In this article we prove weighted norm inequalities and pointwise estimates between the multilinear fractional integral operator and the multilinear fractional maximal. As a consequence of these estimations we obtain weighted weak and strong inequalities for the multilinear fractional integral operator. In particular, we extend some results given in \cite{CPSS} to the multilinear context. On the other hand we prove weighted pointwise estimates between the multilinear fractional maximal operator ${\cal M}_{α,B}$ associated to a Young function $B$ and the multilinear maximal operators ${\cal M}_ψ={\cal M}_{0,ψ}$, $ψ(t)=B(t^{1-α/(nm)})^{{nm}/{(nm-α)}}$. As an application of these estimate we obtain a direct proof of the $L^p-L^q$ boundedness results of ${\cal M}_{α,B}$ for the case $B(t)=t$ and $B_k(t)=t(1+\log^+t)^k$ when $1/q=1/p-α/n$. We also give sufficient conditions on the weights involved in the boundedness results of ${\cal M}_{α,B}$ that generalizes those given in \cite{M} for $B(t)=t$. Finally, we prove some boundedness results in Banach function spaces for a generalized version of the multilinear fractional maximal operator.
We obtain pointwise estimates for solutions of semilinear parabolic equations with a potential on connected domains both of $\mathbb R^n$ and of general Riemannian manifolds.
The purpose of this paper is three-fold: first, we survey on several known pointwise identities involving fractional operators; second, we propose a unified way to deal with those identities; third, we prove some new pointwise identities in different frameworks in particular geometric and infinite-dimensional ones.
For a general dyadic grid, we give a Calderón-Zygmund type decomposition, which is the principle fact about the multilinear maximal function $\mathfrak{M}$ on the upper half-spaces. Using the decomposition, we study the boundedness of $\mathfrak{M}.$ We obtain a natural extension to the multilinear setting of Muckenhoupt's weak-type characterization. We also partially obtain characterizations of Muckenhoupt's strong-type inequalities with one weight. Assuming the reverse Hölder's condition, we get a multilinear analogue of Sawyer's two weight theorem. Moreover, we also get Hytönen-Pérez type w…
We study the behavior near the origin in $\mathbb{R}^n ,n\geq3$, of nonnegative functions \begin{equation}\label{0.1} u\in C^2 (\mathbb{R}^n \backslash \{0\})\cap L^λ(\mathbb{R}^n ) \end{equation} satisfying the Choquard-Pekar type inequalities \begin{equation}\label{0.2} 0\leq-Δu\leq(|x|^{-α}*u^λ)u^σ\quad\text{ in }B_2 (0)\backslash \{0\} \end{equation} where $α\in(0,n),λ>0,$ and $σ\geq0$ are constants and $*$ is the convolution operation in $\mathbb{R}^n$. We provide optimal conditions on $α,λ$, and $σ$ such that nonnegative solutions $u$ satisfy pointwise bounds near the origin.
We prove a general self-improvement property for a family of weighted pointwise inequalities on open sets, including pointwise Hardy inequalities with distance weights. For this purpose we introduce and study the classes of $p$-Poincaré and $p$-Hardy weights for an open set $Ω\subset X$, where $X$ is a metric measure space. We also apply the self-improvement of weighted pointwise Hardy inequalities in connection with usual integral versions of Hardy inequalities.
We investigate nonnegative solutions $u(x,t)$ and $v(x,t)$ of the nonlinear system of inequalities \[0\leq(\partial_t -Δ)^αu\leq v^λ\] \[ 0\leq (\partial_t -Δ)^βv\leq u^σ\] in $\mathbb{R}^n \times\mathbb{R}$, $n\geq 1$, satisfying the initial conditions \[ u=v=0\quad\text{ in }\mathbb{R}^n \times(-\infty,0) \] where $λ,σ,α$, and $β$ are positive constants. Specifically, using the definition of the fractional heat operator $(\partial_t-Δ)^α$ given in \cite{T}, we obtain, when they exist, optimal pointwise upper bounds on $\mathbb{R}^n \times(0,\infty)$ for nonnegative solutions $u$ and $v$ of t…
We study the behavior as $t\to 0^+$ of nonnegative functions \begin{equation}\label{0.1} u\in C^{2,1} (\mathbb{R}^n\times (0,1)) \cap L^λ(\mathbb{R}^n\times (0,1)),\quad n\ge 1, \end{equation} satisfying the parabolic Choquard-Pekar type inequalities \begin{equation}\label{0.2} 0\leq u_t-Δu\leq(Φ^{α/n}*u^λ)u^σ\quad \text{ in }B_1 (0)\times (0,1) \end{equation} where $α\in(0,n+2)$, $λ>0$, and $σ\geq0$ are constants, $Φ$ is the heat kernel, and $*$ is the convolution operation in $\mathbb{R}^n\times (0,1)$. We provide optimal conditions on $α,λ$, and $σ$ such that nonnegative solutions $u$ satis…
We investigate pointwise upper bounds for nonnegative solutions $u(x,t)$ of the nonlinear initial value problem \begin{equation}\label{0.1} 0\leq(\partial_t-Δ)^αu\leq u^λ\quad\text{ in }\mathbb{R}^n \times\mathbb{R},\,n\geq1, \end{equation} \begin{equation}\label{0.2} u=0\quad\text{in }\mathbb{R}^n\times(-\infty,0) \end{equation} where $λ$ and $α$ are positive constants. To do this we first give a definition---tailored for our study of this problem---of fractional powers of the heat operator $(\partial_t-Δ)^α:Y\to X$ where $X$ and $Y$ are linear spaces whose elements are real valued functions …
In this paper we give both an historical and technical overview of the theory of Harnack inequalities for nonlinear parabolic equations in divergence form. We start reviewing the elliptic case with some of its variants and geometrical consequences. The linear parabolic Harnack inequality of Moser is discussed extensively, together with its link to two-sided kernel estimates and to the Li-Yau differential Harnack inequality. Then we overview the more recent developements of the theory for nonlinear degenerate/singular equations, highlighting the differences with the quadratic case and introduci…
We prove fractional Sobolev-Poincaré inequalities, capacitary versions of fractional Poincaré inequalities, and pointwise and localized fractional Hardy inequalities in a metric space equipped with a doubling measure. Our results generalize and extend earlier work where such inequalities have been considered in the Euclidean spaces or in the non-fractional setting in metric spaces. The results concerning pointwise and localized variants of fractional Hardy inequalities are new even in the Euclidean case.
Let (X j , d j , $μ$ j), j = 0, 1,. .. , m be metric measure spaces. Given 0 < p $κ$ $\le$ $\infty$ for $κ$ = 1,. .. , m and an analytic family of multilinear operators T z : L p 1 (X 1) x $\bullet$ $\bullet$ $\bullet$ L p m (X m) $\rightarrow$ L 1 loc (X 0), for z in the complex unit strip, we prove a theorem in the spirit of Stein's complex interpolation for analytic families. Analyticity and our admissibility condition are defined in the weak (integral) sense and relax the pointwise definitions given in [9]. Continuous functions with compact support are natural dense subspaces of Lebesgue s…
Pointwise convergence of the heat and subordinates of the heat semigroups associated with the Laplace operator on homogeneous trees and two weighted $L^p$ maximal inequalities
The paper provides weighted Sobolev inequalities of the Caffarelli-Kohn-Nirenberg type for functions with multi-radial symmetry. Similarly to the previously studied radial case, the range of parameters in CKN inequalities can be extended, sometimes to infinity, providing a pointwise estimate similar to the classical radial estimate. Furthermore, the "multi-radial" weights are a stronger singularity than radial weights of the same homogeneity.
