Mixed Norm Estimates Research Articles

Maximal regularity of parabolic equations associated with a discrete Laplacian

Let Δd be the discrete Laplacian defined on Zd by settingΔdf(n→)=∑j=1d−[f(n→+e→j)+f(n→−e→j)−2f(n→)],n→∈Zd, where {e→j:j=1,…,d} is the standard basis for Rd.In this paper, we prove weighted mixed norm estimates and end-point estimates for the maximal regularity of the discrete parabolic equation{ut+Δdu=f,t∈[0,T)u(0,⋅)=0, where T∈(0,∞).

Localized estimation of electromagnetic sources underlying event-related fields using recurrent neural networks.

To use a recurrent neural network (RNN) to reconstruct neural activity responsible for generating noninvasively measured electromagnetic signals.
Approach: Output weights of an RNN were fixed as the lead field matrix from volumetric source space computed using the boundary element method with co-registered structural magnetic resonance images and magnetoencephalography (MEG). Initially, the network was trained to minimize mean-squared-error loss between its outputs and MEG signals, causing activations in the penultimate layer to converge towards putative neural source activations. Subsequently, L1 regularization was applied to the final hidden layer, and the model was fine-tuned, causing it to favour more focused activations. Estimated source signals were then obtained from the outputs of the last hidden layer. We developed and validated this approach with simulations before applying it to real MEG data, comparing performance with beamformers, minimum-norm estimate, and mixed-norm estimate source reconstruction methods.
Main results: The proposed RNN method had higher output signal-to-noise ratios and comparable correlation and error between estimated and simulated sources. Reconstructed MEG signals were also equal or superior to the other methods regarding their similarity to ground-truth. When applied to MEG data recorded during an auditory roving oddball experiment, source signals estimated with the RNN were generally biophysically plausible and consistent with expectations from the literature.
Significance: This work builds on recent developments of RNNs for modelling event-related neural responses by incorporating biophysical constraints from the forward model, thus taking a significant step towards greater biological realism and introducing the possibility of exploring how input manipulations may influence localized neural activity.

A Mixed-Norm Estimate of the Two-Particle Reduced Density Matrix of Many-Body Schrödinger Dynamics for Deriving the Vlasov Equation

We re-examine the combined semi-classical and mean-field limit in the N-body fermionic Schrödinger equation with pure state initial data using the Husimi measure framework. The Husimi measure equation involves three residue types: kinetic, semiclassical, and mean-field. The main result of this paper is to provide better estimates for the kinetic and mean-field residue than those in Chen et al. (J Stat Phys 182(2):1–41, http://arxiv.org/abs/1910.09892v4, 2021). Especially, the estimate for the mean-field residue is shown to be smaller than the semiclassical residue by a mixed-norm estimate of the two-particle reduced density matrix factorization. Our analysis also updates the oscillation estimate parts in the residual term estimates appeared in Chen et al. (J Stat Phys 182(2):1–41, http://arxiv.org/abs/1910.09892v4, 2021).

Sobolev estimates for fractional parabolic equations with space-time non-local operators

We obtain $$L_p$$ estimates for fractional parabolic equations with space-time non-local operators $$\begin{aligned} \partial _t^\alpha u - Lu + \lambda u= f \quad \text {in } (0,T) \times {\mathbb {R}}^d , \end{aligned}$$ where $$\partial _t^\alpha u$$ is the Caputo fractional derivative of order $$\alpha \in (0,1]$$ , $$T\in (0,\infty )$$ , and $$\begin{aligned} Lu(t,x) {:}{=} \int _{{\mathbb {R}}^d} \bigg ( u(t,x+y)-u(t,x) - y\cdot \nabla _xu(t,x)\chi ^{(\sigma )}(y)\bigg )K(t,x,y)\,dy \end{aligned}$$ is an integro-differential operator in the spatial variables. Here we do not impose any regularity assumption on the kernel K with respect to t and y. We also derive a weighted mixed-norm estimate for the equations with operators that are local in time, i.e., $$\alpha = 1$$ , which extend the previous results in Mikulevičius and Pragarauskas (J Differ Equ 256(4):1581–1626, 2014) and Zhang (Annales l’IHPAnalyse Nonlinéaire 30:573–614, 2013) by using a quite different method.

