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 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...

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.

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...

Given a compact manifold $N^n\subset {ℝ}^{\nu }$ and real numbers $s\ge 1$ and $1\le p\&lt;\infty$, we prove that the class $C^{\infty }({\overline{Q}}^m;N^n)$ of smooth maps on the cube with values into $N^n$ is strongly dense in the fractional Sobolev space $W^{s,p}(Q^m;N^n)$ when $N^n$ is $\lfloor sp\rfloor$ simply connected. For $sp$ integer, we prove weak sequential density of $C^{\infty }({\overline{Q}}^m;N^n)$ when $N^n$ is $sp-1$ simply connected. The proofs are based on the existence of a retraction of ${ℝ}^{\nu }$ onto $N^n$ except for a small subset of $N^n$ and on a pointwise estimate of fractional derivatives of composition of maps in $W^{s,p}\cap W^{1,sp}$.

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...

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 $A_{\alpha }f\left(x\right)={\left|B\right(0,\left|x\right|\left)\right|}^{-\alpha /n}{\int }_{B(0,|x\left|\right)}f\left(t\right)dt$ 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...

Let n be a nonnegative integer and let u ∈ (n,n+1]. We say that f is u-times Peano bounded in the approximate (resp. $L^p$, 1 ≤ p ≤ ∞) sense at $x\in {ℝ}^m$ if there are numbers $f_{\alpha }\left(x\right)$, |α| ≤ n, such that $f(x+h)-{\sum }_{\left|\alpha \right|\le n}{f}_{\alpha }\left(x\right)h^{\alpha }/\alpha !$ is $O\left(h^u\right)$ in the approximate (resp. $L^p$) sense as h → 0. Suppose f is u-times Peano bounded in either the approximate or $L^p$ sense at each point of a bounded measurable set E. Then for every ε > 0 there is a perfect set Π ⊂ E and a smooth function g such that the Lebesgue measure of E∖Π is less than ε and...

We give some rather weak sufficient condition for $L^p$ boundedness of the Marcinkiewicz integral operator ${\mu }_{\Omega }$ on the product spaces $ℝⁿ\times {ℝ}^m$ (1 < p < ∞), which improves and extends some known results.

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 $\theta \in (0,1]$.

We show in two dimensions that if $Kf={\int }_{ℝ₊²}k(x,y)f\left(y\right)dy$, $k(x,y)=\left(e^{i{x}^a·y^b}\right){/\left(\right|x-y|}^{\eta })$, p = 4/(2+η), a ≥ b ≥ 1̅ = (1,1), $v_p\left(y\right)=y^{(p/p^{\text{'}})(1̅-b/a)}$, then ${\left|\right|Kf\left|\right|}_p{\le C\left|\right|f\left|\right|}_{p,v_p}$ if η + α₁ + α₂ < 2, ${\alpha }_j=1-b_j/a_j$, j = 1,2. Our methods apply in all dimensions and also for more general kernels.

We use the genus theory to prove the existence and multiplicity of solutions for the fractional $p$-Kirchhoff problem $$\left\{\begin{array}{c}{\displaystyle -{\left[M\left({\int }_Q{\displaystyle \frac{{|u\left(x\right)-u\left(y\right)|}^p}{{|x-y|}^{N+ps}}}\mathrm{d}x\mathrm{d}y\right)\right]}^{p-1}{(-\Delta )}_p^{s}u=\lambda h(x,u)\text{in}\Omega ,}\hfill \\ u=0\text{on}{ℝ}^N\setminus \Omega ,\hfill \end{array}\right.$$ where $\Omega$ is an open bounded smooth domain of ${ℝ}^N$, $p>1$, $N>ps$ with $s\in (0,1)$ fixed, $Q={ℝ}^{2N}\setminus (C\Omega \times C\Omega )$, $\lambda >0$ is a numerical parameter, $M$ and $h$ are continuous functions.

We prove the $L^p$ boundedness of the Marcinkiewicz integral operators ${\mu }_{\Omega }$ on ${ℝ}^{n₁}\times \cdots \times {ℝ}^{n_k}$ under the condition that $\Omega \in L{\left(logL\right)}^{k/2}{(}^{n₁-1}\times \cdots {\times }^{n_k-1})$. The exponent k/2 is the best possible. This answers an open question posed by Y. Ding.

