We study two regularity problems on Hardy-type analytic tent spaces $\mathcal{AT}^p_{q,\alpha}$ on the unit disk: fractional integration and randomization of Taylor coefficients. For fractional integration, we characterize completely the boundedness and compactness of the Hadamard, Flett, and Riemann--Liouville operators between analytic tent spaces, and obtain parallel results for analytic Triebel--Lizorkin spaces. In particular, the case $t=0$ yields a complete solution to the corresponding embedding problem for analytic tent spaces. For randomization, we characterize completely when the random Taylor series $\mathcal{R}f$ belongs almost surely to an analytic tent space whenever $f\in \mathcal{AT}^p_{q,\alpha}$, and we also obtain the Triebel--Lizorkin counterpart. As part of the proof, we identify the random symbol space associated with $\mathcal{AT}^p_{q,\alpha}$ and solve the embedding problem from analytic tent spaces into mixed norm spaces. These results extend classical theorems of Hardy--Littlewood and Littlewood, as well as their later analogues for Bergman and mixed norm spaces.
Let $\mathscr{H}^2$ denote the Hilbert space of Dirichlet series with square-summable coefficients. It follows from a theorem of Bohr that if $f$ in $\mathscr{H}^2$ has a bounded analytic continuation to the right half-plane, then its norm can be computed from the mean values \[\|f\|_{\mathscr{H}^2}^2 = \lim_{\sigma\to...
Ole Fredrik Brevig, Athanasios Kouroupis· 0 citations
We construct divergence-free, vector-valued approximation functions on the whole space $\mathbb{R}^d$, $d\geq 2$, as well as on general unbounded domains of uniform $\mathrm{C}^{1,1}$-type. These approximations converge simultaneously in both Sobolev and Lebesgue spaces. In the whole-space setting, we employ Bogovski\u...
Akram Khan, Sagar Gautam, Manil T. Mohan· 0 citations
We establish interior maximal $L^{q_c}$-regularity for bounded strong solutions of $u_t-\Delta u+|Du|^\gamma=f$ in $\mathbb{T}^d\times(0,T)$, where $d\geq 2$, $\gamma>2$, and $q_c=(d+2)(\gamma-1)/\gamma$. The estimates are uniform for uniformly bounded families of solutions whose source terms range over a bounded, unif...
We study the area operators $\mathbb{A}_{\mu,l}$, $0<l<\infty$, induced by positive Borel measures on the right half-plane and acting on the Hardy spaces of Dirichlet series $\mathscr H^p$, $0<p<\infty$. We first disprove a conjecture proposed by the present authors in an earlier work by constructing a probability meas...
In this paper we provide a purely metric and derivative-free characterization of continuous mappings in the local supercritical Sobolev space. More specifically, we show that a continuous mapping $f\colon\mathbb{R}^n\to\mathbb{R}^m$ lies in the local Sobolev space $W^{1,p}_\text{loc}(\mathbb{R}^n\colon\mathbb{R}^m)$ if...
R. Alvarado, Efstathios Konstantinos Chrontsios Garitsis· 0 citations
Given a complete doubling metric measure space $(X,\rho,\mu)$ supporting a Poincar\'e inequality, we prove weak-type characterizations of the Sobolev space $\dot{W}^{1,p}(\mu)$ and the space of functions of bounded variation, achieving a full analogy in general Poincar\'e spaces with the Euclidean results of Brezis et...
T. Hytönen, Da-Chun Yang, Wen Yuan et al.· 0 citations
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