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Register a new userAddendum to : Orthonormal bases of regular wavelets in spaces of homogeneous type
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We bring a precision to our cited work concerning the notion of Borel measures, as the choice among different existing definitions impacts on the validity of the results.
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We apply wavelets to identify the Triebel type oscillation spaces with the known Triebel-Lizorkin-Morrey spaces $dot{F}^{gamma_1,gamma_2}_{p,q}(mathbb{R}^{n})$. Then we establish a characterization of $dot{F}^{gamma_1,gamma_2}_{p,q}(mathbb{R}^{n})$ via the fractional heat semigroup. Moreover, we prove the continuity of Calderon-Zygmund operators on these spaces. The results of this paper also provide necessary tools for the study of well-posedness of Navier-Stokes equations.
We establish a connection between the function space BMO and the theory of quasisymmetric mappings on emph{spaces of homogeneous type} $widetilde{X} :=(X,rho,mu)$. The connection is that the logarithm of the generalised Jacobian of an $eta$-quasisymmetric mapping $f: widetilde{X} rightarrow widetilde{X}$ is always in $rm{BMO}(widetilde{X})$. In the course of proving this result, we first show that on $widetilde{X}$, the logarithm of a reverse-H{o}lder weight $w$ is in $rm{BMO}(widetilde{X})$, and that the above-mentioned connection holds on metric measure spaces $widehat{X} :=(X,d,mu)$. Furthermore, we construct a large class of spaces $(X,rho,mu)$ to which our results apply. Among the key ingredients of the proofs are suitable generalisations to $(X,rho,mu)$ from the Euclidean or metric measure space settings of the Calder{o}n--Zygmund decomposition, the Vitali Covering Theorem, the Radon--Nikodym Theorem, a lemma which controls the distortion of sets under an $eta$-quasisymmetric mapping, and a result of Heinonen and Koskela which shows that the volume derivative of an $eta$-quasisymmetric mapping is a reverse-H{o}lder weight.
In the current paper, we study how the speed of convergence of a sequence of angles decreasing to zero influences the possibility of constructing a rare differentiation basis of rectangles in the plane, one side of which makes with the horizontal axis an angle belonging to the given sequence, that differentiates precisely a fixed Orlicz space.
In the present work we provide the bounds for Daubechies orthonormal wavelet coefficients for function spaces $mathcal{A}_k^p:={f: |(i omega)^khat{f}(omega)|_p< infty}$, $kinmathbf{N}cup{0}$, $pin(1,infty)$.
An orthonormal basis consisting of unentangled (pure tensor) elements in a tensor product of Hilbert spaces is an Unentangled Orthogonal Basis (UOB). In general, for $n$ qubits, we prove that in its natural structure as a real variety, the space of UOB is a bouquet of products of Riemann spheres parametrized by a class of edge colorings of hypercubes. Its irreducible components of maximum dimension are products of $2^n-1$ two-spheres. Using a theorem of Walgate and Hardy, we observe that the UOB whose elements are distinguishable by local operations and classical communication (called locally distinguishable or LOCC distinguishable UOB) are exactly those in the maximum dimensional components. Bennett et al, in their in-depth study of quantum nonlocality without entanglement, include a specific 3 qubit example UOB which is not LOCC distinguishable; we construct certain generalized counterparts of this UOB in $n$ qubits.
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