Let I = [a, b] ⊂ R, let 1 < p ≤ q <∞, let u and v be positive functions with u ∈ Lp′ (I), v ∈ Lq(I) and let T : Lp(I)→ Lq(I) be the Hardy-type operator given by

where, p.v. denotes the principal value. It is known that if Φ is of finite type at 0 (see Definition 2.2) and Ω ∈ 1(Sn−1), then TΦ,Ω is bounded on Lp for 1<p <∞ [15]. Moreover, it is known that TΦ,Ω may fail to be bounded on Lp for any p if the finite-type condition is removed. In [8], Fan et al. showed that the Lp boundedness of the operator TΦ,Ω still holds if the condition Ω ∈ 1(Sn−1) is re...

Let 1 < q < p < ∞ and q ≤ r ≤ p. Let X be a reflexive Banach space satisfying a lower-lq-tree estimate and let T be a bounded linear operator from X which satisfies an upper-lp-tree estimate. Then T factors through a subspace of ( ∑ Fn)lr , where (Fn) is a sequence of finite-dimensional spaces. In particular, T factors through a subspace of a reflexive space with an (lp, lq) FDD. Similarly, let...

parameterized by μ > 0, and prove a representation theorem for its solutions using the theory of special functions. This representation is used to obtain Lp–Lq estimates for the solution and for the energy operator corresponding to this Cauchy problem. Especially for the L2 energy estimate we determine the part of the phase space which is responsible for the decay rate. It will be shown that th...

We prove an extrapolation theorem for the nonlinear m-term approximation with respect to a system of functions satisfying very mild conditions. This theorem allows us to prove endpoint Lp − Lq estimates in nonlinear approximation. As a consequence, some known endpoint estimates can be deduced directly and some new estimates are also obtained. Finally, applications of these new estimates are giv...

Artemov’s Logic of Proof, LP, is an explicit proof counterpart of S4. Their formal connection is built through the realization theorem, that every S4 theorem can be converted to an LP theorem by substituting proof terms for provability modals. Instead of the realization of theorems, what is concerned in this paper is the realization of proofs. We will show that only a subclass of S4 proofs, cal...