Weighted estimates for integral operators on local type spaces

We prove the weighted boundedness for a family of integral operators on Lebesgue spaces and local type spaces. To this end we show that can be controlled by the Calderón operator and a local maximal operator. This approach allows us to characterize the power weighted boundedness on Lebesgue spaces.     

Complex interpolation of Herz‐type Triebel–Lizorkin spaces

We study complex interpolation of Herz‐type Triebel–Lizorkin spaces by using the Calderón product method. Additionally we present complex interpolation between Herz‐type Triebel–Lizorkin spaces and Triebel–Lizorkin spaces . Moreover, we apply these results to obtain the complex interpolation of Triebel–Lizorkin spaces equipped with power weights and between (or ) spaces and Herz spaces.     

Hausdorff type operators on the power weighted Hardy spaces

In this paper, we study the high‐dimensional Hausdorff type operators and establish their boundedness on the power weighted Hardy spaces for . As a consequence, we obtain that the Hausdorff type operator is bounded on if Φ is the Gauss function, or the Poisson function.     

Schrödinger type singular integrals: weighted estimates for

A critical radius function ρ assigns to each a positive number in a way that its variation at different points is somehow controlled by a power of the distance between them. This kind of function appears naturally in the harmonic analysis related to a Schrödinger operator with V a non‐negative potential satisfying some specific reverse Hölder condition. For a family of singular integrals associated with such critical radius function, we prove boundedness results in the extreme case . On one side we obtain weighted weak (1, 1) results for a class of weights larger than Muckenhoupt class A₁. On the other side, for the same weights, we prove continuity from appropriate weighted Hardy spaces into weighted L¹. To achieve the latter result we define weighted Hardy spaces by means of a ρ‐localized maximal heat operator. We obtain a suitable atomic decomposition and a characterization via ρ‐localized Riesz Transforms for these spaces. For the case of ρ derived from a Schrödinger operator, we obtain new estimates for many of the operators appearing in  27 .     

Calderón‐Zygmund operators with non‐diagonal singularity

In this paper, we introduce a class of singular integral operators which generalize Calderón‐Zygmund operators to the more general case, where the set of singular points of the kernel need not to be the diagonal, but instead, it can be a general hyper curve. We show that such operators have similar properties as ordinary Calderón‐Zygmund operators. In particular, we prove that they are of weak‐type (1, 1) and strong type for .     

Weighted boundedness of the 2‐fold product of Hardy–Littlewood maximal operators

We study new weighted estimates for the 2‐fold product of Hardy–Littlewood maximal operators defined by . This operator appears very naturally in the theory of bilinear operators such as the bilinear Calderón–Zygmund operators, the bilinear Hardy–Littlewood maximal operator introduced by Calderón or in the study of pseudodifferential operators. To this end, we need to study Hölder's inequality for Lorentz spaces with change of measures Unfortunately, we shall prove that this inequality does not hold, in general, and we shall have to consider a weaker version of it.     

BMO spaces related to Laguerre semigroups

For the system of Laguerre functions we define a suitable BMO space from the atomic version of the Hardy space considered by Dziubański in 7 , where is the maximal operator of the heat semigroup associated to that Laguerre system. We prove boundedness of over a weighted version of that BMO, and we extend such result to other systems of Laguerre functions, namely and . To do that, we work with a more general family of weighted BMO‐like spaces that includes those associated to all of the above mentioned Laguerre systems. In this setting, we prove that the local versions of the Hardy‐Littlewood and the heat‐diffusion maximal operators turn to be bounded over such family of spaces for weights. This result plays a decisive role in proving the boundedness of Laguerre semigroup maximal operators.     

Necessary and sufficient conditions for the boundedness of weighted Hardy operators in Hölder spaces

We study one‐ and multi‐dimensional weighted Hardy operators on functions with Hölder‐type behavior. As a main result, we obtain necessary and sufficient conditions on the power weight under which both the left and right hand sided Hardy operators map, roughly speaking, functions with the Hölder behavior only at the singular point to functions differentiable for and bounded after multiplication by a power weight. As a consequence, this implies necessary and sufficient conditions for the boundedness in Hölder spaces due to the corresponding imbeddings. In the multi‐dimensional case we provide, in fact, stronger Hardy inequalities via spherical means. We also separately consider the case of functions with Hölder‐type behavior at infinity (Hölder spaces on the compactified ).     

Generalized fractional integrals on generalized Morrey spaces

On generalized Morrey spaces with variable exponent and variable growth function the boundedness of generalized fractional integral operators is established, where . The result is a generalization of the theorems of Adams [1] (1975) and Gunawan [11] (2003). Moreover, we prove weak type boundedness. To do this we first prove the boundedness of the Hardy‐Littlewood maximal operator on the generalized Morrey spaces.     

