Search

We give necessary and sufficient conditions for the boundedness of the maximal commutators Mb$$ {M}_b $$ and the commutators of the maximal operator [b,M]$$ \left[b,M\right] $$ in total Morrey spaces Lp,λ,μ(ℝn)$$ {L}^{p,\lambda, \mu}\left({\mathrm{\mathbb{R}}}^n\right) $$ when b$$ b $$ belongs to Lipschitz spaces Λ˙β(ℝn)$$ {\dot{\Lambda}}_{\beta}\left({\mathrm{\mathbb{R}}}^n\right) $$, whereby some new characterizations for certain subclasses of Lipschitz spaces Λ˙β(ℝn)$$ {\dot{\Lambda}}_{\beta}\left({\mathrm{\mathbb{R}}}^n\right) $$ are obtained. [ABSTRACT FROM AUTHOR]

The local potential operator with integral kernel restricted in a ball of radius less than some fixed number r∈(0,∞)$r\in (0,\infty)$ has appeared frequently in the spectral estimates of the Schrödinger operator. In this paper, we establish a good‐λ inequality for this operator and characterize its uniform boundedness on the weighted Orlicz space in both strong and weak senses. The uniformity in r of this boundedness enables us to recover the classical boundedness of "global" potential operator, by letting r→∞$r\rightarrow \infty$. As an application, we establish uniform estimate for the resolvent (λ−L)−1$(\lambda -\mathcal {L})^{-1}$ of some generalized Schrödinger operator L:=L0+V$\mathcal {L}:=\mathcal {L}_0+V$ on the Orlicz space. An explicit representation in its operator norm on the dependence of λ∈(0,∞)$\lambda \in (0,\infty)$ is also given. [ABSTRACT FROM AUTHOR]

We introduce the class of variable anisotropic singular integral operators associated to a continuous multi-level ellipsoid cover Θ of R n introduced by Dahmen et al. (Constr Approx 31:149–194, 2010). This is an extension of the classical isotropic singular integral operators on R n of arbitrary smoothness and their anisotropic analogues for general expansive matrices introduced by the first author Bownik (Mem Am Math Soc 164(781):1–122, 2003). We establish the boundedness of variable anisotropic singular integral operators T on the Hardy spaces with pointwise variable anisotropy H p (Θ) , which were developed by Dekel et al. (J Fourier Anal Appl 17:1066–1107, 2011). In contrast with the general theory of Hardy spaces on spaces of homogenous type, our results work in the full range 0 < p ≤ 1 . [ABSTRACT FROM AUTHOR]

In this paper, the authors introduce a class of mixed-norm Herz spaces, Ė q → α → , p → (ℝ n) , which is a natural generalization of mixed-norm Lebesgue spaces and some special cases of which naturally appear in the study of the summability of Fourier transforms on mixed-norm Lebesgue spaces. The authors also give their dual spaces and obtain the Riesz–Thorin interpolation theorem on Ė q → α → , p → (ℝ n). Applying these Riesz–Thorin interpolation theorem and using some ideas from the extrapolation theorem, the authors establish both the boundedness of the Hardy–Littlewood maximal operator and the Fefferman–Stein vector-valued maximal inequality on Ė q → α → , p → (ℝ n). As applications, the authors develop various real-variable theory of Hardy spaces associated with Ė q → α → , p → (ℝ n) by using the existing results of Hardy spaces associated with ball quasi-Banach function spaces. These results strongly depend on the duality of Ė q → α → , p → (ℝ n) and the non-trivial constructions of auxiliary functions in the Riesz–Thorin interpolation theorem. [ABSTRACT FROM AUTHOR]

We characterize the translation invariant monotone collections of multi-dimensional intervals for which the analogue of Stein's criterion for the integrability of the Hardy--Littlewood maximal function is true. Namely, we characterize the collections B of the mentioned type for which the conditions ∫ [ 0 , 1 ] d M B (f) < ∞ and ∫ [ 0 , 1 ] d | f | log + | f | < ∞ are equivalent for functions f supported on the unit cube [ 0 , 1 ] d . Here M B denotes the maximal operator associated to a collection B . [ABSTRACT FROM AUTHOR]

