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This note is devoted to developing Morrey regularity theory for the following system of Rivière \begin{equation*} -\Delta u=\Omega \cdot \nabla u+f \qquad \text {in }B^{2}, \end{equation*} under the assumption that f belongs to some Morrey space. Our results extend the L^p regularity theory of Sharp and Topping [Trans. Amer. Math. Soc. 365 (2013), pp. 2317–2339], and also generalize a Hölder continuity result of Wang [Calc. Var. Partial Differential Equations 56 (2017), Paper No. 23, 24] on harmonic mappings. Potential applications of our results are also possible in second order conformally invariant geometrical problems as that of Wang [Calc. Var. Partial Differential Equations 56 (2017), Paper No. 23, 24]. [ABSTRACT FROM AUTHOR]

We prove that order convergence on a Boolean algebra turns it into a compact convergence space if and only if this Boolean algebra is complete and atomic. We also show that on an Archimedean vector lattice, order intervals are compact with respect to order convergence if and only the vector lattice is complete and atomic. Additionally we provide a direct proof of the fact that uo convergence on an Archimedean vector lattice is induced by a topology if and only if the vector lattice is atomic. [ABSTRACT FROM AUTHOR]

It was showed by Donner in [ Extension of positive operators and Korovkin theorems , Lecture Notes in Mathematics, vol. 904, Springer-Verlag, Berlin-New York, 1982] that every order complete vector lattice X may be embedded into a cone X^s, called the sup-completion of X. We show that if one represents the universal completion of X as C^\infty (K), then X^s is the set of all continuous functions from K to [-\infty,\infty ] that dominate some element of X. This provides a functional representation of X^s, as well as an easy alternative proof of its existence. [ABSTRACT FROM AUTHOR]

The present paper continues investigation of narrow operators on complex vector lattices started in a recent paper by Dzhusoeva, Huang, Pliev and Sukochev. We answer their questions and strengthen one of their main results by proving that the extension T_{\Bbb {C}}\colon E_{\Bbb {C}}\to F_{\Bbb {C}} of a linear operator T \colon E \to F acting from a Dedekind complete real Riesz space E to a real Banach lattice F to their complexifications is narrow if and only if T is. [ABSTRACT FROM AUTHOR]

We show that a normalized capacity \nu : \mathcal {P}(\mathbf {N})\to \mathbf {R} is invariant with respect to an ideal \mathcal {I} on \mathbf {N} if and only if it can be represented as a Choquet average of \{0,1\}-valued finitely additive probability measures corresponding to the ultrafilters containing the dual filter of \mathcal {I}. This is obtained as a consequence of an abstract analogue in the context of Archimedean Riesz spaces. [ABSTRACT FROM AUTHOR]

Disjoint sequence methods from the theory of Riesz spaces are used to study measures of weak non-compactness in L_{1}(\mu)-spaces. A principal new result of the present paper is the following: Let E be an abstract M-space. Then \begin{align*} \omega (B)&=\sup \{\limsup \limits _{n\rightarrow \infty }\rho _{B}(x_{n}):(x_{n})_{n}\subseteq B_{E} \operatorname {disjoint} \}\\ &=\inf \{\varepsilon >0:\exists x^{*}\in E^{*}_{+} \operatorname {so}\operatorname {that} B\subseteq [-x^{*},x^{*}]+\varepsilon B_{E^{*}}\}\\ &=\sup \{\limsup \limits _{n\rightarrow \infty }\rho _{B}(x_{n}):(x_{n})_{n}\subseteq B_{E} \operatorname {weakly}\operatorname {null} \}\\ &=\sup \{\operatorname {ca}_{\rho _{B}}((x_{n})_{n}):(x_{n})_{n}\subseteq (B_{E})_{+} \operatorname {increasing} \}\\ &=\sup \{\limsup \limits _{n\rightarrow \infty }\|x^{*}_{n}\|:(x^{*}_{n})_{n}\subseteq \operatorname {Sol}(B)\operatorname {disjoint}\}\\ &=\sup \{\limsup \limits _{n\rightarrow \infty }\sup \limits _{x^{*}\in B}|\langle x^{*},x_{n}\rangle |:(x_{n})_{n}\subseteq B_{E}\operatorname {disjoint} \}\\ \end{align*} for every norm bounded subset B of E^{*}. [ABSTRACT FROM AUTHOR]

