We study pointwise convergence of the solution to elastic wave equation initial data which lies in Sobolev spaces. prove that converges along every line almost everywhere whenever regularity is greater than one half. show this optimal.

where dσ is the rotationally invariant measure on Sd−1, normalized such that σ(Sd−1) = 1. Stein [5] showed that limt→0Atf(x) = f(x) almost everywhere, provided f ∈ L(R), p > d/(d − 1) and d ≥ 3. Later Bourgain [1] extended this result to the case d = 2. If p ≤ d/(d − 1) then pointwise convergence fails. However if {tj}j=1 is a fixed sequence converging to 0 then pointwise convergence may hold f...

For any weightable quasi-metric space (X, d) having a maximum with respect to the associated order ≤d, the notion of the quasi-metric of complexity convergence on the the function space (equivalently, the space of sequences) X, is introduced and studied.We observe that its induced quasi-uniformity is finer than the quasi-uniformity of pointwise convergence and weaker than the quasi-uniformity o...

A systematic study is made of spectral measures in locally convex spaces which are countably additive for the topology of uniform convergence on bounded sets, briefly, the bounded convergence topology. Even though this topology is not compatible for the duality with respect to the pointwise convergence topology it turns out, somewhat surprisingly, that the corresponding L'-spaces for the spectr...