Adams inequalities for Riesz subcritical potentials
Luigi Fontana, Carlo Morpurgo
备注：
To appear in Nonlinear Analysis. In v2 several typos, omissions, and punctuation errors were corrected. Some statements in section 4 were corrected, Theorem 14 was restated. Several results in this paper, mostly in weaker forms, originally appeared in arXiv:1504.04678 (unpublished manuscript)
We derive Adams inequalities for potentials on general measure spaces, extending and improving previous results obtained by the authors. The integral operators involved, which we call "Riesz subcritical", have kernels whose decreasing rearrangements are not worse than that of the Riesz kernel on ${\mathbb R}^n$, where the kernel is large, but they behave better where the kernel is small. The new element is a "critical integrability" condition on the kernel at infinity. Typical examples of such kernels are fundamental solutions of nonhomogeneous differential, or pseudo-differential, operators. Another example is the Riesz kernel itself restricted to suitable measurable sets, which we name "Riesz subcritical domains". Such domains are characterized in terms of their growth at infinity. As a consequence of the general results we obtain several new sharp Adams and Moser-Trudinger inequalities on ${\mathbb R}^n$, on the hyperbolic space, on Riesz subcritical domains, and on domains where the Poincaré inequality holds.
We derive sharp Adams inequalities with exact growth condition for the Riesz potential as well as more general Riesz-like potentials on R^n. We also obtain Moser-Trudinger inequalities with exact growth condition for the fractional Laplacian, and for general homogeneous elliptic differential operators with constant coefficients.
In this paper, we establish the sharp critical and subcritical trace Trudinger-Moser and Adams inequalities on the half spaces and prove the existence of their extremals through the method based on the Fourier rearrangement, harmonic extension and scaling invariance. These trace Trudinger-Moser and Adams inequalities can be considered as the borderline case of the Sobolev trace inequalities of first and higher orders. Furthermore, we show the existence of the least energy solutions for a class of bi-harmonic equations with nonlinear Neumann boundary condition associated with the trace Adams in…
In this work we obtain a geometric characterization of the measures $μ$ in $\mathbb{R}^{n+1}$ with polynomial upper growth of degree $n$ such that the $n$-dimensional Riesz transform $\mathcal{R}μ(x) = \int \frac{x-y}{|x-y|^{n+1}}\,dμ(y)$ belongs to $L^2(μ)$, under the assumption that $μ$ satisfies the following Wolff energy estimate, for any ball $B\subset\mathbb{R}^{n+1}$: $$\int_B \int_0^\infty \left(\frac{μ(B(x,r))}{r^{n-\frac38}}\right)^2\,\frac{dr}r\,dμ(x)\leq M\,\bigg(\frac{μ(2B)}{r(B)^{n-\frac38}}\bigg)^2\,μ(2B).$$ More precisely, we show that $μ$ satisfies the following estimate: $$\|…
Though Adams and Hardy-Adams inequalities can be extended to general symmetric spaces of noncompact type fairly straightforwardly by following closely the systematic approach developed in our early works on real and complex hyperbolic spaces, higher order Poincaré-Sobolev and Hardy-Sobolev-Maz'ya inequalities are more difficult to establish. The main purpose of this goal is to establish the Poincaré-Sobolev and Hardy-Sobolev-Maz'ya inequalities on quaternionic hyperbolic spaces and the Cayley hyperbolic plane. A crucial part of our work is to establish appropriate factorization theorems on the…
In this paper we strengthen to Morrey-Lorentz spaces the famous trace principle introduced by Adams. More precisely, we show that Riesz potential $I_α$ is continuous \begin{equation} \Vert I_αf\Vert_{\mathcal{M}_{q, \infty}^{λ_{\ast}}(dμ)}\lesssim \Arrowvertμ\Arrowvert_β^{{1}/{q}}\,\Vert f\Vert_{\mathcal{M}_{p, \infty}^λ(dν)}\nonumber\\[0.02in] \end{equation} if and only if the Radon measure $dμ$ supported in $Ω\subset \mathbb{R}^n$ is controlled by $$\Arrowvertμ\Arrowvert_β=\sup_{x\in\mathbb{R}^n,\,r>0}r^{-β}μ(B(x,r))<\infty$$ provided that $1<p<q<\infty$ satisfies $n-αp<β\leq n,\; α=\frac{n}…
