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In a previous paper, we obtained several "compact versions" of Rubio de Francia's weighted extrapolation theorem, which allowed us to extrapolate the compactness of linear operators from just one space to the full range of weighted Lebesgue spaces, where these operators are bounded. In this paper, we study the extrapolation of compactness for bilinear operators in terms of bilinear Muckenhoupt weights. As applications, we easily recover and improve earlier results on the weighted compactness of commutators of bilinear Calder\'{o}n-Zygmund operators, bilinear fractional integrals and bilinear Fourier multipliers. More general versions of these results are recently due to Cao, Olivo and Yabuta (arXiv:2011.13191), whose approach depends on developing weighted versions of the Fr\'echet--Kolmogorov criterion of compactness, whereas we avoid this by relying on "softer" tools, which might have an independent interest in view of further extensions of the method., Comment: v3: final version, incorporated referee comments, to appear in Indagationes Mathematicae, 27 pages

To extend existing distance metrics in the Lp space, we define a novel distance, named relative Euclidean distance (RED), and prove that it is positive definite, symmetric, and satisfies the triangle inequality. This distance has a value range of [ 0,1] . We prove a uniqueness of it in the Lp space. The technique for order preference by similarity to ideal solution (TOPSIS) is widely used for multiple-attribute decision problems. The commonly used relative closeness measure in the TOPSIS is not a mathematical distance and thus does not have all the nice properties of such a distance. To improve the TOPSIS, particularly its relative closeness measure, we propose two measures, named optimistic distance (OD) and pessimistic distance (PD), based on our RED, and use them to measure the relative closeness in the TOPSIS. Both measures are positive definite, symmetric, and satisfy the triangle inequality. The best choice in OD (or PD) is on the Pareto frontier. Three examples of performance ranking are given to illustrate the usefulness of the TOPSIS with our proposed measures.

In this paper we extend the utilization of the Banach spaces-based formulations, usually employed for solving diverse nonlinear problems in continuum mechanics via primal and mixed finite element methods, to the virtual element method (VEM) framework and its respective applications. More precisely, we propose and analyze an [Formula: see text] spaces-based mixed virtual element method for a pseudostress-velocity formulation of the two-dimensional Navier–Stokes equations with Dirichlet boundary conditions. To this end, a dual-mixed approach determined by the introduction of a nonlinear tensor linking the usual pseudostress for the Stokes equations with the convective term is employed. As a consequence, this new tensor, say [Formula: see text], and the velocity [Formula: see text] of the fluid constitute the unknowns of the formulation, whereas the pressure is computed via a post-processing formula. The simplicity of the resulting VEM scheme is reflected by the absence of augmented terms, on the contrary to previous works on this and related models, and by the incorporation in it of only the projector onto the piecewise polynomial tensors and the usual stabilizer depending on the degrees of freedom of the virtual element subspace approximating [Formula: see text]. In turn, the non-virtual but explicit subspace given by the piecewise polynomial vectors of degree [Formula: see text] is employed to approximate [Formula: see text]. The corresponding solvability analysis is carried out by using appropriate fixed-point arguments, along with the discrete versions of the Babuška–Brezzi theory and the Banach–Nečas–Babuška theorem, both in subspaces of Banach spaces. A Strang-type lemma is applied to derive the a priori error estimates for the virtual element solution as well as for the fully computable approximation of [Formula: see text], the post-processed pressure, and a second post-processed approximation of [Formula: see text]. Finally, several numerical results illustrating the performance of the mixed-VEM scheme and confirming the rates of convergence predicted by the theory are reported.

In this paper, we introduce a new version of the definition of a quasi-norm (in particular, a norm) in Lebesgue spaces with variable order of summability. Using it, we prove an analogue of Holders inequality for such spaces, which is more general and more precise than those known earlier.

In this paper we mainly investigate the inviscid limit for the strong solutions of the finite extensible nonlinear elastic (FENE) dumbbell model. By virtue of the Littlewood-Paley theory, we first obtain a uniform estimate for the solution to the FENE dumbbell model with viscosity in Besov spaces. Moreover, we show that the data-to-solution map is continuous. Finally, we prove that the strong solution of the FENE dumbbell model converges to a Euler system couple with a Fokker-Planck equation. Furthermore, convergence rates in Lebesgue spaces are obtained also.