It is a subtle fact of the theory of regular operators on Banach lattices that every linear operator T \colon L_{1}(\mu)\to L_1(\nu) is norm-bounded if and only if it is regular. We generalize this result to the setting of operators taking values in a Lebesgue–Bochner space. Our main result asserts that every linear operator T \colon L_{1}(\mu)\to L_{1}(\nu,X) is norm-bounded if and only if it is dominated. We show that this result is no longer true for Lebesgue–Bochner domain spaces. As a consequence of the main theorem, we obtain a generalized version of the Grothendieck inequality for linear norm-bounded operators from L_{1}(\mu) to a Lebesgue–Bochner space L_{1}(\nu,X). [ABSTRACT FROM AUTHOR]

Let p be a fixed number with 1 \leq p < \infty. It is shown that every surjective and biseparating linear map between L^p-spaces is continuous when the underlying measure space is non-atomic. We also prove that a separating isomorphism on l^p is both continuous and biseparating. Furthermore, these (bi-)separating maps take the form of a weighted composition operator. Our proofs are direct, elementary and do not invoke deep results about Riesz spaces or Banach lattices. [ABSTRACT FROM AUTHOR]

Given a primitive collection of vectors in the integer lattice, we count the number of ways it can be extended to a basis by vectors with sup-norm bounded by T, producing an asymptotic estimate as T \to \infty. This problem can be interpreted in terms of unimodular matrices, as well as a representation problem for a class of multilinear forms. In the 2-dimensional case, this problem is also connected to the distribution of Farey fractions. As an auxiliary lemma we prove a counting estimate for the number of integer lattice points of bounded sup-norm in a hyperplane in \mathbb R^n. Our main result on counting basis extensions also generalizes to arbitrary lattices in \mathbb R^n. Finally, we establish some basic properties of sparse representations of integers by multilinear forms. [ABSTRACT FROM AUTHOR]

We say that a Riesz space E has the horizontal Egorov property if for every net (ƒα) in E, order convergent to ƒ ∈ E with |ƒα| + |ƒ| ≤ e ∈ E+ for all α, there exists a net (eβ) of fragments of e laterally convergent to e such that for every nβ, the net (|ƒ − ƒα | ∧ eβ)α e-uniformly tends to zero. Our main result asserts that every Dedekind complete Riesz space which satisfies the weak distributive law possesses the horizontal Egorov property. A Riesz space E is said to satisfy the weak distributive law if for every e ∈ E+ \ 0 the Boolean algebra Fe of fragments of e satisfies the weak distributive law; that is, whenever (Πn)n ∈ N is a sequence of partitions of Fe, there is a partition Π of Fe such that every element of Π is finitely covered by each of Πn (e.g., every measurable Boolean algebra is so). Using a new technical tool, we show that for every net (ƒα) order convergent to ƒ in a Riesz space with the horizontal Egorov property there are a horizontally vanishing net (vβ) and a net (uα,β)(α,β) ∈ A × B, which uniformly tends to zero for every fixed β such that |ƒ − ƒα| ≤ uα, β + vβ for all α, β. [ABSTRACT FROM AUTHOR]

We establish an interpolation result for sublinear operators which map into Riesz spaces. This result applies to all interpolation functors including the real interpolation and the complex interpolation. One component of our proof which may be of independent interest is the perhaps already known fact that the generalized versions of the Hahn-Banach theorem due to L. V. Kantorovich and M. M. Day also hold for complex vector spaces. [ABSTRACT FROM AUTHOR]

In this paper, the authors show that a function b ∈ BMO(ℝn) is in CMO(ℝn) if and only if the Riesz transform commutator [b, Ri] is compact on Lwp(ℝn)for i ∈ {1, 2, ..., n}, p ∈ (1, ∞), and w ∈ Ap(ℝn), and if and only if the fractional integral commutator [b, Iα] is compact from ... (ℝn) to ...(ℝn), where α ∈ (0, n), p, q ∈ (1, ∞) with 1/p = 1/q + α/n and w ∈ Ap, q(ℝn). [ABSTRACT FROM AUTHOR]

In this paper, the authors show that a function b ∈ BMO(ℝn) is in CMO(ℝn) if and only if the Riesz transform commutator [b, Ri] is compact on Lwp(ℝn)for i ∈ {1, 2, ..., n}, p ∈ (1, ∞), and w ∈ Ap(ℝn), and if and only if the fractional integral commutator [b, Iα] is compact from ... (ℝn) to ...(ℝn), where α ∈ (0, n), p, q ∈ (1, ∞) with 1/p = 1/q + α/n and w ∈ Ap, q(ℝn). [ABSTRACT FROM AUTHOR]

The commutators of bilinear Calderón-Zygmund operators and pointwise multiplication with a symbol in CMO are bilinear compact operators on products of Lebesgue spaces. We show that, for certain non-degenerate Calderón-Zygmund operators, the symbol being in CMO is not only sufficient but actually necessary for the compactness of the commutators. [ABSTRACT FROM AUTHOR]