The Kato-Ponce inequality with polynomial weights

We consider various versions of fractional Leibniz rules (also known as Kato-Ponce inequalities) with polynomial weights \(\langle x\rangle ^a = (1+|x|^2)^{a/2}\) for \(a\ge 0\). We show that the weighted Kato-Ponce estimate with the inhomogeneous Bessel potential \(J^s = (1- \Delta )^{{s}/{2}}\) holds for the full range of bilinear Lebesgue exponents, for all polynomial weights, and for the sharp range of the degree s. This result, in particular, demonstrates that neither the classical Muckenhoupt weight condition nor the more general multilinear weight condition is required for the weighted Kato-Ponce inequality. We also consider a few other variants such as commutator and mixed norm estimates, and analogous conclusions are derived. Our results contain strong-type inequalities for both \(L^1\) and \(L^\infty \) endpoints, which extend several existing results.

Weighted mixed norm estimates for fractional wave equations with VMO coefficients

This paper is a comprehensive study of Lp estimates for time fractional wave equations of order α∈(1,2) in the whole space, a half space, or a cylindrical domain. We obtain weighted mixed-norm estimates and solvability of the equations in both non-divergence form and divergence form when the leading coefficients have small mean oscillation in small cylinders.

Multiple vector-valued, mixed norm estimates for Littlewood-Paley square functions

We prove that for any $L^Q$-valued Schwartz function $f$ defined on $\mathbb{R}^d$, one has the multiple vector-valued, mixed norm estimate $$ \| f \|_{L^P(L^Q)} \lesssim \| S f \|_{L^P(L^Q)} $$ valid for every $d$-tuple $P$ and every $n$-tuple $Q$ satisfying $0 < P, Q < \infty$ componentwise. Here $S:= S_{d_1}\otimes ... \otimes S_{d_N}$ is a tensor product of several Littlewood-Paley square functions $S_{d_j}$ defined on arbitrary Euclidean spaces $\mathbb{R}^{d_j}$ for $1\leq j\leq N$, with the property that $d_1 + ... + d_N = d$. This answers a question that came up implicitly in our recent works and completes in a natural way classical results of the Littlewood-Paley theory. The proof is based on the \emph{helicoidal method} introduced by the authors.

Boundary Lebesgue mixed-norm estimates for non-stationary Stokes systems with VMO coefficients

We consider Stokes systems with measurable coefficients and Lions-type boundary conditions. We show that, in contrast to the Dirichlet boundary conditions, local boundary mixed-norm -estimates hold for the spatial second-order derivatives of solutions, assuming the smallness of the mean oscillations of the coefficients with respect to the spatial variables in small cylinders. In the un-mixed norm case with the result is still new and provides local boundary Caccioppoli-type estimates. The main challenges in the work arise from the lack of regularity of the pressure and time derivatives of the solutions and from interaction of the boundary with the nonlocal structure of the system. To overcome these difficulties, our approach relies heavily on several newly developed regularity estimates for both divergence and non-divergence form parabolic equations with coefficients that are only measurable in the time variable and in one of the spatial variables.

