In this paper, we obtain a full characterization of the finite positive Borel measures $\mu$ on $\mathbb D$ for which the embedding $$ \operatorname{id}:\mathcal D_{p-1}^p\longrightarrow L^p(\mu),\qquad p>2, $$ is bounded. More precisely, for any dyadic system $\mathcal D$ on $\mathbb T$, we prove that this embedding is bounded if and only if $$ \mathcal C_{p,\mathcal D}(\mu)+\mathcal H_{p,\mathcal D}(\mu)<\infty, $$ where $\mathcal C_{p,\mathcal D}(\mu)$ and $\mathcal H_{p,\mathcal D}(\mu)$ denote the packing energy and the Haar energy of $\mu$, respectively. This resolves a longstanding characterization problem in the theory of Dirichlet-type spaces that arose from Wu's 1999 conjecture, corresponds to the endpoint case not covered by the work of Arcozzi, Rochberg, and Sawyer in 2002, and remained open after the works of Girela and Pel\'aez in 2006 and Galanopoulos, Girela, and Pel\'aez in 2011. We also construct finite measures showing that the two energy conditions are genuinely distinct. The key ingredient in the proof is a reduction of the Dirichlet embedding to a dyadic Whitney embedding, which allows us to combine weighted Hardy inequalities on trees with probabilistic arguments.
In this paper, we obtain non-testing characterizations, in terms of dyadic capacity gauges, of the boundedness and compactness of the differentiation operator $$ \frac{d}{dz}:Q_K\longrightarrow L^q(W\,dA), \qquad 0<q<\infty. $$ We also characterize the limiting case as $q\to0^+$, formulated in terms of a logarithmic geometric mean, while the endpoint $q=\infty$ is treated separately using a standard testing argument. These results greatly extend the previous work on ${\mathcal Q}_p$-spaces to the general setting of $Q_K$-spaces. As applications, we characterize composition operators and Volterra-type integral operators between different $Q_K$-spaces. In particular, the off-diagonal characterization established here, together with the previously established diagonal case, completely resolves Zhao's 2009 open question on composition operators between ${\mathcal Q}_p$-spaces.
We prove that there exist two Bloch functions $f_1$ and $f_2$ on $\mathbb D$ such that $$ |f_1(z)|+|f_2(z)| \geq \left(\log\frac{1}{1-|z|}\right)^{1/2}, \qquad z\in\mathbb D, $$ thereby resolving an open problem posed in 2008 by Girela, Pel\'aez, P\'erez-Gonz\'alez and R\"atty\"a. Our proof is based on a new Szeg\H{o}-type recursion involving $\mathbb C^2$-valued polynomials and their reciprocal polynomials.
Bingyang Hu, Jie Xiao, Xiaojing Zhou· 0 citations
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