On Total Ω-continuity, Strong Ω-continuity and Contra Ω-continuity by N. Rajesh
نویسنده
چکیده
In this paper, ω-closed sets and ω-open sets are used to define and investigate a new class of functions. Relationships between this new class and other classes of functions are established.
منابع مشابه
Absolute Continuity between the Surface Measure and Harmonic Measure Implies Rectifiability
In the present paper we prove that for any open connected set Ω ⊂ R, n ≥ 1, and any E ⊂ ∂Ω with 0 < H(E) < ∞ absolute continuity of the harmonic measure ω with respect to the Hausdorff measure on E implies that ω|E is rectifiable. CONTENTS
متن کاملOn some sharp conditions for lower semicontinuity in L
Let Ω be an open set of R and let f : Ω × R × R be a nonnegative continuous function, convex with respect to ξ ∈ R. Following the well known theory originated by Serrin [14] in 1961, we deal with the lower semicontinuity of the integral F (u,Ω) = ∫ Ω f (x, u(x), Du(x)) dx with respect to the Lloc (Ω) strong convergence. Only recently it has been discovered that dependence of f (x, s, ξ) on the ...
متن کاملHarmonic Measure Is Rectifiable If It Is Absolutely Continuous with Respect to the Co-dimension One Hausdorff Measure
In the present paper we sketch the proof of the fact that for any open connected set Ω ⊂ Rn+1, n ≥ 1, and any E ⊂ ∂Ω with 0 < H(E) < ∞, absolute continuity of the harmonic measure ω with respect to the Hausdorff measure on E implies that ω|E is rectifiable.
متن کامل14.381 F13 Recitation 1 notes: Modes of Convergence
Proof. (⇒) Let Ω0 = {ω : limn Xn(ω) = X(ω)}. Suppose P (Ω0) = 1. Let > 0 be given. →∞ Let Am = ∩k=m{|Xk − X| ≤ }. Then Am ⊂ Am+1 ∀m and limm P (Am) = P (∪m=1Am) by →∞ continuity of probability measure. For each ω0, there exists m(ω0) such that |Xk(ω0)−X(ω0)| ≤ for all k ≥ m(ω0). Therefore, ∀ω0 ∈ Ω0, ω0 ∈ Am for some m and we can conclude that Ω0 ⊂ ∪m=1Am and 1 = P (Ω0) ≤ P (∪m=1Am) = limm P (A ...
متن کاملDouble Sine Series and Higher Order Lipschitz Classes of Functions (communicated by Hüseyin Bor)
Let ω(h, k) be a modulus of continuity, that is, ω(h, k) is a continuous function on the square [0, 2π] × [0, 2π], nondecreasing in each variable, and possessing the following properties: ω(0, 0) = 0, ω(t1 + t2, t3) ≤ ω(t1, t3) + ω(t2, t3), ω(t1, t2 + t3) ≤ ω(t1, t2) + ω(t1, t3). Yu ([3]) introduced the following classes of functions: HH := {f(x, y) : ‖f(x, y)− f(x+ h, y)− f(x, y + k) + f(x+ h,...
متن کامل