Hardy spaces for Bessel–Schrödinger operators

Consider the Bessel operator with a potential on , namely We assume that and is a nonnegative function. By definition, a function belongs to the Hardy space if Under certain assumptions on V we characterize the space in terms of atomic decompositions of local type. In the second part we prove that this characterization can be applied to for with no additional assumptions on the potential V.     

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.     

Musielak–Orlicz–Bochner function spaces which are uniformly noncreasy

In this paper, the criteria for uniform noncreasiness of Musielak–Orlicz–Bochner function spaces are given. Moreover authors also prove that the space (resp ) is uniformly noncreasy if and only if the space (resp ) is uniformly convex or uniformly smooth. As a corollary, the criteria for uniform noncreasiness of Musielak–Orlicz function spaces are given.     

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.     

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.     

A remark on the Schrödinger propagator on Wiener amalgam spaces

In this paper, we study the boundedness of the Schrödinger propagator on Wiener amalgam spaces. In particular, we determine the necessary and sufficient conditions for the propagator to be bounded from to .     

The higher order Riesz transform and BMO type space associated to Schrödinger operators

Let L = ?Δ + V be a Schrödinger operator on $\mathbb {R}^nLet L = ?Δ + V be a Schrödinger operator on $\mathbb {R}^n$ (n ≥ 3), where $V \not\equiv 0$ is a nonnegative potential belonging to certain reverse Hölder class B_s for $s \ge \frac{n}{2}$. In this article, we prove the boundedness of some integral operators related to L, such as L^?1?², L^?1V and L^?1( ? Δ) on the space $BMO_L(\mathbb {R}^n)$.     

Sobolev's inequality for Riesz potentials of functions in grand Musielak–Orlicz–Morrey spaces over nondoubling metric measure spaces

In this paper we are concerned with Sobolev's inequality for Riesz potentials of functions in grand Musielak–Orlicz–Morrey spaces over nondoubling metric measure spaces.     

A compactness criterion and application to the commutators associated with Schrödinger operators

Let T be an integral operator. In this paper, we introduce a ‐compactness criterion of , where . As an application, we apply this criterion to deal with ‐compactness of commutators associated to Schrödinger operators with potentials in the reverse Hölder's class.     

Littlewood–Paley characterizations of higher‐order Sobolev spaces via averages on balls

In this article, the authors characterize higher‐order Sobolev spaces , with , and , or with , and , via the Lusin area function and the Littlewood–Paley ‐function in terms of ball averages, where denotes the maximal integer not greater than . Moreover, the authors also complement the above results in the endpoint cases of p via establishing some weak type estimates. These improve and develop the corresponding known results for Sobolev spaces with smoothness order .     

Traces of functions in Hardy and Besov spaces on Lipschitz domains with applications to compensated compactness and the theory of Hardy and Bergman type spaces

In this paper we study conditions guaranteeing that functions defined on a Lipschitz domain Ω have boundary traces in Hardy and Besov spaces on ∂Ω. In turn these results are used to develop a new approach to the theory of compensated compactness and the theory of non-locally convex Hardy and Bergman type spaces.     

The locally uniformly non‐square points of Orlicz–Bochner sequence spaces

In this paper, we investigate the locally uniformly non‐square point of Orlicz–Bochner sequence spaces endowed with Luxemburg norm. Analysing and combining the generating function M and properties of the real Banach space X , we get sufficient and necessary conditions of locally uniformly non‐square point, which contributes to criteria for locally uniform non‐squareness in Orlicz–Bochner sequence spaces. The results generalize the corresponding results in the classical Orlicz sequence spaces.     

The boundedness of Weyl multiplier on Hardy spaces associated with twisted convolution

In this paper, we will consider the boundedness of Weyl multiplier on Hardy spaces associated with twisted convolution. In order to get our result, we need to give some characterizations of the Hardy space associated with twisted convolution. Including Lusin area integral, Littlewood-Paley g-function.     

On unique continuation for Schrödinger operators of fractional and higher orders

In this note we study the property of unique continuation for solutions of , where V is in a function class of potentials including for . In particular, when , our result gives a unique continuation theorem for the fractional Schrödinger operator in the full range of α values.     

A Calderón–Zygmund operator of higher order Schrödinger type

We consider higher order Schrödinger type operators with nonnegative potentials. We assume that the potential belongs to the reverse Hölder class which includes nonnegative polynomials. We show that an operator of higher order Schrödinger type is a Calderón–Zygmund operator. We also show that there exist potentials which satisfy our assumptions but are not nonnegative polynomials.     

Strong maximal operator and singular integral operators in weighted Morrey spaces on product domains

We introduce the Morrey spaces on product domains and extend the boundedness of strong maximal operator and singular integral operators on product domains to Morrey spaces.     

UMD Banach spaces and square functions associated with heat semigroups for Schrödinger,Hermite and Laguerre operators

In this paper we define square functions (also called Littlewood‐Paley‐Stein functions) associated with heat semigroups for Schrödinger and Laguerre operators acting on functions which take values in UMD Banach spaces. We extend classical (scalar) ‐boundedness properties for the square functions to our Banach valued setting by using γ‐radonifying operators. We also prove that these ‐boundedness properties of the square functions actually characterize the Banach spaces having the UMD property.     

M‐constants in Orlicz–Lorentz function spaces

In this paper some lower and upper estimates of M‐constants for Orlicz–Lorentz function spaces for both, the Luxemburg and the Amemiya norms, are given. Since degenerated Orlicz functions φ and degenerated weighted sequences ω are also admitted, this investigations concern the most possible wide class of Orlicz–Lorentz function spaces. M‐constants were defined in 1969 by E. A. Lifshits, and used by many authors in the study of lattice structures on Banach spaces, as well as in the fixed point theory.     

On Burenkov's extension operator preserving Sobolev–Morrey spaces on Lipschitz domains

We prove that Burenkov's extension operator preserves Sobolev spaces built on general Morrey spaces, including classical Morrey spaces. The analysis concerns bounded and unbounded open sets with Lipschitz boundaries in the n‐dimensional Euclidean space.     

I‐convexity and Q‐convexity in Orlicz–Bochner function spaces endowed with the Orlicz norm

Some characterizations of I‐convexity and Q‐convexity of Banach space are obtained. Moreover, the criteria is shown for Orlicz–Bochner function spaces $L_M(\mu ,X)$ endowed with the Orlicz norm being I‐convex as well as being Q‐convex.