The boundedness of the maximal operator on the upper half-plane \Pi ^{+} is established. Here \Pi ^+ is equipped with a positive Borel measure d\omega (y)dx satisfying the doubling property \omega ((0,2t))\leq C\omega ((0,t)). This result is connected to the Carleson embedding theorem, which we use to characterize the boundedness and compactness of the Volterra type integral operators on the Bergman spaces A_{\omega }^{p}(\Pi ^{+}). [ABSTRACT FROM AUTHOR]

Let (X,ρ,μ)$({\mathcal {X}},\rho ,\mu)$ be a space of homogeneous type in the sense of Coifman and Weiss, and let Y(X)$Y({\mathcal {X}})$ be a ball quasi‐Banach function space on X${\mathcal {X}}$, which supports both a Fefferman–Stein vector‐valued maximal inequality and the boundedness of the powered Hardy–Littlewood maximal operator on its associate space. The authors first introduce the Hardy space HY∗(X)$H_{Y}^*({\mathcal {X}})$, associated with Y(X)$Y({\mathcal {X}})$, via the grand maximal function and then establish its various real‐variable characterizations, respectively, in terms of radial or nontangential maximal functions, atoms or finite atoms, and molecules. As an application, the authors give the dual space of HY∗(X)$H_{Y}^*({\mathcal {X}})$, which proves to be a ball Campanato‐type function space associated with Y(X)$Y({\mathcal {X}})$. All these results have a wide range of generality and, particularly, even when they are applied to variable Hardy spaces, the obtained results are also new. The major novelties of this paper exist in that, to escape both the reverse doubling condition of μ and the triangle inequality of ρ, the authors cleverly construct admissible sequences of balls and fully use the geometrical properties of X${\mathcal {X}}$ expressed by dyadic reference points or dyadic cubes and, to overcome the difficulty caused by the lack of the good dense subset of HY∗(X)$H_{Y}^*({\mathcal {X}})$, the authors further prove that Y(X)$Y({\mathcal {X}})$ can be embedded into the weighted Lebesgue space with certain special weight and then can fully use the known results of the weighted Lebesgue space. [ABSTRACT FROM AUTHOR]

In this paper, we will study local and global Cauchy problems for the semilinear parabolic equations ∂ t u − Δ u = G (u) with initial data in Herz spaces. These spaces unify and generalize many classical function spaces such as Lebesgue spaces of power weights. Our results cover the results obtained with initial data in Lebesgue spaces. Moreover, the results in Herz spaces are a little different from the results in Lebesgue spaces. [ABSTRACT FROM AUTHOR]

In this paper we introduce Besov-type spaces with variable smoothness and integrability. We show that these spaces are characterized by the φ-transforms in appropriate sequence spaces and we obtain atomic decompositions for these spaces. Moreover the Sobolev embeddings for these function spaces are obtained. [ABSTRACT FROM AUTHOR]

Let p,q∈[1,∞], α∈ℝ, and s be a nonnegative integer. In this article, the authors introduce a new function space JN˜(p,q,s)α(X) of John–Nirenberg–Campanato type, where X denotes ℝn or any cube Q0 of ℝn with finite edge length. The authors give an equivalent characterization of JN˜(p,q,s)α(X) via both the John–Nirenberg–Campanato space and the Riesz–Morrey space. Moreover, for the particular case s=0, this new space can be equivalently characterized by both maximal functions and their commutators. Additionally, the authors give some basic properties, a good‐ λ inequality, and a John–Nirenberg‐type inequality for JN˜(p,q,s)α(X). [ABSTRACT FROM AUTHOR]

In this paper we introduce some new Lizorkin–Triebel type spaces of variable smoothness. Here the smoothness lies in a new weighted class. We give some equivalent quasinorms and their characterizations via oscillations. We show that the box splines and tensor-products of -splines are suitable to obtain the stable representations of functions of these spaces. [ABSTRACT FROM AUTHOR]

We introduce local grand variable exponent Lebesgue spaces, where the variable exponent Lebesgue space is "aggrandized" only at a given closed set F of measure zero. It is shown that, for different aggrandizers with positive Matuszewska-Orlicz indices, the corresponding local grand variable exponent Lebesgue spaces coincide. We show that the maximal operator, singular operators, and maximal singular operators are bounded in such spaces. Lastly, an application to a Dirichlet problem for the Poisson equation, where F may be chosen as the boundary of the domain, is provided within the framework of such local grand spaces. [ABSTRACT FROM AUTHOR]

