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Register a new userAn Upbound of Hausdorffs Dimension of the Divergence Set of the fractional Schrodinger Operator on $H^s(mathbb R^n)
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This paper shows $$ sup_{fin H^s(mathbb{R}^n)}dim _Hleft{xinmathbb{R}^n: lim_{trightarrow0}e^{it(-Delta)^alpha}f(x) eq f(x)right}leq n+1-frac{2(n+1)s}{n} text{under} begin{cases} ngeq2; alpha>frac12; frac{n}{2(n+1)}<sleqfrac{n}{2} . end{cases} $$
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Let $p(cdot): mathbb R^nto(0,infty)$ be a variable exponent function satisfying the globally log-Holder continuous condition. In this article, the authors first obtain a decomposition for any distribution of the variable weak Hardy space into good and bad parts and then prove the following real interpolation theorem between the variable Hardy space $H^{p(cdot)}(mathbb R^n)$ and the space $L^{infty}(mathbb R^n)$: begin{equation*} (H^{p(cdot)}(mathbb R^n),L^{infty}(mathbb R^n))_{theta,infty} =W!H^{p(cdot)/(1-theta)}(mathbb R^n),quad thetain(0,1), end{equation*} where $W!H^{p(cdot)/(1-theta)}(mathbb R^n)$ denotes the variable weak Hardy space. As an application, the variable weak Hardy space $W!H^{p(cdot)}(mathbb R^n)$ with $p_-:=mathopmathrm{ess,inf}_{xinrn}p(x)in(1,infty)$ is proved to coincide with the variable Lebesgue space $W!L^{p(cdot)}(mathbb R^n)$.
Let $Omegain L^1{({mathbb S^{n-1}})}$, be a function of homogeneous of degree zero, and $M_Omega$ be the Hardy-Littlewood maximal operator associated with $Omega$ defined by $M_Omega(f)(x) = sup_{r>0}frac1{r^n}int_{|x-y|<r}|Omega(x-y)f(y)|dy.$ It was shown by Christ and Rubio de Francia that $|M_Omega(f)|_{L^{1,infty}({mathbb R^n})} le C(|Omega|_{Llog L({mathbb S^{n-1}})}+1)|f|_{L^1({mathbb R^n})}$ provided $Omegain Llog L {({mathbb S^{n-1}})}$. In this paper, we show that, if $Omegain Llog L({mathbb S^{n-1}})$, then for all $fin L^1({mathbb R^n})$, $M_Omega$ enjoys the limiting weak-type behaviors that $$lim_{lambdato 0^+}lambda|{xin{mathbb R^n}:M_Omega(f)(x)>lambda}| = n^{-1}|Omega|_{L^1({mathbb S^{n-1}})}|f|_{L^1({mathbb R^n})}.$$ This removes the smoothness restrictions on the kernel $Omega$, such as Dini-type conditions, in previous results. To prove our result, we present a new upper bound of $|M_Omega|_{L^1to L^{1,infty}}$, which essentially improves the upper bound $C(|Omega|_{Llog L({mathbb S^{n-1}})}+1)$ given by Christ and Rubio de Francia. As a consequence, the upper and lower bounds of $|M_Omega|_{L^1to L^{1,infty}}$ are obtained for $Omegain Llog L {({mathbb S^{n-1}})}$.
The connection between derivative operators and wavelets is well known. Here we generalize the concept by constructing multiresolution approximations and wavelet basis functions that act like Fourier multiplier operators. This construction follows from a stochastic model: signals are tempered distributions such that the application of a whitening (differential) operator results in a realization of a sparse white noise. Using wavelets constructed from these operators, the sparsity of the white noise can be inherited by the wavelet coefficients. In this paper, we specify such wavelets in full generality and determine their properties in terms of the underlying operator.
Wild sets in $mathbb{R}^n$ can be tamed through the use of various representations though sometimes this taming removes features considered important. Finding the wildest sets for which it is still true that the representations faithfully inform us about the original set is the focus of this rather playful, expository paper that we hope will stimulate interest in cubical coverings as well as the other two ideas we explore briefly: Jones $beta$ numbers and varifolds from geometric measure theory.
In this paper we consider the Hardy-Lorentz spaces $H^{p,q}(R^n)$, with $0<ple 1$, $0<qle infty$. We discuss the atomic decomposition of the elements in these spaces, their interpolation properties, and the behavior of singular integrals and other operators acting on them.
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