Bi-parameter trilinear Fourier multipliers and pseudo-differential operators with flag symbols

The main purpose of this paper is to study \(L^r\) Hölder type estimates for a bi-parameter trilinear Fourier multiplier with flag singularity, and the analogous pseudo-differential operator, when the symbols are in a certain product form. More precisely, for \(f,g,h\in {\mathcal {S}}({\mathbb {R}}^{2})\), the bi-parameter trilinear flag Fourier multiplier operators we consider are defined by $$\begin{aligned} T_{m_1,m_2}(f,g,h)(x):=\int _{{\mathbb {R}}^{6}}m_1(\xi ,\eta ,\zeta )m_2(\eta ,\zeta ){\hat{f}}(\xi ) {\hat{g}}(\eta ){\hat{h}}(\zeta )e^{2\pi i(\xi +\eta +\zeta )\cdot x}d\xi d\eta d\zeta , \end{aligned}$$when \(m_1,m_2\) are two bi-parameter symbols. We study Hölder type estimates: \(L^{p_1}\times L^{p_2}\times L^{p_3} \rightarrow L^r\) for \(1<p_1,p_2,p_3< \infty \) with \(1/p_1+1/p_2+1/p_3=1/r\), and \(0<r<\infty \). We will show that our problem can be reduced to establish the \(L^r\) estimate for the special multiplier \(m_1(\xi _1, \eta _1, \zeta _1) m_2(\eta _2, \zeta _2)\) (see Theorem 1.7). We also study these \(L^r\) estimates for the corresponding bi-parameter trilinear pseudo-differential operators defined by $$\begin{aligned} T_{ab}(f,g,h)(x):=\int _{{\mathbb {R}}^6}a(x,\xi ,\eta ,\zeta )b(x,\eta ,\zeta ){\hat{f}}(\xi ){\hat{g}}(\eta ){\hat{h}}(\zeta )e^{2\pi i x(\xi +\eta +\zeta )}d\xi d\eta d\zeta , \end{aligned}$$where the smooth symbols a, b satisfy certain bi-parameter Hörmander conditions. We will also show that the \(L^r\) estimate holds for \(T_{ab}\) as long as the \(L^r\) estimate for the flag multiplier operator holds when the multiplier has the special form \(m_1(\xi _1, \eta _1, \zeta _1) m_2(\eta _2, \zeta _2)\) (see Theorem 1.10). Using our reduction of the flag multiplier, we also provide an alternative proof of some of the mixed norm estimates recently established by Muscalu and Zhai (Five-linear singular integrals of Brascamp-Lieb type, arXiv:2001.09064m) when the functions g and h are of tensor product forms (Theorem 1.8). Moreover, our method also allows us to establish the weighted mixed norm estimates (Theorem 1.9). The bi-parameter and trilinear flag Fourier multipliers considered in this paper do not satisfy the conditions of the classical bi-parameter trilinear Fourier multipliers considered by Muscalu, Tao, Thiele and the second author (Muscalu et al. in Acta Math 193:269–296, 2004; Rev Mat Iberoam 22(3):963–976, 2006). They may also be viewed as the bi-parameter trilinear variants of estimates obtained for the one-parameter flag paraproducts by Muscalu (Rev Mat Iberoam 23(2):705–742, 2007).

Embeddings of function spaces via the Caffarelli–Silvestre extension, capacities and Wolff potentials

Let Pαf(x,t) be the Caffarelli–Silvestre extension of a smooth function f(x):Rn→R+n+1≔Rn×(0,∞). The purpose of this article is twofold. Firstly, we want to characterize a nonnegative measure μ on R+n+1 such that f(x)→Pαf(x,t) induces bounded embeddings from the Lebesgue spaces Lp(Rn) to the Lq(R+n+1,μ). On one hand, these embeddings will be characterized by using a newly introduced Lp−capacity associated with the Caffarelli–Silvestre extension. In doing so, the mixed norm estimates of Pαf(x,t), the dual form of the Lp−capacity, the Lp−capacity of general balls, and a capacitary strong type inequality will be established, respectively. On the other hand, when p>q>1, these embeddings will also be characterized in terms of the Hedberg–Wolff potential of μ. Secondly, we characterize a nonnegative measure μ on R+n+1 such that f(x)→Pαf(x,t) induces bounded embeddings from the homogeneous Sobolev spaces Ẇβ,p(Rn) to the Lq(R+n+1,μ) in terms of the fractional perimeter of open sets for endpoint cases and the fractional capacity for general cases.

