Sharp one-weight and two-weight bounds for maximal operators

<p>In the setting of a Heisenberg group, we first studied the sharp weak estimate for the $ n $-dimensional fractional Hardy operator from $ L^p $ to $ L^{q, \infty} $. Next, we studied the sharp bounds for the $ m $-linear $ n $-dimensional integral operator with a kernel on weighted Lebesgue spaces. As an application, the sharp bounds for Hardy, Hardy-Littlewood-Pólya, and Hilbert operators on weighted Lebesgue spaces were obtained. Finally, according to the previous steps, we also found the estimate for the Hausdorff operator on weighted $ L^p $ spaces.</p>

Boundedness of the Commutators of Fractional Maximal Operator on Morrey-Herz Space on Spaces of Homogeneous Type

This work studies boundedness properties of the fractional maximal operator on metric measure spaces under standard assumptions on the measure. The main motivation is to show that the fractional maximal operator has similar smoothing and mapping properties as the Riesz potential. Instead of the usual fractional maximal operator, we also consider a so-called discrete maximal operator which has better regularity. We study the boundedness of the discrete fractional maximal operator in Sobolev, Holder, Morrey, and Campanato spaces. We also prove a version of the Coifman– Rochberg lemma for the fractional maximal function.

Our aim in this paper is to deal with the boundedness of the central Hardy-Littlewood maximal operator on Herz spaces with radial variable exponent. As a special case, we show the boundedness of the maximal operator in weighted Lebesgue spaces with radial variable exponent. We also discuss the integrability of the maximal functions in such spaces.

Let 0 ≤ α < n , M α be the fractional maximal operator, M ♯ be the sharp maximal operator and b be the locally integrable function. Denote by [ b , M α ] and [ b , M ♯ ] be the commutators of the fractional maximal operator M α and the sharp maximal operator M ♯ . In this paper, we show some necessary and sufficient conditions for the boundedness of the commutators [ b , M α ] and [ b , M ♯ ] on slice spaces when the function b is the Lipschitz function, by which some new characterizations of the non-negative Lipschitz function are obtained.

We investigate the behaviour of the classical (non-smooth) Hardy-Littlewood maximal operator in the context of Banach lattices. We are mainly concerned with end-point results for p = ∞. Naturally, the main role is played by the space BMO. We analyze the range of the maximal operator in BMOx. This turns out to depend strongly on the convexity of the Banach lattice . We apply these results to study the behaviour of the commutators associated to the maximal operator. We also consider the parallel results for the maximal fractional integral operator.

In this paper a two-weight criterion for multidimensional Hardy type operator and its dual operator acting from weighted Lebesgue spaces into weighted Musielak-Orlicz spaces is proved. As application we prove the boundedness of multidimensional geometric mean operator in the weighted Musielak-Orlicz spaces. In particular, from obtained results implies the boundedness of multidimensional Hardy operator and its dual operator acting from usual weighted Lebesgue spaces into weighted variable Lebesgue spaces. In this paper we establish integral-type neces- sary and sufficient condition on weights, which provides the boundedness of the multidimensional Hardy type operator from weighted Lebesgue spaces into weighted Musielak-Orlicz spaces.

<abstract> In this paper, the author studies the boundedness for a large class of sublinear operators $ T_\alpha, \alpha\in[0, n) $ generated by Calderón-Zygmund operators ($ \alpha=0 $) and generated by fractional integral operator ($ \alpha&gt;0 $) on generalized mixed Morrey spaces $ M^\varphi_{\vec{q}}({\mathbb R}^n) $. Moreover, the boundeness for commutators of $ T_\alpha, \alpha\in[0, n) $ on generalized mixed Morrey spaces $ M^\varphi_{\vec{q}}({\mathbb 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 operator and their commutators on generalzied mixed Morrey spaces. </abstract>

Let [Formula: see text], [Formula: see text] and [Formula: see text] be a singular or fractional integral operator with homogeneous kernel [Formula: see text]. In this paper, a CMO type space [Formula: see text] is introduced and studied. In particular, the relationship between [Formula: see text] and the Lipchitz space [Formula: see text] is discussed. Moreover, a necessary condition of restricted boundedness of the iterated commutator [Formula: see text] on weighted Lebesgue spaces via functions in [Formula: see text], and an equivalent characterization of the compactness for [Formula: see text] via functions in [Formula: see text] are obtained. Some results are new even in the unweighted setting for the first-order commutators.

In this paper the authors give a sufficient condition such that the commutator generated by the weighted Lipschitz function and the fractional integral operator with homogeneous kernel satisfying certain Dini condition is bounded on weighted Lebesgue spaces.

In this paper, we establish BMO estimates for generalized commutators of rough fractional maximal and integral operators on generalized weighted Morrey spaces, respectively.

The Laguerre transform is applied to the convolution product of functions of a real argument (over the time axis) with values in Hilbert spaces. The main results have been obtained by establishing a relationship between the Laguerre and Laplace transforms over the time variable with respect to the elements of Lebesgue weight spaces. This relationship is built using a special generating function. The obtained dependence makes it possible to extend the known properties of the Laplace transform to the case of the Laguerre transform. In particular, this approach concerns the transform of a convolution of functions.&#x0D; The Laguerre transform is determined by a system of Laguerre functions, which forms an orthonormal basis in the weighted Lebesgue space. The inverse Laguerre transform is constructed as a Laguerre series. It is proven that the direct and the inverse Laguerre transforms are mutually inverse operators that implement an isomorphism of square-integrable functions and infinite squares-summable sequences.&#x0D; The concept of a q-convolution in spaces of sequences is introduced as a discrete analogue of the convolution products of functions. Sufficient conditions for the existence of convolutions in the weighted Lebesgue spaces and in the corresponding spaces of sequences are investigated. For this purpose, analogues of Young’s inequality for such spaces are proven. The obtained results can be used to construct solutions of evolutionary problems and time-dependent boundary integral equations.

We prove that if the exponent function p(\cdot) satisfies log-Hölder continuity conditions locally and at infinity, then the fractional maximal operator M_\alpha , 0 &lt; \alpha &lt; n , maps L^{p(\cdot)} to L^{q(\cdot)} , where \frac{1}{p(x)}-\frac{1}{q(x)}=\frac{\alpha}{n} . We also prove a weak-type inequality corresponding to the weak (1,n/(n-\alpha)) inequality for M_\alpha . We build upon earlier work on the Hardy-Littlewood maximal operator by Cruz-Uribe, Fiorenza and Neugebauer [The maximal function on variable L^p spaces. Ann. Acad. Sci. Fenn. Math. 28 (2003), 223-238]. As a consequence of these results for M_\alpha , we show that the fractional integral operator I_\alpha satisfies the same norm inequalities. These in turn yield a generalization of the Sobolev embedding theorem to variable L^p spaces.

We give a characterization of the weights 𝑢(·) and 𝑣(·) for which the fractional maximal operator 𝑀𝑠 is bounded from the weighted Lebesgue spaces 𝐿𝑝(𝑙𝑟, 𝑣𝑑𝑥) into 𝐿𝑞(𝑙𝑟, 𝑢𝑑𝑥) whenever 0 ≤ 𝑠 &lt; 𝑛, 1 &lt; 𝑝, 𝑟 &lt; ∞, and 1 ≤ 𝑞 &lt; ∞.

Vertical square functions and other operators associated with an elliptic operator