Adams inequalities for Riesz subcritical potentials

TL;DR

This paper extends Adams inequalities to Riesz subcritical potentials on general measure spaces, introducing a new critical integrability condition and applying results to various geometric settings.

Contribution

The authors develop new Adams inequalities for Riesz subcritical potentials with a novel integrability condition, broadening applicability to diverse measure spaces and domains.

Findings

Derived sharp Adams inequalities on ${ m f R}^n$ and hyperbolic space.

Established inequalities on Riesz subcritical domains with growth conditions.

Extended results to domains satisfying Poincaré inequalities.

Abstract

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 Poincar\'e inequality holds.

Equations

$$\int_{E}\exp\bigg{[}\gamma|u(x)|^{n\over n-\alpha}\bigg{]}dx\leq C\qquad u\in W_{0}^{\alpha,{n\over\alpha}}(F),\qquad\|Pu\|_{n/\alpha}\leq 1\eqdef{mt1}$$

$$I_{\alpha}f=|x|^{\alpha-n}*f,\qquad c_{\alpha}={\Gamma\big{(}{n-\alpha\over 2}\big{)}\over 2^{\alpha}\pi^{n/2}\Gamma\big{(}{\alpha\over 2}\big{)}}.\eqdef{calpha}$$

$$\int_{E}\exp\bigg{[}{1\over|B_{1}|}|I_{\alpha}f(x)|^{n\over n-\alpha}\bigg{]}dx\leq C|E|\qquad f\in L^{n/\alpha}(E),\qquad\|f\|_{n/\alpha}\leq 1\eqdef{mt2}$$

$$\gamma(\nabla^{\alpha})=\cases{\displaystyle{c_{\alpha}^{-{n\over n-\alpha}}\over|B_{1}|}&if $\alpha$ even\cr\displaystyle{\big{(}(n-\alpha-1)c_{\alpha+1}\big{)}^{-{n\over n-\alpha}}\over|B_{1}|}&if $\alpha$ odd.\cr}$$

$$Tf(x)={\int }_{M}k(x,y)f(y)d\mu (y),x\in N\text{eqdef}T$$

$$\int_{E}\exp\bigg{[}\gamma|Tf(x)|^{n\over n-\alpha}\bigg{]}dx\leq C\qquad f\in L^{n/\alpha}(F),\qquad\|f\|_{n/\alpha}\leq 1\eqdef{mt3}$$

$$k_1^{\ast }(x,t)=[k(x,\cdot ){]}^{\ast }(t),k_2^{\ast }(y,t)=k[(\cdot ,y){]}^{\ast }(t)$$

$$k_1^{\ast }(t):=\underset{x\in N}{\sup }{k}_1^{\ast }(x,t),k_2^{\ast }(t):=\underset{y\in M}{\sup }{k}_2^{\ast }(y,t).$$

$$k_{1}^{*}(t)=t^{-{n-\alpha\over n}}(A^{n-\alpha\over n}+O(|\log t|)^{-\gamma}\big{)},\qquad 0<t\leq 1\eqdef{106a}$$

$$k_2^{\ast }(t)\le B{t}^{-\frac{n-\alpha }{\sigma n}},t>0\text{eqdef}107a$$

$$\gamma=\bigg{(}{1\over n}\sup_{x\in\Omega}\int_{S^{n-1}}\bigg{|}\Big{(}{1\over p_{\alpha}(x,\cdot)}\Big{)}^{\wedge}(\omega)\bigg{|}^{n\over n-\alpha}d\omega\bigg{)}^{-1}\eqdef{gamma}$$

$$\|u\|_{{n/\alpha},q}:=\cases{\big{(}\|u\|_{n/\alpha}^{q{n/\alpha}}+\|\nabla^{\alpha}u\|_{n/\alpha}^{q{n/\alpha}}\big{)}^{\alpha\over qn}&if $1\leq q<\infty$\cr\max\big{\{}\|u\|_{n/\alpha},\|\nabla^{\alpha}u\|_{n/\alpha}\big{\}}&if $q=+\infty$\cr}\eqdef{norm}$$

$$\underset{x\in N}{\sup }{\int }_1^{\infty }{k}_1^{\ast }(x,t{)}^{\frac{n}{n-\alpha }}dt<\infty .\text{eqdef}108a$$

$$k_1^{\ast }(t)\le C{t}^{-\frac{n-\alpha }{n}},k_2^{\ast }(t)\le B{t}^{-\frac{n-\alpha }{\sigma n}},t>0\text{eqdef}conditions$$

$$\underset{x\in \Omega }{\sup }{\int }_1^{\infty }\frac{{\Lambda }_{\Omega }(x,r)}{r^{n+1}}dr<\infty .\text{eqdef}RSC$$

$$\int_{\Omega}\exp_{[{n\over\alpha}-2]}\Big{[}\gamma(\nabla^{\alpha})|u(x)|^{n\over n-\alpha}\Big{]}dx\leq C(1+\|u\|_{L^{n/\alpha}(\Omega)}^{n/\alpha}),\qquad\|\nabla^{\alpha}u\|_{L^{n/\alpha}(\Omega)}\leq 1,\eqdef{regmt}$$

$$Tf(x)={\int }_{M}k(x,y)f(y)d\mu (y),x\in N,\text{eqdef}T$$

$$\cases{\lambda_{1}(x,s)=\mu(\{y\in M:\;|k(x,y)|>s\})&$\;x\in N$\cr\cr\lambda_{2}(y,s)=\nu(\{x\in N:\;|k(x,y)|>s\})&$\;y\in M.$\cr}\eqdef{pdis1}.$$

$$\cases{k_{1}^{*}(x,t)=\inf\{s>0:\,\lambda_{1}(x,s)\leq t\}&$\;x\in N$\cr\cr k_{2}^{*}(y,t)=\inf\{s>0:\,\lambda_{2}(y,s)\leq t\}&$\;y\in M$\cr}\eqdef{pdis2}$$

$$k_1^{\ast }(t)=\underset{x\in N}{\sup }{k}_1^{\ast }(x,t),k_2^{\ast }(t)=\underset{y\in M}{\sup }{k}_2^{\ast }(y,t)\text{eqdef}pdis3$$

$$k_{1}^{*}(t)\leq A^{1\over\beta}t^{-{1\over\beta}}\big{(}1+H(1+|\log t|)^{-\gamma}\big{)},\qquad 0<t\leq\tau\eqdef{106}$$

$$k_2^{\ast }(t)\le B{t}^{-\frac{1}{\sigma \beta }},t>0\text{eqdef}107$$

$$J_{\tau}:={1\over A}\,\sup_{x\in N}\,\int_{\tau}^{\infty}\big{(}k_{1}^{*}(x,t)\big{)}^{\beta}dt<\infty,\eqdef{108}$$

$$\int_{E}\exp\bigg{[}{\sigma\over A}|Tf(x)|^{\beta}\bigg{]}d\nu(x)\leq Ce^{\sigma J_{\tau}}\big{(}\nu(E)+\tau^{\sigma}(1+J_{\tau})\big{)}.\eqdef{109}$$

$$\int_{N}\exp_{[\beta^{\prime}-2]}\bigg{[}{\sigma\over A}|Tf(x)|^{\beta}\bigg{]}d\nu(x)\leq Ce^{\sigma J_{\tau}}\big{(}\|Tf\|_{\beta^{\prime}}^{\beta^{\prime}}+\tau^{\sigma}(1+J_{\tau})\big{)},\eqdef{110}$$

$${\exp }_m(t)=e^t-\sum _{k=0}^m\frac{t^k}{k!},m=0,1,...$$

$$\int_{1}^{\infty}\big{(}k_{1}^{*}(t)\big{)}^{\beta}dt<\infty.\eqdef{stronger}$$

$$\sup_{x\in N}\int_{1}^{\infty}\big{(}k_{1}^{*}(x,t)\big{)}^{\beta}dt<\infty,\qquad\int_{1}^{\infty}\big{(}k_{1}^{*}(t)\big{)}^{\beta}dt=+\infty.\eqdef{supint3}$$

$$k(x,y)=\cases{|y|^{\alpha-n}&if $\displaystyle{|x|\over 2}\leq|y|<|x|$\cr 0&otherwise\cr}$$

$$k_{1}^{*}(x,t)=\cases{\bigg{(}\displaystyle{t\over|B_{1}|}+\Big{(}\displaystyle{|x|\over 2}\Big{)}^{n}\bigg{)}^{-{n-\alpha\over n}}&if $0<t<|B_{1}||x|^{n}(1-2^{-n})$\cr 0&if $t\geq|B_{1}||x|^{n}(1-2^{-n})$\cr}\leq|B_{1}|^{n-\alpha\over n}t^{-{n-\alpha\over n}}=k_{1}^{*}(t)$$

Full text

[TABLE]

[TABLE]

[TABLE]

for some $\beta>1$ and $0<\sigma\leq 1$. Then, $Tf$ is well defined and finite a.e. for $f\in L^{\beta^{\prime}}(M)$, and there exists a constant $C=C(\beta,\sigma,\gamma,A,B,H)$ such that for each $f\in L^{\beta^{\prime}}(M)$ with $\|f\|_{\beta^{\prime}}\leq 1$ and for each measurable $E\subseteq N$ with $\nu(E)<\infty$ we have

[TABLE]

Moreover, for all $f\in L^{\beta^{\prime}}(M)$ such that $\|f\|_{\beta^{\prime}}\leq 1$ we have

[TABLE]

where the “regularized exponential” function $\exp_{m}$ is defined as

[TABLE]

and where $[\lambda]$ denotes the smallest integer greater or equal to $\lambda\in{\hbox{\mathbold\char 82\relax}}.$

Later in Theorem 6 we will show that for reasonable kernels the critical integrability condition \eqref108 is also necessary for the Adams inequality to hold. We note that the earlier version of this theorem appearing in [FM3, Thm. 3], has a slightly stronger assumption, namely

[TABLE]

Assumption \eqrefstronger is in general easier to verify than \eqref108. Moreover Theorem 1 can be deduced from \eqrefstronger by using a milder modification of O’Neil’s Lemma than the one we need (see Lemma 5 below) if we only assume \eqref108 (see Lemma 2 in [FM1] and Note 3 after Lemma 5). On the other hand, there are examples for which \eqref108 holds but \eqrefstronger does not.

