The aim of this paper is twofold. First, we establish a Riesz conjugate theorem for weighted harmonic Bergman spaces. More precisely, we prove that if $f=u+iv$ is a harmonic $K$-quasiregular mapping in $\mathbb{D}$ and the real part $u$ belongs to the weighted harmonic Bergman space $a_\alpha^p$, $0<p<\infty$, then the imaginary part $v$ also belongs to the same space, together with a quantitative norm estimate. Moreover, for $1<p<\infty$, the corresponding constant is shown to be independent of the weight parameter $\alpha$. Second, we establish Riesz--Fej\'er inequalities for weighted harmonic Bergman spaces for $1<p<\infty$. In the special case $p=2$, we further improve the corresponding constant by using the Hilbert space structure and orthogonality techniques. As applications of our main results, we establish Riesz conjugate theorems and Riesz--Fej\'er inequalities for the M\"obius invariant spaces $Q(n,p,\alpha)$ introduced by Zhu [Illinois J. Math. 51 (2007), pp. 977--1002] and their harmonic counterparts $Q_h(n,p,\alpha)$ introduced by Sun, Liu, and Wang [Potential Anal. 65 (2026), Article no. 12].
Let $\mathbb{G}$ be a stratified Lie group and $\mathcal L$ be its sub-Laplacian. We prove that the full horizontal Riesz transform $\nabla_{H} \mathcal L^{-1/2}$ is of weak type $(1,1)$ on real-valued functions, with constant at most $2$. In particular, the constant is independent of the horizontal dimension, the homo...
Inspired by the sharp coefficient estimates established by Cho \emph{et al.}\cite{CKKLS2018} for starlike functions of order $\alpha$ in the unit disk, we investigate the corresponding problems for starlike mappings of order $\alpha$ defined on the unit ball of a complex Banach space. Employing Fr\'echet derivatives to...
Let $H$ be the Hilbert transform, let $h$ be a quasisymmetric homeomorphism of the unit circle $S^1$, and set $V_hu=u\circ h$ and $J_h=V_hHV_h^{-1}$, defined on the Sobolev space $H^{1/2}(S^1)$. We prove that $h\mapsto V_h$ is nowhere continuous in operator norm, although it is continuous in the strong operator topolog...
Let $1<p<\infty$, let $X$ be a UMD Banach space, let $1<p<\infty$, and let $L_d$ be the generator of the Ornstein--Uhlenbeck semigroup $(P_d(t))_{t\geq 0}$ on $L^p(\mathbb R^d,\gamma_d;X)$. We prove that the operators $-L_d$ are $R$-sectorial with a common angle strictly smaller than $\pi/2$ and with bounds independent...
In this paper, we establish a Fefferman--Stein inequality in terms of area function and non-tangential maximal function associated with the Schr\"odinger operator $\mathcal{L} = -\Delta + V$ on stratified Lie groups $\mathcal G$, where $\Delta$ denotes the sub-Laplacian on $\mathcal G$ and $V$ is a nonnegative locally...
Let $\omega$ be a radial $\widehat{\mathcal D}$-weight and $u$ be a bounded function on the unit disk $\mathbb D$. We prove that the Toeplitz operator \(T_{\omega,u}\) is compact on \(A_\omega^2\) if and only if its Berezin transform vanishes at the boundary. Our approach is based on a polynomial frame for $A_\omega^2$...
Yuerang Li, Zi-Peng Wang· 1 citation
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