The expansion of (2D) wavelet functions with respect to Lp(R²) space converging almost everywhere for 1 < p < 1throughout the length of the Lebesgue set points of space functions is investigated in this research. The convergence is established by assuming somewavelet function minimal regularity j1;j2;k1;k2 under the current description of the wavelet projection operator known as 2D Hard Sampling Operator. Note that the feature of fast decline in 2D is derived here. Another condition is used, for instance, the wavelet expansion's boundedness under the Hard Sampling Operator. The bound (limit) is governed in magnitude with respect to the maximal equality of the Hardy-Littlewood maximal operator. Some ideas presented in this work to find a new method to prove the convergence theory for new type of conditional wavelet operator. Propose some conditions for wavelets functions and there expansion can support the operator to be convergence. It also perform a comparison with the identity convergent operator is our method for achieving this convergence. [ABSTRACT FROM AUTHOR]

Let (핏 , d , μ) be a space of homogeneous type in the sense of R. R. Coifman and G. Weiss, and let X ⁢ (핏) be a ball quasi-Banach function space on 핏 . In this article, the authors introduce the weak Hardy space W ⁢ H ~ X ⁢ (핏) associated with X ⁢ (핏) via the Lusin area function. Then the authors characterize W ⁢ H ~ X ⁢ (핏) by the molecule, the grand maximal function, and the Littlewood–Paley g-function and g λ * -function. Moreover, all these results have a wide generality. Particularly, the results of this article are also new even when they are applied, respectively, to weighted Lebesgue spaces, Orlicz spaces, and variable Lebesgue spaces, which actually are new even on RD-spaces (that is, spaces of homogeneous type with additional reverse doubling condition). The main novelties of this article exist in that the authors take full advantage of the geometrical properties of 핏 expressed by both the dyadic cubes and the exponential decay of the approximations of the identity to overcome the difficulties caused by the deficiencies of both the explicit expression of the quasi-norm of X ⁢ (핏) and the reverse doubling condition of μ, and that the authors use the tent space on 핏 × ℤ to characterize W ⁢ H ~ X ⁢ (핏) by the Littlewood–Paley g λ * -function, where the range of λ might be best possible in some cases. [ABSTRACT FROM AUTHOR]

We investigate the Hardy space H^1_L associated with a self-adjoint operator L defined in a general setting by Hofmann, Lu, Mitrea, Mitrea, and Yan [Mem. Amer. Math. Soc. 214 (2011), pp. vi+78]. We assume that there exists an L-harmonic non-negative function h such that the semigroup \exp (-tL), after applying the Doob transform related to h, satisfies the upper and lower Gaussian estimates. Under this assumption we describe an illuminating characterisation of the Hardy space H^1_L in terms of a simple atomic decomposition associated with the L-harmonic function h. Our approach also yields a natural characterisation of the BMO-type space corresponding to the operator L and dual to H^1_L in the same circumstances. The applications include surprisingly wide range of operators, such as: Laplace operators with Dirichlet boundary conditions on some domains in {\mathbb {R}^n}, Schrödinger operators with certain potentials, and Bessel operators. [ABSTRACT FROM AUTHOR]

Recently, both the bilinear decompositions h 1 (ℝ n) × bmo (ℝ n) ⊂ L 1 (ℝ n) + h ∗ Φ (ℝ n) and h 1 (ℝ n) × bmo (ℝ n) ⊂ L 1 (ℝ n) + h log (ℝ n) were established. In this paper, the authors prove in some sense that the former is sharp, while the latter is not. To this end, the authors first introduce the local Orlicz-slice Hardy space which contains h ∗ Φ (ℝ n) , a variant of the local Orlicz Hardy space, introduced by Bonami and Feuto as a special case, and obtain its dual space by establishing its characterizations via atoms, finite atoms, and various maximal functions, which are new even for h ∗ Φ (ℝ n). The relationship h ∗ Φ (ℝ n) ⊊ h log (ℝ n) is also clarified. [ABSTRACT FROM AUTHOR]

We obtain the characterizations of commutators of several versions of maximal functions on spaces of homogeneous type. In addition, with the aid of interpolation theory, we provide weighted version of the commutator theorems by establishing new characterizations of the weighted BMO space. Finally, a concrete example shows that the local version of commutators also has an independent interest. [ABSTRACT FROM AUTHOR]

Let M be the uncentered Hardy-Littlewood maximal operator, or the dyadic maximal operator, and let d ≥ 1. We prove that for a set E ⊂ ∨d of finite perimeter, the bound var M1E ≤ Cd var 1E holds. We also prove this for the local maximal operator. [ABSTRACT FROM AUTHOR]

