Search Results (8)

We obtain the separability of mixed-norm Herz-slice spaces, establish a weak convergence on mixed-norm Herz-slice spaces, and get the boundedness of the Calderón–Zygmund operator T on mixed-norm Herz-slice spaces. Moreover, we get the necessary and sufficient conditions for the boundedness of the commutator $[b,T]$ on mixed-norm Herz-slice spaces, where b is a locally integrable function. Full article

Unstable combustion phenomena such as flame flashback, flame liftoff, extinction and blowout frequently take place during the operation of the bluff-body stabilized burner. Therefore, flame state monitoring is necessary for the safe operation of the bluff-body stabilized burner. In the present study, an electrical capacitance tomography (ECT) system was deployed to detect the permittivity distribution in the premixing channel and further characterize the flame states of stabilization, flashback, liftoff, extinction and blowout. A calderon-based reconstruction method was modified to reconstruct the permittivity distribution in the annular premixing channel. The detection results indicate that the permittivity in the premixing channel increases steeply when the flame flashback takes place and decreases obviously when the flame lifts off from the combustor rim. Based on the varied permittivity distribution at different flame states, a flame state index was proposed to characterize the flame state in quantification. The flame state index is 0, positive, in the range of −0.64–0, and lower than −0.64 when the flame is at the state of stable, flashback, liftoff and blowout, respectively. The flame state index at the flame state of extinction is the same as that at the flame state of liftoff. The extinction state and the blowout state can be distinguished by judging whether the flame flashback takes place before the flame is extinguished. These results reveal that the ECT system is capable of monitoring the flame state, and that the proposed flame state index can be used to characterize the flame state. Full article

Given a compact Riemannian manifold $(M,g)$ with smooth boundary $\partial M$, we give an explicit expression for the full symbol of the thermoelastic Dirichlet-to-Neumann map ${\mathsf{\Lambda }}_g$ with variable coefficients $\lambda ,\mu ,\alpha ,\beta \in C^{\infty }\left(\overline{M}\right)$. We prove that ${\mathsf{\Lambda }}_g$ uniquely determines partial derivatives of all orders of these coefficients on the boundary $\partial M$. Moreover, for a nonempty smooth open subset $\mathsf{\Gamma }\subset \partial M$, suppose that the manifold and these coefficients are real analytic up to $\mathsf{\Gamma }$. We show that ${\mathsf{\Lambda }}_g$ uniquely determines these coefficients on the whole manifold $\overline{M}$. Full article

Let $(\mathcal{X},d)$ be a directionally $(\gamma ,m)$-limited space with every $\gamma \in (0,\infty )$. In this setting, we aim to study an analogue of the classical theory of $A_p\left(\mu \right)$ weights. As an application, we establish some weighted estimates for the Hardy–Littlewood maximal operator. Then, we introduce the relationship between directionally $(\gamma ,m)$-limited spaceand geometric doubling. Finally, we obtain the weighted norm inequalities of the Calderón–Zygmund operator and commutator in non-homogeneous space. Full article

In this paper, we establish the boundedness of the Calderón operator on local Morrey spaces with variable exponents. We obtain our result by extending the extrapolation theory of Rubio de Francia to the local Morrey spaces with variable exponents. The exponent functions of the local Morrey spaces with the exponent functions are only required to satisfy the log-Hölder continuity assumption at the origin and infinity only. As special cases of the main result, we have Hardy’s inequalities, the Hilbert inequalities and the boundedness of the Riemann–Liouville and Weyl averaging operators on local Morrey spaces with variable exponents. Full article

Let $\overrightarrow{p}\in {(0,\infty )}^n$ be an exponent vector and A be a general expansive matrix on ${\mathbb{R}}^n$. Let $H_A^{\overrightarrow{p}}\left({\mathbb{R}}^n\right)$ be the anisotropic mixed-norm Hardy spaces associated with A defined via the non-tangential grand maximal function. In this article, using the known atomic characterization of $H_A^{\overrightarrow{p}}\left({\mathbb{R}}^n\right)$, the authors characterize this Hardy space via molecules with the best possible known decay. As an application, the authors establish a criterion on the boundedness of linear operators from $H_A^{\overrightarrow{p}}\left({\mathbb{R}}^n\right)$ to itself, which is used to explore the boundedness of anisotropic Calderón–Zygmund operators on $H_A^{\overrightarrow{p}}\left({\mathbb{R}}^n\right)$. In addition, the boundedness of anisotropic Calderón–Zygmund operators from $H_A^{\overrightarrow{p}}\left({\mathbb{R}}^n\right)$ to the mixed-norm Lebesgue space $L^{\overrightarrow{p}}\left({\mathbb{R}}^n\right)$ is also presented. The obtained boundedness of these operators positively answers a question mentioned by Cleanthous et al. All of these results are new, even for isotropic mixed-norm Hardy spaces on ${\mathbb{R}}^n$. Full article

We establish one-sided weighted endpoint estimates for the $\mathsf{\varrho }$ -variation ( $\mathsf{\varrho }>2$ ) operators of one-sided singular integrals under certain priori assumption by applying one-sided Calderón–Zygmund argument. Using one-sided sharp maximal estimates, we further prove that the $\mathsf{\varrho }$ -variation operators of related commutators are bounded on one-sided weighted Lebesgue and Morrey spaces. In addition, we also show that these operators are bounded from one-sided weighted Morrey spaces to one-sided weighted Campanato spaces. As applications, we obtain some results for the $\lambda$ -jump operators and the numbers of up-crossings. Our main results represent one-sided extensions of many previously known ones. Full article