Generalized Leray–Schauder principles for general classes of maps in completely regular topological spaces

In this paper, we present new continuation theorems for maps in a very general class.     

Nonlinear Leray–Schauder Alternatives for Decomposable Operators in Dunford–Pettis Spaces and Application to Nonlinear Eigenvalue Problems

We present some new variants of Leray–Schauder type fixed point theorems and eigenvalue results for decomposable single-valued nonlinear weakly compact operators in Dunford-Pettis spaces.     

Schauder estimates for elliptic equations in Banach spaces associated with stochastic reaction–diffusion equations

We consider some reaction–diffusion equations perturbed by white noise and prove Schauder estimates for the elliptic problem associated with the generator of the corresponding transition semigroup, defined in the Banach space of continuous functions. This requires the proof of some new interpolation result.     

New Besov-type spaces and Triebel–Lizorkin-type spaces including Q spaces

Let ${s,\,\tau\in\mathbb{R}}Let s, t ? \mathbbR{s,\,\tau\in\mathbb{R}} and q ? (0,￥]{q\in(0,\infty]} . We introduce Besov-type spaces [(B)\dot]^s, t_p, q(\mathbbRⁿ){{{{\dot B}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}} for p ? (0, ￥]{p\in(0,\,\infty]} and Triebel–Lizorkin-type spaces [(F)\dot]^s, t_p, q(\mathbbRⁿ) for p ? (0, ￥){{{{\dot F}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}\,{\rm for}\, p\in(0,\,\infty)} , which unify and generalize the Besov spaces, Triebel–Lizorkin spaces and Q spaces. We then establish the j{\varphi} -transform characterization of these new spaces in the sense of Frazier and Jawerth. Using the j{\varphi} -transform characterization of [(B)\dot]^s, t_p, q(\mathbbRⁿ) and [(F)\dot]^s, t_p, q(\mathbbRⁿ){{{{\dot B}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}\, {\rm and}\, {{\dot F}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}} , we obtain their embedding and lifting properties; moreover, for appropriate τ, we also establish the smooth atomic and molecular decomposition characterizations of [(B)\dot]^s, t_p, q(\mathbbRⁿ) and [(F)\dot]^s, t_p, q(\mathbbRⁿ){{{{\dot B}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}\,{\rm and}\, {{\dot F}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}} . For s ? \mathbbR{s\in\mathbb{R}} , p ? (1, ￥), q ? [1, ￥){p\in(1,\,\infty), q\in[1,\,\infty)} and t ? [0, \frac1(max{p, q})￠]{\tau\in[0,\,\frac{1}{(\max\{p,\,q\})'}]} , via the Hausdorff capacity, we introduce certain Hardy–Hausdorff spaces B[(H)\dot]^s, t_p, q(\mathbbRⁿ){{{{B\dot{H}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}}} and prove that the dual space of B[(H)\dot]^s, t_p, q(\mathbbRⁿ){{{{B\dot{H}^{s,\,\tau}_{p,\,q}(\mathbb{R}^{n})}}}} is just [(B)\dot]^-s, t_p￠, q￠(\mathbbRⁿ){\dot{B}^{-s,\,\tau}_{p',\,q'}(\mathbb{R}^{n})} , where t′ denotes the conjugate index of t ? (1,￥){t\in (1,\infty)} .     

Non-liftable Calabi–Yau spaces

We construct many new non-liftable three-dimensional Calabi–Yau spaces in positive characteristic. The technique relies on lifting a nodal model to a smooth rigid Calabi–Yau space over some number field as introduced by one of us jointily with D. van Straten.     

