It is known that the energy measures of any two nonconstant harmonic functions on the standard Sierpi\'nski gasket are mutually absolutely continuous. Strichartz and Tse reported numerical evidence for $L^p$-integrability of the corresponding Radon--Nikodym densities in the range \[ 1<p<\frac{\log 15}{\log 9}. \] For arbitrary ordered pairs of nonconstant harmonic functions, we prove uniform boundedness of the associated density-ratio power sums, and hence $L^p$-integrability, in the subinterval \[ 1<p<\frac{\log(35/3)}{\log 9}. \] When the denominator harmonic direction is represented by the boundary values $(0,-1,1)$, we prove boundedness throughout the full conjectured interval.
We establish Morrey-type conditions ensuring that a finite signed Radon measure belongs to the dual of the energy space of solutions to nonlinear parabolic equations of $p$-Laplacian type. More precisely, for the parabolic cylinders $Q_{r,r^p}(z):=B_r(x)\times(t-r^p,t+r^p)$, we prove that \[ |\mu|\bigl(Q_{r,r^p}(z)\cap...
We characterize the positive Borel measures $\mu$ on the unit disc $\mathbb{D}$ for which the M\"obius-invariant space $Q_K$ embeds continuously or compactly into $L^2(\mu)$. The characterization is given in terms of a discrete dyadic capacity $C^{(b)}_{K,\mathcal{R}}(\mu)$ built from a polar dyadic resolution of $\mat...
We prove a H\"ormander multiplier theorem for the Dunkl transform associated with an arbitrary finite reflection group. Uniform $H^\sigma(\mathbb{R}^N)$ bounds for the normalised dyadic pieces of a measurable symbol, with $\sigma>\mathbf N/2$ and $\mathbf N$ the homogeneous dimension, imply $L^p(d\omega)$ boundedness f...
Der-Chen Chang, Ji Li, Chao-Jie Wen et al.· 2 citations· ⚡1
We investigate Kolmogorov width and entropy numbers of the Nikol'skii--Besov classes $B^{\boldsymbol{r}}_{p,\theta}$ and Sobolev classes $W^{\boldsymbol{r}}_{p, \boldsymbol{\alpha}}$ of periodic functions of mixed smoothness of one and many variables in the space $B_{q,1}$. The norm in this space is ``stronger'' than t...
K. Pozharska, A. Romanyuk· Matematychni Studii· 0 citations
Let $\Delta_k$ be the Dunkl Laplacian associated with an arbitrary root system and a nonnegative multiplicity function. For every $0<p\leq1$, a Coifman-Weiss type atomic characterization of the Hardy space $H^p_{\mathrm{Dunkl}}$ is established. More precisely, it is proved that the space $H^p_{\mathrm{Dunkl}}$, which i...
Jacek Dziubański, Agnieszka Hejna-Łyżwa· 0 citations
We prove the constant-one discrete nonlinear Hausdorff-Young inequality. As a consequence, by a discrete-to-continuous limiting argument, we obtain $$ \|(\log|a_f|^{2})^{1/2}\|_{L^{p'}(\mathbb{R})} \le \|f\|_{L^{p}(\mathbb{R})},\quad 1\le p<2, $$ for all $f\in L^{p}(\mathbb{R})$, where $a_f$ denotes the transmission co...
D. Suragan· 0 citations
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