Search

A real number $x$ is normal with respect to an integer base $b \geq 2$ if its digit expansion in this base is ``equitable'', in the sense that for $k \geq 1$, every ordered sequence of $k$ digits from $\{0, 1, \ldots, b-1\}$ occurs in the digit expansion of $x$ with the same limiting frequency. Borel's classical result \cite{b09} asserts that Lebesgue-almost every number $x$ is normal in every base $b \geq 2$. This three-part article considers sets of partial normality. Given any choice of integer bases $\mathscr{B}, \mathscr{B}' \subseteq \{2, 3, \ldots\}$, we investigate measure-theoretic properties of the set $\mathscr{N}(\mathscr{B}, \mathscr{B}')$, whose members are, by definition, normal in the bases of $\mathscr{B}$ and non-normal in the bases of $\mathscr{B}'$. A pair of sets $(\mathscr{B}, \mathscr{B}')$ is compatible if any $(b, b') \in \mathscr{B} \times \mathscr{B}'$ is multiplicatively independent. For compatible $(\mathscr{B}, \mathscr{B}')$ with $\mathscr{B}' \ne \emptyset$, we construct singular probability measures supported on $\mathscr N(\mathscr{B}, \mathscr{B}')$ that are both Frostman and Rajchman, extending prior work of Pollington \cite{p81} and Lyons \cite{l86}. The Rajchman property completely answers a question of Kahane and Salem \cite{Kahane-Salem-64}, identifying $\mathscr N(\mathscr{B}, \mathscr{B}')$ as a set of multiplicity (in the Fourier-analytic sense) if and only if $(\mathscr{B}, \mathscr{B}')$ is compatible. The methodological contribution of the article is the construction of a class of probability measures called skewed measures. These measures depend on a number of parameters that can be independently adjusted to ensure (subsets of) properties such as almost everywhere normality, non-normality, ball conditions and Fourier decay., Comment: 107 pages, 3 figures

A real number $x$ is considered normal in an integer base $b \geq 2$ if its digit expansion in this base is ``equitable'', ensuring that for each $k \geq 1$, every ordered sequence of $k$ digits from $\{0, 1, \ldots, b-1\}$ occurs in the digit expansion of $x$ with the same limiting frequency. Borel's classical result \cite{b09} asserts that Lebesgue-almost every $x \in \mathbb R$ is normal in every base $b \geq 2$. This paper serves as a case study of the measure-theoretic properties of Lebesgue-null sets containing numbers that are normal only in certain bases. We consider the set $\mathscr N(\mathscr{O}, \mathscr{E})$ of reals that are normal in odd bases but not in even ones. This set has full Hausdorff dimension \cite{p81} but zero Fourier dimension. The latter condition means that $\mathscr N(\mathscr{O}, \mathscr{E})$ cannot support a probability measure whose Fourier transform has power decay at infinity. Our main result is that $\mathscr N(\mathscr{O}, \mathscr{E})$ supports a Rajchman measure $\mu$, whose Fourier transform $\widehat{\mu}(\xi)$ approaches 0 as $|\xi| \rightarrow \infty$ by definiton, albeit slower than any negative power of $|\xi|$. Moreover, the decay rate of $\widehat{\mu}$ is essentially optimal, subject to the constraints of its support. The methods draw inspiration from the number-theoretic results of Schmidt \cite{s60} and a construction of Lyons \cite{l86}. As a consequence, $\mathscr N(\mathscr{O}, \mathscr{E})$ emerges as a set of multiplicity, in the sense of Fourier analysis. This addresses a question posed by Kahane and Salem \cite{Kahane-Salem-64} in the special case of $\mathscr N(\mathscr{O}, \mathscr{E})$., Comment: 36 pages

Let $L:=-\Delta+V$ be the Schr\"{o}dinger operator on $\mathbb{R}^n$ with $n\geq 3$, where $V$ is a non-negative potential which belongs to certain reverse H\"{o}lder class $RH_q(\mathbb{R}^n)$ with $q\in (n/2,\,\infty)$. In this article, the authors obtain the quantitative weighted boundedness of Littlewood--Paley functions $g_L$, $S_L$ and $g_{L,\,\lambda}^\ast$, associated to $L$, on weighted Lebesgue spaces $L^p(w)$, where $w$ belongs to the class of Muckenhoupt $A_p$ weights adapted to $L$., Comment: 26 pages; Submitted