As our main result we prove a variant of the fractional Hardy-Sobolev-Maz'ya inequality for half spaces. This result contains a complete answer to a recent open question by Musina and Nazarov. In the proof we apply a new version of the fractional Hardy-Sobolev inequality that we establish also for more general unbounded John domains than half spaces. [ABSTRACT FROM AUTHOR]

As a generalization of almost everywhere convergence to vector lattices, unbounded order convergence has garnered much attention. The concept of boundedly uo-complete Banach lattices was introduced by N. Gao and F. Xanthos, and has been studied in recent papers by D. Leung, V. G. Troitsky, and the aforementioned authors. We will prove that a Banach lattice is boundedly uo-complete iff it is monotonically complete. Afterwards, we study completeness-type properties of minimal topologies; minimal topologies are exactly the Hausdorff locally solid topologies in which uo-convergence implies topological convergence. [ABSTRACT FROM AUTHOR]

The Kato Theorem on similarity for sequences of projections in a Hilbert space is extended to the case when both sequences consist of nonselfadjoint projections. Passing to subspaces, this leads to stability theorems for Riesz bases of subspaces, at least one of which is finite dimensional, and for arbitrary vector Riesz bases. The following is proved as an application. If {φn}∞ n=1 is a Riesz basis and |θn| ≤ C for large n, where the constant C depends only on {φn}∞ n=1, then {φn + θnφn+1}∞ n=1 also forms a Riesz basis. [ABSTRACT FROM AUTHOR]

We prove the existence of Riesz bases of exponentials of L2(Ω), provided that Ω � Rd is a measurable set of finite and positive measure, not necessarily bounded, that satisfies a multi-tiling condition and an arithmetic property that we call admissibility. This property is satisfied for any bounded domain, so our results extend the known case of bounded multi-tiles. We also extend known results for submulti-tiles and frames of exponentials to the unbounded case. [ABSTRACT FROM AUTHOR]

Let Y be a sublattice of a vector lattice X. We consider the problem of identifying the smallest order closed sublattice of X containing Y. It is known that the analogy with topological closure fails. Let ... be the order closure of Y consisting of all order limits of nets of elements from Y. Then ... need not be order closed. We show that in many cases the smallest order closed sublattice containing Y is in fact the second order closure .... Moreover, if X is a σ-order complete Banach lattice, then the condition that ... is order closed for every sublattice Y characterizes order continuity of the norm of X. The present paper provides a general approach to a fundamental result in financial economics concerning the spanning power of options written on a financial asset. [ABSTRACT FROM AUTHOR]

We prove the existence of a minimal action nodal solution for the quadratic Choquard equation...where Ia is the Riesz potential of order α ? (0,N). The solution is constructed as the limit of minimal action nodal solutions for the nonlinear Choquard equations...when p 2. The existence of minimal action nodal solutions for p > 2 can be proved using a variational minimax procedure over a Nehari nodal set. No minimal action nodal solutions exist when p < 2. [ABSTRACT FROM AUTHOR]

Let W, X, Y and Z be Dedekind complete Riesz spaces. For A ε Lr(Y,Z) and B ε Lr(W,X) let MA,B be the two-sided multiplication operator from Lr(X, Y ) into Lr(W, Z) defined by MA,B(T) = ATB. We show that for every 0 ⩽ A0 ε Lrn (Y,Z), ǀMA,B,Bǀ(T) = MA0,ǀBǀ(T) holds for all B ε Lr(W,X) and all T ε Lrn(X, Y ). Furthermore, if W, X, Y and Z are Dedekind complete Banach lattices such that X and Y have order continuous norms, then ǀMA,Bǀ = MǀAǀ, ǀBǀ for all A ε Lr(Y,Z) and all B ε Lr(W,X). Our results generalize the related results of Synnatzschke and Wickstead, respectively. [ABSTRACT FROM AUTHOR]

In a case study on asymptotics of spectral quantities of Schrödinger operators in fractional Sobolev spaces on the circle we show how a nonlinear version of the Riesz-Thorin theorem on the interpolation of linear operators can be applied. [ABSTRACT FROM AUTHOR]

We discuss the existence and stability of Riesz bases of exponential type of L² (T) for special domains T ⊂ R² called trapezoids, and we construct exponential bases on the finite union of rectangles with the same height. We also generalize our main theorems in dimension d ≤ 3. [ABSTRACT FROM AUTHOR]

We prove some nonexistence results of self-similar singular solutions for the surface quasi-geostrophic equation and its generalization by relying on the fundamental local Lp-inequality of the self-similar quantity. [ABSTRACT FROM AUTHOR]