Let Δ S n denote the Laplace-Beltrami operator on the n -dimensional unit sphere S n , n ≥ 2. In this paper we show that ‖ e i t Δ S n f ‖ L 4 ([ 0 , 2 π) × S n) ≤ C ‖ f ‖ W α , 4 (S n) holds if α > (n − 2) / 4. The range of α is sharp in the sense that the estimate fails for α < (n − 2) / 4. As a consequence, we obtain space-time L p -estimates for e i t Δ S n for 2 ≤ p ≤ ∞. We also prove that the maximal operator f → sup 0 ≤ t < 2 π ⁡ | e i t Δ S n f | is bounded from W α , 2 (S n) to L 6 n / (3 n − 2) (S n) for α > 1 / 3 whenever f are zonal functions on S n. [ABSTRACT FROM AUTHOR]

Let L be a one-to-one operator of type w, with w ∈ [ 0 , π / 2 ] , satisfying the Davies–Gaffney estimates. For α ∈ (0 , ∞) and p ∈ (0 , ∞) and under the condition that q (·) : R n ⟶ [ 1 , ∞) satisfies the globally log-Hölder continuity condition, we introduce the Herz-type Hardy space with variable exponents associated to L and establish its molecular decomposition. The atomic characterization and maximal function characterizations of the space are proved under the assumption that L is a non-negative self-adjoint operator on L 2 (R n) whose heat kernels satisfy the Gaussian upper bound estimates. All the results are new even for the constant case. [ABSTRACT FROM AUTHOR]

We establish the boundedness of generalized fractional integral operators I ρ on variable exponent Morrey spaces of an integral form L p (·) , ω (G) , where ρ (x , r) and ω (x , r) are general functions satisfying certain conditions. [ABSTRACT FROM AUTHOR]

We establish a Maz’ya type capacitary inequality which resolves a special case of a conjecture by David R. Adams. As a consequence, we obtain several equivalent norms for Choquet integrals associated to Bessel or Riesz capacities. This enables us to obtain bounds for the Hardy–Littlewood maximal function in a sub- linear setting. [ABSTRACT FROM AUTHOR]

We establish the Gagliardo–Nirenberg and Sobolev interpolation inequalities on Besov spaces. In particular, we improve the recent results of Miyazaki where he proved the inequalities on BMO spaces (see Yoichi Miyazaki [Proc. Amer. Math. Soc. 148 (2020), pp. 4257–4261]). [ABSTRACT FROM AUTHOR]

We study sharp growth conditions for the boundedness of the Hardy-Littlcwood maximal function in the generalized Orlicz spaces. We assume that the generalized Orhcz function 1. Moreover we show that the Hardy-Littlcwood maximal function is bounded from the generalized Orlicz space Lip(Rn) to itself if and only if f is weakly equivalent to a generalized Orlicz function - satisfying (AO), (Al) and (A2) for which -(a>,i)/-p is almost increasing for all t > O and some p > 1. [ABSTRACT FROM AUTHOR]

In this paper we establish inclusions and noninclusions between various Hardy type spaces on noncompact Riemannian manifolds M with Ricci curvature bounded from below, positive injectivity radius and spectral gap. Our first main result states that, if L is the positive Laplace–Beltrami operator on M , then the Riesz–Hardy space H R 1 (M) is the isomorphic image of the Goldberg type space h 1 (M) via the map L 1 / 2 (I + L) − 1 / 2 , a fact that is false in R n. Specifically, H R 1 (M) agrees with the Hardy type space X 1 / 2 (M) recently introduced by the first three authors; as a consequence, we prove that H R 1 (M) does not admit an atomic characterisation. Noninclusions are mostly proved in the special case where the manifold is a Damek–Ricci space S. Our second main result states that H R 1 (S) , the heat Hardy space H H 1 (S) and the Poisson–Hardy space H P 1 (S) are mutually distinct spaces, a fact which is in sharp contrast to the Euclidean case, where these three spaces agree. [ABSTRACT FROM AUTHOR]