It is not too hard to produce kernels for which

[TABLE]

For a somewhat artificial kernel, consider on ${\hbox{\mathbold\char 82\relax}}^{n}$ with the Lebesgue measure

[TABLE]

and let $Tf(x)=\int_{{\hbox{\smathbold\char 82\relax}}^{n}}k(x,y)f(y)dy.$ In this case we have

[TABLE]

which implies \eqrefsupint3, with $\beta={{n\over n-\alpha}}$, and it is easy to check that $k_{2}^{*}(t)\leq Ct^{-{n-\alpha\over n}}$. The Adams inequality trivially holds, since by Hölder’s inequality $|Tf(x)|\leq C\|f\|_{n/\alpha}$ for all $x$. Note in passing that the same example shows that $T$ is not continuous from $L^{{n/\alpha}}({\hbox{\mathbold\char 82\relax}}^{n})$ to any $L^{p}({\hbox{\mathbold\char 82\relax}}^{n})$ (consider the family $f_{r}(y)=|y|^{-\alpha}\chi_{r/2\leq|y|<r}(y))$.

A more interesting example satisfying \eqrefsupint3 is discusssed in \eqrefdominio1, where $k(x,y)=\break|x-y|^{\alpha-n}$ and $M=N=\Omega$ for a suitable $\Omega$ of infinite measure, whose construction is not trivial. Adams’ inequality for $\Omega$ follows immediately from Theorem 1 in the present form, whereas the previous version in [FM3] based on assumption \eqrefstronger is unable to give a sensible answer for $\Omega$.

The right hand sides of \eqref109, \eqref110 are also improvements of previous versions appearing in [FM3, Theorem 3].

If we constrain our functions $f$ to be supported on a given set $F\subseteq M$ with finite measure we obtain the following refinement of Theorem 1 in [FM1]:

[TABLE]

[TABLE]

Then, there exists a constant $C=C(\beta,\sigma,\gamma,A,B,H)$ such that for each measurable $F\subseteq M$ and each $f\in L^{\beta^{\prime}}(F)$ with $\|f\|_{L^{\beta^{\prime}}(F)}\leq 1$ and for each measurable $E\subseteq N$

[TABLE]

Note. When $F\subseteq M$ is measurable, the space $L^{p}(F)$ is defined as the space of those measurable $f:E\to{\hbox{\mathbold\char 82\relax}}$ such that their zero extension to $M$ is in $L^{p}(M)$. Equivalently, it is the space of functions $f\in L^{p}(M)$ which are 0 a.e. outside $F$. **Proof of Corollary 2. ** If $\mu(F)<\infty$ then we can apply Theorem 1 to the measurable space $(F,\mu)$ as a subspace of $M$. If $k_{1,F}^{*}(x,t),\,k_{1,F}^{*}(t)$ denote the rearrangements with respect to $F$ then

[TABLE]

(with $H$ independent of $F$!) and $k_{1,F}^{*}(t)=0$ for $t>\mu(F)$. Hence condition \eqref108 is verified for $\tau=\mu(F)$ which gives the right hand side in \eqref109-a.

[TABLE]

When $T$ is the Riesz potential Corollary 2 gives the following refinement of Adams’ inequality \eqrefmt2 and of Theorem 7 in [FM1]:

[TABLE]

for some $Q>0$ and $\sigma\in(0,1]$. Then, there exists $C=C(n,\alpha,\sigma,Q)$ such that for all $E,F\subseteq{\hbox{\mathbold\char 82\relax}}^{n}$ with $\nu(E)<~{}\infty,\,|F|<~{}\infty$ and for all $f\in L^{{n/\alpha}}(F)$ with $\|f\|_{n/\alpha}\leq 1$ we have

[TABLE]

For given $E$ and $F$, if there exists a ball $B(x_{0},r_{0})$ such that $|B(x_{0},r_{0})\cap F|=|B(x_{0},r_{0})|$, and $\nu(B(x_{0},r)\cap E)\geq c_{1}r^{\sigma n}$ for $r\leq r_{0}$, with $c_{1}>0$, then the exponential constant is sharp.

The condition that $F$ contains a ball (up to a set of zero measure) which has enough mass shared by $E$, is essentially necessary in order to guarantee sharpness in the above corollary. In general the sharp exponential constant will depend on the relative geometry of the sets $E$ and $F$: the less the mass they have in common, the larger the sharp constant. This is a reflection of the fact that the potential becomes “less effective” as the sets $E$ and $F$ get more and more separated (in this regard, see [FM1, Remark 3, p. 5112]).

**Proof of Corollary 3. ** In Corollary 2 take $N={\hbox{\mathbold\char 82\relax}}^{n}$ endowed with the measure $\nu$ as in \eqrefb, $M={\hbox{\mathbold\char 82\relax}}^{n}$ with the Lebesgue measure, $k(x,y)=|x-y|^{\alpha-n}$, $\beta={n\over n-\alpha}$, so that under the assumption \eqrefb we have (see also [FM1, Lemma 9])

[TABLE]

which implies \eqref18.

For the sharpness statement, assume that $F$ contains a ball of radius $r_{0}$ and center $x_{0}$, up to a set of zero measure. We can assume that $x_{0}=0$, and define for $0<\epsilon<r_{0}$.

[TABLE]

Then ${\rm supp}\,\phi_{\epsilon}\subseteq F$, and

[TABLE]

If $\tilde{\phi}_{\epsilon}=\phi_{\epsilon}/\|\phi_{\epsilon}\|_{n/\alpha}$ then by routine computations $|I_{\alpha}\tilde{\phi}_{\epsilon}(x)|^{n\over n-\alpha}\geq|B_{1}|\log{r_{0}^{n}\over\epsilon^{n}}-C$, provided $x\in B(0,\epsilon/2)$. Assuming $\nu(E\cap B(0,\epsilon/2)|\geq C\epsilon^{\sigma n}$ we then get

[TABLE]

if $\gamma>\sigma/|B_{1}|.$

[TABLE]

We now give the proof of Theorem 1. First let us note that the estimate given in \eqref110 will be an immediate consequence of \eqref109, via the following elementary lemma (see also [FM2, Lemma 9]):

[TABLE]

In particular, the functional \int_{N}\exp_{[p-2]}\big{[}\gamma|u|^{p^{\prime}}\big{]} is bounded on a bounded subset $X$ of $L^{p}$, if and only if \int_{\{|u|\geq 1\}}\exp\big{[}\gamma|u|^{p^{\prime}}\big{]} is bounded on $X$. Moreover, for any $m=1,2,3....$ we have

[TABLE]

If the operator $T$ in Theorem 1 is also continuous on $L^{\beta^{\prime}}(M)$, then the regularized exponential integral in \eqref110 is clearly bounded on the unit ball of $L^{\beta^{\prime}}(M)$. For general “well-behaved” kernels $K$ continuity of $T$ in $L^{\beta^{\prime}}(M)$ implies critical integrability, in the form \eqref108 (see Proposition 2 below), but not viceversa. For example, consider on ${\hbox{\mathbold\char 82\relax}}^{n}$ $k(x,y)=K(x-y)$, where

[TABLE]

for any $\delta\in(0,\alpha)$. Here $\beta={n\over n-\alpha},\;{\beta^{\prime}}={n\over\alpha}$. The kernel $K$ behaves like a Riesz kernel at [math], and it satisfies the critical integrability condition. Yet, the convolution operator $K*f$ is not continuous on $L^{n\over\alpha}({\hbox{\mathbold\char 82\relax}}^{n})$, since $K$ is nonnegative and not integrable.

**Proof of Theorem 1. ** As we mentioned in the introduction, the proof of Theorem 1 is accomplished by making suitable modifications and improvements to the proof in [FM1, Theorem 1], in order to take into account the critical integrability condition \eqref108, and by tracking down the various constants a little bit more carefully. For the convenience of the reader we will present here the beginning of the proof in enough details so that the role of \eqref108 is highlighted, relegating the more technical parts to the appendix.

Because of Lemma 4 it is enough to prove \eqref109. Indeed,

[TABLE]

and $|\{|Tf|\geq 1\}|\leq\|Tf\|_{\beta^{\prime}}^{\beta^{\prime}}$.

To prove \eqref109 we show that for each $f\in L^{\beta^{\prime}}(M)$ the function $Tf$ is well-defined, finite a.e., and satisfies

[TABLE]

where $C=C(\beta,\beta_{0},\gamma,H,A,B)$, under the hypotheses \eqref106, \eqref107, \eqref108, and with

[TABLE]

Below, and in the appendix, $C_{j}$ denotes a constant $\geq 1$, depending only on $A,B,\beta,\sigma,p,H,\gamma$. Without loss of generality we can assume that $k$ and $f$ are nonnegative. The first key element of the proof is the following improvement of Lemma 2 in [FM1] (which was itself an improvement of the original lemma due to O’Neil [ON, Lemma 1.5]):

[TABLE]

with $\beta>1$ and $0<\sigma\leq 1$. If

[TABLE]

and

[TABLE]

then there is a constant $C_{0}>0$ such that for any measurable $f:M\to[0,\infty]$

[TABLE]

Notes. 1. When $\sigma=1$ we can take $p=1$ and $q=\beta$ in \eqrefon4.2. Even if $k$ is a kernel with arbitrary sign, conditions \eqrefon2 imply that $T$ is well defined on $L^{p}(M)$ and bounded from $L^{p}(M)$ to $L^{q,\infty}(N)$ (see [A2] and our proof of Lemma 5 in the appendix). 3. The earlier version of the lemma given in [FM1], and also used in [FM3], has an inequality like \eqrefon4, but with second term equal to $\int_{\tau}^{\infty}k_{1}^{*}(u)f^{*}(u)du$, which is larger than the one above. With that version the conclusions of Theorem 1 can be proven under the stronger condition \eqrefstronger.

The proof of Lemma 5 is obtained by suitably modifying the proof given in [FM1]. The details are given in the appendix.

Now we have $k_{1}^{*}(t)\leq A^{{1\over\beta}}(1+H)t^{-{1\over\beta}}$ for $t>0$, so that by the improved O’Neil lemma above, if $p$ is any fixed number satisfying \eqrefon2, and $q$ is as \eqrefon3, then for each $t>0$

[TABLE]

If $t\geq\tau^{\sigma}$ then \eqref1 (combined into a single integral), Hölder’s inequality and \eqref0d imply

[TABLE]

and therefore

[TABLE]

On the interval $[0,\tau^{\sigma}]$ unfortunately this simple argument fails and we need to refine the more sophisticated analysis in [A1] and [FM1]. From \eqref1 followed by the change of variables $t\to\tau^{\sigma}t,\,u\to\tau u^{1\over\sigma}$ we get

[TABLE]

where

[TABLE]

and $C_{2}=C_{0}A^{-{1\over\beta}}$. Note that

[TABLE]

and that

[TABLE]

Now make the further changes

[TABLE]

to obtain that

[TABLE]

where for each fixed $\eta\geq 0$

[TABLE]

[TABLE]

The next technical step is to run the Adams-Garsia machinery to prove that

[TABLE]

The details are given in the appendix.