Let $\alpha\in (0, 1]$, $\beta\in [0, n)$ and $T_{\Omega,\beta}$ be a singular or fractional integral operator with homogeneous kernel $\Omega$. In this article, a CMO type space ${\rm CMO}_\alpha(\mathbb R^n)$ is introduced and studied. In particular, the relationship between ${\rm CMO}_\alpha(\mathbb R^n)$ and the Lipchitz space $Lip_\alpha(\mathbb R^n)$ is discussed. Moreover, a necessary condition of restricted boundedness of the iterated commutator $(T_{\Omega,\beta})^m_b$ on weighted Lebesgue spaces via functions in $Lip_\alpha(\mathbb R^n)$, and an equivalent characterization of the compactness for $(T_{\Omega,\beta})^m_b$ via functions in ${\rm CMO}_\alpha(\mathbb R^n)$ are obtained. Some results are new even in the unweighted setting for the first order commutators., Comment: arXiv admin note: text overlap with arXiv:1712.08292

In this paper we introduce and analyze, up to our knowledge for the first time, Banach spaces-based mixed variational formulations for nearly incompressible linear elasticity and Stokes models. Our interest in this subject is motivated by the respective need that arises from the solvability studies of nonlinear coupled problems in continuum mechanics that involve these equations. We consider pseudostress-based approaches in both cases and apply a suitable integration by parts formula for ad-hoc Sobolev spaces to derive the corresponding continuous schemes. We utilize known and new preliminary results, along with the Babuska-Brezzi theory in Banach spaces, to establish the well-posedness of the formulations for a particular range of the indexes of the Lebesgue spaces involved. Among the aforementioned new results from us, we highlight the construction of a particular operator mapping a tensor Lebesgue space into itself, and the generalization of a classical estimate in L 2 for deviatoric tensors, which plays a key role in the Hilbertian analysis of linear elasticity, to arbitrary Lebesgue spaces. No discrete analysis is performed in this work.

We revisit the well-known work of K. Masuda in 1984 on the weak–strong uniqueness of L ∞ L 3 Leray–Hopf weak solutions of Navier–Stokes equation. We modify the argument, and extend the uniqueness result to the scaling critical anisotropic Lebesgue space with mixed-norms. As a consequence, our results cover the class of initial data and solutions which may be singular or decay with different rates along different spatial variables. The result relies on the establishment of several refined properties of solutions of the Stokes and Navier–Stokes equations in mixed-norm Lebesgue spaces which seem to be of independent interest.

We show that every unconditional basis in a finite direct sum ⊕p∈Aℓp , with A ⊂ (0,∞], splits into unconditional bases of each summand. This settles a 40 years old question raised in 'A. Ortyński, Unconditional bases in ℓp ⊕ ℓq, 0< p < q

In the first part of this paper, we establish the well-posedness for solutions of Caputo stochastic fractional differential equations (for short Caputo SFDE) of order α∈(12,1) in Lp spaces with p≥2...

In this paper, we establish necessary and sufficient conditions on monotone weight functions for the boundedness for Hausdorff operators of special kind in weighted Lebesgue spaces. In particular, we get similar results for important operators of harmonic analysis which are special cases of the Hausdorff operators. The weights are illustrated by examples at the end of the paper.

In this paper, we analyze various classes of weighted Stepanov ergodic spaces, weighted Weyl ergodic spaces and weighted pseudo-ergodic spaces in $${{\mathbb {R}}}^{n}$$ with the help of results from the theory of Lebesgue spaces with variable exponents $$L^{p(x)}$$ . Several structural results of ours seem to be new even in the case of consideration of the constant exponents $$p(x)\equiv p\in [1,\infty ).$$ We especially examine the Stepanov asymptotical almost periodicity at minus infinity and the Weyl asymptotical almost periodicity at minus infinity, providing also some interesting applications to the abstract Volterra integro-differential equations in Banach spaces.

Nonnegative matrix factorization methods have been widely used in many applications in recent years. However, the clustering performances of such methods may deteriorate dramatically in the presence of non-Gaussian noise or outliers. To overcome this problem, in this paper, we propose correntropy-based dual graph regularized NMF with LP smoothness (CDNMFS) for data representation. Specifically, we employ correntropy instead of the Euclidean norm to measure the incurred reconstruction error. Furthermore, we explore the geometric structures of both the input data and the feature space and impose an Lp norm constraint to obtain an accurate solution. In addition, we introduce an efficient optimization scheme for the proposed model and present its convergence analysis. Experimental results on several image datasets demonstrate the superiority of the proposed CDNMFS method.

We investigate the boundedness of Fourier multipliers on a compact Lie group when acting on Triebel-Lizorkin spaces. Criteria are given in terms of the Hörmander-Mihlin-Marcinkiewicz condition. In our analysis, we use the difference structure of the unitary dual of a compact Lie group. Our results cover the sharp Hörmander-Mihlin theorem on Lebesgue spaces and also other historical results on the subject.

