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preprint2017arXiv

On weighted mixed-norm Sobolev estimates for some basic parabolic equations

Novel global weighted parabolic Sobolev estimates, weighted mixed-norm estimates and a.e. convergence results of singular integrals for evolution equations are obtained. Our results include the classical heat equation, the harmonic oscillator evolution equation $$\partial_tu=Δu-|x|^2u+f,$$ and their corresponding Cauchy problems. We also show weighted mixed-norm estimates for solutions to degenerate parabolic extension problems arising in connection with the fractional space-time nonlocal equations $(\partial_t-Δ)^su=f$ and $(\partial_t-Δ+|x|^2)^su=f$, for $0<s<1$.

preprint2017arXiv

L-Dunford-Pettis property in Banach spaces

In this paper, we introduce and study the concept of L-Dunford-Pettis sets and L-Dunford-Pettis property in Banach spaces. Next, we give a characterization of the L-Dunford-Pettis property with respect to some well-known geometric properties of Banach spaces. Finally, some complementability of operators on Banach spaces with the L-Dunford-Pettis property are also investigated

preprint2017arXiv

Some remarks on the structure of Lipschitz-free spaces

We give several structural results concerning the Lipschitz-free spaces $\mathcal F(M)$, where $M$ is a metric space. We show that $\mathcal F(M)$ contains a complemented copy of $\ell_1(Γ)$, where $Γ=\text{dens}(M)$. If $\mathcal N$ is the net in a finite dimensional Banach space $X$, we show that $\mathcal F(\mathcal N)$ is isomorphic to its square. If $X$ contains a complemented copy of $\ell_p, c_0$ then $\mathcal F(\mathcal N)$ is isomorphic to its $\ell_1$-sum. Finally, we prove that for all $X\cong C(K)$ spaces $\mathcal F(\mathcal N)$ are mutually isomorphic spaces with a Schauder basis.

preprint2017arXiv

A study of stability in locally $L^0$-convex modules and a conditional version of James' compactness theorem

Locally $L^0$-convex modules were introduced in [D. Filipovic, M. Kupper, N. Vogelpoth. Separation and duality in locally $L^0$-convex modules. J. Funct. Anal. 256(12), 3996-4029 (2009)] as the analytic basis for the study of conditional risk measures. Later, the algebra of conditional sets was introduced in [S. Drapeau, A. Jamneshan, M. Karliczek, M. Kupper. The algebra of conditional sets and the concepts of conditional topology and compactness. J. Math. Anal. Appl. 437(1), 561-589 (2016)]. In this paper we study locally $L^0$-convex modules, and find exactly which subclass of locally $L^0$-convex modules can be identified with the class of locally convex vector spaces within the context of conditional set theory. Second, we provide a version of the classical James' theorem of characterization of weak compactness for conditional Banach spaces. Finally, we state a conditional version of the Fatou and Lebesgue properties for conditional convex risk measures and, as application of the developed theory, we stablish a version of the so-called Jouini-Schachermayer-Touzi theorem for robust representation of conditional convex risk measures defined on a $L^\infty$-type module.

preprint2017arXiv

Approximation in Morrey spaces

A new subspace of Morrey spaces whose elements can be approximated by infinitely differentiable compactly supported functions is introduced. Consequently, we give an explicit description of the closure of the set of such functions in Morrey spaces. A generalization of known embeddings of Morrey spaces into weighted Lesbesgue spaces is also obtained.

preprint2017arXiv

Using an implicit function to prove common fixed point theorems

In this paper, we prove common fixed point results for a self-mappings satisfying an implicit function which is general enough to cover a multitude of known as well as unknown contractions. Our results modify, unify, extend and generalize many relevant results of the existing literature. Interestingly, unlike several other cases, our main results deduce a nonlinear order-theoretic version of a well-known fixed point theorem (proved for quasi-contraction) due to Ćirić (Proc. Amer. Math. Soc. (54) 267-273, 1974). Finally, in the setting of metric spaces, we drive a sharpened version of Theorem 1 due to Berinde and Vetro (Fixed Point Theory Appl. 2012:105).

preprint2017arXiv

The minus order and range additivity

We study the minus order on the algebra of bounded linear operators on a Hilbert space. By giving a characterization in terms of range additivity, we show that the intrinsic nature of the minus order is algebraic. Applications to generalized inverses of the sum of two operators, to systems of operator equations and to optimization problems are also presented.

preprint2017arXiv

More on functional and quantitative versions of the isoperimetric inequality

This paper deals with the famous isoperimetric inequality. In a first part, we give some new functional form of the isoperimetric inequality, and in a second part, we give a quantitative form with a remainder term involving Wasserstein distance of the classical isopemetric inequality. In both parts, we use optimal transportation. Finally, we use our refined isoperimetric inequality in some classical cases arising in convex geometry.

preprint2016arXiv

Triviality of the generalized Lau product associated to a Banach algebra homomorphism