[TABLE]

For reasonable kernels the integrability condition \eqref108 is essentially necessary in order to obtain exponential integrability or continuity from $L^{n/\alpha}$ to any $L^{p}$:

[TABLE]

[TABLE]

and that for some sequence $x_{m}\in N$

[TABLE]

Suppose additionally that for each $m\in{\hbox{\mathbold\char 78\relax}}$ there is a measurable set $B_{m}\subseteq N$ such that for some $\epsilon_{0}>0$ and all $\epsilon>0$ small enough

[TABLE]

for some $C_{0}>0$, independent of $\epsilon$ (provided that the integral on the left hand side is well defined). Then, there is a sequence of functions $\{\Psi_{m}\}\subseteq L^{\beta^{\prime}}(M)$ with $\|\Psi_{m}\|_{\beta^{\prime}}=1$ such that

[TABLE]

In particular if the $B_{m}$ can be chosen so that $\nu(B_{m})\geq\nu_{0}>0$ for all $m$, then the operator $T$ defined in \eqrefT cannot be continuous from $L^{n/\alpha}$ to any $L^{p}$, and exponential integrability in the form \eqrefmt2 fails for any $\gamma>0$.

**Proof. ** Conditions \eqrefkstar00, \eqrefkstar0 guarantee that

[TABLE]

and that there is $T>0$ such that

[TABLE]

Also, \eqrefkstar00 implies that \eqrefkstar1 holds if and only if for any $\delta>0$

[TABLE]

The level sets

[TABLE]

satisfy, for each given $m$, $0<\mu(F_{\epsilon}^{m})\uparrow\mu(F_{0}^{m})$ as $\epsilon\downarrow 0$, and $\mu(F_{\epsilon_{0}}^{m})\leq T$, some $T>0$. Hence, for fixed $m$ and as $\epsilon\downarrow 0$

[TABLE]

Using \eqrefkstar1 we can find subsequences $m_{k}\uparrow\infty$ and $\epsilon_{k}\downarrow 0$ such that $\Gamma_{\epsilon_{k}}^{m_{k}}\to+\infty$ as $k\to+\infty$. By replacing the original sequence $\{x_{m}\}$ with $\{x_{m_{k}}\}$ we may assume that $\Gamma_{\epsilon_{m}}^{m}\to+\infty$ for a sequence $x_{m}$ and $\epsilon_{m}\downarrow 0$.

If we let

[TABLE]

then

[TABLE]

and, using the hypothesis, $T\Phi_{m}(x)$ is well-defined on some $B_{m}$ with

[TABLE]

The result follows upon taking $\Psi_{m}=\Phi_{m}\|\Phi_{m}\|_{\beta^{\prime}}^{-1}$.

[TABLE]

Note. If one of the conditions \eqrefkstar00, \eqrefkstar0 is not satisfied (regardless of \eqrefkstar1), then the conclusion of the theorem still holds, provided that $\mu$ is semifinite and \eqrefreg holds.

Certainly condition \eqrefkstar1 is met for some sequence $\{x_{m}\}$ if \eqref108 fails. It does seem unavoidable to impose some sort of “regularity” condition equivalent to \eqref108, however. On ${\hbox{\mathbold\char 82\relax}}^{n}$, for example, condition \eqrefreg is typically verified when $B_{m}$ is a small ball around $x_{m}$ and for $x\in B_{m}$, $k(x,y)-k(x_{m},y)$ decays better than $k(x_{m},y)$ as $y\to+\infty$. For convolution kernels which are radially decreasing (even just outside a large ball) $k(x,y)=K(|x-y|)\geq 0$ condition \eqrefreg is easily verified, since both $K(|x-y|)$ and $K(|y|)$ are greater than $K(2|y|)$, for $|x|\leq 1$ and $|y|\geq 2$

3. Riesz subcritical kernels and domains in ${\hbox{\mathbold\char 82\relax}}^{n}$

The most obvious application of Theorems 1 and 6 is to convolution operators on ${\hbox{\mathbold\char 82\relax}}^{n}$:

[TABLE]

[TABLE]

for some $\epsilon,R>0$, where $g\in L^{{n\over n-\alpha}}(S^{n-1})$. Then $K*f$ is finite a.e for $f\in L^{{n\over\alpha}}({\hbox{\mathbold\char 82\relax}}^{n})$, and there exists $C>0$ such that for all $f\in L^{{n\over\alpha}}({\hbox{\mathbold\char 82\relax}}^{n})$ with $\|f\|_{n/\alpha}\leq 1$, and for each measurable $E\subseteq{\hbox{\mathbold\char 82\relax}}^{n}$ with $|E|<\infty$

[TABLE]

where

[TABLE]

Moreover,

[TABLE]

for all $f\in L^{n\over\alpha}({\hbox{\mathbold\char 82\relax}}^{n})$ such that $\|f\|_{n/\alpha}\leq 1$. If $g$ is smooth, then the exponential constant in \eqref112a (if $|E|>0$) and in \eqref113 is sharp.

Note that the “big O” notation in \eqref111 means that $|O(|x|^{\alpha-n+\epsilon})|\leq C|x|^{\alpha-n+\epsilon}$, for $|x|\leq R$.

**Proof of Theorem 7. ** In Theorem 1 let $N=E$ and $M={\hbox{\mathbold\char 82\relax}}^{n}$ with the Lebesgue measure, and let $k(x,y)=K(x-y)$, $\beta={n\over n-\alpha}$. Note first that $k_{1}^{*}(x,t)=k_{1}^{*}(t)=k_{2}^{*}(t)$, for all $x,t$. Let $\tau=|B(0,R)|$ and suppose that $|K(x)|\leq M$ when $|x|\geq R$. If $\lambda(s)=|\{x\in{\hbox{\mathbold\char 82\relax}}^{n}:|K(x)|>s\}|$ denotes the distribution function of $K$ relative to ${\hbox{\mathbold\char 82\relax}}^{n}$, then for $s>M$ the distribution function of $K$ relative to the ball of radius $R$ coincides with $\lambda(s)$. This means that $k_{1}^{*}(t)$ is the same as the corresponding rearrangement relative to $B(0,R)$, when $t<\tau$. The proof that \eqref111 implies \eqref106 and \eqref107 for small $t$, on sets of finite measure and therefore for $t\in(0,\tau]$, has been done in [FM1, Lemma 9]. Note that the proof there was done in the case $g$ bounded on the sphere, but it works even in our more general hypothesis.

It is enough to check that \eqref112 implies \eqref108 (from which \eqref107 follows for all $t$, since $k_{2}^{*}=k_{1}^{*}$ is decreasing and finite). The proof of this fact is straightforward. Let $|K(x)|\leq M$ for $|x|\geq R$, and let

[TABLE]

If $\lambda(s)$ and $\widetilde{\lambda}(s)$ denote the distribution functions of $K,\widetilde{K}$ respectively, then $\widetilde{\lambda}(s)\geq\lambda(s)$ for $s<M$, and $\widetilde{\lambda}(M^{-})\geq|B_{1}|R^{n}$. Hence, if $k_{1}^{*},\,\widetilde{k}_{1}^{*}$ denote the rearrangements of $K,\,\widetilde{K}$ resp., then $\widetilde{k}_{1}^{*}(t)\geq k_{1}^{*}(t)$ for $t\geq|B_{1}|R^{n}$. Obviously, $\widetilde{K}\in L^{n\over n-\alpha}({\hbox{\mathbold\char 82\relax}}^{n})$, so \eqref108 follows with $\tau=|B_{1}|R^{n}$. This proves inequality \eqref112a, and therefore \eqref113.

The proof of the sharpness statement is the same as that of [FM1, Theorem 8]. In particular, assuming, without loss of generality, that $|E\cap B(0,\epsilon)|\geq c_{0}\epsilon^{n}$ for small $\epsilon$ and for some $c_{0}>0$, one can take the extremal family of functions

[TABLE]

and show that along the normalized family $\phi_{\epsilon}/\|\phi_{\epsilon}\|_{n/\alpha}$ the exponential integrals in \eqref112a and \eqref113 are saturated.

[TABLE]

Generally speaking, non-homogeneous invertible elliptic operators will have kernels satisfying \eqref112, and for those operators a sharp Moser-Trudinger inequality will hold. As a first example, consider the Bessel potential $(I-\Delta)^{\alpha\over 2}$, whose fundamental solution $G_{\alpha}(x)$ behaves like the Riesz potential locally, and decays exponentially at infinity. In fact, the aforementioned results by [A, Thm. 3], [LL2] and [RS] are immediate consequences of Theorem 7, and the fact that $\|G_{\alpha}*f\|_{p}\leq\|f\|_{p}$:

[TABLE]

we have

[TABLE]

and the exponential constant is sharp.

In this paper we define $W^{\alpha,{n\over\alpha}}({\hbox{\mathbold\char 82\relax}}^{n})$ for $\alpha>0,p>1$ to be the space of Bessel potentials:

[TABLE]

In section 4, Theorem 11, we will exhibit a class of non-homogeneous, elliptic, invertible, linear partial differential operators $P$ whose inverses have kernels satisfying \eqref111 and \eqref112, and therefore a sharp Moser-Trudinger inequality holds for such $P$ (Theorem 14).

We point out that Theorem 7 can be formulated so as to accommodate more general (non-convolution) kernels satisfying

[TABLE]

together with suitable integrability and boundedness conditions at infinity, in the same spirit as in [FM1, Thm. 8].

We also remark that Theorem 7 could have been stated in the slightly more general situation where the convolution operator is acting on $L^{n\over\alpha}(\Omega)$, where $\Omega\subseteq{\hbox{\mathbold\char 82\relax}}^{n}$ is an arbitrary measurable set of ${\hbox{\mathbold\char 82\relax}}^{n}$. In this case the conclusion holds provided that $K\in L^{\infty}\cap L^{n\over n-\alpha}(\{x\in\Omega:|x|\geq R\})$. We find that the latter condition is of little applicability if $\Omega\neq{\hbox{\mathbold\char 82\relax}}^{n}$, in which case one is better off checking out the corresponding critical integrability condition on $k_{1}^{*}(x,t)$, the rearrangement of $K(x-\cdot)$ with respect to $\Omega$.

When $K$ is the Riesz kernel, however, \eqref108 leads to an interesting geometric condition on $\Omega$, under which inequality \eqref112a holds under the sole condition that $\|f\|_{L^{n/\alpha}(\Omega)}\leq 1$, as expressed in the next theorem.

