On weak and Schauder decompositions

Schauder Decompositions and Completeness

00 It x = S QnIf» m addition, the projections Pn = £ Q,are equicontinuous, then n = 1 « = 1 (£n)^°=1 is said to be an equi-Schauder decomposition of E. It is obvious that a Schauder basis is equivalent to a Schauder decomposition in which each subspace is one-dimensional, and that it is equi-Schauder if and only if the corresponding decomposition is equi-Schauder. For more information on Schaud...

Schauder decompositions in locally convex spaces

Upper and Lower Estimates for Schauder Frames and Atomic Decompositions

We prove that a Schauder frame for any separable Banach space is shrinking if and only if it has an associated space with a shrinking basis, and that a Schauder frame for any separable Banach space is shrinking and boundedly complete if and only if it has a reflexive associated space. To obtain these results, we prove that the upper and lower estimate theorems for finite dimensional decompositi...

Notes on Schauder Estimates

Proof. Let g(x) = u(x) − sup∂Br(y) u − r2 − |x − y|2 2n supBr(y) f . We have ∆g = ∆u + supBr(y) f ≥ − f + supBr(y) f ≥ 0, that is, g is subharmonic in Br(y). Then supBr(y) g = sup∂Br(y) g = 0, so g ≤ 0 in Br(y) and the lemma follows. Lemma 2. If u is a solution to ∆u = f in Br(y) and v solves ∆v = 0 and v = u on ∂Br(y), then r2 − |x − y|2 2n inf Br(y) f ≤ v(x) − u(x) ≤ r 2 − |x − y|2 2n sup Br(...

Schauder Estimates on Lie Groups