Strichartz Estimates for Non-selfadjoint Operators and Applications

Abstract

We prove Strichartz type estimates for non-selfadjoint pseudo-differential operators of order m with a curvature condition on the characteristic set. Local solvability results for the corresponding equations with L ^n/m potential (where n > m stands for the number of variables) and for the corresponding semi-linear equations with a non-linearity of order m/n ≤ α ≤ 2m/(n ? m) follow. This is achieved using a parametrix construction and a Brenner inequality for Fourier integral operators with complex phase.     

Properties of m-EP operators

In this paper, we study the characteristics of m-EP operators and the properties of the particular case of m-EP operators that are Drazin invertible. The relations between the Moore–Penrose inverses and Drazin inverses are built by m-EP and many closely equivalent relations are investigated by using appropriate idempotents.     

Pseudo-differential operators on Sobolev and Lipschitz spaces

In this paper, by discovering a new fact that the Lebesgue boundedness of a class of pseudo- differential operators implies the Sobolev boundedness of another related class of pseudo-differential operators, the authors establish the boundedness of pseudo-differential operators with symbols in Sρ,δ^m on Sobolev spaces, where ∈ R, ρ≤ 1 and δ≤ 1. As its applications, the boundedness of commutators generated by pseudo-differential operators on Sobolev and Bessel potential spaces is deduced. Moreover, the boundedness of pseudo-differential operators on Lipschitz spaces is also obtained.     

Clifford-Hermite-monogenic operators

In this paper we consider operators acting on a subspace ℳ of the space L ₂ (ℝ^m; ℂ_m) of square integrable functions and, in particular, Clifford differential operators with polynomial coefficients. The subspace ℳ is defined as the orthogonal sum of spaces ℳ_s,k of specific Clifford basis functions of L ₂(ℝ^m; ℂm). Every Clifford endomorphism of ℳ can be decomposed into the so-called Clifford-Hermite-monogenic operators. These Clifford-Hermite-monogenic operators are characterized in terms of commutation relations and they transform a space ℳ_s,k into a similar space ℳ_s′,k′. Hence, once the Clifford-Hermite-monogenic decomposition of an operator is obtained, its action on the space ℳ is known. Furthermore, the monogenic decomposition of some important Clifford differential operators with polynomial coefficients is studied in detail.     

Some Problems in Operator Theory on Bounded Symmetric Domains

We discuss some problems concerning the operators K _m (Z,Z ^*), where K _m is the reproducing kernel of the Peter–Weyl space with signature m and Z stands for the commuting tuple of operators of multiplication by the coordinate variables on a bounded symmetric domain. These problems have applications for construction of operator models, as well as for a possible generalization of the recent dilation theory for row contractions due to Arveson.     

Parabolic Marcinkiewicz integrals along surfaces on product domains

In this paper, parabolic Marcinkiewicz integral operators along surfaces on the product domain ℝ ⁿ × ℝ ^m (n,m ⩾ 2) are introduced. L ^p bounds of such operators are obtained under weak conditions on the kernels.     

Sobolev spaces associated to a polyhedron and Fourier integral operations inR n

The theory of the Sobolev spacesH _m ^p (R ⁿ ) (mR,p polyhedron in R ²ⁿ )of [BG]is revisited here in the frame of new classes of pseudodifferential operators related to the same polyhedron p.These operators generalize to corresponding classes of Fourier integral operators, for which we present the main lines of a symbolic calculus and results of continuity on the H _m ^p (R ⁿ ) spaces.     

The Estimates for Sharp Maximal Functions of Multilinear Strongly Singular Integral Operators

In this paper, the author obtains that the multilinear operators of strongly singular integral operators and their dual operators are bounded from some L^p（R^n） to L^p（R^n） when the m-th order derivatives of A belong to L^p（R^n） for r large enough. By this result, the author gets the estimates for the Sharp maximal functions of the multilinear operators with the m-th order derivatives of A being Lipschitz functions. It follows that the multilinear operators are （L^p, L^p）-type operators for 1 〈 p 〈 ∞.     

Boundedness of Hausdorff operators on some product Hardy type spaces

We study Hausdorff operators on the product Besov space B01,1 (Rn × Rm) and on the local product Hardy space h1 (Rn × Rm).We establish some boundedness criteria for Hausdorff operators on these functio...     