For a measurable set $\Omega\subseteq{\hbox{\mathbold\char 82\relax}}^{n}$ define for $r>0$

[TABLE]

[TABLE]

[TABLE]

Then, for $0<\alpha<n$, there exists $C>0$ such that for all $f\in L^{{n\over\alpha}}(\Omega)$ with $\|f\|_{n/\alpha}\leq 1$, and for each measurable $E\subseteq\Omega$ with $|E|<\infty$

[TABLE]

Moreover,

[TABLE]

for all $f\in L^{n\over\alpha}(\Omega)$ such that $\|f\|_{n/\alpha}\leq 1$. If there is $x_{0}\in\Omega$ and $r_{0}>0$ such that $|\Omega\cap B(x_{0},r_{0})|=|B(x_{0},r_{0})|$, and if $|E|>0$, then the exponential constant in \eqrefG1 and \eqrefG2 is sharp. Conversely, if \eqrefG is not satisfied then \eqrefG1 cannot hold, in fact there is a sequence of functions $\Psi_{m}\in L^{n/\alpha}(\Omega)$, with $\|\Psi_{m}\|_{n/\alpha}=1$ and $r_{1},\delta>0$ such that with $B_{m}=B(x_{m},r_{1})\cap\Omega$ and $|B_{m}|\geq\delta$ we have

[TABLE]

Condition \eqrefG is independent of $\alpha$, and expresses the Riesz subcriticality of the Riesz potential restricted to the measurable set $\Omega$.

We also note that \eqrefG is implied by the stronger condition

[TABLE]

with $\Lambda_{\Omega}(r)$ defined in \eqrefsupG. As the following example shows, it is possible to construct an $\Omega$ such that \eqrefG (and hence \eqrefG1and \eqrefG2) holds but \eqrefstronger fails. Let $c_{m},\,R_{m},\,\epsilon_{m}$ be positive sequences with $\epsilon_{m}<{{1\over 2}}$, $c_{m}\uparrow+\infty,\;R_{m}\uparrow+\infty,$ $c_{m}-c_{m-1}>>R_{m}$. If $k=(k_{1},...,k_{n})\in{\hbox{\mathbold\char 90\relax}}_{+}^{n}$ and $x_{m}=c_{m}{\bf e}_{1}$, let

[TABLE]

In other words, $\Omega_{m}$ is the union of all the balls of radius $\epsilon_{m}$ centered at the integer points contained in the ball of radius $R_{m}$ and center $x_{m}$.

With this in mind, it is possible to show that there is $C$ independent of $m$ such that for all $x\in\Omega_{m}$

[TABLE]

and

[TABLE]

Choosing for example

[TABLE]

we have that the series in \eqrefsupint1 is finite and the one in \eqrefsupint2 is infinite. Note that the Adams inequality for $L^{n\over\alpha}(\Omega_{*})$ is guaranteed by Theorem 9, whereas it would be hard to determine this fact using previously known methods.

**Proof of Theorem 9. ** Let us apply Theorem 1 with $M=N=\Omega$ and $\mu=\nu=$ Lebesgue measure. Since $I_{\alpha}$ has kernel $|x-y|^{\alpha-n}$ we have $k_{1}^{*}=k_{2}^{*}$, where rearrangement is with respect to $\Omega$. For simplicity let us drop the index “1”: and let

[TABLE]

[TABLE]

[TABLE]

Observe here that

[TABLE]

from which it follows

[TABLE]

(Equation \eqreflambda holds for general kernels $k$, and it was stated in [FM1, p. 5073, “Fact 3”]. The proof is based on the fact that for each $s>0$ and each $x\in\Omega$ there exists $T_{x,s}>0$ such that $\{t:k_{1}^{*}(x,t)>s\}=(0,T_{x,s})$.)

From general facts about rearrangements we have that $\lambda(x,\cdot),k^{*}(x,\cdot)$ are decreasing and right-continuous. Moreover $k^{*}(x,\lambda(x,s))\leq s$ and $\lambda(x,k^{*}(x,t))\leq t$. However, in this case we have also that $\lambda(x,s)$ is actually continuous (in fact locally Lipschitz) in $s$, hence $\lambda(x,k^{*}(x,t))=t$ for all $t>0$, and all $x\in\Omega$.

Obviously, $\Lambda_{\Omega}(x,r)\leq|B_{1}|r^{n}$ for every $x\in\Omega$, and $r>0$, hence

[TABLE]

and condition \eqref106 is verified with $A=|B_{1}|$ and $\beta={n\over n-\alpha}$.

Now note that if $\phi:(0,\infty)\to[0,\infty)$ is decreasing and right-continuous, and if $\lambda_{\phi}(s)$ denote its distribution function, then for each $t_{0}>0$

[TABLE]

(this is a consequence of Fubini’s theorem). Hence, applying this to $\phi(t)=k^{*}(x,t)$, $p={n\over n-\alpha}$, $t_{0}=\lambda(x,1)$ we get

[TABLE]

which implies that condition \eqref108 holds for any $\tau>\lambda(1)$.

This shows that the conditions of Theorem 1 are met, and the exponential inequalities follow. The sharpness of the exponential constant follows exactly as in the proof of Lemma 3.

Conversely, suppose that \eqrefG does not hold. Then we can find a sequence $\{x_{m}\}\subseteq\Omega$ such that

[TABLE]

We would like to apply Theorem 6. Conditions \eqrefkstar00, \eqrefkstar0 of Theorem 6 are a consequence of \eqrefstimak1. To show \eqrefkstar1, apply formula \eqrefformula, with $\phi(t)=k^{*}(x_{m},t)$ and $t_{0}=1$, together with \eqrefstimak1

[TABLE]

where we let

[TABLE]

If $\delta_{m}\geq 1$ for infinitely many $m$, then for a subsequence $\{x_{m_{k}}\}$

[TABLE]

so condition \eqrefkstar1 is satisfied up to passing to a subsequence.

If instead $\delta_{m}=k^{*}(x_{m},1)<1$ for all $m$ large enough, since $\lambda(x_{m},\delta_{m})=1$ (owing to the continuity of $\lambda(x,\cdot)$) we have $\Lambda_{\Omega}(x_{m},\delta_{m}^{-{1\over n-\alpha}})=1$ and therefore (since $\Lambda_{\Omega}(x_{m},\cdot)$ is increasing)

[TABLE]

[TABLE]

and condition \eqrefkstar1 is satisfied even in this case.

To verify the integral condition in \eqrefreg we proceed as follows. Define

[TABLE]

Clearly the relations in \eqreflambda, \eqreflambda1 continue to hold with $\tilde{\lambda},\,\tilde{k}$ in place of $\lambda,\,k_{1}^{*}$, and \eqrefkstar1 implies $\tilde{k}(t)>0$ for all $t$. Pick any $t_{1}>0$. Since $\tilde{k}(t)\to 0$ as $t\to+\infty$ (from \eqrefstimak1), we can find $t_{0}>t_{1}$ such that $0<\tilde{k}(t_{0})<\tilde{k}(t_{1})$. Let

[TABLE]

With this choice of $\epsilon_{0}$ the inequality in \eqrefkstar1 can be written as

[TABLE]

The inequality is certainly verified if $x$ belongs to the set

[TABLE]

since $|y-x|\leq|y-x_{m}|+r_{1}\leq 2|y-x_{m}|$, if $|y-x_{m}|\geq r_{0},\;|x-x_{m}|<r_{1}$.

Moreover, $|B_{m}|=\lambda(x_{m},\epsilon_{1})$, and since $\tilde{\lambda}(\epsilon_{1})=t_{1}$, we can pass to a subsequence of $x_{m}$ in order to guarantee that $|B_{m}|\geq t_{1}/2$, for all $m$.

[TABLE]

Remark. Under the hypothesis \eqrefG, estimate \eqrefG1 actually holds for any measurable set $E\subseteq{\hbox{\mathbold\char 82\relax}}^{n}$. To see this, note that for each fixed $x\in{\hbox{\mathbold\char 82\relax}}^{n}$ if $\Omega$ has positive measure then $\Lambda_{\Omega}(x,r)$ is increasing in $r$ and eventually positive. If $R_{x}:=\inf\{r>1:\Lambda_{\Omega}(x,r)>~{}0\}$ then $\Lambda_{\Omega}(x,r)\leq\Lambda_{\Omega}(y,3r)$ for any $y\in A_{x}:=B(x,2R_{x})\cap\Omega$ (of positive measure) and for any $r\geq R_{x}$. Hence for all $y\in A_{x}$

[TABLE]

which implies that

[TABLE]

and one can proceed as in the proof of Theorem 9 with $N={\hbox{\mathbold\char 82\relax}}^{n}$ instead of $N=\Omega$.

Theorem 9 gives a sufficient condition independent of $\alpha$, i.e. \eqrefG, under which Adams’ original result \eqrefmt2 holds for domains of infinite measure. In a sense, the condition says that the domain $\Omega$ misses enough dimensions at infinity. Examples of such domains are “strips” namely $\Omega=\{x\in{\hbox{\mathbold\char 82\relax}}^{n}:x_{j}\in[a_{j},b_{j}],j=1,...,k\}$, ($k<n$), in which case it is easy to see that $\Lambda_{\Omega}(r)\sim Cr^{n-k}$ as $r\to+\infty$. It is not hard to construct domains of infinite measure so that the corresponding $\Lambda_{\Omega}(r)$ has prescribed order of growth, within the upper bound $|B_{1}|r^{n}$. Take any smooth $h:[0,\infty)\to[0,\infty)$, $h(0)=0$, strictly increasing to $+\infty$ and with $h(x+1)-h(x)$ increasing (for example $h$ convex), and let

[TABLE]

where $\delta_{0}>0$ is chosen so that all $B(C_{\bf m},10\delta_{0})$ are pairwise disjoint. Then, one can check that for some $c_{1},c_{2},c_{3}>0$

[TABLE]

and

[TABLE]

some $r_{0}$ large enough. For the details of the proof see the Appendix.

Estimates \eqrefstimaG1 and \eqrefstimaG2 give

[TABLE]

for all $r$ large enough. With this in mind, one can, for example, produce an $h$ as above so that $\Lambda_{\Omega}(r)$ grows like $r^{n-\delta}(\log r)^{-q}$, for large $r$, for any $\delta,q$ with $0\leq\delta<n,\,q\geq 0$.

[TABLE]

and the exponential constant is sharp.

From the above estimate it is clear that if the Poincaré inequality $\|u\|_{n/\alpha}\leq C\|\nabla^{\alpha}u\|_{n/\alpha}$ holds in $W_{0}^{\alpha,{n/\alpha}}(\Omega)$, then there is uniformity on the right hand side of \eqrefMTS0. We shall return to this connection with the Poincaré inequality in Section 6.

