The notions of Lipschitz and bilipschitz mappings provide classes of mappings connected to the geometry of metric spaces in certain ways. A notion between these two is given by regular mappings (reviewed in Section 1), in which some non-bilipschitz behavior is allowed, but with limitations on this, and in a quantitative way. In this paper we look at a class of mappings called (s, t)-regular mappings. These mappings are the same as ordinary regular mappings when s = t, but otherwise they behave somewhat...

In this paper, the authors introduce a kind of local Hardy spaces in Rn associated with the local Herz spaces. Then the authors investigate the regularity in these local Hardy spaces of some nonlinear quantities on superharmonic functions on R2. The main results of the authors extend the corresponding results of Evans and Müller in a recent paper.

In this paper we consider the regularity problem for the commutators $\left(\right[b,R_k]{)}_{1\le k\le n}$ where $b$ is a locally integrable function and $(R_j{)}_{1\le j\le n}$ are the Riesz transforms in the $n$-dimensional euclidean space ${ℝ}^n$. More precisely, we prove that these commutators $\left(\right[b,R_k]{)}_{1\le k\le n}$ are bounded from $L^p$ into the Besov space ${\dot{B}}_p^{s,p}$ for $1\<p\<+\infty$ and $0\<s\<1$ if and only if $b$ is in the $BMO$-Triebel-Lizorkin space ${\dot{F}}_{\infty }^{s,p}$. The reduction of our result to the case $p=2$ gives in particular that the commutators $\left(\right[b,R_k]{)}_{1\le k\le n}$ are bounded form $L^2$ into the Sobolev space ${\dot{H}}^s$ if and only if $b$ is in the $BMO$-Sobolev...

For s>0, we consider bounded linear operators from $D\left({ℝ}^n\right)$ into $D^{\text{'}}\left({ℝ}^n\right)$ whose kernels K satisfy the conditions $|{\partial }_x^{\gamma }K(x,y)|\le C_{\gamma }{|x-y|}^{-n+s-\left|\gamma \right|}$ for x≠y, |γ|≤ [s]+1, $|{\nabla }_y{\partial }_x^{\gamma }K(x,y)|\le C_{\gamma }{|x-y|}^{-n+s-\left|\gamma \right|-1}$ for |γ|=[s], x≠y. We establish a new criterion for the boundedness of these operators from $L^2\left({ℝ}^n\right)$ into the homogeneous Sobolev space ${Ḣ}^s\left({ℝ}^n\right)$. This is an extension of the well-known T(1) Theorem due to David and Journé. Our arguments make use of the function T(1) and the BMO-Sobolev space. We give some applications to the Besov and Triebel-Lizorkin spaces as well as some other potential...

Let $M^{\circ }$ be a complete noncompact manifold of dimension at least 3 and $g$ an asymptotically conic metric on $M^{\circ }$, in the sense that $M^{\circ }$ compactifies to a manifold with boundary $M$ so that $g$ becomes a scattering metric on $M$. We study the resolvent kernel ${(P+k^2)}^{-1}$ and Riesz transform $T$ of the operator $P={\Delta }_g+V$, where ${\Delta }_g$ is the positive Laplacian associated to $g$ and $V$ is a real potential function smooth on $M$ and vanishing at the boundary.In our first paper we assumed that $P$ has neither zero modes nor a zero-resonance and showed...

In the first part we consider restriction theorems for hypersurfaces Γ in Rn, with the affine curvature KΓ1/(n+1) introduced as a mitigating factor. Sjölin, [19], showed that there is a universal restriction theorem for all convex curves in R2. We show that in dimensions greater than two there is no analogous universal restriction theorem for hypersurfaces with non-negative curvature.In the second part we discuss decay estimates for the Fourier transform of the density KΓ1/2 supported on the surface...

We show that the $L^p$ boundedness, p > 2, of the Riesz transform on a complete non-compact Riemannian manifold with upper and lower Gaussian heat kernel estimates is equivalent to a certain form of Sobolev inequality. We also characterize in such terms the heat kernel gradient upper estimate on manifolds with polynomial growth.

We study the validity of the $L^p$ inequality for the Riesz transform when $p\>2$ and of its reverse inequality when $1\<p\<2$ on complete riemannian manifolds under the doubling property and some Poincaré inequalities.

We prove $L^p$-bounds for the Riesz transforms associated to the Hodge-Laplacian equipped with absolute and relative boundary conditions in a Lipschitz subdomain of a (smooth) Riemannian manifold for $p$ in a certain interval depending on the Lipschitz character of the domain.

Assume that $M_0$ is a complete Riemannian manifold with Ricci curvature bounded from below and that $M_0$ satisfies a Sobolev inequality of dimension $\nu \>3$. Let $M$ be a complete Riemannian manifold isometric at infinity to $M_0$ and let $p\in (\nu /(\nu -1),\nu )$. The boundedness of the Riesz transform of $L^p\left(M_0\right)$ then implies the boundedness of the Riesz transform of $L^p\left(M\right)$