We consider the class of semi-stable positive solutions to semilinear equations −∆u = f(u) in a bounded domain Ω ⊂ Rn of double revolution, that is, a domain invariant under rotations of the first m variables and of the last n−m variables. We assume 2 ≤ m ≤ n− 2. When the domain is convex, we establish a priori Lp and H1 0 bounds for each dimension n, with p = ∞ when n ≤ 7. These estimates lead...

It is shown that for a parabolic problem with maximal Lp-regularity (for 1 < p < ∞), the time discretization by a linear multistep method or Runge–Kutta method has maximal `p-regularity uniformly in the stepsize if the method is A-stable (and satisfies minor additional conditions). In particular, the implicit Euler method, the Crank–Nicolson method, the second-order backward difference formula ...

is the most fundamental example. Much like in the classical case of the Lebesgue di erentiation theorem, pointwise convergence almost everywhere of the inverse Fourier transform to f ∈ Lp (R) can be reduced to Lp bounds for the maximal operator C. Weak type L2 bounds were rst obtained by Lennart Carleson in 1966 [8], thus providing a surprising a rmative solution to the question of pointwise co...

Let Ω be an open subset of Rn. Denote by LpΩ(R n) the closure in Lp(Rn) of the set of all functions ε ∈ L1(Rn)∩Lp(Rn) whose Fourier transform has compact support contained in Ω. The subspaces of the form LpΩ(R n) are called the spectral subspaces of Lp(Rn). It is easily seen that each spectral subspace is translation invariant; i.e., f(x + a) ∈ LpΩ(R) for all f ∈ L p Ω(R n) and a ∈ Rn. Sufficie...

where U is the class of all h∈ L2(R+,r−1dr) with ‖h‖L2(R+,r−1dr) ≤ 1. The operator Ω was introduced by Chen and Lin [7]. They showed that Ω is bounded on Lp(Rn) for all p > 2n/(2n− 1) provided that Ω ∈ (Sn−1). Recently, we have been able to show that the Lp(Rn) boundedness of Ω still holds for all p ≥ 2 if the condition Ω ∈ (Sn−1) is replaced by the more natural and weaker condition Ω ∈ L(logL)...