Algebraic Order Bounded Disjointness Preserving Operators and Strongly Diagonal Operators

Let T be an order bounded disjointness preserving operator on an Archimedean vector lattice. The main result in this paper shows that T is algebraic if and only if there exist natural numbers m and n such that n ≥ m, and T^n!, when restricted to the vector sublattice generated by the range of T^m, is an algebraic orthomorphism. Moreover, n (respectively, m) can be chosen as the degree (respectively, the multiplicity of 0 as a root) of the minimal polynomial of T. In the process of proving this result, we define strongly diagonal operators and study algebraic order bounded disjointness preserving operators and locally algebraic orthomorphisms. In addition, we introduce a type of completeness on Archimedean vector lattices that is necessary and sufficient for locally algebraic orthomorphisms to coincide with algebraic orthomorphisms.     

m-Berezin Transform on the Polydisk

m-Berezin transforms are introduced for bounded operators on the Bergman space of the polydisk. We show several properties of m-Berezin transform and use them to show that a radial operator in the Toeplitz algebra is compact iff its Berezin transform vanishes on the boundary of the polydisk. A useful and sharp approximate identity of its m-Berezin transforms is obtained for a bounded operator.     

Annihilators for cusp forms of weight 2 and level 4p m

We obtain operators, given essentially by formal sums of Hecke operators, that annihilate spaces of cusp forms of weight 2 for Γ ₁(p ^m )∩Γ(4), whose dimensions will be specified. Moreover, we obtain the principal part (mod p), over the cusps, of certain meromorphic modular functions of level 4p ^m .     

The existence and uniqueness of solution and the convergence of a multi-step iterative algorithm for a system of variational inclusions with (<Emphasis Type="Italic">A</Emphasis>, <Emphasis Type="Italic">η</Emphasis>, <Emphasis Type="Italic">m</Emphasis>)-accretive operators

In this paper, we introduce and study a new system of variational inclusions with (A, η, m)-accretive operators which contains variational inequalities, variational inclusions, systems of variational inequalities and systems of variational inclusions in the literature as special cases. By using the resolvent technique for the (A, η, m)-accretive operators, we prove the existence and uniqueness of solution and the convergence of a new multi-step iterative algorithm for this system of variational inclusions in real q-uniformly smooth Banach spaces. The results in this paper unifies, extends and improves some known results in the literature.      

One-dimensional nonlinear Laplacians under a 3-point boundary condition

We consider a three-point boundary value problem for operators such as the one-dimensional p-Laplacian, and show when they have solutions or not, and how many. The inverse operators are given by various formulas involving zeros of a real-valued function. They are shown to be orderpreserving, for some parameter values, and non-singleton valued for others. The operators are shown to be m-dissipative in the space of continuous functions.     

On the stability of the index of unbounded nonlocal operators in Sobolev spaces

Unbounded operators corresponding to nonlocal elliptic problems on a bounded region G ⊂ ℝ² are considered. The domain of these operators consists of functions in the Sobolev space W ₂ ^m (G) that are generalized solutions of the corresponding elliptic equation of order 2m with the right-hand side in L ₂(G) and satisfy homogeneous nonlocal boundary conditions. It is known that such unbounded operators have the Fredholm property. It is proved that lower order terms in the differential equation do not affect the index of the operator. Conditions under which nonlocal perturbations on the boundary do not change the index are also formulated. Published in Russian in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2006, Vol. 255, pp. 116–135.     

Boundedness of Calderón-Zygmund operators in product Hardy spaces

By means of vector-valued product Calderón-Zygmund operators and some subtle estimates,the boundedness in product Hardy spaces on R^n × R^m of Calderón-Zygmund operators introduced by J.L. Journé is established.     

Characterizations of m-EP elements in rings

Let R be a ring with involution. In this paper, we extend the notions of m-EP matrices and m-EP operators to an arbitrary ring case. A number of new characterizations of m-EP elements in rings are presented. In particular, the existence criteria for 1-EP (i.e. EP) elements are obtained by means of the group inverse, Moore–Penrose inverse, and core inverse. Some properties of 2-EP are also given.     

On conditions for and the order of approximation of functions by operators of classS 2m

Following P. P. Korovkin, we study conditions for the convergence of operators of classesS _2m to continuous functions and the asymptotics of approximation by such operators to differentiable functions. Translated fromMatematicheskie Zametki, Vol. 67, No. 5, pp. 654–661, May, 2000.     

A Lyapunov Lemma for Elliptic Systems

We study a possible extension to the infinite-dimensional case of the classicalLyapunov lemma for matrices. More precisely, for a fixed elliptic system A ofdifferential operators of order m, we consider the operator equationTA + A ^* T = Q, where Q is any given classical pseudodifferential system oforder m, and T is sought as a classical pseudodifferential system of order0.     

The Fourier integral operators on hardy spaces associated with Herz spaces

In this paper, it is proved that the Fourier integral operators of order m, with −n < m ⩽ −(n−1)/2, are bounded from three kinds of Hardy spaces associated with Herz spaces to their corresponding Herz spaces.