable exponent applying some properties of varible exponent and $\BMO$ function.
Characterizations for multilinear fractional maximal and integral operators and their commutators on generalized weighted Morrey spaces and applications
In this paper, boundedness of Hausdorff operator on weak central Morrey space is obtained. Furthermore, we investigate the weak bounds of p- adic fractional Hausdorff Operator on weighted p-adic weak Lebesgue Space. We also obtain the sufficient condition of commutators of p-adic fractional Hausdorff Operator by taking symbol function from Lipschitz space. Moreover, strong type estimates for fractional Hausdorff Operator and its commutator on weighted p-adic Lorentz space are also acquired.
In this paper we investigate the boundedness of classical operators, namely the Hardy-Littlewood maximal operator, fractional integral operators, and Calderon-Zygmund operators, on generalized weighted Morrey spaces and generalized weighted weak Morrey spaces. We prove that the operators are bounded on these spaces, under certain assumptions.
It was well known that geometric considerations enter in a decisive way in many questions of harmonic analysis. The main purpose of this paper is to provide the criterion of the boundedness for singular integrals on the Hardy spaces and as well as on its dual, particularly on $\bmo$ for spaces of homogeneous type $(X, d,μ)$ in the sense of Coifman and Weiss, where the quasi-metric $d$ may have no regularity and the measure $μ$ satisfies only the doubling property. We make no additional geometric assumptions on the quasi-metric or the doubling measure and thus, the results of this paper extend …
This paper includes new bounds concepting the vanishing generalized weighted Morrey space. In this sense, it is outlined improved bounds about the a class of fractional type rough higher order commutators on vanishing generalized weighted Morrey spaces.
Many authors have considered and investigated generalized fractional differential operators. The main object of this present paper is to define a new generalized fractional differential operator $\mathfrak{T}^{β,τ,γ},$ which generalized the Srivastava-Owa operators. Moreover, we investigate of the geometric properties such as univalency, starlikeness, convexity for their normalization. Further, boundedness and compactness in some well known spaces, such as Bloch space for last mention operator also are considered. Our tool is based on the generalized hypergeometric function.
In this paper, we prove strong type, weak type inequalities of Hardy-Littlewood maximal operator and fractional Hardy-Littlewood maximal operator on variable sequence spaces lp(Z). This is achieved using Calderon-Zygmund decomposition for sequences, properties of modular functional and Log Holder continuity.
In this paper, we introduced the local and global mixed Morrey-type spaces, and some properties of these spaces are also studied. After that, the necessary conditions of the boundedness of fractional integral operators $I_α$ are studied respectively in mixed-norm Lebesgue spaces and the local mixed Morrey-type spaces. Last but not least, the boundedness of $I_α$ is obtained by Hardy operators' boundedness in weighted Lebesgue spaces. As corollaries, we acquire the boundedness of $I_α$ in mixed Morrey spaces(\hyperlink{Corollary 5.6}{Corollary 5.6}) and mixed Lebesgue spaces(\hyperlink{Corollar…
The boundedness and compactness of Toeplitz operator from $A_ω^p$ to $A_ω^q$ with doubling weights $ω$ are studied in this paper. The characterizations of Schatten class Toeplitz operators and Volterra operators on $A_ω^2$ are also investigated.
In the present paper, we will characterize the boundedness of the generalized fractional integral operators $I_ρ$ and the generalized fractional maximal operators $M_ρ$ on Orlicz spaces, respectively. Moreover, we will give a characterization for the Spanne-type boundedness and the Adams-type boundedness of the operators $M_ρ$ and $I_ρ$ on generalized Orlicz--Morrey spaces, respectively. Also we give criteria for the weak versions of the Spanne-type boundedness and the Adams-type boundedness of the operators $M_ρ$ and $I_ρ$ on generalized Orlicz--Morrey spaces.
In this article, the authors introduce the Newton-Morrey-Sobolev space on a metric measure space $(\mathscr{X},d,μ)$. The embedding of the Newton-Morrey-Sobolev space into the Hölder space is obtained if $\mathscr{X}$ supports a weak Poincaré inequality and the measure $μ$ is doubling and satisfies a lower bounded condition. Moreover, in the Ahlfors $Q$-regular case, a Rellich-Kondrachov type embedding theorem is also obtained. Using the Hajłasz gradient, the authors also introduce the Hajłasz-Morrey-Sobolev spaces, and prove that the Newton-Morrey-Sobolev space coincides with the Hajłasz-Morr…
The aim of this note is to give the boundedness conditions for Hausdorff operators on Hardy spaces $H^{1}$ with the norm defined via $(1,q)$ atoms over homogeneous spaces of Lie groups with doubling property and to apply results we obtain to generalized Delsarte operators and to Hausdorff operators over multidimensional spheres.