Several papers have, as their raison d'etre, the exploration of the \emph{generalized Lau product} associated to a homomorphism $T:B\to A$ of Banach algebras. In this short note, we demonstrate that the generalized Lau product is isomorphic as a Banach algebra to the usual direct product $A\oplus B$. We also correct some misleading claims made about the relationship between this generalized Lau product, and an older construction of Monfared (Studia Mathematica, 2007).

preprint2017arXiv

A classification of the cofinal structures of precompacta

We provide a complete classification of the possible cofinal structures of the families of precompact (totally bounded) sets in general metric spaces, and compact sets in general complete metric spaces. Using this classification, we classify the cofinal structure of local bases in the groups $\C(X,\bbR)$ of continuous real-valued functions on complete metric spaces $X$, with respect to the compact-open topology.

preprint2017arXiv

Euler-Maclaurin formulas for functions of bounded variation

The first-order Euler-Maclaurin formula relates the sum of the values of a smooth function on an interval of integers with its integral on the same interval on $\mathbb R$. We formulate here the analogue for functions that are just of bounded variation.

preprint2016arXiv

On first and second eigenvalues of Riesz transforms in spherical and hyperbolic geometries

In this note we prove an analogue of the Rayleigh-Faber-Krahn inequality, that is, that the geodesic ball is a maximiser of the first eigenvalue of some convolution type integral operators, on the sphere $\mathbb{S}^{n}$ and on the real hyperbolic space $\mathbb{H}^{n}$. It completes the study of such question for complete, connected, simply connected Riemannian manifolds of constant sectional curvature. We also discuss an extremum problem for the second eigenvalue on $\mathbb{H}^{n}$ and prove the Hong-Krahn-Szegö type inequality. The main examples of the considered convolution type operators are the Riesz transforms with respect to the geodesic distance of the space.

preprint2017arXiv

Decomposition of functions between Banach spaces in the orthogonality equation

Let $E,F$ be Banach spaces. In the case that $F$ is reflexive we give a description for the solutions $(f,g)$ of the Banach-orthogonality equation $$\langle f(x),g(α)\rangle=\langle x,α\rangle\hspace{10mm}\forall x\in E,\forall α\in E^*,$$ where $f:E\rightarrow F,g:E^*\rightarrow F^*$ are two maps. Our result generalizes the recent result of Łukasik and Wójcik in the case that $E$ and $F$ are Hilbert spaces.

preprint2017arXiv

Ando dilations and inequalities on noncommutative domains

We obtain intertwining dilation theorems for noncommutative regular domains D_f and noncommutative varieties V_J of n-tuples of operators, which generalize Sarason and Sz.-Nagy--Foias commutant lifting theorem for commuting contractions. We present several applications including a new proof for the commutant lifting theorem for pure elements in the domain D_f (resp. variety V_J) as well as a Schur type representation for the unit ball of the Hardy algebra associated with the variety V_J. We provide Ando type dilations and inequalities for bi-domains D_f \times D_g and bi-varieties V_J \times V_I. In particular, we obtain extensions of Ando's results and Agler-McCarthy's inequality for commuting contractions to larger classes of commuting operators.

preprint2016arXiv

Elastic-net regularization versus $\ell^1$-regularization for linear inverse problems with quasi-sparse solutions

We consider the ill-posed operator equation $Ax=y$ with an injective and bounded linear operator $A$ mapping between $\ell^2$ and a Hilbert space $Y$, possessing the unique solution \linebreak $x^†=\{x^†_k\}_{k=1}^\infty$. For the cases that sparsity $x^†\in \ell^0$ is expected but often slightly violated in practice, we investigate in comparison with the $\ell^1$-regularization the elastic-net regularization, where the penalty is a weighted superposition of the $\ell^1$-norm and the $\ell^2$-norm square, under the assumption that $x^†\in \ell^1$. There occur two positive parameters in this approach, the weight parameter $η$ and the regularization parameter as the multiplier of the whole penalty in the Tikhonov functional, whereas only one regularization parameter arises in $\ell^1$-regularization. Based on the variational inequality approach for the description of the solution smoothness with respect to the forward operator $A$ and exploiting the method of approximate source conditions, we present some results to estimate the rate of convergence for the elastic-net regularization. The occurring rate function contains the rate of the decay $x^†_k \to 0$ for $k \to \infty$ and the c

preprint2017arXiv

Weak Averaging of Semilinear Stochastic Differential Equations with Almost Periodic Coefficients

An averaging result is proved for stochastic evolution equations with highly oscillating coefficients. This result applies in particular to equations with almost periodic coefficients. The convergence to the solution of the averaged equation is obtained in distribution, as in previous works by Khasminskii and Vrko{\v c}.This version corrects two minor errors from our paper published in J. Math. Anal. Appl. 427(1):336--364, 2015.