Let ℳ be a hyperfinite finite von Nemann algebra and ${\left({ℳ}_k\right)}_{k\ge 1}$ be an increasing filtration of finite-dimensional von Neumann subalgebras of ℳ. We investigate abstract fractional integrals associated to the filtration ${\left({ℳ}_k\right)}_{k\ge 1}$. For a finite noncommutative martingale $x={\left(x_k\right)}_{1\le k\le n}\subseteq L₁\left(ℳ\right)$ adapted to ${\left({ℳ}_k\right)}_{k\ge 1}$ and 0 < α < 1, the fractional integral of x of order α is defined by setting $I^{\alpha }x={\sum }_{k=1}^n{\zeta }_k^{\alpha }d{x}_k$ for an appropriate sequence ${\left({\zeta }_k\right)}_{k\ge 1}$ of scalars. For the case of a noncommutative dyadic martingale in L₁() where is the type II₁ hyperfinite factor...

Let $W$ be the subspace of ${\mathbb{N}}^{*}$ consisting of all weak $P$-points. It is not hard to see that $W$ is a pseudocompact space. In this paper we shall prove that this space has stronger pseudocompact properties. Indeed, it is shown that $W$ is a $p$-pseudocompact space for all $p\in {\mathbb{N}}^{*}$.

In this article, we obtain sufficient conditions so that every solution of neutral delay differential equation $${(y\left(t\right)-\sum _{i=1}^k{p}_i\left(t\right)y\left(r_i\left(t\right)\right))}^{\left(n\right)}+v\left(t\right)G\left(y\left(g\left(t\right)\right)\right)-u\left(t\right)H\left(y\left(h\left(t\right)\right)\right)=f\left(t\right)$$ oscillates or tends to zero as $t\to \infty$, where, $n\ge 1$ is any positive integer, $p_i$, $r_i\in C^{\left(n\right)}([0,\infty ),ℝ)$  and $p_i$ are bounded for each $i=1,2,\cdots ,k$. Further, $f\in C\left(\right[0,\infty ),ℝ)$, $g$, $h$, $v$, $u\in C\left(\right[0,\infty ),[0,\infty \left)\right)$, $G$ and $H\in C(ℝ,ℝ)$. The functional delays $r_i\left(t\right)\le t$, $g\left(t\right)\le t$ and $h\left(t\right)\le t$ and all of them approach $\infty$ as $t\to \infty$. The results hold when $u\equiv 0$ and $f\left(t\right)\equiv 0$. This article extends, generalizes and improves some recent results, and further answers some unanswered questions from the literature. ...

Let $X$ be the Tychonoff product ${\prod }_{\alpha <\tau }{X}_{\alpha }$ of $\tau$-many Tychonoff non-single point spaces $X_{\alpha }$. Let $p\in X^{*}$ be a point in the closure of some $G\subset X$ whose weak Lindelöf number is strictly less than the cofinality of $\tau$. Then we show that $\beta X\setminus \left\{p\right\}$ is not normal. Under some additional assumptions, $p$ is a butterfly-point in $\beta X$. In particular, this is true if either $X={\omega }^{\tau }$ or $X=R^{\tau }$ and $\tau$ is infinite and not countably cofinal.

Let ${}_s$ be the family of open rectangles in the plane ℝ² with a side of angle s to the x-axis. We say that a set S of directions is an R-set if there exists a function f ∈ L¹(ℝ²) such that the basis ${}_s$ differentiates the integral of f if s ∉ S, and $D{̅}_{s}f\left(x\right)=limsu{p}_{diam\left(R\right)\to 0,x\in R{\in }_s}{\left|R\right|}^{-1}{\int }_{R}f=\infty$ almost everywhere if s ∈ S. If the condition $D{̅}_{s}f\left(x\right)=\infty$ holds on a set of positive measure (instead of a.e.) we say that S is a WR-set. It is proved that S is an R-set (resp. a WR-set) if and only if it is a $G_{\delta }$ (resp. a $G_{\delta \sigma }$).

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...

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.