Genuinely multilinear weighted estimates for singular integrals in product spaces

We prove genuinely multilinear weighted estimates for singular integrals in product spaces. The estimates complete the qualitative weighted theory in this setting. Such estimates were previously known only in the one-parameter situation. Extrapolation gives powerful applications – for example, a free access to mixed-norm estimates in the full range of exponents.

Rubio de Francia extrapolation theorem and related topics in the theory of elliptic and parabolic equations. A survey

This is a brief historical overview of the Sobolev mixed norm theory of linear elliptic and parabolic equations and the recent development in this theory based on the Rubio de Francia extrapolation theorem. A self-contained proof of this theorem along with other relevant tools of Real Analysis are also presented as well as an application to mixed norm estimates for fully nonlinear equations.

An approach for weighted mixed-norm estimates for parabolic equations with local and non-local time derivatives

We give a unified approach to weighted mixed-norm estimates and solvability for both the usual and time fractional parabolic equations in nondivergence form when coefficients are merely measurable in the time variable. In the spatial variables, the leading coefficients locally have small mean oscillations. Our results extend the previous result in [5] for unmixed Lp-estimates without weights.

End-point estimates, extrapolation for multilinear Muckenhoupt classes, and applications

In this paper we present the results announced in the recent work by the first, second, and fourth authors of the current paper concerning Rubio de Francia extrapolation for the so-called multilinear Muckenhoupt classes. Here we consider the situations where some of the exponents of the Lebesgue spaces appearing in the hypotheses and/or in the conclusion can be possibly infinity. The scheme we follow is similar, but, in doing so, we need to develop a one-variable end-point off-diagonal extrapolation result. This complements the corresponding “finite” case obtained by Duoandikoetxea, which was one of the main tools in the aforementioned paper. The second goal of this paper is to present some applications. For example, we obtain the full range of mixed-norm estimates for tensor products of bilinear Calderón-Zygmund operators with a proof based on extrapolation and on some estimates with weights in some mixed-norm classes. The same occurs with the multilinear Calderón-Zygmund operators, the bilinear Hilbert transform, and the corresponding commutators with BMO functions. Extrapolation along with the already established weighted norm inequalities easily give scalar and vector-valued inequalities with multilinear weights and these include the end-point cases.

Parabolic equations involving Laguerre operators and weighted mixed-norm estimates

<p style='text-indent:20px;'>In this paper, we study evolution equation <inline-formula><tex-math id="M1">\begin{document}$ \partial_t u = -L_\alpha u+f $\end{document}</tex-math></inline-formula> and the corresponding Cauchy problem, where <inline-formula><tex-math id="M2">\begin{document}$ L_\alpha $\end{document}</tex-math></inline-formula> represents the Laguerre operator <inline-formula><tex-math id="M3">\begin{document}$ L_\alpha = \frac 12(-\frac{d^2}{dx^2}+x^2+\frac 1{x^2}(\alpha^2-\frac 14)) $\end{document}</tex-math></inline-formula>, for every <inline-formula><tex-math id="M4">\begin{document}$ \alpha\geq-\frac 12 $\end{document}</tex-math></inline-formula>. We get explicit pointwise formulas for the classical solution and its derivatives by virtue of the parabolic heat-diffusion semigroup <inline-formula><tex-math id="M5">\begin{document}$ \{ e^{-\tau(\partial_t+L_\alpha)}\}_{\tau&gt;0} $\end{document}</tex-math></inline-formula>. In addition, we define the Poisson operator related to the fractional power <inline-formula><tex-math id="M6">\begin{document}$ (\partial_t+L_\alpha)^s $\end{document}</tex-math></inline-formula> and reveal weighted mixed-norm estimates for revelent maximal operators.

