Singular Integrals on Product Spaces with Mixed Norms

In this paper,we prove the boundedness of the singular integral operators on product spaces with mixed norms and obtain the endpoint weak-type estimates.

DERIVATIVES OF HARMONIC MIXED NORM AND BLOCH-TYPE SPACES IN THE UNIT BALL OF R^n

Let H（B） be the set of all harmonic functions f on the unit ball B of Rn. For 0 p, q ≤ ∞ and a normal weight φ, the mixed norm space Hp,q, φ（B） consists of all functions f in H（B） for which the mixed norm p,q, φ ∞. In this article, we obtain some characterizations in terms of radial, tangential, and partial derivative norms in Hp,q, φ（B）. The parallel results for the Bloch-type space are also obtained. As an application, the analogous problems for polyharmonic functions are discussed.

A FIXED POINT APPROACH TO THE STABILITY OF A GENERAL MIXED ADDITIVE-CUBIC EQUATION ON BANACH MODULES

Using a fixed-point method, we establish the generalized Hyers-Ulam stability of a general mixed additive-cubic equation： f（kx ＋ y） ＋ f（kx - y） = kf（x ＋ y） ＋ kf（x - y） ＋ 2f（kx） - 2kf（x） in Banach modules over a unital Banach algebra.

Image denoising method with tree-structured group sparse modeling of wavelet coefficients

In order to enhance the image contrast and quality, inspired by the interesting observation that an increase in noise intensity tends to narrow the dynamic range of the local standard deviation (LSD) of an image, a tree-structured group sparse optimization model in the wavelet domain is proposed for image denoising. The compressed dynamic range of LSD caused by noise leads to a contrast reduction in the image, as well as the degradation of image quality. To equalize the LSD distribution, sparsity on the LSD matrix is enforced by employing a mixed norm as a regularizer in the image denoising model. This mixed norm introduces a coupling between wavelet coefficients and provides a tree-structured group scheme. The alternating direction method of multipliers (ADMM) and the fast iterative shrinkage-thresholding algorithm (FISTA) are applied to solve the group sparse model based on different cases. Several experiments are conducted to verify the effectiveness of the proposed model. The experimental results indicate that the proposed group sparse model can efficiently equalize the LSD distribution and therefore can improve the image contrast and quality.

BERGMAN TYPE OPERATOR ON MIXED NORM SPACES WITH APPLICATIONS

The authors investigate the conditions for the boundedness of Bergman type operators Ps,t in mixed norm space L_(p),q(φ)on the unit ball of C^(n)(n≥1),and obtain a sufficient condition and a necessary condition for general normal functionφ,and a sufficient and necessary condition forφ(r)=(1-r 2)αlogβ(2(1-r)-1)(α>0,β≥0).This generalizes the result of Forelli Rudin on Bergman operator in Bergman space.As applications,a more natural method is given to compute the duality of the mixed norm space,solve the Gleason's problem for mixed norm space and obtain the characterization of mixed norm space in terms of partial derivatives.Moreover,it is proved thatf∈L(0)∞,q(φ)iff all the functions(1-|z|2)|α||α|fzα(z)∈L(0)∞,q(φ)for holomorphic functionf,1≤q≤∞.

Coefficient multipliers of mixed norm space in the ball Dedicated to Professor Sheng GONG on the occasion of his 75th birthday

In the paper, we characterize the coefficient multiplier spaces of mixed norm spaces (Hp,q((?)1),Hu,v((?)2)) for the values of p, q, u, v in three cases: (i)0<p≤u≤∞, 0 < q≤min(1,v)≤frr. (ii) v =∞,0<p≤u≤∞, 1≤u, q≤∞. (iii) 1≤v≤2≤q≤∞, and 0<p≤u≤∞or 1≤p, u≤∞. The first case extends the result of Blasco, Jevtic, and Pavlovic in one variable. The third case generalizes partly the results of Jevtic, Jovanovic, and Wojtaszczyk to higher dimensions.

Carleson Type Measures for Harmonic Mixed Norm Spaces with Application to Toeplitz Operators

Let Ω be a bounded domain in Rnwith a smooth boundary, and let h p,q be the space of all harmonic functions with a finite mixed norm. The authors first obtain an equivalent norm on h p,q, with which the definition of Carleson type measures for h p,q is obtained. And also, the authors obtain the boundedness of the Bergman projection on h p,q which turns out the dual space of h p,q. As an application, the authors characterize the boundedness(and compactness) of Toeplitz operators T μ on h p,q for those positive finite Borel measures μ.

An Image Denoising Model via the Reconciliation of the Sparsity and Low-Rankness of the Dictionary Domain Coefficients

Sparse coding has achieved great success in various image restoration tasks.However,if the sparse representation coefficients of the structure(low-frequency information)and texture(high-frequency information)components of the image are under the same penalty constraint,the restoration effect may not be ideal.In this paper,an image denoising model combining mixed norm and weighted nuclear norm as regularization terms is proposed.The proposed model simultaneously exploits the group sparsity of the high frequency and lowrankness of the low frequency in dictionary-domain.The mixed norm is used to constrain the high frequency part and the weighted nuclear norm is used to constrain the low frequency part.In order to make the proposed model easy to solve under the framework of alternative direction multiplier method(ADMM),iterative shrinkage threshold method and weighted nuclear norm minimization method are used to solve the two sub-problems.The validity of the model is verified experimentally through comparison with some state-of-the-art methods.

Compact Ceshro Operators from Spaces H（p,q,u） to H（p, q, v）

Let μ and v be normal functions and let Tg be the extended Ceshso operator in terms of the symbol g. In this paper, we will characterize those g so that Tg is bounded （or compact） from mixed norm spaces H（p, q, μ） to H（p, q, v） in the unit ball of C^n, Furthermore, as applications, some analogous results are also given on weighted Bergman spaces and Dirichlet type spaces.

An Image Denoising Method Based on Group Sparsity and Low Rank

In this paper,we propose an image denoising method combining the priors of non-local self similarity(NSS),low rank and group sparsity.In the proposed scheme,the image is decomposed into overlapping patches,and then these patches are classified by the K-means clustering.Patches in each cluster are stacked into a matrix and then are decomposed into low frequency component and high frequency component through 2-D wavelet transform.Intuitively,the low frequency component should be a low rank matrix.We show that the high frequency component can be recovered by weighted mixed norm minimization which is also known as group sparse model.Then we propose an image denoising model using nuclear norm and weighted mixed norm as regularizers to enforce the priors on the low and high frequency.The proposed model can be solved efficiently in the framework of alternating direction multiplier method(ADMM)algorithm.Several experiments are carried out to verify the performance of the proposed model.