Norm Spaces Research Articles

Principal frequency of Δ∞ as limit of Rayleigh quotients in Orlicz spaces

The asymptotic behavior of Rayleigh quotients involving Luxemburg norms in Orlicz spaces is studied as , when is a sequence of N-functions. In a particular case, we recover a well-known result of Juutinen, Lindqvist & Manfredi and Fukagai, Ito & Narukawa regarding the limit, as of the minima of Rayleigh quotients associated to the eigenvalue problem for the p-Laplacian.

Poisson—Bergman Type Operators on Lipschitz and Mixed Norm Spaces in the Real Ball

Boundedness of some Poisson-Bergman type operators is stated over the unit ball in ℝn. Forelli-Rudin type theorems are proved and bounded harmonic projections are found on Lipschitz and mixed norm spaces.

Discrete Littlewood–Paley–Stein characterization of multi-parameter local Hardy spaces

Abstract In our recent work [W. Ding, G. Lu and Y. Zhu, Multi-parameter local Hardy spaces, Nonlinear Anal. 184 2019, 352–380], the multi-parameter local Hardy space h p ⁢ ( ℝ n 1 × ℝ n 2 ) {h^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} has been introduced by using the continuous inhomogeneous Littlewood–Paley–Stein square functions. In this paper, we will first establish the new discrete multi-parameter local Calderón’s identity. Based on this identity, we will define the local multi-parameter Hardy space h dis p ⁢ ( ℝ n 1 × ℝ n 2 ) {h_{\mathrm{dis}}^{p}(\mathbb{R}^{n_{1}}\times\mathbb{R}^{n_{2}})} by using the discrete inhomogeneous Littlewood–Paley–Stein square functions. Then we prove that these two multi-parameter local Hardy spaces are actually the same. Moreover, the norms of the multi-parameter local Hardy spaces under the continuous and discrete Littlewood–Paley–Stein square functions are equivalent. This discrete version of the multi-parameter local Hardy space is also critical in establishing the duality theory of the multi-parameter local Hardy spaces.

Sobolev embeddings, rearrangement-invariant spaces and Frostman measures

Sobolev embeddings, of arbitrary order, are considered into function spaces on domains of Rn endowed with measures whose decay on balls is dominated by a power d of their radius. Norms in arbitrary rearrangement-invariant spaces are contemplated. A comprehensive approach is proposed based on the reduction of the relevant n-dimensional embeddings to one-dimensional Hardy-type inequalities. Interestingly, the latter inequalities depend on the involved measure only through the power d. Our results allow for the detection of the optimal target space in Sobolev embeddings, for broad families of norms, in situations where customary techniques do not apply. In particular, new embeddings, with augmented target spaces, are deduced even for standard Sobolev spaces.

Norms of sub-exponential random vectors

We discuss various forms of the Luxemburg norm in spaces of random vectors with coordinates belonging to the classical Orlicz spaces of exponential type. We prove equivalent relations between some kinds of these forms. We also show when the so-called uniform norm is majorized by norms of coordinates up to some constants. We give an application of other norm to study of chaos in random vectors with sub-exponential coordinates.

Derivatives of Blaschke products in weighted mixed norm spaces

For $1/2\lt p\lt \infty $, $0\lt q\lt \infty $ and a certain two-sided doubling weight $\omega $, we give a condition for the zeros of a Blaschke product $B$ which guarantees that $$ \|B'\|_{A^{p,q}_\omega }^q=\int _0^1 \bigg (\int _0^{2\pi } |B'(re^{i\theta })|^p d\theta \bigg )^{q/p} \omega (r)\,dr\lt \infty . $$ In addition, it is shown that the condition is necessary if the zero-sequence is a finite union of separated sequences.

Semigroups of composition operators and integral operators on mixed norm spaces

We characterize the semigroups of composition operators that are strongly continuous on the mixed norm spaces $$H(p,q,\alpha )$$ . First, we study the separable spaces $$H(p,q,\alpha )$$ with $$q<\infty ,$$ that behave as the Hardy and Bergman spaces. In the second part we deal with the spaces $$H(p,\infty ,\alpha ),$$ where polynomials are not dense. To undertake this study, we introduce the integral operators, characterize its boundedness and compactness, and use its properties to find the maximal closed linear subspace of $$H(p,\infty ,\alpha )$$ in which the semigroups are strongly continuous. In particular, we obtain that this maximal space is either $$H(p,0,\alpha )$$ or non-separable, this result being the deepest one in the paper.

On a certain class of norms in semimodular spaces and their monotonicity properties

Let X be a vector space over K, K=R or K=C and let ρi be an s-convex semimodular defined on X for i∈{1,…,n−1}, where n≥2. Put ρ=max1≤i≤n−1⁡{ρi} and Xρ={x∈X:ρ(dx)<∞ for some d>0}. In this paper we define a new class of s-norms (norms if s=1) on Xρ. In particular, our definition generalizes in a natural way the Orlicz-Amemiya and Luxemburg norms defined for s-convex semimodulars. Then, we investigate order continuous, the Fatou Property and various monotonicity properties of s-semimodular spaces equipped with these s-norms.

New inertial algorithm for solving split common null point problem in Banach spaces

Inspired by the works of Alvarez and Attouch (Set-Valued Anal. 9:3–11, 2001), López et al. (Inverse Probl. 28:ID085004, 2012), Takahashi (Arch. Math. (Basel) 104(4):357–365, 2015) and Suantai et al. (Appl. Gen. Topol. 18(2):345–360, 2017), as well as Promluang and Kuman (J. Inform. Math. Sci. 9(1):27–44, 2017), we propose a new inertial algorithm for solving split common null point problem without the prior knowledge of the operator norms in Banach spaces. Under mild and standard conditions, the weak and strong convergence theorems of the proposed algorithms are obtained. Also the split minimization problem is considered as the application of our results. Finally, the performances and computational examples are presented, and a comparison with related algorithms is provided to illustrate the efficiency and applicability of our new algorithm.