Let L be a homogeneous divergence form higher order elliptic operator with complex bounded measurable coefficients on Rn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {R}}^n$$\end{document} and X a ball quasi-Banach function space on Rn\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$${\mathbb {R}}^n$$\end{document} satisfying some mild assumptions. Denote by HX,L(Rn)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H_{X,\, L}({\mathbb {R}}^n)$$\end{document} the Hardy space, associated with both L and X, which is defined via the Lusin area function related to the semigroup generated by L. In this article, the authors establish both the maximal function and the Riesz transform characterizations of HX,L(Rn)\documentclass[12pt]{minimal}\usepackage{amsmath}\usepackage{wasysym}\usepackage{amsfonts}\usepackage{amssymb}\usepackage{amsbsy}\usepackage{mathrsfs}\usepackage{upgreek}\setlength{\oddsidemargin}{-69pt}\begin{document}$$H_{X,\, L}({\mathbb {R}}^n)$$\end{document}. The results obtained in this article have a wide range of generality and can be applied to the weighted Hardy space, the variable Hardy space, the mixed-norm Hardy space, the Orlicz–Hardy space, the Orlicz-slice Hardy space, and the Morrey–Hardy space, associated with L. In particular, even when L is a second order divergence form elliptic operator, both the maximal function and the Riesz transform characterizations of the mixed-norm Hardy space, the Orlicz-slice Hardy space, and the Morrey–Hardy space, associated with L, obtained in this article, are completely new. [ABSTRACT FROM AUTHOR]

We establish the strong L p -inequality for the maximal function over hyper-surface based on Euclidean ball sup t > 0 1 | B n | ∫ B n f (x - t u , y - t 2 | u | 2) d u . We prove the L p -boundedness of the maximal function for p > 1 . Specially, when p = 2 , we prove the L 2 -boundedness of the maximal function can be controlled by C n 7 8 , where C is independent of n. [ABSTRACT FROM AUTHOR]

Let (X,d,μ) be a space of homogeneous type, with upper dimension μ, in the sense of R. R. Coifman and G. Weiss. Let η be the Hölder regularity index of wavelets constructed by P. Auscher and T. Hytönen. In this article, the authors introduce the local Hardy space h∗,p(X) via local grand maximal functions and also characterize h∗,p(X) via local radial maximal functions, local non‐tangential maximal functions, local atoms and local Littlewood–Paley functions. Furthermore, the authors establish the relationship between the global and the local Hardy spaces. Finally, the authors also obtain the finite atomic characterizations of h∗,p(X). As an application, the authors give the dual spaces of h∗,p(X) when p∈(ω/(ω+η),1), which further completes the result of G. Dafni and H. Yue on the dual space of h∗,1(X). This article also answers the question of R. R. Coifman and G. Weiss on the nonnecessity of any additional geometric assumptions except the doubling condition for the radial maximal function characterization of Hcw1(X) when μ(X)<∞. [ABSTRACT FROM AUTHOR]

In this paper, we investigated the commutators of Hardy–Littlewood maximal function with symbol function b in generalized BMO spaces. Some characterizations of generalized BMO spaces are obtained via the strong and weak boundedness of the commutators on Orlicz–Morrey space. [ABSTRACT FROM AUTHOR]

We introduce the variable integral and the smooth exponent Besov spaces associated to non-negative self-adjoint operators. Then we give the equivalent norms via the Peetre type maximal functions and atomic decomposition of these spaces. [ABSTRACT FROM AUTHOR]

In this paper, by using the idea of linearizing maximal operators originated by Charles Fefferman and the TT* method of Stein-Wainger, we establish a weighted inequality for vector valued maximal Carleson type operators with singular kernels proposed by Andersen and John on the weighted Lorentz spaces with vector-valued functions. [ABSTRACT FROM AUTHOR]

In a metric measure space (X, d, µ), our first aim in this paper is to discuss the weak estimates for the maximal and Riesz potential operators in the non-homogeneous central Morrey type space M1,q,a(X) (about x0 ∈ X) of all measurable functions f on X satisfying ‖f‖M1,q,a(X) = (∞ 1 r −a ‖f‖L1(B(x0,r)))q dr r (1/q < ∞ for a ≥ 0 and 0 < q < ∞; when q = ∞, we apply a necessary modification. To do this, we consider the family WMϕ,q,a(X) of all measurable functions f ∈ L 1 loc(X) such that ‖f‖WMϕ,q,a(X) = sup λ>0 λ ∞ 1 r −aϕ −1B(x0,r) χEf (λ)(x) dµ(x))q dr/r)1/q < ∞, where ϕ is a general function satisfying certain conditions and χEf (λ) denotes the characteristic function of Ef (λ) = {x ∈ X : |f(x)| > λ}. In connection with M1,q,a(X), we treat the complementary space N∞,q,a(X) of all measurable functions f on X satisfying ‖f‖N∞,q,a(X) = ‖f‖L∞(B(x0,2)) + ) ∞ 1 r a ‖f‖L1(X\B(x0,r))(q dr r )1/q < ∞. [ABSTRACT FROM AUTHOR]

