We prove well-posedness results for the Dirichlet problem in $\mathbb{R}^{n}_{+}$ for homogeneous, second order, constant complex coefficient elliptic systems with boundary data in generalized Hölder spaces $\mathscr{C}^ω(\mathbb{R}^{n-1},\mathbb{C}^M)$ and in generalized Morrey-Campanato spaces $\mathscr{E}^{ω,p}(\mathbb{R}^{n-1},\mathbb{C}^M)$ under certain assumptions on the growth function $ω$. We also identify a class of growth functions $ω$ for which $\mathscr{C}^ω(\mathbb{R}^{n-1},\mathbb{C}^M)=\mathscr{E}^{ω,p}(\mathbb{R}^{n-1},\mathbb{C}^M)$ and for which the aforementioned well-posedness results are equivalent, in the sense that they have the same unique solution, satisfying natural regularity properties and estimates.
A characterization of rough fractional type integral operators and Campanato estimates for their commutators on the variable exponent vanishing generalized Morrey spaces
Interpolation inequalities between Sobolev and Morrey-Campanato spaces: A common gateway to concentration-compactness and Gagliardo-Nirenberg interpolation inequalities
Parabolic sublinear operators with rough kernel generated by parabolic Calderón-Zygmund operators and parabolic local Campanato space estimates for their commutators on the parabolic generalized local Morrey spaces
Let $p,q\in [1,\infty]$, $α\in{\mathbb{R}}$, and $s$ be a non-negative integer. In this article, the authors introduce a new function space $\widetilde{JN}_{(p,q,s)_α}(\mathcal{X})$ of John-Nirenberg-Campanato type, where $\mathcal{X}$ denotes $\mathbb{R}^n$ or any cube $Q_{0}$ of $\mathbb{R}^n$ with finite edge length. The authors give an equivalent characterization of $\widetilde{JN}_{(p,q,s)_α}(\mathcal{X})$ via both the John-Nirenberg-Campanato space and the Riesz-Morrey space. Moreover, for the particular case $s=0$, this new space can be equivalently characterized by both maximal functio…
Let $X$ be a ball quasi-Banach function space on ${\mathbb R}^n$ satisfying some mild assumptions. In this article, the authors first find a reasonable version $\widetilde{T}$ of the Calderón--Zygmund operator $T$ on the ball Campanato-type function space $\mathcal{L}_{X,q,s,d}(\mathbb{R}^n)$ with $q\in[1,\infty)$, $s\in\mathbb{Z}_+^n$, and $d\in(0,\infty)$. Then the authors prove that $\widetilde{T}$ is bounded on $\mathcal{L}_{X,q,s,d}(\mathbb{R}^n)$ if and only if, for any $γ\in\mathbb{Z}^n_+$ with $|γ|\leq s$, $T^*(x^γ)=0$, which is hence sharp. Moreover, $\widetilde{T}$ is proved to be th…
Interpolation, extrapolation, Morrey spaces and local energy control for the Navier--Stokes equations
Barker最近证明了新的弱强唯一性结果的Navier-斯托克斯方程的基础上的标准，涉及Besov空间和证明通过插值之间的Besov-H{<$}lder空间和L 2。通过考虑Besov-Morrey空间和Besov-Morrey空间与L2 uloc之间的插值，我们改进了他的结果.设u 0是R3上的无发散向量场.我们将考虑满足能量估计的Navier-斯托克斯方程的Cauchy初值问题的弱解。微分纳维-斯托克斯方程读作$\partial$ t u + u。$\nabla$ u = $Δ$ u -- $\nabla$ p div u = 0 u（0，.）= u 0 *
Let $X$ be a ball quasi-Banach function space on ${\mathbb R}^n$ satisfying some mild assumptions and let $α\in(0,n)$ and $β\in(1,\infty)$. In this article, when $α\in(0,1)$, the authors first find a reasonable version $\widetilde{I}_α$ of the fractional integral $I_α$ on the ball Campanato-type function space $\mathcal{L}_{X,q,s,d}(\mathbb{R}^n)$ with $q\in[1,\infty)$, $s\in\mathbb{Z}_+^n$, and $d\in(0,\infty)$. Then the authors prove that $\widetilde{I}_α$ is bounded from $\mathcal{L}_{X^β,q,s,d}(\mathbb{R}^n)$ to $\mathcal{L}_{X,q,s,d}(\mathbb{R}^n)$ if and only if there exists a positive c…
Our aim is to characterize the homogeneous fractional Sobolev-Slobodecki\uı spaces $\mathcal{D}^{s,p} (\mathbb{R}^n)$ and their embeddings, for $s \in (0,1]$ and $p\ge 1$. They are defined as the completion of the set of smooth and compactly supported test functions with respect to the Gagliardo-Slobodecki\uı seminorms. For $s\,p < n$ or $s = p = n = 1$ we show that $\mathcal{D}^{s,p}(\mathbb{R}^n)$ is isomorphic to a suitable function space, whereas for $s\,p \ge n$ it is isomorphic to a space of equivalence classes of functions, differing by an additive constant. As one of our main tools, we…
In this paper we study the regularity of weak solutions to an elliptic-parabolic system modeling natural network formation. The system is singular and involves cubic nonlinearity. Our investigation reveals that weak solutions are Hölder continuous when the space dimension $N$ is $2$. This is achieved via an inequality associated with the Stummel-Kato class of functions and refinement of a lemma originally due to S. Campanato and C. B. Morrey (\cite{G}, p. 86).
In this paper we are interested on solvability of the problem \begin{align*} \begin{cases} -Δu=0 & \text{in} \;\;\;\mathbb{R}^{n+1}_{+}\;\;\;\;\;\;\;\;\;\\ \;\;\displaystyle{\frac{\partial u}{\partial ν}} = V(x)u+b \vert u\vert^{ρ-1}u+f \; & \text{on} \;\;\partial\mathbb{R}^{n+1}_+\;\;\;\;\;\;\;\;\,\, \end{cases} \end{align*} %Laplace equation in the upper half-space with nonlinear Neumann boundary with high singular data $f$ and potential $V$ on boundary $\partial\mathbb{R}^{n+1}_+$ of half-space $ \mathbb{R}^{n+1}_{+}=\{(x,t)\in\mathbb{R}^{n+1}\,\vert\, t>0\}$ for $n\geq 2$. More precisely, …