In this article, the authors consider the Schr\"{o}dinger type operator $L:=-{\rm div}(A\nabla)+V$ on $\mathbb{R}^n$ with $n\geq 3$, where the matrix $A$ is symmetric and satisfies uniformly elliptic condition and the nonnegative potential $V$ belongs to the reverse H\"{o}lder class $RH_q(\mathbb{R}^n)$ with $q\in(n/2,\,\infty)$. Let $p(\cdot):\ \mathbb{R}^n\to(0,\,1]$ be a variable exponent function satisfying the globally $\log$-H\"{o}lder continuous condition. The authors introduce the variable Hardy space $H_L^{p(\cdot)}(\mathbb{R}^n)$ associated to $L$ and establish its atomic characterization. The atoms here are closer to the atoms of variable Hardy space $H^{p(\cdot)}(\mathbb{R}^n)$ in spirit, which further implies that $H^{p(\cdot)}(\mathbb{R}^n)$ is continuously embedded in $H_L^{p(\cdot)}(\mathbb{R}^n)$., Comment: arXiv admin note: text overlap with arXiv:1811.10768

In this article, the authors consider the Schr\"{o}dinger type operator $L:=-{\rm div}(A\nabla)+V$ on $\mathbb{R}^n$ with $n\geq 3$, where the matrix $A$ satisfies uniformly elliptic condition and the nonnegative potential $V$ belongs to the reverse H\"{o}lder class $RH_q(\mathbb{R}^n)$ with $q\in(n/2,\,\infty)$. Let $p(\cdot):\ \mathbb{R}^n\to(0,\,\infty)$ be a variable exponent function satisfying the globally $\log$-H\"{o}lder continuous condition. When $p(\cdot):\ \mathbb{R}^n\to(1,\,\infty)$, the authors prove that the operators $VL^{-1}$, $V^{1/2}\nabla L^{-1}$ and $\nabla^2L^{-1}$ are bounded on variable Lebesgue space $L^{p(\cdot)}(\mathbb{R}^n)$. When $p(\cdot):\ \mathbb{R}^n\to(0,\,1]$, the authors introduce the variable Hardy space $H_L^{p(\cdot)}(\mathbb{R}^n)$, associated to $L$, and show that $VL^{-1}$, $V^{1/2}\nabla L^{-1}$ and $\nabla^2L^{-1}$ are bounded from $H_L^{p(\cdot)}(\mathbb{R}^n)$ to $L^{p(\cdot)}(\mathbb{R}^n)$.

Let $L$ be a one-to-one operator of type $\omega$ in $L^2(\mathbb{R}^n)$, with $\omega\in[0,\,\pi/2)$, which has a bounded holomorphic functional calculus and satisfies the Davies-Gaffney estimates. Let $p(\cdot):\ \mathbb{R}^n\to(0,\,1]$ be a variable exponent function satisfying the globally log-H\"{o}lder continuous condition. In this article, the authors introduce the variable Hardy space $H^{p(\cdot)}_L(\mathbb{R}^n)$ associated with $L$. By means of variable tent spaces, the authors establish the molecular characterization of $H^{p(\cdot)}_L(\mathbb{R}^n)$. Then the authors show that the dual space of $H^{p(\cdot)}_L(\mathbb{R}^n)$ is the BMO-type space ${\rm BMO}_{p(\cdot),\,L^\ast}(\mathbb{R}^n)$, where $L^\ast$ denotes the adjoint operator of $L$. In particular, when $L$ is the second order divergence form elliptic operator with complex bounded measurable coefficients, the authors obtain the non-tangential maximal function characterization of $H^{p(\cdot)}_L(\mathbb{R}^n)$ and show that the fractional integral $L^{-\alpha}$ for $\alpha\in(0,\,\frac12]$ is bounded from $H_L^{p(\cdot)}(\mathbb{R}^n)$ to $H_L^{q(\cdot)}(\mathbb{R}^n)$ with $\frac1{p(\cdot)}-\frac1{q(\cdot)}=\frac{2\alpha}{n}$ and the Riesz transform $\nabla L^{-1/2}$ is bounded from $H^{p(\cdot)}_L(\mathbb{R}^n)$ to the variable Hardy space $H^{p(\cdot)}(\mathbb{R}^n)$., Comment: 54 pages; Proc. Edinb. Math. Soc. (2) (to appear)

Let $w$ be a Muckenhoupt $A_2(\mathbb{R}^n)$ weight and $L_w:=-w^{-1}\mathop\mathrm{div}(A\nabla)$ the degenerate elliptic operator on the Euclidean space $\mathbb{R}^n$. In this article, the authors establish the Riesz transform characterization of the Hardy space $H_{L_w}^p(\mathbb{R}^n)$ associated with $L_w$, for $w\in A_{q}(\mathbb{R}^n)$ and $w^{-1}\in A_{2-\frac{2}{n}}(\mathbb{R}^n)$ with $n\geq 3$, $q\in[1,2]$ and $p\in(q(\frac{1}{r}+\frac{q-1}{2}+\frac{1}{n})^{-1},1]$ if, for some $r\in[1,\,2)$, $\{tL_w e^{-tL_w}\}_{t\geq 0}$ satisfies the weighted $L^r-L^2$ full off-diagonal estimate., Comment: 28 pages; Submitted