Characterization of Triebel–Lizorkin type spaces with variable exponents via maximal functions,local means and non‐smooth atomic decompositions

In this paper we study the maximal function and local means characterizations and the non‐smooth atomic decomposition of the Triebel–Lizorkin type spaces with variable exponents . These spaces were recently introduced by Yang et al. and cover the Triebel–Lizorkin spaces with variable exponents as well as the classical Triebel–Lizorkin spaces , even the case when . Moreover, covered by this scale are also the Triebel–Lizorkin‐type spaces with constant exponents which, in turn cover the Triebel–Lizorkin–Morrey spaces. As an application we obtain a pointwise multiplier assertion for those spaces.     

Fourier Multipliers and Littlewood‐Paley for modulation spaces

In this paper we have studied Fourier multipliers and Littlewood‐Paley square functions in the context of modulation spaces. We have also proved that any bounded linear operator from modulation space into itself possesses an l₂‐valued extension. This is an analogue of a well known result due to Marcinkiewicz and Zygmund on classical ‐spaces.     

Boundedness of composition operators with mixed homogeneities

Suppose that T₁ is a Calderón–Zygmund operator with isotropic homogeneity and T₂ is a Calderón–Zygmund operator with non‐isotropic homogeneity. In this note, the boundedness of the composition operator on the Hardy space is presented. The results in this paper extend earlier related results on convolution operators to non‐convolution setting.     

Sobolev‐type inequalities for potentials in grand variable exponent Lebesgue spaces

We introduce a new scale of grand variable exponent Lebesgue spaces denoted by . These spaces unify two non‐standard classes of function spaces, namely, grand Lebesgue and variable exponent Lebesgue spaces. The boundedness of integral operators of Harmonic Analysis such as maximal, potential, Calderón–Zygmund operators and their commutators are established in these spaces. Among others, we prove Sobolev‐type theorems for fractional integrals in . The spaces and operators are defined, generally speaking, on quasi‐metric measure spaces with doubling measure. The results are new even for Euclidean spaces.     

Weak Musielak–Orlicz Hardy spaces and applications

Let satisfy that , for any given , is an Orlicz function and is a Muckenhoupt weight uniformly in . In this article, the authors introduce the weak Musielak–Orlicz Hardy space via the grand maximal function and then obtain its vertical or its non–tangential maximal function characterizations. The authors also establish other real‐variable characterizations of , respectively, in terms of the atom, the molecule, the Lusin area function, the Littlewood–Paley g‐function or ‐function. All these characterizations for weighted weak Hardy spaces (namely, and with and ) are new and part of these characterizations even for weak Hardy spaces (namely, and with ) are also new. As an application, the boundedness of Calderón–Zygmund operators from to in the critical case is presented.     

Duality theorem for B‐valued martingale Orlicz–Hardy spaces associated with concave functions

The dual space of B ‐valued martingale Orlicz–Hardy space with a concave function Φ, which is associated with the conditional p‐variation of B ‐valued martingale, is characterized. To obtain the results, a new type of Campanato spaces for B ‐valued martingales is introduced and the classical technique of atomic decompositions is improved. Some results obtained here are connected closely with the p‐uniform smoothness and q‐uniform convexity of the underlying Banach space.     

Strong convergence of two‐dimensional Vilenkin‐Fourier series

We prove that certain means of the quadratical partial sums of the two‐dimensional Vilenkin‐Fourier series are uniformly bounded operators from the Hardy space to the space for We also prove that the sequence in the denominator cannot be improved.     

Boundedness and compactness of Hardy operators on Lorentz‐type spaces

Given non‐negative measurable functions on we study the high dimensional Hardy operator between Orlicz–Lorentz spaces , where f is a measurable function of and is the ball of radius t in . We give sufficient conditions of boundedness of and . We investigate also boundedness and compactness of between weighted and classical Lorentz spaces. The function spaces considered here do not need to be Banach spaces. Specifying the weights and the Orlicz functions we recover the existing results as well as we obtain new results in the new and old settings.     

Paley–Littlewood decomposition for sectorial operators and interpolation spaces

We prove Paley–Littlewood decompositions for the scales of fractional powers of 0‐sectorial operators A on a Banach space which correspond to Triebel–Lizorkin spaces and the scale of Besov spaces if A is the classical Laplace operator on We use the ‐calculus, spectral multiplier theorems and generalized square functions on Banach spaces and apply our results to Laplace‐type operators on manifolds and graphs, Schrödinger operators and Hermite expansion. We also give variants of these results for bisectorial operators and for generators of groups with a bounded ‐calculus on strips.     

Some integral‐type operators from spaces to mixed‐norm spaces on the unit ball

We discuss the boundedness and compactness of some integral‐type operators acting from spaces to mixed‐norm spaces on the unit ball of .     

Stability of integral operators on a space of homogeneous type

In this paper, we consider integral operators T on compact spaces of homogeneous type with finite diameter, whose kernels have certain Hölder regularity and mild singularity near the diagonal. We show that given any , the ‐stability of the operator is equivalent for different , where I stands for the identity operator.