4. Riesz subcritical fundamental solutions of elliptic differential operators

with constant coefficients on ${\hbox{\mathbold\char 82\relax}}^{n}$

Let us consider an elliptic differential operator of order $\alpha<n$ with constant complex coefficients and acting on $C^{\infty}_{c}({\hbox{\mathbold\char 82\relax}}^{n})$

[TABLE]

where $k=(k_{1},...,k_{n})$ denotes a nonnegative multiindex in ${\hbox{\mathbold\char 90\relax}}^{n}$. We will let

[TABLE]

and define the strictly homogeneous principal symbol of $P$ as

[TABLE]

For simplicity we only consider the case $a_{k}\in{\hbox{\mathbold\char 82\relax}}$, in which case $\alpha$ is even and “$P$ elliptic” means that

[TABLE]

for some $c_{0}>0$.

It is well known that $P$ has a fundamental solution, given by a function ${K_{P}}$ which is $C^{\infty}$ outside the origin, and which is formally the inverse Fourier transform of $1/p(\xi)$ i.e.

[TABLE]

With this notation we have that

[TABLE]

In what follow we will consider the case $p(x)\neq 0$ for $x\neq 0$, so that $K_{P}$ has a singularity only at [math]. Indeed, a formula using classical integrals for ${K_{P}}$ can be written for example as follows:

[TABLE]

for $x\neq 0$, were $\eta$ is a smooth cutoff which is 1 for $|x|\leq 1$ and 0 for $|x|\geq 2$, for any $\ell>{n-\alpha\over 2}$. This follows by writing e^{2\pi ix\cdot\xi}=(2\pi|x|)^{-2\ell}\Delta^{\ell}\big{(}e^{2\pi ix\cdot\xi}\big{)} and integrating by parts.

The first term in \eqrefK1 is in $C^{\infty}({\hbox{\mathbold\char 82\relax}}^{n})$ and controls the decay of $K_{P}$ at infinity. The second term is $C^{\infty}({\hbox{\mathbold\char 82\relax}}^{n}\setminus 0)$ and controls the singularity of $K_{P}$ at the origin. To analyze the singularity, write $p=p_{\alpha}+p_{m}$, for some $p_{m}$ polynomial of order $m<\alpha$ and

[TABLE]

The last term above is integrable outside a ball if $\ell$ large, whereas the other terms can be arranged into a finite sum, where the first term is $1/p_{\alpha}$ and the other terms are all homogeneous of order $<-\alpha$. From this one obtains that

[TABLE]

where $x^{*}=x/|x|$, and where

[TABLE]

in the sense of distributions (see also [FM1], formulas (67), (69)). This is precisely the local asymptotic expansion \eqref111 of Theorem 7, which has already been used in a more general context in [FM1] to prove the sharp Moser-Trudinger inequality for $P$ on bounded domains. The validity of the sharp Moser-Trudinger inequality for $P$ in the form \eqrefmt1 on the whole of ${\hbox{\mathbold\char 82\relax}}^{n}$ is therefore relying on the critical integrability of $K_{P}$ at infinity.

Question: Which non-homogeneous elliptic differential operators with constant coefficients on ${\hbox{\mathbold\char 82\relax}}^{n}$ have a fundamental solution which is Riesz subcritical?

Note that if $P$ is homogeneous of order $\alpha$ then $K_{P}$ is homogeneous of order $\alpha-n$ and critical integrability does not hold.

The precise asymptotic behavior of $K_{P}$ for large value of $x$ is not so obvious to figure out. It is well known that if $p$ never vanishes then $K_{P}$ decays exponentially at infinity. For the case $p(0)=0$ not much seems to be known in the literature other than a few special cases. From \eqrefK1 we see that $K_{P}(x)\to 0$ as $x\to\infty$, and in particular it is bounded outside a ball centered at 0.

If the lowest order terms of $P$ form an elliptic homogeneous operator of order $m<\alpha$, then one can show that $|K_{P}(x)|\leq C|x|^{m-n}$ for large $x$, so critical integrability holds. This can be seen by writing $p=p_{\alpha}^{\prime}+p_{m}^{\prime}$, for some $p_{m}^{\prime}$ elliptic homogeneous or order $m$, $p_{\alpha}^{\prime}$ is elliptic of order $\alpha$, and using formula \eqrefexpan with $p_{m}^{\prime}$ in place of $p_{\alpha}$ and $p_{\alpha}^{\prime}$ in place of $p_{m}$.

If P=\big{(}\nabla^{T}{\bf A}\nabla+b^{T}\nabla\big{)}^{\alpha/2}, where ${\bf A}$ is a real, symmetric and positive definite matrix and $b\in{\hbox{\mathbold\char 82\relax}}^{n}\setminus{\bf 0}$, then the fundamental solution $K_{P}$ can be explicitly computed via linear transformations from the one for the Bessel operator (see [OW, (2.5.3)] and [Lo], formula (7)). For example, when ${\bf A}=I$, the identity matrix, we have

[TABLE]

where $K_{\nu}$ is the modified Bessel function of the second kind. From this formula one obtains that for $|x|\geq R/|b|$ we have $|K_{P}(x)|\leq C_{b}|x|^{\alpha-n-1\over 2}e^{-{|x||b|+b\cdot x\over 2}}$, and a little computation in polar coordinates reveals that $K_{P}$ is in $L^{p}(|x|\geq 1)$ for $p>{n+1\over n+1-\alpha}$, which includes the case $p={n\over n-\alpha}$. For general ${\bf A}$ critical integrability follows for the same reasons, since the fundamental solution is obtained from \eqrefFS1 by multiplying it by $(\det{\bf A})^{-{{1\over 2}}}$ and by replacing $x$ with ${\bf A}^{-1/2}x$ and $b$ with ${\bf A}^{-1/2}b$. For $\alpha\geq n/2$ we can prove the following result:

[TABLE]

then its fundamental solution $K_{P}$ is in $L^{{n\over n-\alpha}}(|x|\geq 1)$.

Note that condition \eqrefellip1 implies the ellipticity condition \eqrefellip0. As it turns out there are elliptic operators with $p(\xi)\neq 0$, $p(0)=0$ and whose fundamental solution does not satisfy the critical integrability condition. See Remark 2 after the proof of Lemma 13. The proof of Theorem 11 is accomplished by showing that the first term in \eqrefK1 is in $L^{n\over n-\alpha}({\hbox{\mathbold\char 82\relax}}^{n})$. From the Hausdorff-Young inequality it is enough to prove that $\eta/p\in L^{n\over\alpha}({\hbox{\mathbold\char 82\relax}}^{n})$ i.e. $1/p\in L^{{n\over\alpha}}(B_{1})$:

**Proof of Lemma 12. ** Obviously, if $p$ is homogeneous of order $\alpha$ then $1/p$ cannot be in $L^{n\over\alpha}(B_{1})$. Suppose $p$ is not homogeneous and assume, without loss of generality, that $p$ is real-valued and positive, away from [math] (otherwise consider $|p(x)|^{2}$ instead of $p(x)$, and $2\alpha$ instead of $\alpha$.) Let $p_{\alpha},p_{{\kappa}}$ be the highest and lowest order homogeneous parts of $p$, of orders $\alpha$ and ${\kappa}<\alpha$ respectively. Then we can write $p=p_{\alpha}+q+p_{\kappa}$ and the hypotheses imply that for all $x\in{\hbox{\mathbold\char 82\relax}}^{n}$

[TABLE]

for some constant $c_{0}>0$.

Note also that $p_{\kappa}(\omega)=0$ on a set of zero measure on $S^{n-1}$.

Write

[TABLE]

To ease a bit the notation, for any given $\omega\in S^{n-1}$ write $p_{\alpha}=p_{\alpha}(\omega),\,p_{\kappa}=p_{\kappa}(\omega)$. We then have

[TABLE]

We can now choose $r_{0}>0$ such that for all $r\leq r_{0}$ and all $\omega\in S^{n-1}$

[TABLE]

(recall that ${\kappa}+1\leq\alpha$ and the lowest homogeneous part of $q$ has order greater than ${\kappa}$). Hence we can write

[TABLE]

Now, the function $\log p_{\kappa}(\omega)$ is integrable on the sphere. By homogeneity it is easy to check that this is equivalent to the local integrability of $\log p_{\kappa}(x)$, which follows from this general lemma:

We have not seen this result in the literature, so we will give here a short proof.

**Proof of Lemma 13. ** Suppose $p$ has degree $m$. By a linear transformation $x_{1}=x_{1}^{\prime}$, $x_{j}=x_{j}^{\prime}+\lambda_{j}x_{1}^{\prime}$, $j=2,3..,n$, we can assume that $p(x)=x_{1}^{m}+a_{m-1}x_{1}^{m-1}+...+a_{1}x_{1}+a_{0}$, where the $a_{j}$ are polynomials in $x_{2},...,x_{n}.$ If $Q=[a,b]\times Q^{\prime}$ is any cube in ${\hbox{\mathbold\char 82\relax}}^{n}$, then for fixed $x_{2},...,x_{n}\in Q^{\prime}$ the polynomial $p(x)$ has $m$ complex roots $\rho_{k}=\rho_{k}(x_{2},...,x_{n})$, $k=1,...,m$, which are all contained inside a fixed ball of radius $R$. Then the result follows from Fubini’s theorem, since

[TABLE]

[TABLE]

Remarks.1. After the first version of this manuscript was completed, Fulvio Ricci pointed out to us that Lemma 13 could also be seen as a consequence of the estimate (2.1) in [RiSt], which in our notation reads

[TABLE]

for all $\epsilon<1/\alpha$. Indeed, one checks easily that the constant $A_{\epsilon}$ in that estimate is of type $C^{\epsilon}$ and this implies the local integrability of $\log|p(x)|$. We thank Fulvio Ricci, and we also thank Peter Wagner who first pointed out to us the Proposition on p.182 of [RiSt], which contains estimate (2.1). 2. Lemma 12 is not valid under the weaker hypothesis $p$ elliptic, $p(0)=0$ and $p(x)\neq 0$ for all $x\neq 0$. An example is the $4-$th order elliptic polinomial in ${\hbox{\mathbold\char 82\relax}}^{8}$

[TABLE]

With a little calculation one shows that $p(x)^{-1}$ is not in $L^{2}(B_{1})$. In fact, by Plancherel’s formula this also shows that the Fourier transform of $\eta/p$, where $\eta$ is a smooth cutoff equal 1 on $B_{1}$, cannot be in $L^{2}({\hbox{\mathbold\char 82\relax}}^{n})$. This means, in view of \eqrefK1, that the fundamental solution of the $4$-th order operator in ${\hbox{\mathbold\char 82\relax}}^{8}$ whose symbol is $p$, cannot be Riesz subcritical.Taking into account Theorem 11 and the discussion preceding it, we can now state the following:

i) P=\big{(}\nabla^{T}{\bf A}\nabla+b^{T}\nabla\big{)}^{\alpha/2}, where ${\bf A}$ is a real, symmetric and positive definite matrix and $b\in{\hbox{\mathbold\char 82\relax}}^{n}\setminus{\bf 0}$;

ii) the total symbol of $P$ satisfies $p(\xi)\neq 0$ for all $\xi\neq 0$ and the lowest order terms of $P$ form an elliptic and homogeneous operator of order $m<\alpha$;

iii) the total symbol of $P$ satisfies \eqrefellip1, and $\alpha\geq\displaystyle{n\over 2}$. Then, there exists $C$ such that for all measurable $E\subseteq{\hbox{\mathbold\char 82\relax}}^{n}$ and for all $u\in W^{\alpha,{n\over\alpha}}({\hbox{\mathbold\char 82\relax}}^{n})$ with $\|Pu\|_{n/\alpha}\leq 1$ we have

[TABLE]

with

[TABLE]

(and with $g_{P}$ as in \eqrefgp). Moreover, for all $u\in W^{\alpha,{n\over\alpha}}({\hbox{\mathbold\char 82\relax}}^{n})$ with $\|Pu\|_{n/\alpha}\leq 1$

[TABLE]

The exponential constants in both of the above inequalities are sharp.

