Bergman空间上分数导数的Littlewood-Paley不等式
Littlewood-Paley inequalities for fractional derivative on Bergman spaces
AI总结：
本文研究了Bergman空间上分数导数的Littlewood-Paley不等式，证明了在特定条件下的等价关系和不等式成立的充要条件。
摘要：
对于任意一对(n,p)，n∈N且0<p<∞，最近已证明，复平面上单位圆盘D上的径向权ω满足Littlewood-Paley等价关系：∫_D |f(z)|^p ω(z) dA(z) ≈ ∫_D |f^{(n)}(z)|^p (1-|z|)^{np} ω(z) dA(z) + ∑_{j=0}^{n-1} |f^{(j)}(0)|^p，对于任意在D内解析的函数f，当且仅当ω∈D=\widehat{D} ∩ \check{D}。一个径向权ω属于类\widehat{D}如果sup_{0≤r<1} [∫_r^1 ω(s) ds / ∫_{(1+r)/2}^1 ω(s) ds] < ∞，且ω∈\check{D}如果存在k>1使得inf_{0≤r<1} [∫_r^1 ω(s) ds / ∫_{1-(1-r)/k}^1 ω(s) ds] > 1。在本文中，我们将此结果扩展到分数导数的设置。具体而言，对于解析函数f(z)=∑_{n=0}^∞ \widehat{f}(n) z^n，我们考虑由径向权μ∈D诱导的分数导数D^μ(f)(z)=∑_{n=0}^∞ [\widehat{f}(n)/μ_{2n+1}] z^n，其中μ_{2n+1}=∫_0^1 r^{2n+1} μ(r) dr。然后我们证明，对于任何p∈(0,∞)，Littlewood-Paley等价关系∫_D |f(z)|^p ω(z) dA(z) ≈ ∫_D |D^μ(f)(z)|^p [∫_{|z|}^1 μ(s) ds]^p ω(z) dA(z) 对于任何在D内解析的函数f成立当且仅当ω∈D。我们还证明，对于任何p∈(0,∞)，不等式∫_D |D^μ(f)(z)|^p [∫_{|z|}^1 μ(s) ds]^p ω(z) dA(z) ≲ ∫_D |f(z)|^p ω(z) dA(z) 对于任何在D内解析的函数f成立当且仅当ω∈\widehat{D}。
英文摘要：
For any pair $(n,p)$, $n\in\mathbb{N}$ and $0<p<\infty$, it has been recently proved that a radial weight $ω$ on the unit disc of the complex plane $\mathbb{D}$ satisfies the Littlewood-Paley equivalence $$ \int_{\mathbb{D}}|f(z)|^p\,ω(z)\,dA(z)\asymp\int_\mathbb{D}|f^{(n)}(z)|^p(1-|z|)^{np}ω(z)\,dA(z)+\sum_{j=0}^{n-1}|f^{(j)}(0)|^p,$$ for any analytic function $f$ in $\mathbb{D}$, if and only if $ω\in\mathcal{D}=\widehat{\mathcal{D}} \cap \check{\mathcal{D}}$. A radial weight $ω$ belongs to the class $\widehat{\mathcal{D}}$ if $\sup_{0\le r<1} \frac{\int_r^1 ω(s)\,ds}{\int_{\frac{1+r}{2}}^1ω(s)\,ds}<\infty$, and $ω\in \check{\mathcal{D}}$ if there exists $k>1$ such that $\inf_{0\le r<1} \frac{\int_{r}^1ω(s)\,ds}{\int_{1-\frac{1-r}{k}}^1 ω(s)\,ds}>1$. In this paper we extend this result to the setting of fractional derivatives. Being precise, for an analytic function $f(z)=\sum_{n=0}^\infty \widehat{f}(n) z^n$ we consider the fractional derivative $ D^μ(f)(z)=\sum\limits_{n=0}^{\infty} \frac{\widehat{f}(n)}{μ_{2n+1}} z^n $ induced by a radial weight $μ\in \mathcal{D}$, where $μ_{2n+1}=\int_0^1 r^{2n+1}μ(r)\,dr$. Then, we prove that for any $p\in (0,\infty)$, the Littlewood-Paley equivalence $$\int_{\mathbb{D}} |f(z)|^p ω(z)\,dA(z)\asymp \int_{\mathbb{D}}|D^μ(f)(z)|^p\left[\int_{|z|}^1μ(s)\,ds\right]^pω(z)\,dA(z)$$ holds for any analytic function $f$ in $\mathbb{D}$ if and only if $ω\in\mathcal{D}$. We also prove that for any $p\in (0,\infty)$, the inequality $$\int_{\mathbb{D}} |D^μ(f)(z)|^p \left[\int_{|z|}^1μ(s)\,ds\right]^pω(z)\,dA(z) \lesssim \int_{\mathbb{D}} |f(z)|^p ω(z)\,dA(z) $$ holds for any analytic function $f$ in $\mathbb{D}$ if and only if $ω\in \widehat{\mathcal{D}}$.
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