We investigate the best order of smoothness of L p (L q). We prove in particular that there exists a C ∞-smooth bump function on L p (L q) if and only if p and q are even integers and p is a multiple of q.

In this paper, optimal Lp − Lq estimates are obtained for operators which average functions over polynomial submanifolds, generalizing the k-plane transform. An important advance over previous work (e.g., [9]) is that full Lp − Lq estimates are obtained by methods which have traditionally yielded only restricted weak-type estimates. In the process, one is lead to make coercivity estimates for c...

We study the Cauchy problem for the damped wave equation. In a previous paper [16] the author has shown the Lp-Lq estimates between the solutions of the damped wave equation and the solutions of the corresponding heat equation. In this paper, we show new Lp-Lq estimates for the damped wave equation with odd initial data.

New criteria of Lp − Lq boundedness of Hardy-Steklov type operator (1.1) with both increasing on (0, ∞) boundary functions a(x) and b(x) are obtained for 1 < p ≤ q < ∞ and 0 < q < p < ∞, p > 1. This result is applied for two-weighted Lp − Lq characterization of the corresponding geometric Steklov operator (1.3) and other related problems.

We study the lattice finite representability of the Bochner space Lp(μ1, Lq(μ2)) in `p{`q}, 1 ≤ p, q < ∞, and then we characterize the ideal of the operators which factor through a lattice homomorphism between L∞(μ) and Lp(μ1, Lq(μ2)).

1. General setting Let T be a nonempty set, Σ be the σ -algebra of subsets of T , and μ be a nonnegative σ -additive measure on Σ . We denote by Lp(T , Σ, μ) (or simply Lp(T , μ)) the set of all Σ-measurable functions with values in R or in C for which ∥x(·)∥Lp(T ,μ) =  T |x(t)|p dμ(t) 1/p < ∞, 1 ≤ p < ∞, ∥x(·)∥L∞(T ,μ) = ess sup t∈T |x(t)| < ∞, p = ∞. Put W = {x(·) ∈ Lp(T , μ) : ∥φ(·)x(·)∥L...

We prove that certain Sobolev-type norms, slightly stronger than those given by energy conservation, stay bounded uniformly in time and N. This allows one to extend the local existence results of second third author globally time. The proof is based on interaction Morawetz-type estimates Strichartz (including some new end-point results) for equation {1i?t??x??y+1NVN(x?y)}?(t,x,y)=F mixed coordi...

For p ∈ [2,∞) the metric Xp inequality with sharp scaling parameter is proven here to hold true in Lp. The geometric consequences of this result include the following sharp statements about embeddings of Lq into Lp when 2 < q < p <∞: the maximal θ ∈ (0, 1] for which Lq admits a bi-θ-Hölder embedding into Lp equals q/p, and for m,n ∈ N the smallest possible bi-Lipschitz distortion of any embeddi...