Remark. In case i) the sharp exponential constant is $\gamma(\nabla^{\alpha})(\det{\bf A})^{\alpha\over 2(n-\alpha)}$ (cf. [FM1, Corollary 11]). This can be seen directly from the explicit formula for $K_{P}$ given in [OW, (2.5.3)], since $K_{P}(x)=c_{\alpha}(\det{\bf A})^{-{{1\over 2}}}|{\bf A}^{-{{1\over 2}}}x|^{\alpha-n}(1+O(|{\bf A}^{-{{1\over 2}}}x|^{\sigma})$, as $x\to 0$ for some $\sigma>0$, from which it follows that $(K_{P})^{*}(t)=c_{\alpha}(\det{\bf A})^{-{\alpha\over 2n}}t^{-{n-\alpha\over n}}(1+O(t^{\epsilon})$), as $t\to 0$, from some $\epsilon>0$. In cases ii) or iii) the exponential constant can sometimes be computed explicitly, especially when $\alpha=n/2$, via the Plancherel formula (cf. [FM1, formula (85)]).

**Proof. ** We know that for a $P$ of the above types $K_{P}$ is Riesz subcritical, so that Theorem 7 applies. To prove the sharpness of the exponential constants, one would like to take the family of function $\psi_{\epsilon}=\phi_{\epsilon}/\|\phi_{\epsilon}\|_{n/\alpha}$ , where the $\phi_{\epsilon}$ are defined in \eqrefextremals, and then consider $u_{\epsilon}=K_{P}*\psi_{\epsilon}$. The only problem is that one can guarantee that $K_{P}*\psi_{\epsilon}\in L^{n/\alpha}(|x|\geq 1)$ only when $\alpha<n/2$. For higher values of $\alpha$ one needs to first normalize the $\phi_{\epsilon}$ in order to have enough vanishing moments. This is accomplished in [FM2, sect 6].

[TABLE]

5. Moser-Trudinger inequalities in hyperbolic space

In this section we obtain the sharp Moser-Trudinger inequalities for the higher order gradients on the hyperbolic space ${\hbox{\mathbold\char 72\relax}}^{n}$, as a consequence of Theorem 1. Below, ${\hbox{\mathbold\char 72\relax}}^{n}$ will denote the hyperbolic space modeled by the forward sheet of the hyperboloid

[TABLE]

endowed with the metric induced by the form

[TABLE]

and with distance function $d(x,y)={\rm{arccosh}}[x,y]$. One can introduce polar coordinates on ${\hbox{\mathbold\char 72\relax}}^{n}$ via

[TABLE]

and in these coordinates the metric and the volume element are written as

[TABLE]

The Laplace-Beltrami operator on ${\hbox{\mathbold\char 72\relax}}^{n}$ is denoted as $\Delta_{{{\hbox{\smathbold\char 72\relax}}^{n}}}$, and in polar coordinates is written as

[TABLE]

whereas the gradient $\nabla_{{\hbox{\ssmathbold\char 72\relax}}^{n}}$ is given by

[TABLE]

The Sobolev space $W^{\alpha,p}({\hbox{\mathbold\char 72\relax}}^{n})$ of integer order $\alpha$ is defined in the standard way via the covariant derivatives $\nabla^{k}$: it is the closure of the space of $C^{\infty}$ functions $\phi$ such that

[TABLE]

where $\|\cdot\|_{p}$ denotes the norm in $L^{p}({\hbox{\mathbold\char 72\relax}}^{n},\nu)$. As it turns out, on ${\hbox{\mathbold\char 72\relax}}^{n}$ it is enough to use the highest order derivatives in order to characterize the Sobolev space. In particular, if we define the higher order gradient on ${\hbox{\mathbold\char 72\relax}}^{n}$ as

[TABLE]

then one has that $\|\nabla_{{\hbox{\ssmathbold\char 72\relax}}^{n}}^{\alpha}u\|_{p}$ is an equivalent norm on $W^{\alpha,p}({\hbox{\mathbold\char 72\relax}}^{n})$. In particular, note that we have the Poincaré-Sobolev inequality

[TABLE]

This inequality is proved in [Mancini-Sandeep-Tintarev] in the case of the gradient in the ball model (really a consequence of Hardy’s inequality) and for even $\alpha$ in [Tat].

In this setup sharp versions of the Moser-Trudinger inequality for $W^{\alpha,{n\over\alpha}}({\hbox{\mathbold\char 72\relax}}^{n})$ are only known in the case $\alpha=1$ for the gradient ([MS], [MST], [LT1], [LT2]), and with the same sharp constant as in the Euclidean case. In the following theorem we give the general version of this result for arbitrary $\alpha:$

[TABLE]

and

[TABLE]

and the constant $\gamma_{n,\alpha}$ is sharp.

**Proof. ** If $\alpha$ is even, the operator $(-\Delta_{{\hbox{\ssmathbold\char 72\relax}}^{n}})^{\alpha\over 2}$ has a fundamental solution given by a kernel of type H_{\alpha}\big{(}d(x,y)\big{)}, where $H_{\alpha}$ is positive and satisfies

[TABLE]

(with the same $c_{\alpha}$ as in the Euclidean Riesz potential), and

[TABLE]

some $c_{\alpha}^{\prime}>0$. These asymptotic estimates follow in a straightforward manner from the known formula for the fundamental solution of the Laplacian (see for example [CK])

[TABLE]

using iterated integrations and the known addition formulas for the Riesz potential on ${\hbox{\mathbold\char 82\relax}}^{n}$. (In [BGS] asymptotic formulas are derived for general $\alpha$, using the Fourier transform.)

It is now easy to check that \eqrefZ3 implies that in the measure space $({\hbox{\mathbold\char 72\relax}}^{n},\nu)$ we have

[TABLE]

while \eqrefZ4 implies that H_{\alpha}\big{(}d(\cdot,O)\big{)}\in L^{{n\over n-\alpha}}\cap L^{\infty}\big{(}\{x:d(x,O)\geq 1\},\nu\big{)} (where $O=(1,0,...,0)$) and hence

[TABLE]

Thus, we are in a position to apply Theorem 7 in order to obtain \eqrefZ2 for $\alpha$ even, simply by writing $u(x)=\int H_{\alpha}(d(x,y))f(y)d\nu(y)$, with $f=\nabla_{{\hbox{\ssmathbold\char 72\relax}}^{n}}^{\alpha}u$, for any $u\in C_{0}^{\infty}({\hbox{\mathbold\char 72\relax}}^{n})$.

If $\alpha$ is an odd integer, then we write

[TABLE]

and use asymptotic estimates for $|\nabla_{{\hbox{\ssmathbold\char 72\relax}}^{n}}H_{\alpha+1}|$, which turn out to be the same exact estimates as in \eqrefZ3, \eqrefZ4, with $(n-\alpha-1)c_{\alpha+1}$ instead of $c_{\alpha}$.

The proof of the sharpness statement is identical to the one in the Euclidean case, namely we let $v_{\epsilon}$ to be a smoothing of the radial function

[TABLE]

Using local calculations as in [F, Prop. 3.6] it is a routine task to check that if $\alpha$ is even then

[TABLE]

whereas if $\alpha$ is odd then the same estimate holds with $(n-\alpha-1)c_{\alpha+1}$ in place of $c_{\alpha}$. From this estimate it is then clear that the exponential integral evaluated at the functions $u_{\epsilon}=v_{\epsilon}/\|\nabla_{{\hbox{\ssmathbold\char 72\relax}}^{n}}^{\alpha}v_{\epsilon}\|_{n/\alpha}$ can be made arbitrarily large if the exponential constant is larger than $\gamma_{n,\alpha}$. Note also that $\|v_{\epsilon}\|_{n/\alpha}\leq C$, so that $\|u_{\epsilon}\|_{n/\alpha}\to 0$ with $\epsilon$, and the sharpness statement for the regularized inequality on ${\hbox{\mathbold\char 72\relax}}^{n}$ follows as well.

[TABLE]

6. Connections with the Poincaré inequality:

The Moser-Trudinger inequality on Agmon-Souplet domains

In this section we are concerned with the validity of the Moser-Trudinger inequality

[TABLE]

where $\alpha$ is an integer in $(0,n)$, and $\Omega$ an open set in ${\hbox{\mathbold\char 82\relax}}^{n}$. From \eqrefMTS0 of Corollary 10 we know that if $\Omega$ is Riesz subcritical and the Poincarè inequality holds in the form

[TABLE]

then \eqrefMTS also holds. On the other hand, for $\alpha=1$ Battaglia and Mancini [BM] proved that \eqrefMTS holds if and only if \eqrefP1 holds.

One direction of this result is in some sense an artifact of the exponential regularization. Indeed it is clear from \eqrefexp2 of Lemma 4 that if \eqrefMTS holds under the hypothesis $\|\nabla^{\alpha}u\|_{n/\alpha}\leq~{}1$, then $\|u\|_{n/\alpha}\leq C$ provided that ${n/\alpha}$ is an integer. The interesting part is the reverse implication: when is it true that the Poincaré inequality for $p={n/\alpha}$ implies the Moser-Trudinger inequality \eqrefMTS? We know this fact for $\alpha=1$, by the above-mentioned result in [BM], and for general $\alpha$ in the case of Riesz subcritical domains. The question remains open for the general case $\alpha>1$, since there are are indeed domains $\Omega$ which are not Riesz subcritical and on which the Poincarè inequality holds for any $p>1$:

[TABLE]

To describe such domains in some generality we recall a few definitions.