In this paper, we generalize the usual multiplicative ∊ -indicator I ∊ and additive ∊ -indicator I ∊ + to the forms with L p -norm, i.e. I ∊ p -indicator and I ∊ + p -indicator, and investigate their properties under different p. From the analysis, we find that both I ∊ p -indicator and I ∊ + p -indicator have a good performance on measuring the population convergence and diversity, especially when p is set to be infinity. In order to show their differences more intuitively, we implement a series of comparative experiments by the proposed I ∊ p -indicator and I ∊ + p -indicator based evolutionary algorithm. The experimental results confirm that all these ∊ -indicators have good performances, particulary as p is set to be infinity. We also compare the proposed indicator-based evolutionary algorithm with five existing algorithms. The experimental results demonstrate that our algorithm is effective. In addition, considering that the usual IGD metric is not accurate in measuring the convergence of a population, we adopt the measure based on I ∊ + p -indicator and propose a more effective performance metric, which is denoted as IGD ∊ + .

In this paper, we prove the boundedness of the fractional maximal and the fractional integral operator in the p -adic variable exponent Lebesgue spaces. As an application, we show the existence and uniqueness of the solution for a nonhomogeneous Cauchy problem in the p -adic variable exponent Lebesgue spaces.

This article is devoted to presenting a general interpolation result on mixed-norm function spaces generated by quasi-Banach lattices. Under certain conditions, the authors show that such mixed-norm function spaces are closed under the Calderon product and the ± interpolation method. As applications, the authors obtain some new interpolation results for mixed-norm variable Lebesgue spaces, mixed-norm Lorentz spaces, and mixed-norm Morrey spaces.

Let $n\ge2$ and $\Omega$ be a bounded Lipschitz domain in $\mathbb{R}^n$. In this article, the authors investigate global (weighted) estimates for the gradient of solutions to Robin boundary value problems of second order elliptic equations of divergence form with real-valued, bounded, measurable coefficients in $\Omega$. More precisely, let $p\in(n/(n-1),\infty)$. Using a real-variable argument, the authors obtain two necessary and sufficient conditions for $W^{1,p}$ estimates of solutions to Robin boundary value problems, respectively, in terms of a weak reverse H\"older inequality with exponent $p$ or weighted $W^{1,q}$ estimates of solutions with $q\in(n/(n-1),p]$ and some Muckenhoupt weights. As applications, the authors establish some global regularity estimates for solutions to Robin boundary value problems of second order elliptic equations of divergence form with small $\mathrm{BMO}$ coefficients, respectively, on bounded Lipschitz domains, $C^1$ domains or (semi-)convex domains, in the scale of weighted Lebesgue spaces, via some quite subtle approach which is different from the existing ones and, even when $n=3$ in case of bounded $C^1$ domains, also gives an alternative correct proof of some know result. By this and some technique from harmonic analysis, the authors further obtain the global regularity estimates, respectively, in Morrey spaces, (Musielak--)Orlicz spaces and variable Lebesgue spaces, Comment: 54 pages; Submitted

In this paper, the authors study the boundedness properties of a class of multilinear strongly singular integral operator with generalized kernels on product of weighted Lebesgue spaces and product of variable exponent Lebesgue spaces, respectively. Moreover, the types $ L^{\infty}\times \dots \times L^{\infty}\rightarrow BMO $ and $ BMO\times \dots \times BMO\rightarrow BMO $ endpoint estimates are also obtained.

In [3], Chen, Deng, Ding and Fan proved that the fractional power dissipative operator is bounded on Lebesgue spaces Lp(ℝn), Hardy spaces Hp(ℝn) and general mixed norm spaces, which implies almost everywhere convergence of such operator. In this paper, we study the rate of convergence on fractional power dissipative operator on some sobolev type spaces.

The study of operator algebras on Hilbert spaces, and C ∗ -algebras in particular, is one of the most active areas within Functional Analysis. A natural generalization of these is to replace Hilbert spaces (which are L 2 -spaces) with L p -spaces, for p ∈ [ 1 , ∞ ) . The study of such algebras of operators is notoriously more challenging, due to the very complicated geometry of L p -spaces by comparison with Hilbert spaces. We give a modern overview of a research area whose beginnings can be traced back to the 50’s, and that has seen renewed attention in the last decade through the infusion of new techniques. The combination of these new ideas with old tools was the key to answer some long standing questions. Among others, we provide a description of all unital contractive homomorphisms between algebras of p -pseudofunctions of groups.

This paper deals with the existence and regularity of solutions for $$p(\cdot )$$ -Laplacian problem with singular nonlinearity having variable exponent. The model example is $$\begin{aligned} \left\{ \begin{array}{ll}{} -\Delta _{p(x)}u=\frac{f(x)}{u^{\alpha (x)}}, &{} x \in \Omega ,\\ u=0, &{} x \in \partial \Omega , \end{array}\right. \end{aligned}$$ where $$\alpha (x)>0$$ and f is a nonnegative function belonging to the Lebesgue space $$L^{m}(\Omega )$$ for some suitable $$m \ge 1$$ . The results show the dependence of the summability of f in some Lebesgue spaces and on the values of $$\alpha (x)$$ .

