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.     

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.     

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.     

Two inequalities for pseudo-differential operators

Изучается ограничен ность псевдодиффере нциальных операторов на \(L^2 (R^n )\) и на пр остранствах Харди в \(R^n \) . Пусть \(D_k = \{ \xi \in R^n :2^{k - 1} \leqq \left| \xi \right|< 2^k \} , k = 1,2,3, \ldots ,\) и \(D_0 = \{ \xi \in R^n :\left| \xi \right|< 1\} \) . Псевдодиффер енциальный операторP с символом p определяется соотно шением $$Pf(x) = \int\limits_{R^n } {e^{ix \cdot \xi } p(x,\xi )\hat f(\xi )d\xi ,x \in R^n .} $$ Будем говорить, что p пр инадлежит классу \(\bar S_{\varrho ,} {}_\delta (M,N), 0 \leqq \delta ,\varrho \leqq 1\) , ес ли $$\left| {D_x^a p(x,\xi )} \right| \leqq C_a (1 + \left| \xi \right|)^{\delta \left| a \right|} , x,\xi \in R^n ,\left| a \right| \leqq M,$$ и $$\int\limits_{D_k } {\left| {D_x^a D_\xi ^\beta p(x,\xi )} \right|d\xi \leqq C_{a\beta } 2^{kn} 2^{k(\delta |a| - \varrho |\beta |)} , x} \in R^n , k = 0,1,2, \ldots ;|a| \leqq M, |\beta | \leqq N.$$ Изучаются условия, ко торым должны удовлет ворять ?. δ,M иN, чтобы для каждого символа \(p \in \bar S_\varrho , {}_\delta (M,N)\) соответствующий оп ераторP был ограниче н на \(L^2 (R^n )\) . Далее, пусть \(p \in S_\varrho , {}_\delta \) , если дл я всех мультииндексо в а и β выполнено условие $$|D_x^a D_\xi ^\beta p(x,\xi )| \leqq C_{a\beta } (1 + |\xi |)^{\delta |\alpha | - \varrho |\beta |} , x,\xi \in R^n .$$ Доказывается, что при 0≦δ<1 операторP отображ ает пространство Харди \(H^p (R^n )\) в локальное пространство Харди ? ^p , если символp принадл ежит классуS ₁, δ.     

Numerical solution of inverse spectral problems for Sturm-Liouville operators with discontinuous potentials

We consider Sturm-Liouville differential operators on a finite interval with discontinuous potentials having one jump. As the main result we obtain a procedure of recovering the location of the discontinuity and the height of the jump. Using our result, we apply a generalized Rundell-Sacks algorithm of Rafler and Böckmann for a more effective reconstruction of the potential and present some numerical examples.     

Construction of time-inhomogeneous Markov processes via evolution equations using pseudo-differential operators

For a pseudo-differential operator with symbol which is time-and space-dependent, elliptic and continuous negative definite,the corresponding evolution equation is solved. Further, itis shown that the solution defines a Markov process. In general,this will be a time- and spaceinhomogeneous jump process. Tosolve the evolution equation, we combine a fixed-point methodwith the symbolic calculus for negative definite symbols developedby Hoh. The properties of the fundamental solution which ensurethe existence of a corresponding Markov process are proved alongthe lines of Eidelman, Ivasyshen and Kochubei. However, insteadof hyper-singular integral representations, we use the pseudo-differentialoperator representation together with the positive maximum principleto obtain the required properties.     

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 .     

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.