The inradius of a domain (nonemtpy, open) $\Omega\subseteq{\hbox{\mathbold\char 82\relax}}^{n}$ is defined as

[TABLE]

whereas the strict inradius is defined as

[TABLE]

Note that the set appearing in the definition of $\rho^{\prime}(\Omega)$ is an open interval of type $(0,a)$, so that

[TABLE]

The notion of strict inradius as stated is due to Souplet ([So], [QS]), who proved that a sufficient condition for the validity of the Poincarè inequality \eqrefPoincare is that $\rho^{\prime}(\Omega)<\infty$ (Souplet proved it for $\alpha=1$, but from there it is easy to extend it to any $\alpha\in{\hbox{\mathbold\char 78\relax}}$). This result was due to Agmon in [Ag] in the case $p=2,\alpha=1$ under the condition

[TABLE]

which is equivalent to $\rho^{\prime}(\Omega)<\infty$ on ${\hbox{\mathbold\char 82\relax}}^{n}$. In this paper we will call $\Omega$ an Agmon-Souplet domain if $\rho^{\prime}(\Omega)<\infty$, i.e. if $\Omega$ satisfies condition \eqrefsi3.

Examples of Agmon-Souplet domains include bounded smooth domains (in which case $\rho(\Omega)=\rho^{\prime}(\Omega)$), domains contained in a “strip”, complements of periodic nets of balls whose radii is bounded below by a positive number. It is worth observing that in such domains we cannot expect the Riesz potential to be continuous, nor the Adams inequality to hold. For example, if $\Omega={\hbox{\mathbold\char 82\relax}}^{n}\setminus\bigcup_{{\bf m}\in{\hbox{\mbi\char 90\relax}}^{n}}B({\bf m},\epsilon_{0})$, some fixed small $\epsilon_{0}>0$, then the Agmon-Souplet condition is verified, the Poincare’ inequality holds, however, in the notation of section 3, $\Lambda_{\Omega}(x,r)\geq Cr^{n}$ for any fixed $x\in\Omega$ and for $r\geq r_{0}>0$, hence condition \eqrefG of Theorem 9 is certainly not met, hence the set is not Riesz subcritical and Adams’ inequality fails. However, we are able to show, as a nice application of Theorem 1, that in domains satisfying the Agmon-Souplet condition the Moser-Trudinger inequality actually holds:

**Proof. ** Let us consider first the case $\alpha$ even. If $u\in C_{c}^{\infty}(\Omega)$ with $\|\nabla u\|_{n/\alpha}\leq 1$ then $u=c_{\alpha}I_{\alpha}f$, where $f=\nabla^{\alpha}u=(-\Delta)^{\alpha/2}u$, compactly supported inside $\Omega$. The point is that it is possible to normalize the Riesz kernel so that the critical integrability condition at infinity is satisfied, without interfering with the local asymptotics. Indeed, we can cover $\Omega$ with countably many balls $B(x_{j},R_{1})$, with $x_{j}\in\Omega$. If for each $j$ we pick $x_{j}^{*}\in\Omega^{c}$ according to \eqrefsi3, then $B(x_{j},R_{1})\subseteq B(x_{j}^{*},2R_{1})$ and $B(x^{*}_{j},\epsilon_{1})\subseteq\Omega^{c}$. Since $I_{\alpha}f(x_{j}^{*})=0$ for all $j$, then for each $j$ we can write

[TABLE]

Let

[TABLE]

then $\bigcup_{j}B_{j}\cap\Omega=\Omega$ and the $B_{j}$ are disjoint. For each $(x,y)\in\Omega\times\Omega$ define

[TABLE]

then $u(x)=c_{\alpha}I_{\alpha}f(x)=\int_{\Omega}K(x,y)f(y)dy$ for each $x\in{\rm supp}\,f$, so it is enough to show that $K_{1}^{*}$ and $K_{2}^{*}$ satisfy conditions \eqref106, \eqref107, \eqref108 of Theorem 1.

If $|x-y|\leq\epsilon_{1}$ then $|K(x,y)|\leq c_{\alpha}|x-y|^{\alpha-n}$. On the other hand, for $x\in B_{j}\cap\Omega$ and $y\in\Omega$ we have

[TABLE]

from which we deduce that if $|x-y|\geq\epsilon_{1}$, then $|K(x,y)|\leq C\epsilon_{1}^{\alpha-n-1}R_{1}$, and if $|x-y|\geq 2R_{1}$ then $|x_{j}^{*}-y|\geq{{1\over 2}}|x-y|$ and $|K(x,y)|\leq CR_{1}|x-y|^{\alpha-n-1}$. From these estimates it is straightforward to check that $K_{1}^{*}$ and $K_{2}^{*}$ satisfy \eqref106, \eqref107, and \eqref108 of Theorem 1, with $\beta={n\over n-\alpha}$, $A^{-1}=\gamma(\nabla^{\alpha})$, $\sigma=1$. The proof in the case $\alpha$ odd is similar, starting from the identity

[TABLE]

In this case we normalize the kernel of $J_{\alpha}$ by letting, for $(x,y)\in\Omega\times\Omega$

[TABLE]

If $|x-y|\leq\epsilon_{1}$ then use $|K(x,y)|\leq c_{\alpha+1}(n-\alpha-1)(|x-y|^{\alpha-n}+\epsilon_{1}^{\alpha-n})$, whereas if $|x-y|\geq\epsilon_{1}$ then use for $x\in\Omega\cap B_{j}$, $y\in\Omega$

[TABLE]

and the same estimates as in the case $\alpha$ even apply.

The proof of the sharpness statement is the same one as in the classical case of bounded domains.

[TABLE]

Note that it is possible to construct domains for which the Poincaré inequality \eqrefPoincare holds for all $p\geq 1$ and which are not satisfying the Agmon-Souplet condition. Here’s an outline of this construction. First, we note the following variation of Theorem 9:

[TABLE]

then the Riesz potential is continuous from $L^{p}(\Omega)$ to $L^{p}(\Omega)$. In particular, if $\Omega$ is open, and \eqrefGG holds for at least $\alpha=1$, \eqrefPoincare holds for any integer $\alpha$ and any $p\geq 1$.

**Proof. ** By a standard result the continuity follows from

[TABLE]

which is the same as \eqrefGG, since

[TABLE]

[TABLE]

Condition \eqrefGG is meaningful only for large $r$, say $r\geq 1$, (since the integrand is integrable around 0), and it is clearly stronger than \eqrefG. For example, when $\Omega=(a,b)\times{\hbox{\mathbold\char 82\relax}}^{n-1}$ then \eqrefG holds but \eqrefGG holds only for $0<\alpha<1$.

Now consider the open subset $A\subseteq(0,\infty)$ obtained by removing from each $(k,k+1)$ $2^{k}-1$ equally spaced intervals of length $\delta_{k}<2^{-k}$. Specifically, let

[TABLE]

[TABLE]

If we let S_{m}=\sum_{0}^{m}\big{(}1-(2^{k}-1)\delta_{k}\big{)}, and assuming that $(2^{k}-1)\delta_{k}$ is increasing, then it is easy to see that in ℝ

[TABLE]

Now condition \eqrefGG holds if and only if it holds with $\Lambda_{\Omega}(x,r)$ replaced by

[TABLE]

where $Q(x,r)$ is the cube centered at $x$ and with sidelength $2r$. If we set

[TABLE]

then $\Omega$ is an open set in ${\hbox{\mathbold\char 82\relax}}^{n}$ for which the Agmon-Souplet condition is not satisfied (for each $R,\epsilon>0$ a point $x$ can be chosen far enough inside $\Omega$ so that $B(x,R)\cap\Omega^{c}$ is not empty, but it does not contain any ball of radius $\epsilon$). Yet, Poincaré’s inequality holds in such domain, since

[TABLE]

and one can choose $\delta_{k}$ satisfying the above conditions in a way that \eqrefGG is satisfied for all $\alpha<n$, with $|\Omega|$ either finite or infinite (for example by choosing $\delta_{k}$ so that $(1-(2^{k}-1)\delta_{k})=Ak^{-\gamma}$, for $\gamma>0$ and suitable $A>0$).

6. Appendix

Proof of Lemma 5 (Improved O’Neil’s Lemma)

The given hypothesis \eqrefon1 are equivalent to the weak-type estimates

[TABLE]

and a result due to Adams [A2] gives

[TABLE]

or

[TABLE]

In particular this means that $Tf$ is well defined on $L^{p}(M)$ and bounded from $L^{p}(M)$ to $L^{q,\infty}(N)$.

[TABLE]

[TABLE]

Estimate \eqref(23) is an improvement of the corresponding estimate given in [FM1, Lemma 2], which was given as $(Tf)^{**}(t)\leq\alpha zk_{1}^{**}(z)$. Assuming the Claim, the proof of the lemma proceeds as follows. For fixed $t,\tau>0$, pick $\{y_{n}\}_{-\infty}^{\infty}$ such that $y_{0}=f^{*}(\tau),\,y_{n}\leq y_{n+1},\,y_{n}\to+\infty$ as $n\to+\infty$, and $y_{n}\to 0$ as $n\to-\infty$. Then

[TABLE]

Observe that suppf_{n}\subseteq E_{n}:=\big{\{}y:f(y)>y_{n-1}\big{\}}, $\;\mu(E_{n})=\lambda_{f}(y_{n-1})$, with

[TABLE]

and also $\;0\leq f_{n}(y)\leq y_{n}-y_{n-1}$. Write

[TABLE]

By the subadditivity of $(\cdot)^{**}$ ([BS], Ch. 2, Thm. 3.4), and using \eqref(24) we obtain

[TABLE]

so that taking the inf over all such $\{y_{n}\}$ we get

[TABLE]

(The last inequality follows since $f\in L^{\beta^{\prime}}\Longrightarrow t^{{1\over\beta^{\prime}}}f^{*}(t)\to 0$, as $t\to 0$.)

Using \eqref(23), for $\nu-$a.e. $x\in N$ we have

[TABLE]

and so, arguing as above,

[TABLE]

Finally,

[TABLE]

which, together with \eqref25b, implies (19).

Proof of Claim. Estimate \eqref(24) is an immediate consequence of the weak-type estimate \eqref(22). To show \eqref(23), let $r>0$ and set

[TABLE]

so that

[TABLE]

Then, for every given $x$

[TABLE]

[TABLE]

so that letting $r=k_{1}^{*}(x,z)$ in \eqref(26) and \eqref(27) leads to

[TABLE]

which is \eqref(23).

