In this paper, we study the boundedness of fractional multilinear integral operators with rough kernels [...] TΩ,αA1,A2,…,Ak, $T_{\Omega ,\alpha }^{A_1,A_2,...,A_k},$ which is a generalization of the higher-order commutator of the rough fractional integral on the generalized weighted Morrey spaces Mp,ϕ (w). We find the sufficient conditions on the pair (ϕ1, ϕ2) with w ∈ Ap,q which ensures the boundedness of the operators [...] TΩ,αA1,A2,…,Ak, $T_{\Omega ,\alpha }^{A_1,A_2,...,A_k},$ from [...] Mp,φ1wptoMp,φ2wq $M_{p,{\varphi }_1}\left(w^p\right)\mathrm{to}{M}_{p,{\varphi }_2}\left(w^q\right)$ for 1 < p < q < ∞. In all cases the conditions...

The following iterated commutators $T_{\ast ,\Pi b}$ of the maximal operator for multilinear singular integral operators and $I_{\alpha ,\Pi b}$ of the multilinear fractional integral operator are introduced and studied: $T_{\ast ,\Pi b}\left(f⃗\right)\left(x\right)=su{p}_{\delta >0}\left|[b₁,[b₂,\dots {[b_{m-1},[bₘ,T_{\delta }]ₘ]}_{m-1}\cdots ]₂]₁\left(f⃗\right)\left(x\right)\right|$, $I_{\alpha ,\Pi b}\left(f⃗\right)\left(x\right)=[b₁,[b₂,\dots {[b_{m-1},[bₘ,I_{\alpha }]ₘ]}_{m-1}\cdots ]₂]₁\left(f⃗\right)\left(x\right)$, where $T_{\delta }$ are the smooth truncations of the multilinear singular integral operators and $I_{\alpha }$ is the multilinear fractional integral operator, $b_i\in BMO$ for i = 1,…,m and f⃗ = (f1,…,fm). Weighted strong and L(logL) type end-point estimates for the above iterated commutators associated with two classes of multiple...

Let $\mu$ be a nonnegative Borel measure on ${ℝ}^d$ satisfying that $\mu \left(Q\right)\le l{\left(Q\right)}^n$ for every cube $Q\subset {ℝ}^n$, where $l\left(Q\right)$ is the side length of the cube $Q$ and $0<n\le d$. We study the class of pairs of weights related to the boundedness of radial maximal operators of fractional type associated to a Young function $B$ in the context of non-homogeneous spaces related to the measure $\mu$. Our results include two-weighted norm and weak type inequalities and pointwise estimates. Particularly, we give an improvement of a two-weighted result...

We introduce function spaces $B^{p,\lambda }$ with Morrey-Campanato norms, which unify $B^{p,\lambda }$, $CM{O}^{p,\lambda }$ and Morrey-Campanato spaces, and prove the boundedness of the fractional integral operator $I_{\alpha }$ on these spaces.

We introduce the generalized fractional integrals $I{̃}_{\alpha ,d}$ and prove the strong and weak boundedness of $I{̃}_{\alpha ,d}$ on the central Morrey spaces $B^{p,\lambda }\left(ℝⁿ\right)$. In order to show the boundedness, the generalized λ-central mean oscillation spaces ${\Lambda }_{p,\lambda }^{\left(d\right)}\left(ℝⁿ\right)$ and the generalized weak λ-central mean oscillation spaces $W{\Lambda }_{p,\lambda }^{\left(d\right)}\left(ℝⁿ\right)$ play an important role.

Let $A_{\alpha }f\left(x\right)={\left|B\right(0,\left|x\right|\left)\right|}^{-\alpha /n}{\int }_{B(0,|x\left|\right)}f\left(t\right)dt$ be the n-dimensional fractional Hardy operator, where 0 < α ≤ n. It is well-known that $A_{\alpha }$ is bounded from $L^p$ to $L^{p_{\alpha }}$ with $p_{\alpha }=np/(\alpha p-np+n)$ when n(1-1/p) < α ≤ n. We improve this result within the framework of Banach function spaces, for instance, weighted Lebesgue spaces and Lorentz spaces. We in fact find a ’source’ space $S_{\alpha ,Y}$, which is strictly larger than X, and a ’target’ space $T_Y$, which is strictly smaller than Y, under the assumption that $A_{\alpha }$ is bounded from X into Y and the Hardy-Littlewood...

For α ∈ (1,2) we consider the equation ${\partial }_{t}u={\Delta }^{\alpha /2}u+b·\nabla u$, where b is a time-independent, divergence-free singular vector field of the Morrey class $M{₁}^{1-\alpha }$. We show that if the Morrey norm ${\left|\right|b\left|\right|}_{M{₁}^{1-\alpha }}$ is sufficiently small, then the fundamental solution is globally in time comparable with the density of the isotropic stable process.