In a metric measure space (X, d, µ), our first aim in this paper is to discuss the weak estimates for the maximal and Riesz potential operators in the non-homogeneous central Morrey type space M1,q,a(X) (about x0 = X) of all measurable functions f on X satisfying ... for a = 0 and 0 < q < 8; when q = 8, we apply a necessary modification. To do this, we consider the family WM',q,a(X) of all measurable functions f = L¹loc(X) such that ... where' is a general function satisfying certain conditions and Ef () denotes the characteristic function of Ef () = {x = X: |f(x)| > }. In connection with M1,q,a(X), we treat the complementary space N1,q,a(X) of all measurable functions f on X satisfying ... [ABSTRACT FROM AUTHOR]

In this paper, we study Triebel–Lizorkin-type spaces with variable smoothness and integrability. We show that our space is well-defined, i.e., independent of the choice of basis functions and we obtain their atomic characterization. Moreover, the Sobolev embeddings for these function spaces are obtained. [ABSTRACT FROM AUTHOR]

We prove in this paper one weight norm inequalities for some positive Bergman-type operators. [ABSTRACT FROM AUTHOR]

We give an abstract argument that an a priori Fourier restriction estimate for a certain choice of exponents automatically implies maximal and variational Fourier restriction estimates. These, in turn, provide pointwise and quantitative interpretations of restriction of the Fourier transform to the hypersurface in question. [ABSTRACT FROM AUTHOR]

We introduce a class of operators on abstract measure spaces that unifies the Calderón-Zygmund operators on spaces of homogeneous type, the maximal functions, the martingale transforms, and Carleson operators. We prove that such operators can be dominated by simple sparse operators with a definite form of the domination constant. Applying these estimates, we improve on several results obtained by different authors in recent years. [ABSTRACT FROM AUTHOR]

It is shown that the radial averaging operator T ω ⁢ (f) ⁢ (z) = ∫ | z | 1 f ⁢ (s ⁢ z | z | ) ⁢ ω ⁢ (s) ⁢ 𝑑 s ω ^ ⁢ (z) , ω ^ ⁢ (z) = ∫ | z | 1 ω ⁢ (s) ⁢ 𝑑 s , T_{\omega}(f)(z)=\frac{\int_{|z|}^{1}f\bigl{(}s\frac{z}{|z|}\bigr{)}\omega(s)% \,ds}{\widehat{\omega}(z)},\quad\widehat{\omega}(z)=\int_{|z|}^{1}\omega(s)\,ds, induced by a radial weight ω on the unit disc 𝔻 {\mathbb{D}} , is bounded from the weighted Bergman space A ν p {A^{p}_{\nu}} , where 0 < p < ∞ {0 0 , \eta(\{z\in\mathbb{D}:|T_{\omega}(f)(z)|\geq\lambda\})\lesssim\lambda^{-p}\|f% \|_{L^{p}_{\nu}}^{p},\quad\lambda>0, are established for arbitrary radial weights ω, ν and η. Moreover, differences and interrelationships between the cases A ν p → L ν p {A^{p}_{\nu}\to L^{p}_{\nu}} , L ν p → L ν p {L^{p}_{\nu}\to L^{p}_{\nu}} and L ν p → L ν p , ∞ {L^{p}_{\nu}\to L^{p,\infty}_{\nu}} are analyzed. [ABSTRACT FROM AUTHOR]