[TABLE]

Proof of inequality \eqrefAG.

Let

[TABLE]

In what follows we will repeatedly make use of the following inequalities

[TABLE]

[TABLE]

Note that if $0\leq z_{1}\leq z_{2}\leq\eta$ we have

[TABLE]

Also, for $z\geq\eta\geq 0$ we have

[TABLE]

and

[TABLE]

Next, we note that for $\eta>0$ Hölder’s inequality implies

[TABLE]

From now on let

[TABLE]

Now let for $\lambda\in{\hbox{\mathbold\char 82\relax}}$

[TABLE]

and let us prove that there exists $C_{5}$ such that

[TABLE]

Proceeding as in [A1] and [FM1], it is enough to prove that there exists $C_{6}$ such that for any $\lambda\in{\hbox{\mathbold\char 82\relax}}$

[TABLE]

indeed, if that is the case, then

[TABLE]

which implies \eqref3a.

We now prove \eqref3. If $\eta,\eta^{\prime}\in E_{\lambda}$ and $|\lambda|<\eta<\eta^{\prime}$, then $F(\eta^{\prime})\leq\lambda$, so that

[TABLE]

(note that $g(x,\xi,\eta)=g(x,\xi,\eta^{\prime})$ if $\xi\leq\eta\leq\eta^{\prime}$) from which we deduce

[TABLE]

Now we show that there exists $C_{7}$ such that

[TABLE]

Indeed, proceeding as above

[TABLE]

or

[TABLE]

Letting $z=(\eta+d^{*})^{1\over\beta^{\prime}}L(\eta)$ the last inequality can be written as

[TABLE]

which proves \eqref5. Back to \eqref4

[TABLE]

so that

[TABLE]

Taking $\eta>C_{6}(|\lambda|+d^{*}):=(2\beta^{\prime}C_{11})^{{\beta^{\prime}}\over\beta}(|\lambda|+d^{*})$ gives \eta^{\prime}-\eta\leq 2\beta^{\prime}C_{9}\big{(}|\lambda|+d^{*}\big{)}, which is \eqref3.

To complete the proof we now estimate

[TABLE]

[TABLE]

Proof of estimates \eqrefstimaG1 and \eqrefstimaG2

First note that for each real $t>-\delta_{0}$ there is a unique integer $m_{t}\geq 0$ such that

[TABLE]

If $\widetilde{\Lambda}_{\Omega}(x,r)=|\Omega\cap Q(x,r)|$ where $Q(x,r)$ is the cube of center $x$ and side length $2r$, then $\widetilde{\Lambda}_{\Omega}(x,r/\sqrt{n})\leq\Lambda_{\Omega}(x,r)\leq\widetilde{\Lambda}_{\Omega}(x,r)$. If $r>\delta_{0}$ then the definition of $m_{r}$ gives

[TABLE]

Choosing $r>h(4)-\delta_{0}$ we also have $m_{r}\geq 3$ and

[TABLE]

which gives \eqrefstimaG1.

To prove \eqrefstimaG2 note that if $r>\delta_{0}$ and $t>-\delta_{0}$ we have $t+r\leq h(m_{t}+m_{r}+2)-2\delta_{0}$ (if not, then $r>h(m_{t}+m_{r}+2)-t-2\delta_{0}\geq h(m_{t}+m_{r}+2)-h(m_{t}+1)-\delta_{0}\geq h(m_{r}+1)-\delta_{0}$, using that $h(m+1)-h(m)$ is increasing).

Likewise, if $t>h(m_{r}+1)-\delta_{0}$ then $m_{t}\geq m_{r}+1$ and

[TABLE]

so that the number of integers $m$ such that the interval \big{(}h(m)-\delta_{0},h(m)+\delta_{0}\big{)} is inside $(t-r,t+r)$ does not exceed $(m_{t}+m_{r}+2)-(m_{t}-m_{r}-1)=2m_{r}+3$. The same is true if $-\delta_{0}<t\leq h(m_{r}+1)-\delta_{0}$, (which is less than $r$) since $m_{t}+m_{r}+2\leq 2m_{r}+2$. It follows that the number of open cubes centered at $C_{\bf m}$ and with side length $2\delta_{0}$ inside a cube of center $x$ and side length $2r$ does not exceed $(2m_{r}+3)^{n}$. This implies that for $r>h(4)-\delta_{0}$

[TABLE]

which is \eqrefstimaG2.

References

[A1] Adams D.R. A sharp inequality of J. Moser for higher order derivatives, Ann. of Math. 128 (1988), no. 2, 385–398.

[A2] Adams D.R., A trace inequality for generalized potentials, Studia Math. 48 (1973), 99–105.

[Ag] Agmon S., Lectures on Elliptic Boundary Value Problems, Van Nostrand, Princeton, N.J., 1965.

[AT] Adachi S., Tanaka K., Trudinger type inequalities in ${\hbox{\mathbold\char 82\relax}}^{N}$ and their best exponents, Proc. Amer. Math. Soc. 128 (2000), 2051-2057.

[BGS] Banica V., González M.d.M., Sáez M., Some constructions for the fractional Laplacian on noncompact manifolds, Rev. Mat. Iberoam. 31 (2015), 681-712.

[BM] Battaglia L., Mancini G., Remarks on the Moser-Trudinger inequality, Adv. Nonlinear Anal. 2 (2013), 389-425.

[BS] Bennett C., Sharpley R., Interpolation of Operators, Pure and Applied Math., 129, Academic Press, Inc., Boston, MA, 1988.

[C] Cao D. M., Nontrivial solution of semilinear elliptic equation with critical exponent in ${\hbox{\mathbold\char 82\relax}}^{2}$, Comm. Partial Differential Equations 17 (1992), 407-435.

[CK] Cohl H.S., Kalnins E.G., Fourier and Gegenbauer expansions for a fundamental solution of the Laplacian in the hyperboloid model of hyperbolic geometry, J. Phys. A: Math. Theor. 45 (2012), 1-33.

[doÓ] do Ó J.M.B. $N$-Laplacian equations in ${\hbox{\mathbold\char 82\relax}}^{N}$ with critical growth, Abstr. Appl. Anal. 2 (1997), 301-315.

[FM1] Fontana L., Morpurgo C., Adams inequalities on measure spaces, Adv. Math. 226 (2011), 5066-5119.

[FM2] Fontana L., Morpurgo C., Sharp exponential integrability for critical Riesz potentials and fractional Laplacians on ${\hbox{\mathbold\char 82\relax}}^{n}$, Nonlinear Anal. 167 (2018), 85-122.

[FM3] Fontana L., Morpurgo C., Sharp Adams and Moser-Trudinger inequalities on ${\hbox{\mathbold\char 82\relax}}^{n}$ and other spaces of infinite measure, preprint (2015), arXiv:1504.04678.

[LL] Lam N., Lu G., A new approach to sharp Moser-Trudinger and Adams type inequalities: a rearrangement-free argument, J. Differential Equations 255 (2013), 298-325.

[LR] Li Y., Ruf B., A sharp Trudinger-Moser type inequality for unbounded domains in $R^{n}$, Indiana Univ. Math. J. 57 (2008), 451-480.

[Lo] Lorenzi A., On elliptic equations with piecewise constant coefficients. II, Ann. Scuola Norm. Sup. Pisa Cl. Sci. 26 (1972), 839-870.

[LT1] Lu G., Tang H., Best constants for Moser-Trudinger inequalities on high dimensional hyperbolic spaces, Adv. Nonlinear Stud. 13 (2013), 1035-1052.

[LT2] Lu G., Tang H., Sharp Moser-Trudinger Inequalities on Hyperbolic Spaces with Exact Growth Condition, J. Geom. Anal. 26 (2016), 837-857.

[MS] Mancini G., Sandeep K., Moser-Trudinger inequality on conformal discs, Commun. Contemp. Math. 12 (2010), 1055-1068.

[MST] Mancini G., Sandeep K., Tintarev C., Trudinger-Moser inequality in the hyperbolic space ${\hbox{\mathbold\char 72\relax}}^{N}$, Adv. Nonlinear Anal. 2 (2013), 309-324.

[MaS] Masmoudi N., Sani F., Higher order Adams’ inequality with the exact growth condition, Commun. Contemp. Math. 20 (2018), 33 pp.

[OW] Ortner N., Wagner P., Fundamental solutions of linear partial differential operators, Springer 2015.

[P] Panda R. Nontrivial solution of a quasilinear elliptic equation with critical growth in ${\hbox{\mathbold\char 82\relax}}^{n}$, Proc. Indian Acad. Sci. Math. Sci. 105 (1995), 425-444.

[QS] Quittner P., Souplet P., Superlinear parabolic problems. Blow-up, global existence and steady states. Birkhäuser Advanced Texts: Basler Lehrbücher. Birkhäuser Verlag, Basel, 2007.

[RiSt] Ricci F., Stein E.M., Harmonic analysis on nilpotent groups and singular integrals I. Oscillatory integrals, J. Funct. Anal. 73 (1987), 179-194.

[R] Ruf B., A sharp Trudinger–Moser type inequality for unbounded domains in ${\hbox{\mathbold\char 82\relax}}^{2}$, J. Funct. Anal. 219 (2005), 340-367.

[RS] Ruf B., Sani F., Sharp Adams-type inequalities in ${\hbox{\mathbold\char 82\relax}}^{n}$, Trans. Amer. Math. Soc. 365 (2013), 645-670.

[S] Stein E.M., Singular integrals and differentiability properties of functions, Princeton Mathematical Series 30, Princeton University Press, 1970.

[So] Souplet P., Geometry of unbounded domains, Poincaré inequalities and stability in semilinear parabolic equations, Comm. Partial Differential Equations 24 (1999), 951-973.

[Tar] Tarsi C., Adams’ inequality and limiting Sobolev embeddings into Zygmund spaces, Potential Anal. 37 (2012), 353-385.

[Tat] Tataru D., Strichartz estimates in the hyperbolic space and global existence for the semilinear wave equation, Trans. Amer. Math. Soc. 353 (2001), 795-807.

[Tr] Trudinger N.S., On imbeddings into Orlicz spaces and some applications, J. Math. Mech. 17 (1967), 473-483.

[Y] Yang Y., Trudinger-Moser inequalities on complete noncompact Riemannian manifolds, J. Funct. Anal. 263 (2012), 1894-1938.

Luigi Fontana                                                          Carlo Morpurgo

Dipartimento di Matematica ed Applicazioni                  Department of Mathematics

Universitá di Milano-Bicocca                                       University of Missouri

Via Cozzi, 53                                                           Columbia, Missouri 65211

20125 Milano - Italy                                                  USA luigi.fontana@unimib.it                                              morpurgoc@missouri.edu