The Poisson's problem for the Laplacian with Robin boundary condition in non-smooth domains.
The power boundedness and resolvent conditions for functions of the classical Volterra operator
Let ϕ(z) be an analytic function in a disk |z| < ρ (in particular, a polynomial) such that ϕ(0) = 1, ϕ(z)≢ 1. Let V be the operator of integration in , 1 ≤ p ≤ ∞. Then ϕ(V) is power bounded if and only if ϕ’(0) < 0 and p = 2. In this case some explicit upper bounds are given for the norms of ϕ(V)ⁿ and subsequent differences between the powers. It is shown that ϕ(V) never satisfies the Ritt condition but the Kreiss condition is satisfied if and only if ϕ’(0) < 0, at least in the polynomial...
The proof of the Nirenberg-Treves conjecture
We prove the Nirenberg-Treves conjecture : that for principal type pseudo-differential operators local solvability is equivalent to condition (). This condition rules out certain sign changes of the imaginary part of the principal symbol along the bicharacteristics of the real part. We obtain local solvability by proving a localizable estimate for the adjoint operator with a loss of two derivatives (compared with the elliptic case). The proof involves a new metric in the Weyl (or Beals-Fefferman)...
The pseudodifferential operator .
The Sobolev-Il'in theorem for the -Riesz potential.
The solvability of non solvable operators
The spectra and singular values of multidimensional integral operators with bihomogeneous kernels.
The Spectra of Hyponormal Integral Operators
The spectrum of the transport operator with a potential term under the spatial periodicity condition
The stationary phase method in Gevrey classes and Fourier integral operators on ultradistributions
The symbol of a function of a pseudo-differential operator
We give an explicit formula for the symbol of a function of an operator. Given a pseudo-differential operator on with symbol and a smooth function , we obtain the symbol of in terms of . As an application, Bohr-Sommerfeld quantization rules are explicitly calculated at order 4 in .
The trajectory-coherent approximation and the system of moments for the Hartree type equation.
The transmission property.
The verification of the Nirenberg-Treves conjecture
In a series of recent papers, Nils Dencker proves that condition implies the local solvability of principal type pseudodifferential operators (with loss of derivatives for all positive ), verifying the last part of the Nirenberg-Treves conjecture, formulated in 1971. The origin of this question goes back to the Hans Lewy counterexample, published in 1957. In this text, we follow the pattern of Dencker’s papers, and we provide a proof of local solvability with a loss of derivatives.
The Vlasov equation with strong mangnetic field and oscillating electric field as a model for isotop resonant separation.
The weak type L1 convergence of eigenfunction expansions for pseudodifferential operators.
The Weyl asymptotic formula by the method of Tulovskiĭ and Shubin
Let A be a pseudodifferential operator on whose Weyl symbol a is a strictly positive smooth function on such that for some ϱ>0 and all |α|>0, is bounded for large |α|, and . Such an operator A is essentially selfadjoint, bounded from below, and its spectrum is discrete. The remainder term in the Weyl asymptotic formula for the distribution of the eigenvalues of A is estimated. This is done by applying the method of approximate spectral projectors of Tulovskiĭ and Shubin.
Time regularity and functions of the Volterra operator
Our aim is to prove that for any fixed 1/2 < α < 1 there exists a Hilbert space contraction T such that σ(T) = 1 and . This answers Zemánek’s question on the time regularity property.
Time-frequency analysis of Sjöstrand's class.
We investigate the properties an exotic symbol class of pseudodifferential operators, Sjöstrand's class, with methods of time-frequency analysis (phase space analysis). Compared to the classical treatment, the time-frequency approach leads to striklingly simple proofs of Sjöstrand's fundamental results and to far-reaching generalizations.
Trace class pseudodifferential calculus with operator valued symbols and unusual index formulas
For several classes of pseudodifferential operators with operator-valued symbol analytic index formulas are found. The common feature is that usual index formulas are not valid for these operators. Applications are given to pseudodifferential operators on singular manifolds.