Hardy–Steklov Operators and the Duality Principle in Weighted First-Order Sobolev Spaces on the Real Axis

Estimates of the norms of spaces associated to weighted first-order Sobolev spaces with various weight functions and summation parameters are established. As the main technical tool, boundedness criteria for the Hardy–Steklov integral operator with variable limits of integration in Lebesgue spaces on the real axis are used.

Structure-Texture Image Decomposition using Deep Variational Priors.

Most variational formulations for structure-texture image decomposition force structure images to have small norm in some functional spaces, and share a common notion of edges, i.e., large-gradients or -intensity differences. However, such definition makes it difficult to distinguish structure edges from oscillations that have fine spatial scale but high contrast. In this paper, we introduce a new model by learning deep variational prior for structure images without explicit training data. An alternating direction method of multiplier (ADMM) algorithm and its modular structure are adopted to plug deep variational priors into an iterative smoothing process. The central observations are that convolution neural networks (CNNs) can replace the total variation prior, and are indeed powerful to capture the natures of structure and texture. We show that our learned priors using CNNs successfully differentiate highamplitude details from structure edges, and avoid halo artifacts. Different from previous data-driven smoothing schemes, our formulation provides another degree of freedom to produce continuous smoothing effects. Experimental results demonstrate the effectiveness of our approach on various computational photography and image processing applications, including texture removal, detail manipulation, HDR tone-mapping, and nonphotorealistic abstraction.

Relationship with modular and norm of vector valued sequences spaces Cesàro-φ

In this paper, we introduce the generalization of Cesàro-Örlicz sequence space i.e Cesàro-φ vector valued sequence space, cesφ(X), and give the necessary and suffiecient conditions that the space is nontrivial. Furthermore, we will prove the relationship between modular and norm on this space.

Wavelets in weighted norm spaces

We give a complete characterization of the classes of weight functions for which the higher rank Haar wavelet systems are unconditional bases in weighted norm Lebesgue spaces. Particulary it follows that higher rank Haar wavelets are unconditional bases in the weighted norm spaces with weights which have strong zeros at some points. This shows that the class of weight functions for which higher rank Haar wavelets are unconditional bases is much richer than it was supposed.

On Hardy Inequality in Variable Lebesgue Spaces with Mixed Norm

In this paper a two-weight boundedness of multidimensional Hardy operator and its dual operator acting from one weighted variable Lebesgue spaces with mixed norm into other weighted variable Lebesgue spaces with mixed norm spaces is proved. In particular, a new type two-weight criterion for multidimensional Hardy operator is obtained.

Characterization of generalized Orlicz spaces

The norm in classical Sobolev spaces can be expressed as a difference quotient. This expression can be used to generalize the space to the fractional smoothness case. Because the difference quotient is based on shifting the function, it cannot be used in generalized Orlicz spaces. In its place, we introduce a smoothed difference quotient and show that it can be used to characterize the generalized Orlicz–Sobolev space. Our results are new even in Orlicz spaces and variable exponent spaces.

Sharp Norm Estimates for Weighted Bergman Projections in the Mixed Norm Spaces

In this paper, we show that the norm of the Bergman projection on Lp,q-spaces in the upper half-plane is comparable to csc(π/q). Then we extend this result to a more general class of domains, known as the homogeneous Siegel domains of type II.

ANALYSIS IN THE MULTI‐DIMENSIONAL BALL

We study the heat semigroup maximal operator associated with a well-known orthonormal system in the -dimensional ball. The corresponding heat kernel is shown to satisfy Gaussian bounds. As a consequence, we can prove weighted estimates, as well as some weighted inequalities in mixed norm spaces, for this maximal operator.

N-Normed Spaces with Norms of Its Quotient Spaces

The concept of n-normed spaces is a generalization of the concept of normed spaces. Some characteristics of n-normed spaces have been discussed by many researchers. These spaces are usually observed using a set of n linear independent vectors. In this paper, we will construct quotient spaces of an n-normed space with respect to n linear independent vectors. We define a norm in each quotient space by using the n-norm that we have. Norms of these quotient spaces will be a new viewpoint in observing characteristics of n-normed spaces.

Wavelets and Bidemocratic Pairs in Weighted Norm Spaces

A complete characterization of weight functions for which the higher-rank Haar wavelets are greedy bases in weighted Lp spaces is given. The proof uses the new concept of a bidemocratic pair for a Banach space and also pairs (Φ, Φ), where Φ is an orthonormal system of bounded functions in the spaces Lp, p≠2.

Derivatives of Inner Functions in Weighted Mixed Norm Spaces

For $$0<p,q<\infty $$ , we characterize those radial weights $$\omega $$ satisfying a two-sided doubling condition for which the asymptotic equation $$\begin{aligned} \Vert \Theta '\Vert _{A^{p,q}_\omega }^q= \int _0^1 M_p^q(r,\Theta ')\,\omega (r)\,\mathrm{d}r \asymp \int _0^1 \left( \int _0^{2\pi } \left( \frac{1-|\Theta (re^{i\theta })|}{1-r}\right) ^p \mathrm{d}\theta \right) ^{q/p} \omega (r)\, \mathrm{d}r \end{aligned}$$ is valid for all inner functions $$\Theta $$ . As a consequence of this result, we obtain a sharp condition which guarantees that the only inner functions whose derivative belongs to the weighted mixed norm space $$A^{p,q}_\omega $$ are Blaschke products. Moreover, a condition which implies that the only inner functions whose derivative belongs to $$A^{p,q}_\omega $$ are finite Blaschke products is proved.