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 $\left|\right|T^{n+1}-Tⁿ\left|\right|\asymp \left(n\ge 1\right)$. This answers Zemánek’s question on the time regularity property.

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.

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.

Certain weighted norm inequalities for integral operators with non-negative, monotone kernels are shown to remain valid when the weight is replaced by a monotone majorant or minorant of the original weight. A similar result holds for operators with quasi-concave kernels. To prove these results a careful investigation of the functional properties of the monotone envelopes of a non-negative function is carried-out. Applications are made to function space embeddings of the cones of monotone functions...

In this paper we show an asymptotic formula for the number of eigenvalues of a pseudodifferential operator. As a corollary we obtain a generalization of the result by Shubin and Tulovskiĭ about the Weyl asymptotic formula. We also consider a version of the Weyl formula for the quasi-classical asymptotics.

New sufficient conditions on the weight functions u(.) and v(.) are given in order that the fractional maximal [resp. integral] operator Ms [resp. Is], 0 ≤ s &lt; n, [resp. 0 &lt; s &lt; n] sends the weighted Lebesgue space Lp(v(x)dx) into Lp(u(x)dx), 1 &lt; p &lt; ∞. As a consequence a characterization for this estimate is obtained whenever the weight functions are radial monotone.

In [2] and [3] upper and lower estimates and asymptotic results were obtained for the approximation numbers of the operator $T:L^p\left({ℝ}^{+}\right)\to L^p\left({ℝ}^{+}\right)$ defined by $\left(Tf\right)\left(x\right)≔v\left(x\right){ʃ}_0^{\infty }u\left(t\right)f\left(t\right)dt$ when 1 < p < ∞. Analogous results are given in this paper for the cases p = 1,∞ not included in [2] and [3].

We consider the Volterra integral operator $T:L^p\left({ℝ}^{+}\right)\to L^p\left({ℝ}^{+}\right)$ defined by $\left(Tf\right)\left(x\right)=v\left(x\right){ʃ}_0^{x}u\left(t\right)f\left(t\right)dt$. Under suitable conditions on u and v, upper and lower estimates for the approximation numbers $a_n\left(T\right)$ of T are established when 1 < p < ∞. When p = 2 these yield $li{m}_{n\to \infty }n{a}_n\left(T\right)={\pi }^{-1}{ʃ}_0^{\infty }\left|u\left(t\right)v\left(t\right)\right|dt$. We also provide upper and lower estimates for the ${\ell }^{\alpha }$ and weak ${\ell }^{\alpha }$ norms of (an(T)) when 1 < α < ∞.

Necessary and sufficient conditions governing two-weight $L^p$ norm estimates for multiple Hardy and potential operators are presented. Two-weight inequalities for potentials defined on nonhomogeneous spaces are also discussed. Sketches of the proofs for most of the results are given.