Bilinear Calderón–Zygmund theory on product spaces

We develop a wide general theory of bilinear bi-parameter singular integrals T. This includes general Calderón–Zygmund type principles in the bilinear bi-parameter setting: easier bounds, like estimates in the Banach range, imply boundedness in the full bilinear range Lp×Lq→Lr, 1/p+1/q=1/r, 1<p,q≤∞, 1/2<r<∞, weighted estimates, mixed-norm estimates, and so on. We build this Calderón–Zygmund theory using the very useful perspective of dyadic representation theorems that hold under testing conditions. New weighted estimates are developed and used effectively throughout. We also develop commutator decompositions and use them to show estimates in the full range for commutators and iterated commutators, like [b1,T]1 and [b2,[b1,T]1]2, where b1 and b2 are little BMO functions. Our commutator method contains several novelties – one is that the new method can be used to simplify and improve linear commutator proofs, even in the two-weight Bloom setting. We also quickly show commutator lower bounds by using and developing the recent median method.

Lebesgue mixed norm estimates for Bergman projectors: from tube domains over homogeneous cones to homogeneous Siegel domains of type II

In this paper, the weight functions are fundamental compound functions of the homogeneous cone. First, we use the recent Bourgain-Demeter $$l^2$$ -decoupling theorem to obtain the optimal weighted Lebesgue mixed norm estimates for weighted Bergman projectors on tube domains over Lorentz cones. This settles a conjecture of D. Debertol. Secondly, we present a transference principle of weighted Lebesgue mixed norm estimates for weighted Bergman projectors from tube domains over homogeneous cones to homogeneous Siegel domains of type II associated to the same cones. So results of C. Nana for homogeneous Siegel domains of type II can be deduced from earlier results of C. Nana and B. Trojan for tube domains over homogeneous cones. Combining our two theorems, we improve these estimates for homogeneous Siegel domains of type II associated with Lorentz cones, e.g. the Pyateckii-Shapiro Siegel domain of type II.

Recent trends and advances in solving the inverse problem for EEG source localization

ABSTRACTThis paper addresses the recent advancements and trends in the field of electroencephalography (EEG) using inverse problem solutions. Using the EEG data of the brain to gather the information regarding the neuronal current source distribution has been a persisting challenge. Since the EEG inverse problem is ill-posed in nature; therefore, it does not offer a unique result. A trivial and precise solution yields a detailed insight regarding the electrical activity as well as the damaged tissue in the brain. Ordinarily, this problem is solved using the regularization techniques, such as minimum norm estimates, mixed-norm estimate, low-resolution electrical tomography, artificial neural networks, and their modified variants. In this paper, the latest algorithmic developments in solving the EEG inverse problem are reviewed. The optimization rendered by these techniques in accurately solving the neural source localization problem is also discussed. The comparative performance analysis of the recent techniques has been presented. Furthermore, a number of future enhancements have also been proposed to further improve the performance of these state-of-the-art techniques.

Vector-valued extensions for fractional integrals of Laguerre expansions

We prove some vector-valued inequalities for fractional integrals defined for several orthonormal systems of Laguerre functions. On the one hand, we obtain weighted Lp-Lq vector-valued extensions, in a multidimensional setting, for negative powers of the operator related to so-called Laguerre expansions of Hermite type. On the other hand, we give necessary and sufficient conditions for vector-valued Lp-Lq estimates related to negative powers of the Laguerre operator associated to expansions of convolution type, in a one-dimensional setting. Both types of vector-valued inequalities are based on estimates of the kernel with precise control of the parameters involved. As an application, mixed norm estimates for fractional integrals related to the harmonic oscillator are deduced. © Instytut Matematyczny PAN, 2018.

Smoothing Estimates for the Kinetic Transport Equation at the Critical Regularity

We prove smoothing estimates for velocity averages of the kinetic transport equation in hyperbolic Sobolev spaces at the critical regularity, leading to a complete characterisation of the allowable regularity exponents. Such estimates will be deduced from some mixed-norm estimates for the cone multiplier operator at a certain critical index. Our argument is not particular to the geometry of the cone and we illustrate this by establishing analogous estimates for the paraboloid.