ISSN:
1588-273X
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract Quasi-normed Lorentz spaces Λψ, q of 2π-periodic functions with quasinorms $$\left\| f \right\|_{\psi ,q} = \left\{ {\int\limits_0^{2\pi } {\psi ^q (t)\left[ {\frac{1}{t}\int\limits_0^t {f * (x)} dx} \right]} ^q \frac{{dt}}{t}} \right\}^{{1 \mathord{\left/ {\vphantom {1 q}} \right. \kern-\nulldelimiterspace} q}} $$ (0〈q〈∞,ω(t): [0,2π]→R is a continuous concave function with finite derivative everywhere on (0, 2gp)) and classes of functions $$H_{\psi ,q}^\omega \equiv \{ f(x):f(x) \in \Lambda _{\psi ,q} ;\mathop {\sup }\limits_{0 \leqq h \leqq \delta } \left\| {f(x + h) - f(x)} \right\|_{\psi ,q} = O\{ \omega (\delta )\} , \delta \to + 0\} $$ (ω(δ) — modulus of continuity) are studied. Precise embedding conditions of classes H ψ, q ω into Lorentz spaces and into each other are obtained: $$\begin{array}{*{20}c} {H_{\psi ,q_1 }^\omega \subset \Lambda _{\psi ,q_2 } ;} & {H_{\psi ,q_1 }^\omega \subset {\rm H}_{\psi ,q_2 }^{\omega * } ,} & {0〈 q_2〈 q_1〈 \infty ,} \\ \end{array} $$ under conditions $$\mathop {\lim }\limits_{t \to \infty } \frac{{\psi (2t)}}{{\psi (t)}} 〉 1,\mathop {\overline {\lim } }\limits_{x \to \infty } \frac{{\psi (2t)}}{{\psi (t)}}〈 2$$ andω(δ)=O{ω(δ 2)},δ→+0, andω * (δ) is an arbitrary modulus of continuity.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF02074616
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