Relations among Besov-type spaces,Triebel–Lizorkin-type spaces and generalized Carleson measure spaces

In this article, the authors construct some counterexamples to show that the generalized Carleson measure space and the Triebel–Lizorkin-type space are not equivalent for certain parameters, which was claimed to be true in Lin and Wang [C.-C. Lin and K.Wang, Equivalency between the generalized Carleson measure spaces and Triebel–Lizorkin-type spaces, Taiwanese J. Math. 15 (2011), pp. 919–926]. Moreover, the authors show that for some special parameters, the generalized Carleson measure space, the Triebel–Lizorkin-type space and the Besov-type space coincide with certain Triebel–Lizorkin space, which answers a question posed in Remark 6.11(i) of Yuan et al. [W. Yuan, W. Sickel and D. Yang, Morrey and Campanato Meet Besov, Lizorkin and Triebel, Lecture Notes in Mathematics 2005, Springer-Verlag, Berlin, 2010]. In conclusion, the Triebel–Lizorkin-type space and the Besov-type space become the classical Besov spaces, when the fourth parameter is sufficiently large.     

The Fourier–Faber–Schauder Series Unconditionally Divergent in Measure

We prove that, for every ε ∈ (0, 1), there is a measurable set E ? [0, 1] whose measure |E| satisfies the estimate |E| > 1?ε and, for every function f ∈ C_[0,1], there is ? f ∈ C_[0,1] coinciding with f on E whose expansion in the Faber–Schauder system diverges in measure after a rearrangement.     

Sharp Hardy–Leray inequality for axisymmetric divergence-free fields

We show that the sharp constant in the classical n-dimensional Hardy–Leray inequality can be improved for axisymmetric divergence-free fields, and find its optimal value. The same result is obtained for n = 2 without the axisymmetry assumption.     

Quasiunconditional Basis Property of the Faber–Schauder System

Ukrainian Mathematical Journal - We prove that, for any 0 < ?? < 1, there exists a measurable set E?? ? [0, 1], mes (E??) > 1...     

Factorization in generalized Calogero–Moser spaces

Using a recent construction of Bezrukavnikov and Etingof, [R. Bezrukavnikov, P. Etingof, Induction and restriction functors for rational Cherednik algebras, arXiv: 0803.3639], we prove that there is a factorization of the Etingof–Ginzburg sheaf on the generalized Calogero–Moser space associated to a complex reflection group. In the case $W=S_n$, this confirms a conjecture of Etingof and Ginzburg, [P. Etingof, V. Ginzburg, Symplectic reflection algebras, Calogero–Moser space, and deformed Harish-Chandra homomorphisms, Invent. Math 147 (2002) 243–348].     

Composition operators in Orlicz–Sobolev spaces

We obtain necessary and sufficient conditions for a homeomorphism of domains in a Euclidean space to generate a bounded embedding operator of the Orlicz–Sobolev spaces defined by a special class of N-functions.     

Tubes in three–dimensional projective spaces

A partial tube in PG(3, q) is a pair , where is a collection of mutually disjoint lines of PG(3, q) with the property that for each plane π of PG(3, q) through L, the intersection of π with the lines of is an arc. Here, we generalize the notion of partial tube allowing the ground field to be any algebraically closed field. To a generalized partial tube we will associate an irreducible surface of degree d in providing upper bounds on d. The authors were partially supported by MIUR and GNSAGA of INdAM (Italy).     

Backward–backward splitting in Hadamard spaces

The backward–backward algorithm is a tool for finding minima of a regularization of the sum of two convex functions in Hilbert spaces. We generalize this setting to Hadamard spaces and prove the convergence of an error-tolerant version of the backward–backward method.     

Dixmier classes on generalized Segal–Bargmann–Fock spaces

We give criteria for the membership of Toeplitz operators and products of Hankel operators, with symbols of a certain type, in the Dixmier class, and formulas for their Dixmier trace, on a variety of weighted Segal–Bargmann–Fock spaces on the complex plane.     

Complex interpolation of Besov spaces and Triebel–Lizorkin spaces with variable exponents

This paper concerns the complex interpolation of Besov spaces and Triebel–Lizorkin spaces with variable exponents.     

Interpolation operators in Orlicz–Sobolev spaces

We study classical interpolation operators for finite elements, like the Scott–Zhang operator, in the context of Orlicz–Sobolev spaces. Furthermore, we show estimates for these operators with respect to quasi-norms which appear in the study of systems of p-Laplace type.