We extend to the multilinear setting classical inequalities of Marcinkiewicz and Zygmund on \(\ell ^r\)-valued extensions of linear operators. We show that for certain \(1 \le p, q_1, \dots , q_m, r \le \infty \), there is a constant \(C\ge 0\) such that for every bounded multilinear operator \(T:L^{q_1}(\mu _1) \times \cdots \times L^{q_m}(\mu _m) \rightarrow L^p(\nu )\) and functions \(\{f_{k_1}^1\}_{k_1=1}^{n_1} \subset L^{q_1}(\mu _1), \dots , \{f_{k_m}^m\}_{k_m=1}^{n_m} \subset L^{q_m}(\mu _m)\), the following inequality holds
$$\begin{aligned} \left\| \left( \sum _{k_1, \dots , k_m} |T(f_{k_1}^1, \dots , f_{k_m}^m)|^r\right) ^{1/r} \right\| _{L^p(\nu )} \le C \Vert T\Vert \prod _{i=1}^m \left\| \left( \sum _{k_i=1}^{n_i} |f_{k_i}^i|^r\right) ^{1/r} \right\| _{L^{q_i}(\mu _i)}. \end{aligned}$$ (1)In some cases we also calculate the best constant \(C\ge 0\) satisfying the previous inequality. We apply these results to obtain weighted vector-valued inequalities for multilinear Calderón-Zygmund operators.
相似文献We consider the problem of characterizing the bounded linear operator multipliers on \(L^{2}(\mathbb{R})\) that map Gabor frame generators to Gabor frame generators. We prove that a functional matrix \(M(t)=[f_{ij}(t)]_{m \times m}\) (where \(f_{ij}\in L^{\infty}(\mathbb{R})\)) is a multiplier for Parseval Gabor multi-frame generators with parameters \(a, b >0\) if and only if \(M(t)\) is unitary and \(M^{*}(t)M(t+\frac{1}{b})= \lambda(t)I\) for some unimodular \(a\)-periodic function \(\lambda(t)\). As a special case (\(m =1\)) this recovers the characterization of functional multipliers for Parseval Gabor frames with single function generators.
相似文献We study non reflexive Orlicz spaces \(L^\varPsi \) and their Morse subspace \(M^\varPsi \), i.e. the closure of \(L^\infty \) in \(M^\varPsi \) to determine when \((M^\varPsi ,L^\varPsi )\) can be described as having an o–O type structure with respect to an equivalent norm on \(L^\varPsi \). Examples of classes of Young functions for which the answer is affirmative are provided, but also examples are given to show that this is not possible for all non-reflexive Orlicz spaces. An equivalent expression of the distance in \(L^\varPsi \) to \(M^\varPsi \), induced by the new norm, is also provided.
相似文献We study a multilinear version of the Hörmander multiplier theorem, namely
$$ \Vert T_{\sigma}(f_{1},\dots,f_{n})\Vert_{L^{p}}\lesssim \sup_{k\in\mathbb{Z}}{\Vert \sigma(2^{k}\cdot,\dots,2^{k}\cdot)\widehat{\phi^{(n)}}\Vert_{L^{2}_{(s_{1},\dots,s_{n})}}}\Vert f_{1}\Vert_{H^{p_{1}}}\cdots\Vert f_{n}\Vert_{H^{p_{n}}}. $$We show that the estimate does not hold in the limiting case \(\min \limits {(s_{1},\dots ,s_{n})}=d/2\) or \({\sum}_{k\in J}{({s_{k}}/{d}-{1}/{p_{k}})}=-{1}/{2}\) for some \(J \subset \{1,\dots ,n\}\). This provides the necessary and sufficient condition on \((s_{1},\dots ,s_{n})\) for the boundedness of Tσ.
相似文献Bounds are obtained for the \(L^p\) norm of the torsion function \(v_{\varOmega }\), i.e. the solution of \(-\varDelta v=1,\, v\in H_0^1(\varOmega ),\) in terms of the Lebesgue measure of \(\varOmega \) and the principal eigenvalue \(\lambda _1(\varOmega )\) of the Dirichlet Laplacian acting in \(L^2(\varOmega )\). We show that these bounds are sharp for \(1\le p\le 2\).
相似文献We study integrals of the form
$$\begin{aligned} \int _{-1}^1(C_n^{(\lambda )}(x))^2(1-x)^\alpha (1+x)^\beta {{\,\mathrm{\mathrm {d}}\,}}x, \end{aligned}$$where \(C_n^{(\lambda )}\) denotes the Gegenbauer-polynomial of index \(\lambda >0\) and \(\alpha ,\beta >-1\). We give exact formulas for the integrals and their generating functions, and obtain asymptotic formulas as \(n\rightarrow \infty \).