In this paper, the sharp maximal theorem is generalized to mixed-norm ball Banach function spaces, which is defined as Definition 2.7. As an application, we give a characterization of BMO via the boundedness of commutators of fractional integral operators on mixed-norm Lebesgue spaces. Moreover, the characterization of homogeneous Lipschitz space is also given by the boundedness of commutators of fractional integral operators on mixed-norm Lebesgue spaces. Finally, two applications of Corollary 6.4 are given.
In this paper, we investigate the boundedness of maximal operator and its commutators in generalized Orlicz-Morrey spaces on the spaces of homogeneous type. As an application of this boundedness, we give necessary and sufficient condition for the Adams type boundedness of fractional integral and its commutators in these spaces. We also discuss criteria for the boundedness of these operators in Orlicz spaces.
We consider the global Morrey-type spaces with variable exponents and general function defining these spaces. In the case of unbounded sets, we prove boundedness of the Hardy-Littlewood maximal operator, potential type operator in these spaces.
A necessary and sufficient condition for fractional Orlicz-Sobolev spaces to be continuously embedded into $L^\infty(\mathbb R^n)$ is exhibited. Under the same assumption, any function from the relevant fractional-order spaces is shown to be continuous. Improvements of this result are also offered. They provide the optimal Orlicz target space, and the optimal rearrangement-invariant target space in the embedding in question. These results complement those already available in the subcritical case, where the embedding into $L^\infty(\mathbb R^n)$ fails. They also augment a classical embedding t…
We give necessary and sufficient conditions for the boundedness of generalized fractional integral and maximal operators on Orlicz-Morrey and weak Orlicz-Morrey spaces. To do this we prove the weak-weak type modular inequality of the Hardy-Littlewood maximal operator with respect to the Young function. Orlicz-Morrey spaces contain $L^p$ spaces ($1\le p\le\infty$), Orlicz spaces and generalized Morrey spaces as special cases. Hence we get necessary and sufficient conditions on these function spaces as corollaries.
Let $({\mathcal X},d,μ)$ be a metric measure space satisfying both the upper doubling and the geometrically doubling conditions. In this paper, the authors establish some equivalent characterizations for the boundedness of fractional integrals over $({\mathcal X},d,μ)$. The authors also prove that multilinear commutators of fractional integrals with ${\mathop\mathrm{\,RBMO(μ)}}$ functions are bounded on Orlicz spaces over $({\mathcal X},d,μ)$, which include Lebesgue spaces as special cases. The weak type endpoint estimates for multilinear commutators of fractional integrals with functions in t…
We consider generalized Orlicz-Morrey spaces $M_{Φ,φ}(\mathbb{R}^{n})$ including their weak versions $WM_{Φ,φ}(\mathbb{R}^{n})$. We find the sufficient conditions on the pairs $(φ_{1},φ_{2})$ and $(Φ, Ψ)$ which ensures the boundedness of the fractional maximal operator $M_α$ from $M_{Φ,φ_1}(\mathbb{R}^{n})$ to $M_{Ψ,φ_2}(\mathbb{R}^{n})$ and from $M_{Φ,φ_1}(\mathbb{R}^{n})$ to $WM_{Ψ,φ_2}(\mathbb{R}^{n})$. As applications of those results, the boundedness of the commutators of the fractional maximal operator $M_{b,α}$ with $b \in BMO(\mathbb{R}^{n})$ on the spaces $M_{Φ,φ}(\mathbb{R}^{n})$ is …
The boundedness of $P_ω:L^\infty(\mathbb{B})\to \mathcal{B}(\mathbb{B})$ and $P_ω(P_ω^+):L^p(\mathbb{B},\upsilon dV)\to L^p(\mathbb{B},\upsilon dV)$ on the unit ball of $\mathbb{C}^n$ with $p>1$ and $ω,\upsilon\in \mathcal{D}$ are investigated in this paper.
We establish various $L^{p}$ estimates for the Schrödinger operator $-Δ+V$ on Riemannian manifolds satisfying the doubling property and a Poincaré inequality, where $Δ$ is the Laplace-Beltrami operator and $V$ belongs to a reverse Hölder class. At the end of this paper we apply our result on Lie groups with polynomial growth.
