About The Nature Of The Two Different Types Of Lorentz Space

This dissertations consists of two parts.In first chapter,we mainly get two results.The first is the relations between some weights,as follows:and so on. The second is the following theorem: Ifwhere Tg{t) = u(s)g(st)ds,T is an operator acting on functions 0 g ,and = sup (u Xq)*u(u(Q)t),0 < t < ∞,where the supremum istaken over all cubes Q ∈ Rⁿ, thenM :where M is the H-L maximal operator.In second chapter,we discuss the properties of two indexes of two dimensions Lorentz space ∧₂^p,q(w), which can be stated as follows. Theorem 2.2.1.7 当 0 < r₁,r₂ < ∞,0 < p,q < ∞ 时, (a) For r₁ > r₂ , the following conditions are equivalent:where D_f,t = {x : f(x) > t}.(b) For r₁ > r₂ , the following conditions are equivalent:(u>i(Df ()) d I — (w2(Dftt)Y ) ^ °°i where Dj>t — {x : f(x) > t}.(iv) sup T*kez{w\(Dk+i))^r [w-iiDk+iY2 w2(DkY2'q) < oo. (c) For n > r2 , the following conditions are equivalent: j r/ri■{V(\Dk+1\)-V(\Dk\))dt 0 so thatw(2x)dx 9(w)becomes a complete quasinormed space.Theorem 2.2.3.2 For l* h-2'q(w) xs bounded .(iii)if in/in = \\s?f;yw)\\L,.<,then HI ? HI is a norm equavilent to || ■ HA'-'fu;) ?(iv)Quasinorm \\f\\%L(w) = \\S2{f*x)\\Lr,,{w) = ||/**|Up.,(u;) is equivalent to...