In this paper, the authors first introduce a class of Orlicz-slice spaces which generalize the slice spaces recently studied by Auscher et al. Based on these Orlicz-slice spaces, the authors then introduce a new kind of Hardy-type spaces, the Orlicz-slice Hardy spaces, via the radial maximal functions. This new scale of Orlicz-slice Hardy spaces contains the variant of the Orlicz–Hardy space of Bonami and Feuto as well as the Hardy-amalgam space of de Paul Ablé and Feuto as special cases. Their characterizations via the atom, the molecule, various maximal functions, the Poisson integral and the Littlewood–Paley functions are also obtained. As an application of these characterizations, the authors establish their finite atomic characterizations, which further induce a description of their dual spaces and a criterion on the boundedness of sublinear operators from these Orlicz-slice Hardy spaces into a quasi-Banach space. Then, applying this criterion, the authors obtain the boundedness of δ -type Calderón–Zygmund operators on these Orlicz-slice Hardy spaces. All these results are new even for slice Hardy spaces and, moreover, for Hardy-amalgam spaces, the Littlewood–Paley function characterizations, the dual spaces and the boundedness of δ -type Calderón–Zygmund operators on these Hardy-type spaces are also new. [ABSTRACT FROM AUTHOR]

Let p(⋅) : ℝn → [0, 1] be a variable exponent function satisfying the globally log-Hölder continuous condition, and L a one-to-one operator of type ω in L2 ⁢ (ℝ n), with ω ∈ [0, π /2), which has a bounded holomorphic functional calculus and satisfies the Davies–Gaffney estimates. In this article, we introduce the variable weak Hardy space WHLp(⋅) (ℝnn), associated with L via the corresponding square function. Its molecular characterization is then established by means of the atomic decomposition of the variable weak tent space WTp (⋅) ⁢ (ℝ+n+1), which is also obtained in this article. In particular, when L is non-negative and self-adjoint, we obtain the atomic characterization of WHLp(⋅) (ℝnn). As an application of the molecular characterization, when L is the second-order divergence form elliptic operator with complex bounded measurable coefficients, we prove that the associated Riesz transform ∇ ⁡ L−1/2 is bounded from WHLp(⋅) (ℝnn) to the variable weak Hardy space WTp (⋅) ⁢ (ℝ+n+1). Moreover, when L is non-negative and self-adjoint with the kernels of {e−tL} t > 0 satisfying the Gaussian upper bound estimates, the atomic characterization of WHLp(⋅) (ℝnn) is further used to characterize this space via non-tangential maximal functions. [ABSTRACT FROM AUTHOR]

Abstract We prove endpoint bounds for derivatives of fractional maximal functions with either smooth convolution kernel or lacunary set of radii in dimensions n ≥ 2. We also show that the spherical fractional maximal function maps L p into a first order Sobolev space in dimensions n ≥ 5. [ABSTRACT FROM AUTHOR]

We obtain the exponential integrability of the maximal function, the quadratic variation and the conditional quadratic variation of bounded martingales and exponential integrable martingales. [ABSTRACT FROM AUTHOR]

Abstract A now classical result in the theory of variable Lebesgue spaces due to Lerner (2005) is that a modular inequality for the Hardy–Littlewood maximal function in L p (⋅) (R n) holds if and only if the exponent is constant. We generalize this result and give a new and simpler proof. We then find necessary and sufficient conditions for the validity of the weaker modular inequality ∫ Ω M f (x) p (x) d x ⩽ c 1 ∫ Ω | f (x) | q (x) d x + c 2 , where c 1 , c 2 are non-negative constants and Ω is any subset of R n. As a corollary we get sufficient conditions for the modular inequality ∫ Ω | T f (x) | p (x) d x ⩽ c 1 ∫ Ω | f (x) | q (x) d x + c 2 , where T is any operator that is bounded on L p (Ω) , 1 < p < ∞. [ABSTRACT FROM AUTHOR]

We extend the main result of Harrow, Kolla, and Schulman—the existence of dimension-free L² -bounds for the spherical maximal function in the hypercube, {0, 1}N—to all Lp, p > 1. Our approach is motivated by the spectral technique developed by Nevo and Stein, and by Stein, in the context of pointwise ergodic theorems on general groups. We provide an example which demonstrates that no dimension-free weak-type 1 − 1 bound exists at the endpoint. [ABSTRACT FROM AUTHOR]

We explore some variants of the Boman covering lemma, and their relationship to the boundedness properties of the maximal operator. Let 1 < p < ∞ and let q be its conjugate exponent. We prove that the strong type ( q , q ) of the uncentered maximal operator, by itself, implies certain generalizations of the Boman covering lemma for the exponent p , and in turn, these generalizations entail the weak type ( q , q ) of the centered maximal operator. We show by example that it is possible for the uncentered maximal operator to be unbounded for all 1 < s < ∞ , while the conclusion of the lemma holds for every 1 < p < ∞ ; thus, the latter condition is much weaker. Also, the boundedness of the centered maximal operator entails weak versions of the lemma. [ABSTRACT FROM AUTHOR]

