The author investigates the solubility in rationals of equations of the form $$a_0X_0^4 + a_1X_1^4 + a_2X_2^4 + a_3X_3^4 = 0 $$ where $a_0a_1a_2a_3$ is a square, building on the ideas which Colliot-Thène, Skorobogatov and he have developed; see {\em Invent. Math.} 134 (1998) 579--650. He obtains sufficient conditions for solubility, which appear to be related to the absence of a Brauer--Manin obstruction. This represents the first large family of K3 surfaces which almost satisfy the Hasse principle, in the sense that the auxiliary condition which ensures that local solubility everywhere implies global solubility is nearly always satisfied. 1991 Mathematics Subject Classification: 10B10.

Let $\mathbf G$ be a connected reductive group defined over a finite field with $q$ elements and let $F : {\mathbf G} \to {\mathbf G}$ be the corresponding Frobenius endomorphism. We prove that the Mackey formula for Lusztig induction and restriction holds in ${\mathbf G}^F$ if all the simple components of ${\mathbf G}$ are of type A. Our result holds for every Frobenius endomorphism (split or non-split) and for every $q$. 1991 Mathematics Subject Classification: 20G05, 20G40.

If $G$ is a projective special linear group $\text{PSL}(3,q)$ with $q \equiv 4 \; \text{or} \; 7 \pmod{9}$, then a Sylow 3-subgroup of $G$ is elementary abelian of order 9. We show that the principal 3-blocks of any two such groups are Morita equivalent. This result and Okuyama's theorem for $\text{PSL}(3,4)$ prove the Broué conjecture for these blocks. 1991 Mathematics Subject Classification: 20C05, 20C20.

In this paper we study the problem of $L^p$-independence of the spectrum of second-order elliptic operators with measurable coefficients. A new technique is developed to treat the problem in cases when the kernel of the corresponding semigroup does not satisfy upper bounds of Gaussian type and the semigroup itself exists only in a certain interval of the $L^p$-scale. This allows the authors to treat second-order elliptic operators of divergence type with singular coefficients in the main part and singular lower-order terms. A criterion of $L^p$-independence obtained in the paper applies also to certain classes of higher-order elliptic operators. The problem of stability of the essential spectrum is also studied. It is shown, under some mild conditions on the coefficients of the elliptic operators, that its essential spectrum is the same as the one of the Laplacian. 1991 Mathematics Subject Classification: 35P99, 35B25, 35J15.

We consider Hilbert transforms along curves $\Gamma$ on the Heisenberg group. In particular, necessary and sufficient conditions for the $L^2$-boundedness are given for $\Gamma(t)=(t,\gamma(t),0)$ when $\gamma$ is even or odd and convex on ${\Bbb R}^{+}$. 1991 Mathematics Subject Classification: 42B20,42B25.

In this paper we consider Fourier multipliers for $L^p$ $(p>1)$ on Chébli-Trimèche hypergroups and establish a version of Hörmander's multiplier theorem. As applications we give some results concerning the Riesz potentials and oscillating multipliers.

1991 Mathematics Subject Classification: 43A62, 43A15, 43A32.

We derive Sobolev--Poincaré inequalities that estimate the $L^q(d\mu)$ norm of a function on a metric ball when $\mu$ is an arbitrary Borel measure. The estimate is in terms of the $L^1(d\nu)$ norm on the ball of a vector field gradient of the function, where $d\nu/dx$ is a power of a fractional maximal function of $\mu$. We show that the estimates are sharp in several senses, and we derive isoperimetric inequalities as corollaries. 1991 Mathematics Subject Classification: 46E35, 42B25.

The Laplacian on a metric tree is $\Delta u = u''$ on its edges, with the appropriate compatibility conditions at the vertices. We study the eigenvalue problem on a rooted tree $\Gamma$: $$-\lambda \Delta u = V u \quad \text{on } \Gamma, \qquad u(o)=0.$$ Here $V \ge 0$ is a given ‘weight function’ on $\Gamma$, and $o$ is the root of $\Gamma$. The eigenvalues for such a problem decay no faster than $\lambda_n = O(n^{-2})$, this last case being typical for one-dimensional problems. We obtain estimates for the eigenvalues in the classes $l_p$, with $p > \frac12$, and their weak analogues $l_{p,\infty}$ with $p \ge \frac12$. The results for $p<1$ and $p>1$ are of different character. The case $p<1$ is studied in more detail. To analyse this case, we use two methods; in both of them the problem is reduced to a family of one-dimensional problems. One of the methods is based upon a useful orthogonal decomposition of the Dirichlet space on $\Gamma$. For a particular class of trees and weights, this method leads to the complete spectral analysis of the problem. We illustrate this in several examples, where we were able to obtain the asymptotics of the eigenvalues. 1991 Mathematics Subject Classification: primary 47E05, 05C05; secondary 34L15, 34L20.

Asymptotic expansions are obtained for finite-dimensional symmetric stable distributions. They are used to prove the existence of continuous transition probability densities of stable and stable-like jump-diffusions, and to construct local multiplicative asymptotics and global two-sided estimates for these densities. As a consequence, the distribution of the first passage times for stable jump-diffusions is estimated and the integral test for the limsup behaviour of their sample paths as $t\to 0$ is provided. 1991 Mathematics Subject Classification: 60E07, 60G17, 60J35, 47D07.