arXiv 2110.136982021-10-26math.FA

广义Lorentz空间与经典Lorentz空间之间极大函数的范数

Norms of maximal functions between generalized and classical Lorentz spaces

Rza Mustafayev, Nevin Bilgiçli, Merve Yılmaz

备注：: 30 pages. arXiv admin note: substantial text overlap with arXiv:2109.06745

摘要：

本文用广义极大算子$M_{ϕ,Λ^α(b)}$和函数$b,\,ϕ: (0,\infty) \rightarrow (0,\infty)$定义了广义极大算子$M_{ϕ,Λ^α(b)}$的范数，并通过\begin{equation*} M_{ϕ,Λ^α(b)}f(x) : = \sup_{Q \ni x} \frac{\|f χ_Q\|_{Λ^α(b)}}{ϕ(|Q|)}, \qquad x \in {\mathbb R}^n, \end{equation*}从${\operatorname{GΓ}}(p,m,v)$到$Λ^q(w)$对${\mathbb R}^n$上的所有可测函数$f$进行了计算.这里$Λ^α(b)$和${\operatorname{GΓ}}(p,m,w)$是经典的和广义的洛伦兹空间，定义为定义在${\mathbb R}^n$上的所有可测函数$f$的集合，其中分别为$$ \|f\|_{Λ^α(b)} = \bigg( \int_0^{\infty} [f^*(s)]^α b(s)\,ds \bigg)^{\frac{1}α} < \infty \quad \mbox{and} \quad \|f\|_{\operatorname{GΓ}(p,m,w)} = \bigg( \int_0^{\infty} \bigg( \int_0^x [f^* (τ)]^p\,dτ\bigg)^{\frac{m}{p}} v(x)\,dx \bigg)^{\frac{1}{m}} < \infty, $$。我们把问题归结为不等式\begin{equation*} \bigg( \int_0^{\infty} \big[ T_{u,b}f^* (x)\big]^q \, w(x)\,dx\bigg)^{\frac{1}{q}} \le C \, \bigg( \int_0^{\infty} \bigg( \int_0^x [f^* (τ)]^p\,dτ\bigg)^{\frac{m}{p}} v(x)\,dx \bigg)^{\frac{1}{m}} \end{equation*}的解，其中$w$和$v$是$(0,\infty)$上的权函数。这里$f^*$是定义在${\mathbb R}^n$上的$f$的非递增重排，$T_{u,b}$是涉及上界的迭代Hardy型算子，其由$$ (T_{u,b} g)(t) : = \sup_{τ\in [t,\infty)} \frac{u(τ)}{B(τ)} \int_0^τ g(s)b(s)\,ds,\qquad t \in (0,\infty), $$针对$(0,\infty)$上的可测量非负函数$f$定义，其中$u$和$b$是$(0,\infty)$上的适当权重函数，并且函数$B(t) : = \int_0^t b(s)\,ds$对于每个$t \in (0,\infty)$都满足$0 < B(t) < \infty$。

英文摘要：

In this paper we calculate the norm of the generalized maximal operator $M_{ϕ,Λ^α(b)}$, defined with $0 < α< \infty$ and functions $b,\,ϕ: (0,\infty) \rightarrow (0,\infty)$ for all measurable functions $f$ on ${\mathbb R}^n$ by \begin{equation*} M_{ϕ,Λ^α(b)}f(x) : = \sup_{Q \ni x} \frac{\|f χ_Q\|_{Λ^α(b)}}{ϕ(|Q|)}, \qquad x \in {\mathbb R}^n, \end{equation*} from ${\operatorname{GΓ}}(p,m,v)$ into $Λ^q(w)$. Here $Λ^α(b)$ and ${\operatorname{GΓ}}(p,m,w)$ are the classical and generalized Lorentz spaces, defined as a set of all measurable functions $f$ defined on ${\mathbb R}^n$ for which $$ \|f\|_{Λ^α(b)} = \bigg( \int_0^{\infty} [f^*(s)]^α b(s)\,ds \bigg)^{\frac{1}α} < \infty \quad \mbox{and} \quad \|f\|_{\operatorname{GΓ}(p,m,w)} = \bigg( \int_0^{\infty} \bigg( \int_0^x [f^* (τ)]^p\,dτ\bigg)^{\frac{m}{p}} v(x)\,dx \bigg)^{\frac{1}{m}} < \infty, $$ respectively. We reduce the problem to the solution of the inequality \begin{equation*} \bigg( \int_0^{\infty} \big[ T_{u,b}f^* (x)\big]^q \, w(x)\,dx\bigg)^{\frac{1}{q}} \le C \, \bigg( \int_0^{\infty} \bigg( \int_0^x [f^* (τ)]^p\,dτ\bigg)^{\frac{m}{p}} v(x)\,dx \bigg)^{\frac{1}{m}} \end{equation*} where $w$ and $v$ are weight functions on $(0,\infty)$. Here $f^*$ is the non-increasing rearrangement of $f$ defined on ${\mathbb R}^n$ and $T_{u,b}$ is the iterated Hardy-type operator involving suprema, which is defined for a measurable non-negative function $f$ on $(0,\infty)$ by $$ (T_{u,b} g)(t) : = \sup_{τ\in [t,\infty)} \frac{u(τ)}{B(τ)} \int_0^τ g(s)b(s)\,ds,\qquad t \in (0,\infty), $$ where $u$ and $b$ are appropriate weight functions on $(0,\infty)$ and the function $B(t) : = \int_0^t b(s)\,ds$ satisfies $0 < B(t) < \infty$ for every $t \in (0,\infty)$..

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