Littlewood-Paley characterizations of anisotropic Hardy-Lorentz spaces

Let p ∈(0, 1], q ∈(0, ∞] and A be a general expansive matrix on R~n. Let H_A~(p,q )(R~n) be the anisotropic Hardy-Lorentz spaces associated with A defined via the nontangential grand maximal function. In this article, the authors characterize H_A~(p,q )(R~n) in terms of the Lusin-area function, the Littlewood-Paley g-function or the Littlewood-Paley g~*_λ-function via first establishing an anisotropic Fefferman-Stein vector-valued inequality in the Lorentz space L_(p,q)(R~n). All these characterizations are new even for the classical isotropic Hardy-Lorentz spaces on R~n. Moreover, the range of λ in the g~*_λ-function characterization of H_A~(p,q )(R~n) coincides with the best known one in the classical Hardy space H~p(R~n) or in the anisotropic Hardy space H_A~p (R~n).     

Nonlinear fractional magnetic Schrödinger equation: Existence and multiplicity

In this paper we focus our attention on the following nonlinear fractional Schrödinger equation with magnetic field

${\epsilon }^{2s}{(?\mathrm{\Delta })}_{A/\epsilon }^{s}u+V(x)u=f(|u{|}^2)u\text{ in }{\mathbb{R}}^N,$

where $\epsilon >0$ is a parameter, $s\in (0,1)$, $N\ge 3$, ${(?\mathrm{\Delta })}_A^s$ is the fractional magnetic Laplacian, $V:{\mathbb{R}}^N\to \mathbb{R}$ and $A:{\mathbb{R}}^N\to {\mathbb{R}}^N$ are continuous potentials and $f:{\mathbb{R}}^N\to \mathbb{R}$ is a subcritical nonlinearity. By applying variational methods and Ljusternick–Schnirelmann theory, we prove existence and multiplicity of solutions for ε small.     

On the gentle properties of anisotropic Besov spaces

In this paper, we prove that anisotropic homogeneous Besov spaces ${\stackrel{?}{B}}_{p,q}^{s,u}\left({\mathbb{R}}^d\right)$ are gentle spaces, for all parameters $s,p,q$ and all anisotropies $u$. Using the Littlewood–Paley decomposition, we study their completeness, separability, duality and homogeneity. We then define the notion of anisotropic orthonormal wavelet basis of $L^2\left({\mathbb{R}}^d\right)$, and we show that the homogeneous version of Triebel families of anisotropic orthonormal wavelet bases associated to the tensor product of Lemarié–Meyer (resp. Daubechies) wavelets are particular examples. We characterize the ${\stackrel{?}{B}}_{p,q}^{s,u}\left({\mathbb{R}}^d\right)$ spaces using Lemarié–Meyer wavelets. In fact, we show that these bases will be either unconditional bases or unconditional $1$-weak bases of ${\stackrel{?}{B}}_{p,q}^{s,u}\left({\mathbb{R}}^d\right)$, depending on whether ${\stackrel{?}{B}}_{p,q}^{s,u}\left({\mathbb{R}}^d\right)$ is separable or not. By introducing an anisotropic version of the class of almost diagonal matrices related to anisotropic orthonormal wavelet bases, we prove that these spaces are stable under changes of anisotropic orthonormal wavelet bases. As a consequence, we extend the characterization of ${\stackrel{?}{B}}_{p,q}^{s,u}\left({\mathbb{R}}^d\right)$ using Daubechies wavelets.     

Weighted pseudo-almost periodic solutions of a class of semilinear fractional differential equations

We study the existence and uniqueness of a weighted pseudo-almost periodic (mild) solution to the semilinear fractional equation ${?}_t^{\alpha }u=Au+{?}_t^{\alpha ?1}f(?,u)$, $1<\alpha <2$, where $A$ is a linear operator of sectorial negative type. This article also deals with the existence of these types of solutions to abstract partial evolution equations.     

On a p-Laplace equation with multiple critical nonlinearities

Using the Mountain-Pass Theorem of Ambrosetti and Rabinowitz we prove that $?{\mathrm{\Delta }}_{p}u?\mu {|x|}^{?p}{u}^{p?1}={|x|}^{?s}{u}^{p^{?}(s)?1}+u^{p^{?}?1}$ admits a positive weak solution in ${\mathbb{R}}^n$ of class $D_1^p({\mathbb{R}}^n)\cap C^1({\mathbb{R}}^n?\{0\})$, whenever $\mu <{\mu }_1$, and ${\mu }_1={[(n?p)/p]}^p$. The technique is based on the existence of extremals of some Hardy–Sobolev type embeddings of independent interest. We also show that if $u\in D_1^p({\mathbb{R}}^n)$ is a weak solution in ${\mathbb{R}}^n$ of $?{\mathrm{\Delta }}_{p}u?\mu {|x|}^{?p}{|u|}^{p?2}u={|x|}^{?s}{|u|}^{p^{?}(s)?2}u+{|u|}^{q?2}u$, then $u\equiv 0$ when either $1<q<p^{?}$, or $q>p^{?}$ and u is also of class $L_{\mathrm{loc}}^{\infty }({\mathbb{R}}^n?\{0\})$.     

Quasi-linear PDEs and forward–backward stochastic differential equations: Weak solutions

In this paper, we study the existence, uniqueness and the probabilistic representation of the weak solutions of quasi-linear parabolic and elliptic partial differential equations (PDEs) in the Sobolev space $H_{\rho }^1({\mathbb{R}}^d)$. For this, we study first the solutions of forward–backward stochastic differential equations (FBSDEs) with smooth coefficients, regularity of solutions and their connection with classical solutions of quasi-linear parabolic PDEs. Then using the approximation procedure, we establish their convergence in the Sobolev space to the solutions of the FBSDES in the space $L_{\rho }^2({\mathbb{R}}^d;{\mathbb{R}}^d)?L_{\rho }^2({\mathbb{R}}^d;{\mathbb{R}}^k)?L_{\rho }^2({\mathbb{R}}^d;{\mathbb{R}}^{k\times d})$. This gives a connection with the weak solutions of quasi-linear parabolic PDEs. Finally, we study the unique weak solutions of quasi-linear elliptic PDEs using the solutions of the FBSDEs on infinite horizon.     

Notes on the convergence order of gradient schemes for time fractional differential equations

We apply the GDM (Gradient Discretization Method), developed recently, as discretization in space to time-fractional diffusion and diffusion-wave equations with a fractional derivative of Caputo type in any space dimension.In the case of time-fractional diffusion equations, we establish an implicit scheme, and we prove an $L^{\infty }(L^2)$-error estimate. A similar result in a discrete $L^{\infty }(H_0^1)$–norm is also stated.To construct the numerical scheme for the time-fractional diffusion-wave equation, we write the equation in the form of a system of two low-order equations. We state an a prior estimate result that helps us to derive error estimates in discrete semi-norms of $L^{\infty }(H^1)$ and $H^1(L^2)$. The convergence is unconditional. Another gradient scheme is also suggested. We state its convergence results, which improve the convergence order proved recently for a SUSHI scheme.These results hold then for all the schemes within the framework of GDM: conforming and nonconforming finite element, mixed finite element, hybrid mixed mimetic family, some Multi-Point Flux approximation finite volume schemes, and some discontinuous Galerkin schemes.