The localized single-valued extension property is stable under commuting Riesz perturbations. [ABSTRACT FROM AUTHOR]

Suppose Ω ⊆ ℝd is a bounded and measurable set and Λ ⊆ ℝd is a lattice. Suppose also that Ω tiles multiply, at level κ, when translated at the locations Λ. This means that the Λ-translates of Ω cover almost every point of ℝd exactly κ times. We show here that there is a set of exponentials exp(2π it· x), t ε T, where T is some countable subset of ℝd, which forms a Riesz basis of L²(Ω). This result was recently proved by Grepstad and Lev under the extra assumption that Ω has boundary of measure 0, using methods from the theory of quasicrystals. Our approach is rather more elementary and is based almost entirely on linear algebra. The set of frequencies T turns out to be a finite union of shifted copies of the dual lattice Λ*. It can be chosen knowing only Λ and κ and is the same for all Ω that tile multiply with Λ. [ABSTRACT FROM AUTHOR]

Let G = - ... be the Grushin operator on Rn X R. We prove that the Riesz transforms associated to this operator are bounded on Lp (Rn+1), 1 < p < ∞, and their norms are independent of dimension n. [ABSTRACT FROM AUTHOR]

For any function f belonging to Qp,λ, a certain proper subspace of the classical Morrey space Lp,λ, a sharp capacity weak-type estimate is obtained for its Riesz potential Iαf. This extends a 1969 result due to Franco Conti. [ABSTRACT FROM AUTHOR]

For one-dimensional Dirac operators ..., subject to periodic or antiperiodic boundary conditions, we give necessary and sufficient conditions which guarantee that the system of root functions contains Riesz bases in L²([0, π],C²). In particular, if the potential matrix v is skew-symmetric (i.e., ￣Q = -P), or more generally if ￣Q ≠ tP for some real t= 0, then there exists a Riesz basis that consists of root functions of the operator L. [ABSTRACT FROM AUTHOR]

Sharp constant versions of two endpoint inequalities for Riesz transforms are derived using probabilistic methods. [ABSTRACT FROM AUTHOR]

Let S be the union of finitely many disjoint intervals on R. Suppose that there are two real numbers α, β such that the length of each interval belongs to Zα + Zβ. We use quasicrystals to construct a discrete set Λ ⊂ R such that the system of exponentials {exp 2πiλx, λ ∈ Λ} is a Riesz basis in the space L2(S). [ABSTRACT FROM AUTHOR]

In this paper, the general Clifford $R_{n,s}$-valued Hardy spaces and conjugate Hardy spaces are characterized. In particular, each function in $R_n$-valued Hardy space can be determined by half of its function components through Riesz transform, and the explicit determining formulas are given. The products of two functions in the Hardy space give six kinds of compensated quantities, which correspond to six paracommutators, and their boundedness, compactness and Schatten-von Neumann properties are given. [ABSTRACT FROM AUTHOR]

We show that for $\delta$ below certain critical indices there are functions whose Jacobi or Laguerre expansions have almost everywhere divergent Cesàro and Riesz means of order $\delta$. [ABSTRACT FROM AUTHOR]

Let $L$ be a vector lattice of real functions on a set $\Om$ with $\boldsymbol{1}\in L$, and let $P$ be a linear positive functional on $L$. Conditions are given which imply the representation $P(f)=\int fd\pi$, $f\in L$, for some bounded charge $\pi$. As an application, for any bounded charge $\pi$ on a field $\mathcal F$, the dual of $L^1(\pi)$ is shown to be isometrically isomorphic to a suitable space of bounded charges on $\mathcal F$. In addition, it is proved that, under one more assumption on $L$, $P$ is the integral with respect to a $\sigma$-additive bounded charge. [ABSTRACT FROM AUTHOR]

For the operator of the Riesz potential type we estimate the second term in the asymptotic behavior of the spectrum and find its ``regularized'' trace. [ABSTRACT FROM AUTHOR]

In this paper we introduce whales in the collection of subsets of the Boolean algebra of bands in a Dedekind complete Riesz space and employ them to give a short (and constructive) proof of the existence of universal completions for Archimedean Riesz spaces. [ABSTRACT FROM AUTHOR]

In this note we show that the standard convolution regularization of the Haar system generates Riesz bases of smooth functions for $L^2(\mathbb R)$, providing in this way an alternative to the approach given by Govil and Zalik [Proc. Amer. Math. Soc. \textbf{125} (1997), 3363--3370]. [ABSTRACT FROM AUTHOR]

In this paper we give a characterization of order ideals in Riesz spaces. [ABSTRACT FROM AUTHOR]