相似文献We consider primal-dual pairs of semidefinite programs and assume that they are singular, i.e., both primal and dual are either weakly feasible or weakly infeasible. Under such circumstances, strong duality may break down and the primal and dual might have a nonzero duality gap. Nevertheless, there are arbitrary small perturbations to the problem data which would make them strongly feasible thus zeroing the duality gap. In this paper, we conduct an asymptotic analysis of the optimal value as the perturbation for regularization is driven to zero. Specifically, we fix two positive definite matrices, \(I_p\) and \(I_d\), say, (typically the identity matrices), and regularize the primal and dual problems by shifting their associated affine space by \(\eta I_p\) and \(\varepsilon I_d\), respectively, to recover interior feasibility of both problems, where \(\varepsilon \) and \(\eta \) are positive numbers. Then we analyze the behavior of the optimal value of the regularized problem when the perturbation is reduced to zero keeping the ratio between \(\eta \) and \(\varepsilon \) constant. A key feature of our analysis is that no further assumptions such as compactness or constraint qualifications are ever made. It will be shown that the optimal value of the perturbed problem converges to a value between the primal and dual optimal values of the original problems. Furthermore, the limiting optimal value changes “monotonically” from the primal optimal value to the dual optimal value as a function of \(\theta \), if we parametrize \((\varepsilon , \eta )\) as \((\varepsilon , \eta )=t(\cos \theta , \sin \theta )\) and let \(t\rightarrow 0\). Finally, the analysis leads us to the relatively surprising consequence that some representative infeasible interior-point algorithms for SDP generate sequences converging to a number between the primal and dual optimal values, even in the presence of a nonzero duality gap. Though this result is more of theoretical interest at this point, it might be of some value in the development of infeasible interior-point algorithms that can handle singular problems.
相似文献Consider the following nonparametric model: \(Y_{ni}=g(x_{ni})+ \varepsilon _{ni},1\le i\le n,\) where \(x_{ni}\in {\mathbb {A}}\) are the nonrandom design points and \({\mathbb {A}}\) is a compact set of \({\mathbb {R}}^{m}\) for some \(m\ge 1\), \(g(\cdot )\) is a real valued function defined on \({\mathbb {A}}\), and \(\varepsilon _{n1},\ldots ,\varepsilon _{nn}\) are \(\rho ^{-}\)-mixing random errors with zero mean and finite variance. We obtain the Berry–Esseen bounds of the weighted estimator of \(g(\cdot )\). The rate can achieve nearly \(O(n^{-1/4})\) when the moment condition is appropriate. Moreover, we carry out some simulations to verify the validity of our results.
相似文献This paper deals with the regularity of weak solutions to the 3D magneto-micropolar fluid equations in Besov spaces. It is shown that for \(0\le\alpha\le1\) if \(u\in L^{\frac{2}{1+\alpha}}(0,T; \dot{B}_{\infty,\infty}^{\alpha})\), then the weak solution \((u,\omega ,b)\) is regular on \((0,T]\).
相似文献This paper is devoted to the study of the \(p\)-fractional Schrödinger–Kirchhoff equations with electromagnetic fields and critical nonlinearity. By using the variational methods, we obtain the existence of mountain pass solutions \(u_{\varepsilon }\) which tend to the trivial solutions as \(\varepsilon \rightarrow 0\). Moreover, we get \(m^{\ast }\) pairs of solutions for the problem in absence of magnetic effects under some extra assumptions.
相似文献The problem of the minimax testing of the Poisson process intensity \({\mathbf{s}}\) is considered. For a given intensity \({\mathbf{p}}\) and a set \(\mathcal{Q}\), the minimax testing of the simple hypothesis \(H_{0}: {\mathbf{s}} = {\mathbf{p}}\) against the composite alternative \(H_{1}: {\mathbf{s}} = {\mathbf{q}},\,{\mathbf{q}} \in \mathcal{Q}\) is investigated. The case, when the 1-st kind error probability \(\alpha \) is fixed and we are interested in the minimal possible 2-nd kind error probability \(\beta ({\mathbf{p}},\mathcal{Q})\), is considered. What is the maximal set \(\mathcal{Q}\), which can be replaced by an intensity \({\mathbf{q}} \in \mathcal{Q}\) without any loss of testing performance? In the asymptotic case (\(T\rightarrow \infty \)) that maximal set \(\mathcal{Q}\) is described.
相似文献In previous papers we introduced a class of polynomials which follow the same recursive formula as the Lucas–Lehmer numbers, studying the distribution of their zeros and remarking that this distribution follows a sequence related to the binary Gray code. It allowed us to give an order for all the zeros of every polynomial \(L_n\). In this paper, the zeros, expressed in terms of nested radicals, are used to obtain two formulas for \(\pi \): the first can be seen as a generalization of the known formula
$$\begin{aligned} \pi =\lim _{n\rightarrow \infty } 2^{n+1}\cdot \sqrt{2-\underbrace{\sqrt{2+\sqrt{2+\sqrt{2+\cdots +\sqrt{2}}}}}_{n}}, \end{aligned}$$related to the smallest positive zero of \(L_n\); the second is an exact formula for \(\pi \) achieved thanks to some identities valid for \(L_n\).
相似文献We prove a density lower bound for some functionals involving bulk and interfacial energies. The bulk energies are convex functions with p-power growth not subjected to any further structure conditions. The interface \(\partial E\) is the boundary of a set \(E\subset \Omega \) such that \(|E|=d\) is prescribed. Then we get \(\mathcal {H}^{n-1}((\partial E{\setminus }\partial E^*)\cup \Omega )=0\).
相似文献In this article, I explore in a unified manner the structure of uniform slash and \(\alpha \)-slash distributions which, in the continuous case, are defined to be the distributions of Y / U and \( Y_\alpha /U^{1/\alpha }\) where Y and \(Y_\alpha \) follow any distribution on \(\mathbb {R}^+\) and, independently, U is uniform on (0, 1). The parallels with the monotone and \(\alpha \)-monotone distributions of \( Y \times U\) and \(Y_\alpha \times U^{1/\alpha }\), respectively, are striking. I also introduce discrete uniform slash and \(\alpha \)-slash distributions which arise from a notion of negative binomial thinning/fattening. Their specification, although apparently rather different from the continuous case, seems to be a good one because of the close way in which their properties mimic those of the continuous case.
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