In this paper, we introduce weighted Morrey-Herz spaces $ M\dot K^{\alpha, \lambda}_{q, p(\cdot)}(w~^{p(\cdot)}) $ with variable exponent $ p(\cdot) $. Then we prove the boundedness of multilinear Calderón-Zygmund singular operators on weighted Lebesgue spaces and weighted Morrey-Herz spaces with variable exponents.

In this paper we consider existence, nonexistence and multiplicity of solutions for a class of indefinite quasilinear elliptic problems in the upper half-space involving weights in anisotropic Lebesgue spaces. One of our basic tools consists in a Hardy type inequality proved in the present paper that allows us to establish Sobolev embeddings into Lebesgue spaces with weights in anisotropic Lebesgue spaces.

The Cauchy problem for the parabolic--elliptic Keller--Segel system in the whole $n$-dimensional space is studied. For this model, every constant $A \in \mathbb{R}$ is a stationary solution. The main goal of this work is to show that $A < 1$ is a stable steady state while $A > 1$ is unstable. Uniformly local Lebesgue spaces are used in order to deal with solutions that do not decay at spatial variable on the unbounded domain., 20 pages

We are concerned with basic properties of the Helmholtz decomposition of the Muckenhoupt Ap-weighted Lp-space ( $$L_w^p(\Omega )$$ )n for any domain Ω in ℝn (n ∈ ℤ, n ≥ 2), 1 < p < ∞ and Muckenhoupt Ap-weight w ∈ Ap. Set $${p^\prime}: = {p \over {p - 1}}$$ and $${w^\prime}: = {w^{ - {1 \over {p - 1}}}}$$ . Then the consistency of the Helmholtz decomposition of $${(L_w^p(\Omega ))^n}$$ and $${(L_{{w^\prime}}^{{p^\prime}}(\Omega ))^n}$$ are equivalent to the validity of the variational estimate of $$L_{w,\pi}^p(\Omega ) + L_{{w^\prime},\pi}^{{p^\prime}}(\Omega )$$ . The variational estimate of $$L_{w,\pi}^p(\Omega )$$ and $$L_{{w^\prime},\pi}^{{p^\prime}}(\Omega )$$ plays an important role in the proof. Furthermore, we can replace $$L_{w,\pi}^p(\Omega )$$ , $$L_{{w^\prime},\pi}^{{p^\prime}}(\Omega )$$ and $$L_{w,\pi}^p(\Omega ) + L_{{w^\prime},\pi}^{{p^\prime}}(\Omega )$$ by $$L_{w,\sigma}^p(\Omega ),\,\,L_{{w^\prime},\sigma}^{{p^\prime}}(\Omega )$$ and $$L_{w,\sigma}^p(\Omega ) + L_{{w^\prime},\sigma}^{{p^\prime}}(\Omega )$$ respectively.

Bu çalışmada. sağ yan fonksiyonları birinci değişkenlerine göre singülerliğe sahip ve başlangıç koşulu homojen olmayan Riemann-Liouville kesirli diferansiyel denklemlerinin bazı sınıfları göz önüne alınmıştır. İlk önce, bu başlangıç-değer probleminin bir lokal sürekli çözümünün varlığını hangi koşular altında gerçekleştiği kısaca ifade edilmiştir. Daha sonra ise, sırasıyla Krasnosel’skii-Krein, Kooi, Roger ve Banaś-Rivero tiplerinde teklik teoremleri ortaya çıkarılmıştır. Bu teoremler daha önceden elde edilen sonuçları geliştirken, bu teoremlerin ispatları için, daha önceden var olan teknikler Lebesgue uzaylarının araçları ile zenginleştirilmiştir., In this study, some classes of Riemann-Liouville fractional differential equations with right-hand side functions having a singularity with respect to their first variable and with a nonhomogeneous initial condition are considered. First, it is briefly stated that under which conditions the existence of a local continuous solution of this initial value problem occurs. Later, uniqueness theorems were developed in types of Krasnosel’skii-Krein, Kooi, Roger and Banaś-Rivero, respectively. These theorems improve the previously obtained results, and for their proofs pre-existing techniques are enriched by the tools of Lebesgue spaces.

Tartar gave an alternative proof of the Riesz–Thorin interpolation theorem for operators of strong types (1, 1) and $$(\infty ,\infty )$$ . His method characterizes the $$L^{p}$$ norm in terms of the Lebesgue spaces $$L^{1}$$ and $$L^{\infty }$$ , and works not only for complex Lebesgue spaces but also for real Lebesgue spaces. The aim of this paper is to extend the proof for operators of strong types $$(p_{1},q_{1})$$ and $$(\infty ,\infty )$$ with $$1\le p_{1}\le q_{1}