Shah and Debnath [Tight wavelet frames on local fields, Analysis, 33 (2013), 293-307] have derived an explicit method for constructing tight wavelet frames on local fields using the machinery of unitary extension principle. Continuing our investigation of wavelet frames on local fields, this paper deals with the establishment of complete characterization of functions in Lebesgue spaces Lp(K), 1 < p < ∞, in terms of their wavelet frame coefficients. [ABSTRACT FROM AUTHOR]

The goal of this article is to establish L p -estimates for maximal functions associated with nonisotropic dilations of hypersurfaces in R 3 . Several results have already been obtained by Greenleaf [5] , Iosevich–Sawyer [9] , Ikromov–Kempe–Müller [6] and Zimmermann [28] , but for some situations such as the hypersurface parameterized as the graph of a smooth function Φ ( x 1 , x 2 ) = x 2 d ( 1 + O ( x 2 m ) ) near the origin, where d ≥ 2 , m ≥ 1 , and associated dilations δ t ( x ) = ( t a x 1 , t x 2 , t d x 3 ) for an arbitrary real number a > 0 , the question was open until recently. In fact, such problems do arise already in lower dimensions. For instance, we consider the curve γ ( x ) = ( x , x 2 ( 1 + ϕ ( x ) ) ) and associated dilations δ t ( x ) = ( t x 1 , t 2 x 2 ) . If ϕ ≡ 0 , then the corresponding maximal function is the maximal function along parabolas in the plane, which is very well understood due to the work by Nagel–Riviere–Wainger [17] and others. If ϕ ≠ 0 and ϕ ( x ) = O ( x m ) , m ≥ 1 , the problem was open until recently. We observe that in the study of the maximal function related to the mentioned curve γ ( x ) and associated dilations, we will consider a family of corresponding Fourier integral operators which fail to satisfy the “cinematic curvature condition” uniformly, which means that classical local smoothing estimates established by Mockenhaupt–Seeger–Sogge could not be directly applied to our problem. In this article, we develop new ideas to establish sharp L p -estimates for the maximal function related to the curve γ ( x ) with associated dilations in the plane. Later, we generalize the result to curves of finite type d ( d ≥ 2 ) and associated dilations δ t ( x ) = ( t x 1 , t d x 2 ) . Furthermore, we also obtain L p -estimates for the maximal function related to the mentioned hypersurface Φ ( x 1 , x 2 ) in R 3 with associated dilations. [ABSTRACT FROM AUTHOR]

Abstract: Consider the Bessel operator with a potential on L 2 ( ( 0 , ∞ ) , x α d x ), namely L f ( x ) = − f ′ ′ ( x ) − α x f ′ ( x ) + V ( x ) f ( x ) .We assume that α > 0 and V ∈ L l o c 1 ( ( 0 , ∞ ) , x α d x ) is a nonnegative function. By definition, a function f ∈ L 1 ( ( 0 , ∞ ) , x α d x ) belongs to the Hardy space H 1 ( L ) if sup t > 0 e − t L f ∈ L 1 ( ( 0 , ∞ ) , x α d x ) .Under certain assumptions on V we characterize the space H 1 ( L ) in terms of atomic decompositions of local type. In the second part we prove that this characterization can be applied to L for α ∈ ( 0 , 1 ) with no additional assumptions on the potential V. [ABSTRACT FROM AUTHOR]

Let, (χ, d, μ) be a metric measure space of homogeneous type and p(.): χ→(0.1] a variable exponent function satisfying the globally log-Holder continuous condition. Assume that L is a one-to-one operator of type w on L2(χ), with w ∊ [0, π/2), which has a bounded holomorphic functional calculus, and whose heat, kernel satisfies the Davies-Gaffney estimates. In this article, the authors introduce the variable Hardy space Hp(.)L(χ) associated with L. Then the authors establish the molecular characterization of Hp(.)L(χ) via the atomic decomposition of variable tent spaces and show that the dual space of Hp(.)L(χ) is the BMO-type space BMOP(.)L*(χ), where L* denotes the adjoint operator of L on L2(χ). In particular, if L is a non-negative self-adjoint operator whose heat kernel has a Gaussian upper bound, the authors then obtain the lion-tangential and the radial maximal function characterizations of Hp(.)L(χ) via establishing its atomic characterization. [ABSTRACT FROM AUTHOR]