The boundedness of the small Hankel operator $h_f^ν(g)=P_ν(f\bar{g})$, induced by an analytic symbol $f$ and the Bergman projection $P_ν$ associated to $ν$, acting from the weighted Bergman space $A^p_\om$ to $A^q_ν$ is characterized on the full range $0<p,q<\infty$ when $ω,ν$ belong to the class $\mathcal{D}$ of radial weights admitting certain two-sided doubling conditions. Certain results obtained are equivalent to the boundedness of bilinear Hankel forms, which are in turn used to establish the weak factorization $A_η^{q}=A_ω^{p_{1}}\odot A_ν^{p_{2}}$, where $1<q,p_{1},p_{2}<\infty$ such t…
One defines a non-homogeneous space $(X, μ)$ as a metric space equipped with a non-doubling measure $μ$ so that the volume of the ball with center $x$, radius $r$ has an upper bound of the form $r^n$ for some $n> 0$. The aim of this paper is to study the boundedness of a Calderón-Zygmund operator $T$ as well as the boundedness of certain related singular integrals associated with $T$ on various function spaces on $(X, μ)$ such as the Hardy spaces, the $L^p$ spaces and the regularized BMO spaces. This article thus extends the work of X. Tolsa \cite{T1} on the non-homogeneous space $(\mathbb R^n…
On the Boundedness of Generalized Fractional Integral Operators in Morrey Spaces and Camapanato Spaces associated with the Dunkl Operator on the Real line
Let $ϕ: \Bbb R^n \mapsto \Bbb R$ be a strictly convex and smooth function, and $μ= \text{det}\,D^2 ϕ$ be the Monge-Ampère measure generated by $ϕ.$ For $x\in \Bbb R^n$ and $t>0$, let $S(x,t):=\{y\in \Bbb R^n: ϕ(y)<ϕ(x)+\nabla ϕ(x)\cdot(y-x)+t\}$ denote the section. If $μ$ satisfies the doubling property, Caffarelli and Gutiérrez (Trans. AMS 348:1075--1092, 1996) provided a variant of the Calderón-Zygmund decomposition and a John-Nirenberg-type inequality associated with sections. Under a stronger uniform continuity condition on $μ$, they also (Amer. J. Math. 119:423--465, 1997) proved an invar…
In this article, the authors establish a general (two-weight) boundedness criterion for a pair of functions, $(F,f)$, on $\mathbb{R}^n$ in the scale of weighted Lebesgue spaces, weighted Lorentz spaces, (Lorentz--)Morrey spaces, and variable Lebesgue spaces. As applications, the authors give a unified approach to prove the (two-weight) boundedness of Calderón--Zygmund operators, Littlewood--Paley $g$-functions, Lusin area functions, Littlewood--Paley $g^\ast_λ$-functions, and fractional integral operators, in the aforementioned function spaces. Moreover, via applying the above (two-weight) bou…
In this paper, we show the strong and weak type boundedness of $T_{Ω,α}^A$ and $M_{Ω,α}^A$, the multilinear fractional integral operators and the corresponding fractional maximal operators, on the two weights weighted Morrey space. Furthermore, we also obtain the mulitilinear singular integral operators $T_Ω^A$ and the corresponding maximal operators $M_Ω^A$ are strong bounded operators on the same spaces.
We prove that $b$ is in $Lip_{\bz}(\bz)$ if and only if the commutator $[b,L^{-α/2}]$ of the multiplication operator by $b$ and the general fractional integral operator $L^{-α/2}$ is bounded from the weighed Morrey space $L^{p,k}(ω)$ to $L^{q,kq/p}(ω^{1-(1-α/n)q},ω)$, where $0<β<1$, $0<α+β<n, 1<p<{n}/({α+β})$, ${1}/{q}={1}/{p}-{(α+β)}/{n},$ $0\leq k<{p}/{q},$ $ω^{{q}/{p}}\in A_1$ and $ r_ω> \frac{1-k}{p/q-k},$ and here $r_ω$ denotes the critical index of $ω$ for the reverse Hölder condition.
Density of Lipschitz functions in Newtonian spaces based on quasi-Banach function lattices is discussed. Newtonian spaces are first-order Sobolev-type spaces on abstract metric measure spaces defined via (weak) upper gradients. Our main focus lies on metric spaces with a doubling measure that support a $p$-Poincaré inequality. Absolute continuity of the function lattice quasi-norm is shown to be crucial for approximability by (locally) Lipschitz functions. The proof of the density result uses, among others, that a suitable maximal operator is locally weakly bounded. In particular, various suff…
In this paper, the fractional Hardy-type operator of variable order $β(x)$ is shown to be bounded from the variable exponent Herz-Morrey spaces $M\dot{K}_{p_{_{1}},q_{_{1}}(\cdot)}^{α(\cdot),λ}(\R^{n})$ into the weighted space $M\dot{K}_{p_{_{2}},q_{_{2}}(\cdot)}^{α(\cdot),λ}(\R^{n},ω)$, where $α(x)\in L^{\infty}(\mathbb{R}^{n})$ be log-Hölder continuous both at the origin and at infinity, $ω=(1+|x|)^{-γ(x)}$ with some $γ(x)>0$ and $ 1/q_{_{1}}(x)-1/q_{_{2}}(x)=β(x)/n$ when $q_{_{1}}(x)$ is not necessarily constant at infinity.
Let $Γ$ be a doubling graph satisfying some pointwise subgaussian estimates of the Markov kernel. We introduce a space $H^1(Γ)$ of functions and a space $H^1(T_Γ)$ of 1-forms and give various characterizations of them. We prove the $H^1$-boundedness of the Riesz transform, from which we deduce the $L^p$ boundedness of the Riesz transform for any $p\in (1,2)$. In a previous work, we showed a $H^1_w$-boundedness of the Riesz transform under weaker assumptions, but the $L^p$ boundedness was not established.