preprint2017arXiv

K-energy on polarized compactifications of Lie groups

In this paper, we study Mabuchi's K-energy on a compactification M of a reductive Lie group G, which is a complexification of its maximal compact subgroup K. We give a criterion for the properness of K-energy on the space of K \times K-invariant Kahler potentials. In particular, it turns to give an alternative proof of Delcroix's theorem for the existence of Kahler-Einstein metrics in case of Fano manifolds M . We also study the existence of minimizers of K-energy for general Kahler classes of M.

preprint2017arXiv

Orlicz Spaces with Bicomplex Scalars

In this paper we define bicomplex Orlicz space with hyperbolic valued Luxemburg norm and discussed some of their properties. We have also partially characterize an integral representation of a $\mathbb{D}$-valued convex function. Further we have shown that the spectrum of the unilateral shift operator on $l^2(\mathbb{BC})$ is the null cone $\mathcal{NC}$.

preprint2017arXiv

Products of general Menger spaces

We study products of general topological spaces with Menger's covering property, and its refinements based on filters and semifilters. To this end, we extend the projection method from the classic real line topology to the Michael topology. Among other results, we prove that, assuming \CH{}, every productively Lindelöf space is productively Menger, and every productively Menger space is productively Hurewicz. None of these implications is reversible.

preprint2017arXiv

Hyperbolic geometry on noncommutative polyballs

This paper is an introduction to the hyperbolic geometry of noncommutative polyballs B_n of bounded linear operators on Hilbert spaces. We use the theory of free pluriharmonic functions on polyballs and noncommutative Poisson kernels on tensor products of full Fock spaces to define hyperbolic type metrics on B_n, study their properties, and obtain hyperbolic versions of Schwarz-Pick lemma for free holomorphic functions on polyballs. As a consequence, the polyballs can be viewed as noncommutative hyperbolic spaces. When specialized to the operatorial polydisk D_k, our hyperbolic metric is complete and invariant under the group of all free holomorphic automorphisms of D_k, and the topology induced on D_k is the usual operator norm topology.

preprint2016arXiv

Pointwise ergodic theorems in symmetric spaces of measurable functions

For a Dunford-Schwartz operator in a fully symmetric space of measurable functions of an arbitrary measure space, we prove pointwise convergence of the conventional and weighted ergodic averages.

preprint2017arXiv

Completely bounded bimodule maps and spectral synthesis

We initiate the study of the completely bounded multipliers of the Haagerup tensor product $A(G)\otimes_{\rm h} A(G)$ of two copies of the Fourier algebra $A(G)$ of a locally compact group $G$. If $E$ is a closed subset of $G$ we let $E^{\sharp} = \{(s,t) : st\in E\}$ and show that if $E^{\sharp}$ is a set of spectral synthesis for $A(G)\otimes_{\rm h} A(G)$ then $E$ is a set of local spectral synthesis for $A(G)$. Conversely, we prove that if $E$ is a set of spectral synthesis for $A(G)$ and $G$ is a Moore group then $E^{\sharp}$ is a set of spectral synthesis for $A(G)\otimes_{\rm h} A(G)$. Using the natural identification of the space of all completely bounded weak* continuous $VN(G)'$-bimodule maps with the dual of $A(G)\otimes_{\rm h} A(G)$, we show that, in the case $G$ is weakly amenable, such a map leaves the multiplication algebra of $L^{\infty}(G)$ invariant if and only if its support is contained in the antidiagonal of $G$.

preprint2016arXiv

Derivatives at the Boundary for Analytic Lipschitz Functions

We consider the behaviour of holomorphic functions on a bounded open subset of the plane, satisfying a Lipschitz condition with exponent $α$, with $0<α<1$, in the vicinity of an exceptional boundary point where all such functions exhibit some kind of smoothness. Specifically, we consider the relation between the abstract idea of a bounded point derivation on the algebra of such functions and the classical complex derivative evaluated as a limit of difference quotients. We show that whenever such a bounded point derivation exists at a boundary point $b$, it may be evaluated by taking a limit of classical difference quotients, for approach from a set having full area density at $b$.

preprint2016arXiv

Fredholm theory connected with a Douglis-Nirenberg system of differential equations over $\mathbb{R}^n$

We consider a spectral problem over $\mathbb{R}^n$ for a Douglis-Nirenberg system of differential operators under limited smoothness assumptions and under the assumption of parameter-ellipticity in a closed sector $\mathcal{L}$ in the complex plane with vertex at the origin. We pose the problem in an $L_p$ Sobolev-Bessel potential space setting, $1 < p < \infty$, and denote by $A_p$ the operator induced in this setting by the spectral problem. We then derive results pertaining to the Fredholm theory for $A_p$ for values of the spectral parameter $λ$ lying in $\mathcal{L}$ as well as results pertaining to the invariance of the Fredholm domain of $A_p$ with $p$.

399 works