Littlewood–Paley inequalities for fractional derivative on Bergman spaces
For any pair \((n,p)\), \(nN\) and \(0<p<∞\), it has been recently proved by Peláez and Rättyä (2021) that a radial weight \(ω\) on the unit disc of the complex plane \(D\) satisfies the Littlewood-Paley equivalence \(_D |f(z)|^p ω(z) dA(z)_D|f^(n)(z)|^p(1-|z|)^npω(z) dA(z)+_j=0^n-1|f^(j)(0)|^p,\) for any analytic function \(f\) in \(D\), if and only if \(ωD=D D \). A radial weight \(ω\) belongs to the class \(D \) if \(_0r<1 _r^1 ω(s) ds_1+r2 ^1ω(s) ds<∞\), and \(ω D \) if there exists \(k>1\) such that \(_0r<1 _r^1ω(s) ds_1-1-rk ^1 ω(s) ds>1.\) In this paper we extend this result to the setting of fractional derivatives. Being precise, for an analytic function \(f(z)=_n=0^∞ f(n) z^n\) we consider the fractional derivative \(D^μ(f)(z)=_n=0^∞ f(n)μ_2n+1 z^n\) induced by a radial weight \(μ D\) where \(μ_2n+1=_0^1 r^2n+1μ(r) dr\). Then, we prove that for any \(p(0,∞)\), the Littlewood-Paley equivalence \(_D |f(z)|^p ω(z) dA(z)_D|D^μ(f)(z)|^p[_|z|^1μ(s) ds]^pω(z) dA(z)\) holds for any analytic function \(f\) in \(D\) if and only if \(ωD\). We also prove that for any \(p(0,∞)\), the inequality \(_D|D^μ(f)(z)|^p[_|z|^1μ(s) ds]^pω(z) dA(z)_D |f(z)|^p ω(z) dA(z)\) holds for any analytic function \(f\) in \(D\) if and only if \(ωD\). , number=2, journal=Annales Fennici Mathematici, author=Peláez, José Ángel and de la Rosa, Elena, year=2022, month=Sep., pages=1109–1130