Our aim is to establish Sobolev type inequalities for fractional maximal functions $M_{ℍ,\nu }f$ and Riesz potentials $I_{ℍ,\alpha }f$ in weighted Morrey spaces of variable exponent on the half space $ℍ$. We also obtain Sobolev type inequalities for a $C^1$ function on $ℍ$. As an application, we obtain Sobolev type inequality for double phase functionals with variable exponents $\Phi (x,t)=t^{p\left(x\right)}+{\left(b\left(x\right)t\right)}^{q\left(x\right)}$, where $p(·)$ and $q(·)$ satisfy log-Hölder conditions, $p\left(x\right)<q\left(x\right)$ for $x\in ℍ$, and $b(·)$ is nonnegative and Hölder continuous of order $\theta \in (0,1]$.

Let d > 0 and θ ∈ (0,1]. We consider homogeneous type spaces, ${(X,\varrho ,\mu )}_{d,\theta }$, which are variants of the well known homogeneous type spaces in the sense of Coifman and Weiss. We introduce fractional integrals and derivatives, and prove that the Besov spaces $B_{pq}^s\left(X\right)$ and Triebel-Lizorkin spaces $F_{pq}^s\left(X\right)$ have the lifting properties for |s| < θ. Moreover, we give explicit representations for the inverses of these fractional integrals and derivatives. By using these representations, we prove that the fractional...

Given a two-dimensional fractional multiplicative process ${\left(F_t\right)}_{t\in [0,1]}$ determined by two Hurst exponents $H_1$ and $H_2$, we show that there is an associated uniform Hausdorff dimension result for the images of subsets of $[0,1]$ by $F$ if and only if $H_1=H_2$.

Let G = (V,E) be a graph. A function g:V → [0,1] is called a global dominating function (GDF) of G, if for every v ∈ V, $g\left(N\left[v\right]\right)={\sum }_{u\in N\left[v\right]}g\left(u\right)\ge 1$ and $g\left(\overline{N\left(v\right)}\right)={\sum }_{u\notin N\left(v\right)}g\left(u\right)\ge 1$. A GDF g of a graph G is called minimal (MGDF) if for all functions f:V → [0,1] such that f ≤ g and f(v) ≠ g(v) for at least one v ∈ V, f is not a GDF. The fractional global domination number ${\gamma }_{fg}\left(G\right)$ is defined as follows: ${\gamma }_{fg}\left(G\right)$ = min|g|:g is an MGDF of G where $\left|g\right|={\sum }_{v\in V}g\left(v\right)$. In this paper we initiate a study of this parameter.

Let $b>a>0$. We prove the following asymptotic formula $$\sum _{n⩾0}|\{x/(n+a)\}-\{x/(n+b)\}|=\frac{2}{\pi }\zeta (3/2)\sqrt{cx}+O\left(c^{2/9}{x}^{4/9}\right),$$ with $c=b-a$, uniformly for $x⩾40c^{-5}{(1+b)}^{27/2}$.

In this paper, we consider the following boundary value problem $$\left\{\begin{array}{c}{D}_{T^{-}}^{\alpha }\left(D_{0^{+}}^{\alpha }\left(D_{T^{-}}^{\alpha }\left(D_{0^{+}}^{\alpha }u\left(t\right)\right)\right)\right)=f(t,u\left(t\right)),t\in [0,T],\hfill \\ u\left(0\right)=u\left(T\right)=0\hfill \\ D_{T^{-}}^{\alpha }\left(D_{0^{+}}^{\alpha }u\left(0\right)\right)=D_{T^{-}}^{\alpha }\left(D_{0^{+}}^{\alpha }u\left(T\right)\right)=0,\hfill \end{array}\right.$$ where $0<\alpha \le 1$ and $f:[0,T]\times ℝ\to ℝ$ is a continuous function, $D_{0^{+}}^{\alpha }$, $D_{T^{-}}^{\alpha }$ are respectively the left and right fractional Riemann–Liouville derivatives and we prove the existence of at least one solution for this problem.

We consider highly accurate schemes for nonlinear time fractional Schrödinger equations (NTFSEs). While an $L1$ strategy is employed for approximating the Caputo fractional derivative in the temporal direction, compact CCD finite difference approaches are incorporated in the space. A highly effective hybrid $L1$-CCD method is implemented successfully. The accuracy of this linearized scheme is order six in space, and order $2-\gamma$ in time, where $0<\gamma <1$ is the order of the Caputo fractional derivative...

The aim of this work is to study the existence and uniqueness of solutions of the fractional integro-differential equations $\frac{d}{dt}[x\left(t\right)-L\left(x_t\right)]=A[x\left(t\right)-L\left(x_t\right)]+G\left(x_t\right)+\frac{1}{\Gamma \left(\alpha \right)}{\int }_{-\infty }^t{(t-s)}^{\alpha -1}\left({\int }_{-\infty }^{s}a(s-\xi )x\left(\xi \right)d\xi \right)ds+f\left(t\right)$, ($\alpha >0$) with the periodic condition $x\left(0\right)=x\left(2\pi \right)$, where $a\in L^1\left({ℝ}_{+}\right)$ . Our approach is based on the R-boundedness of linear operators $L^p$-multipliers and UMD-spaces.

