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Register a new userUncertainty Principles Associated to Sets Satisfying the Geometric Control Condition
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In this paper, we study forms of the uncertainty principle suggested by problems in control theory. First, we prove an analogue of the Paneah-Logvinenko-Sereda Theorem characterizing sets which satisfy the Geometric Control Condition (GCC). This result is applied to get a uniqueness result for functions with spectrum supported on sufficiently flat sets. One corollary is that a function with spectrum in an annulus of a given thickness can be bounded, in $L^2$-norm, from above by its restriction to any open GCC set, independent of the radius of the annulus. This result is applied to the energy decay rates for damped fractional wave equations.
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Let $p(cdot): mathbb R^nto(0,1]$ be a variable exponent function satisfying the globally log-Holder continuous condition and $L$ a one to one operator of type $omega$ in $L^2({mathbb R}^n)$, with $omegain[0,,pi/2)$, which has a bounded holomorphic functional calculus and satisfies the Davies-Gaffney estimates. In this article, the authors introduce the variable weak Hardy space $W!H_L^{p(cdot)}(mathbb R^n)$ associated with $L$ via the corresponding square function. Its molecular characterization is then established by means of the atomic decomposition of the variable weak tent space $W!T^{p(cdot)}(mathbb R^n)$ which is also obtained in this article. In particular, when $L$ is non-negative and self-adjoint, the authors obtain the atomic characterization of $W!H_L^{p(cdot)}(mathbb R^n)$. As an application of the molecular characterization, when $L$ is the second-order divergence form elliptic operator with complex bounded measurable coefficient, the authors prove that the associated Riesz transform $ abla L^{-1/2}$ is bounded from $W!H_L^{p(cdot)}(mathbb R^n)$ to the variable weak Hardy space $W!H^{p(cdot)}(mathbb R^n)$. Moreover, when $L$ is non-negative and self-adjoint with the kernels of ${e^{-tL}}_{t>0}$ satisfying the Gauss upper bound estimates, the atomic characterization of $W!H_L^{p(cdot)}(mathbb R^n)$ is further used to characterize the space via non-tangential maximal functions.
In this paper we prove Hormander-Mihlin multiplier theorems for pseudo-multipliers associated to the harmonic oscillator (also called the Hermite operator). Our approach can be extended to also obtain the $L^p$-boundedness results for multilinear pseudo-multipliers. By using the Littlewood-Paley theorem associated to the harmonic oscillator we also give $L^p$-boundedness and $L^p$-compactness properties for multipliers. $(L^p,L^q)$-estimates for spectral pseudo-multipliers also are investigated.
Let $L$ be a one-to-one operator of type $omega$ in $L^2(mathbb{R}^n)$, with $omegain[0,,pi/2)$, which has a bounded holomorphic functional calculus and satisfies the Davies-Gaffney estimates. Let $p(cdot): mathbb{R}^nto(0,,1]$ 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 $alphain(0,,frac12]$ is bounded from $H_L^{p(cdot)}(mathbb{R}^n)$ to $H_L^{q(cdot)}(mathbb{R}^n)$ with $frac1{p(cdot)}-frac1{q(cdot)}=frac{2alpha}{n}$ and the Riesz transform $ abla L^{-1/2}$ is bounded from $H^{p(cdot)}_L(mathbb{R}^n)$ to the variable Hardy space $H^{p(cdot)}(mathbb{R}^n)$.
We prove bounds in the local $ L^2 $ range for exotic paraproducts motivated by bilinear multipliers associated with convex sets. One result assumes an exponential boundary curve. Another one assumes a higher order lacunarity condition.
We study the uncertainty principles related to the generalized Logan problem in $mathbb{R}^{d}$. Our main result provides the complete solution of the following problem: for a fixed $min mathbb{Z}_{+}$, find [ sup{|x|colon (-1)^{m}f(x)>0}cdot sup {|x|colon xin mathrm{supp},widehat{f},}to inf, ] where the infimum is taken over all nontrivial positive definite bandlimited functions such that $int_{mathbb{R}^d}|x|^{2k}f(x),dx=0$ for $k=0,dots,m-1$ if $mge 1$. We also obtain the uncertainty principle for bandlimited functions related to the recent result by Bourgain, Clozel, and Kahane.
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