Suppose $\Phi$ is a nonnegative, locally integrable, radial function on ${\mathbf{R}}^n$, which is nonincreasing in $\left|x\right|$. Set $\left(Tf\right)\left(x\right)={\int }_{R^n}\Phi (x-y)f\left(y\right)dy$ when $f\ge 0$ and $x\in {\mathbf{R}}^n$. Given $1\<p\<\infty$ and $v\ge 0$, we show there exists $C\>0$ so that ${\int }_{{\mathbf{R}}^n}\left(Tf\right)(x{)}^{p}v\left(x\right)dx\le C{\int }_{{\mathbf{R}}^n}f(x{)}^{p}dx$ for all $f\ge 0$, if and only if $C^{\text{'}}\>0$ exists with ${\int }_{Q}T(x_{Q}v\left)\right(x{)}^{p^{\text{'}}}dx\le C^{\text{'}}{\int }_{Q}v\left(x\right)dx\<\infty$ for all dyadic cubes Q, where $p^{\text{'}}=p/(p-1)$. This result is used to refine recent estimates of C.L. Fefferman and D.H. Phong on the distribution of eigenvalues of Schrödinger operators.

We study a geometric generalization of the time-dependent Schrödinger equation for the harmonic oscillator$$\left(D_t+\frac{1}{2}\Delta +V\right)\psi =0(0.1)$$where $\Delta$ is the Laplace-Beltrami operator with respect to a “scattering metric” on a compact manifold $M$ with boundary (the class of scattering metrics is a generalization of asymptotically Euclidean metrics on ${ℝ}^n$, radially compactified to the ball) and $V$ is a perturbation of $\frac{1}{2}{\omega }^2{x}^{-2}$, with $x$ a boundary defining function for $M$ (e.g. $x=1/r$ in the compactified Euclidean case). Using the quadratic-scattering...

The unique solvability of the problem Δu = 0 in G⁺ ∪ G¯, u₊ - au_ = f on ∂G⁺, n⁺·∇u₊ - bn⁺·∇u_ = g on ∂G⁺ is proved. Here a, b are positive constants and g is a real measure. The solution is constructed using the boundary integral equation method.

The theory of Markov processes and the analysis on Lie groups are used to study the eigenvalue asymptotics of Dirichlet forms perturbed by scalar potentials.

This paper deals with the linear approximation scheme to approximate a singular parabolic problem: the two-phase Stefan problem on a domain consisting of two components with imperfect contact. The results of some numerical experiments and comparisons are presented. The method was used to determine the temperature of steel in the process of continuous casting.

The vanishing viscosity method is adapted to the infinite dimensional case, by showing that the value function of a deterministic optimal control problem can be approximated by the solutions of suitable parabolic equations in Hilbert spaces.

In a series of recent papers, Nils Dencker proves that condition $\left(\psi \right)$ implies the local solvability of principal type pseudodifferential operators (with loss of $\frac{3}{2}+ϵ$ derivatives for all positive $ϵ$), 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 $\frac{3}{2}$ derivatives.