We collect and extend results on the limit of ${\sigma }^{1-k}{(1-\sigma )}^k{\left|v\right|}_{l+\sigma ,p,\Omega }^p$ as σ → 0⁺ or σ → 1¯, where Ω is ℝⁿ or a smooth bounded domain, k ∈ 0,1, l ∈ ℕ, p ∈ [1,∞), and ${\left|·\right|}_{l+\sigma ,p,\Omega }$ is the intrinsic seminorm of order l+σ in the Sobolev space $W^{l+\sigma ,p}\left(\Omega \right)$. In general, the above limit is equal to $c{\left[v\right]}^p$, where c and [·] are, respectively, a constant and a seminorm that we explicitly provide. The particular case p = 2 for Ω = ℝⁿ is also examined and the results are then proved by using the Fourier transform.

Let $L$ be a non-negative self-adjoint operator acting on $L^2\left({ℝ}^n\right)$ satisfying a pointwise Gaussian estimate for its heat kernel. Let $w$ be an $A_r$ weight on ${ℝ}^n\times {ℝ}^n$, $1<r<\infty$. In this article we obtain a weighted atomic decomposition for the weighted Hardy space $H_{L,w}^p({ℝ}^n\times {ℝ}^n)$, $0<p\le 1$ associated to $L$. Based on the atomic decomposition, we show the dual relationship between $H_{L,w}^1({ℝ}^n\times {ℝ}^n)$ and ${\mathrm{BMO}}_{L,w}({ℝ}^n\times {ℝ}^n)$.

We study the duality theory of the weighted multi-parameter Triebel-Lizorkin spaces ${\dot{F}}_p^{\alpha ,q}(\omega ;{ℝ}^{n_1}\times {ℝ}^{n_2})$. This space has been introduced and the result $${\left({\dot{F}}_p^{\alpha ,q}(\omega ;{ℝ}^{n_1}\times {ℝ}^{n_2})\right)}^{*}={\mathrm{CMO}}_p^{-\alpha ,q^{\text{'}}}(\omega ;{ℝ}^{n_1}\times {ℝ}^{n_2})$$ for $0<p\le 1$ has been proved in Ding, Zhu (2017). In this paper, for $1<p<\infty$, $0<q<\infty$ we establish its dual space ${\dot{H}}_p^{\alpha ,q}(\omega ;{ℝ}^{n_1}\times {ℝ}^{n_2})$.

We consider a double analytic family of fractional integrals $S_z^{\gamma ,\alpha }$ along the curve ${t\mapsto \left|t\right|}^{\alpha }$, introduced for α = 2 by L. Grafakos in 1993 and defined by $\left(S_z^{\gamma ,\alpha }f\right)(x₁,x₂):=1/\Gamma (z+1/2){\int \int |u-1|}^z\psi (u-1){f(x₁-t,x₂-u|t|}^{\alpha }{\left)du\right|t|}^{\gamma }dt/t$, where ψ is a bump function on ℝ supported near the origin, $f{\in }_c^{\infty }\left(ℝ²\right)$, z,γ ∈ ℂ, Re γ ≥ 0, α ∈ ℝ, α ≥ 2. We determine the set of all (1/p,1/q,Re z) such that $S_z^{\gamma ,\alpha }$ maps $L^p\left(ℝ²\right)$ to $L^q\left(ℝ²\right)$ boundedly. Our proof is based on product-type kernel arguments. More precisely, we prove that the kernel $K_{-1+i\theta }^{i\varrho ,\alpha }$ is a product kernel on ℝ², adapted to the curve ${t\mapsto \left|t\right|}^{\alpha }$; as a consequence, we show...

Let $B_p$ be the Ariõ-Muckenhoupt weight class which controls the weighted $L^p$-norm inequalities for the Hardy operator on non-increasing functions. We replace the constant p by a function p(x) and examine the associated $L^{p\left(x\right)}$-norm inequalities of the Hardy operator.

In the present paper, we prove nonexistence results for the following nonlinear evolution equation, see works of T. Cazenave and A. Haraux (1990) and S. Zheng (2004), $$u_{tt}+f\left(x\right)u_t+{(-\Delta )}^{\alpha /2}\left(u^m\right)=h(t,x){\left|u\right|}^p,$$ posed in $(0,T)\times {ℝ}^N,$ where ${(-\Delta )}^{\alpha /2},0<\alpha \le 2$ is $\alpha /2$-fractional power of $-\Delta .$ Our method of proof is based on suitable choices of the test functions in the weak formulation of the sought solutions. Then, we extend this result to the case of a $2\times 2$ system of the same type.

We consider the problem of determining the unknown source term $f=f(x,t)$ in a space fractional diffusion equation from the measured data at the final time $u(x,T)=\psi \left(x\right)$. In this way, a methodology involving minimization of the cost functional $J\left(f\right)={\int }_0^l{(u(x,t;f){|}_{t=T}-\psi \left(x\right))}^2\mathrm{d}x$ is applied and shown that this cost functional is Fréchet differentiable and its derivative can be formulated via the solution of an adjoint problem. In addition, Lipschitz continuity of the gradient is proved. These results help us to prove the monotonicity and...