The Cauchy problem of the modified Helmholtz-type equation is severely ill-posed, i.e., the solution does not depend continuously on the given Cauchy data. Thus the regularization methods are required to recover the numerical stability. In this paper, we propose a quasi-reversibility regularization method to deal with this ill-posed problem. Convergence estimates are obtained under a-priori bound assumptions for the exact solution and the selection of regularization parameter. Some numerical results are given to show that this method is stable and feasible.

Let T be the singular integral operator with variable kernel de ned byand D γ ( ≤ γ ≤ ) be the fractional di erentiation operator. Let T * and T ♯ be the adjoint of T and the pseudo-adjoint of T, respectively. The aim of this paper is to establish some boundedness for TD γ − D γ T and via the convolution operator T m,j and Calderón-Zygmund operator, and then establish their boundedness on these spaces. The boundedness on HMKMoreover, the authors also establish various norm characterizations for the product T T and the pseudo-product T ○ T .

The aim of this paper is to deal with the boundedness of the θ-type Calderón-Zygmund operators and their commutators on Herz spaces with two variable exponents p(⋅), q(⋅). It is proved that the θ-type Calderón-Zygmund operators are bounded on the homogeneous Herz space with variable exponents $\begin{array}{} \displaystyle \dot{K}^{\alpha,q(\cdot)}_{p(\cdot)}(\mathbb{R}^{n}). \end{array}$ Furthermore, the boundedness of the corresponding commutators generated by BMO function and Lipschitz function is also obtained respectively.

We obtain a class of commutators of bilinear pseudo-differential operators on products of Hardy spaces by applying the accurate estimates of the Hörmander class. And we also prove another version of these types of commutators on Herz-type spaces.

Let be the singular integral operator with variable kernel defined by ( ) = p.v. ∫ R (Ω( , − )/| − | ) ( )d and let (0 ≤ ≤ 1) be the fractional differentiation operator. Let * and ♯ be the adjoint of and the pseudoadjoint of , respectively. In this paper, the authors prove that − and ( * − ♯ ) are bounded, respectively, from Morrey-Herz spaceṡ, ,1 (R ) to the weak Morrey-Herz spaceṡ, ,1 (R ) by using the spherical harmonic decomposition. Furthermore, several norm inequalities for the product 1 2 and the pseudoproduct 1 ∘ 2 are also given.