Trace theorems for pseudo-differential operators

An integral operator with smooth kernel can always be restricted to a hypersurfaceS. Acutally, it is again an integral operator and its kernel is the restriction (in both variables) of the original one toS. Here we study restrictions of pseudo-differential operators of arbitrary order. We find sufficient and (to some extent) necessary conditions on the symbol ensuring existence of the restriction. These conditions require the vanishing of some geometrical invariants defined on the conormal bundle of the hypersurface. In particular, for a pseudo-differential operator of orderm, the principal symbol should vanish of order [m]+2 and the subprincipal symbol of order [m]+1. These classical invariants are sufficient to treat the problem for the casem<1, but in the general case we need to introduce new higher order invariants related to the operator and the hypersurface.     

Non-local pseudo-differential operators

The notion of non-local pseudo-differential operators, as well as their symbols and the operation on holomorphic functions, is established and the invertibility theorem for such operators is proved.     

Bounded pseudo-differential operators

This lecture gives an inside look into the proof of the continuity of pseudo-differential operators of orderm and typep, δ₁, δ₂ for 0≦p≦δ₁=1, 0≦p≦δ₂<1, andm/n≦p≦(δ₁+δ₂)/2. Applications are mentioned.     

Initial value problem for pseudo-differential operators

The algebra of generalized Weyl symbols is used in the proof of the continuity of the semigroupexptĤ in the Schwartz space of test functions. Fundamental results on algebras of differentiable Weyl symbols are presented. New examples of σ-temperate Riemannian metrics are constructed. Such metrics form a basis for construction of algebras of differentiable Weyl symbols. Conditions for the existence of semigroups of operators, conditions for pseudo-differential operators to be sectorial, and conditions for the continuity of such semigroups in spaces of test functions and distributions are established. Initial value problems for second-order differential operators are considered. Bibliography: 16 titles. Translated fromProblemy Matematicheskogo Analiza, No. 18, 1998, pp. 3–42.     

L p bounds for pseudo-differential operators

SharpL ^p boundedness results are proven for pseudo-differential operators in the classS _pδ ^m .     

Commutators of pseudo-differential operators

In this paper, the boundedness of commutators generated by pseudo-differential operators and BMO functions is discussed on Lebesgue spaces.     

Propagation of singularities for non-real pseudo-differential operators

The purpose of this work is to prove a theorem of propagation of singularities for a class of non-real pseudo-differential operator with multiple characteristics. The main tools are L² estimates on the time-dependent Schrödinger equation related toP. We extend here the results of [6]; we improve the results announced by the second author in [7]. The second part of this work consists in an extension of the result of [5] to complex-valued symbols.     

Maximal operators of pseudo-differential operators with rough symbols

Consider a pseudo-differential operator■ where the symbol a is in the rough H¨ormander class L^∞S^m_ρwith m∈R andρ∈[0,1].In this note,when 1≤p≤2,if ■and a∈L^∞S^m_ρ,then for any f∈S（R~n） and x∈R~n,we prove that ■ where M is the Hardy-Littlewood maximal operator.Our theorem improves the known results and the bound on m is sharp,in the sense that■~ncan not be replaced by a larger constant.     

Endpoint estimates for the commutator of pseudo-differential operators

It is well known that the commutator Tb of the Calderón-Zygmund singular integral operator is bounded on L^p(Rⁿ) for 1 < p < +∞ if and only if b ∈ BMO [1]. On the other hand, the commutator T_b is bounded from H¹(Rⁿ) into L¹(Rⁿ) only if the function b is a constant [2]. In this article, we will discuss the boundedness of commutator of certain pseudo-differential operators on Hardy spaces H¹. Let T^σ be the operators that its symbol is S⁰_1,δ with 0 ≤ δ < 1, if b ∈ LMO_∞, then, the commutator [b, T^σ] is bounded from H¹(Rⁿ) into L¹(Rⁿ) and from L¹(Rⁿ) into BMO(Rⁿ); If [b, T^σ] is bounded from H¹(Rⁿ) into L¹(Rⁿ) or L¹(Rⁿ) into BMO(Rⁿ), then, b ∈ LMO_loc.     

Microlocal energy methods and pseudo-differential operators

A functionk(x, u) of the(n+n)-variables is said to be a positive Hermite kernel if , and the matrix(k(x _i, x_j))_i,j is positive semi-definite for every integerN and everyx ₁, ..., x_N. In this paper, we prove that this positive structure can be microlocalized in the category of microfunctions. Further we obtain a useful theorem concerning the positivity of pseudo-differential operators. This theory will play important roles in the study of analytic singularities of solutions of boundary value problems.     

Two nontrivial solutions for hemivariational inequalities driven by nonlocal elliptic operators

In this paper, we establish the existence of two nontrivial solutions to a class of nonlocal hemivariational inequalities depending on two parameters. Our methods are based on critical point theory for non-differentiable functionals.