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Orlicz Spaces and Moduli Spaces: A Comprehensive Guide (PDF)


0 (st - \Phi(s)).$$ The norm on $L^\Phi(X,\mu)$ is then given by $$\f\_L^\Phi(X,\mu) = \inf\left\\lambda > 0 : \int_X \Psi\left(\frac\lambda\right) d\mu(x) \leq 1\right\.$$ Alternatively, we can use another equivalent norm called the Luxemburg norm, which is defined by $$[f]_L^\Phi(X,\mu) = \inf\left\\lambda > 0 : \int_X \Phi\left(\frac\lambda\right) d\mu(x) \leq 1\right\.$$ A modulus space or a modular space is a function space that is defined by a convex function called a modular. A modular is a function $\rho : [0,\infty) \to [0,\infty)$ that satisfies $\rho(0) = 0$, $\rho(t) > 0$ for $t > 0$, $\rho(t)/t$ is nondecreasing, and $\rho(t)$ is convex. Given a modular $\rho$ and a measure space $(X,\mu)$, the modulus space $L_\rho(X,\mu)$ is the set of all measurable functions $f : X \to \mathbbR$ such that $$\int_X \rho(f(x)) d\mu(x) 0$ for $t > 0$, $\Phi(t)/t \to \infty$ as $t \to \infty$, and $\Phi(t)$ is convex. ### Relation between Orlicz spaces and Lp spaces Orlicz spaces are a generalization of Lp spaces that allow more flexibility in choosing the Young function that defines the size of a function. In fact, if we take $\Phi(t) = t^p$ for some $1 0$ for $t > 0$, $\rho(t)/t$ is nondecreasing, and $\rho(t)$ is convex. Given a modular $\rho$ and a measure space $(X,\mu)$, the modular on the vector space of measurable functions $f : X \to \mathbbR$ is given by $$\rho(f) = \int_X \rho(f(x)) d\mu(x).$$ This modular may not satisfy the homogeneity or the triangle inequality of a norm, but it satisfies some weaker properties that make it a useful tool for studying the properties of the vector space. ### Relation between moduli spaces and normed spaces Moduli spaces are a generalization of normed spaces that allow more flexibility in choosing the modular that defines the size of a function. In fact, if we take $\rho(t) = t^p$ for some $1 \leq p




Orlicz Spaces And Moduli Spaces Pdf Download

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