Wolff's inequality for intrinsic nonlinear potentials and quasilinear elliptic equations

TL;DR

This paper establishes a Wolff's inequality analogue for intrinsic nonlinear potentials linked to quasilinear elliptic equations, providing new criteria for positive solutions in the sub-natural growth case, with implications for measure data problems.

Contribution

It introduces a novel Wolff's inequality for intrinsic nonlinear potentials and characterizes discrete Littlewood-Paley spaces, advancing understanding of quasilinear elliptic equations with measure data.

Findings

Derived necessary and sufficient conditions for positive solutions in L^r spaces.

Extended Wolff's inequality to intrinsic nonlinear potentials.

Provided new characterizations of Littlewood-Paley spaces.

Abstract

We prove an analogue of Wolff's inequality for the so-called intrinsic nonlinear potentials associated with the quasilinear elliptic equation \[ -\Delta_{p} u = \sigma u^{q} \quad \text{in} \;\; \mathbb{R}^n, \] in the sub-natural growth case $0<q< p-1$, where $\Delta_{p}u = \text{div}( |\nabla u|^{p-2} \nabla u )$ is the $p$-Laplacian, and $\sigma$ is a nonnegative measurable function (or measure) on $\mathbb{R}^n$. As an application, we give a necessary and sufficient condition for the existence of a positive solution $u \in L^{r}(\mathbb{R}^{n})$ ($0<r<\infty$) to this problem, which was open even in the case $p=2$. Our version of Wolff's inequality for intrinsic nonlinear potentials relies on a new characterization of discrete Littlewood-Paley spaces $f^{p, q}(\sigma)$ defined in terms of characteristic functions of dyadic cubes in $\mathbb{R}^n$.

Equations

$$-{\Delta }_{p}u=\sigma u^q\text{in}{\mathbb{R}}^n,$$

$${\mathbf{W}}_{\alpha ,r}\sigma (x)={\int }_0^{\infty }{\left[\frac{\sigma (B(x,\rho ))}{s^{n-\alpha r}}\right]}^{\frac{1}{r-1}}\frac{d\rho }{\rho },x\in {\mathbb{R}}^n,$$

$${\mathbf{I}}_{\beta }\sigma (x)={\int }_{{\mathbb{R}}^n}\frac{d\sigma (y)}{|x-y{|}^{n-\beta }},x\in {\mathbb{R}}^n,$$

$${\mathbf{W}}_{\alpha ,r}^d\sigma (x):=\sum _{Q\in \mathcal{Q}}{\left[\frac{\sigma (Q)}{|Q{|}^{1-\frac{\alpha r}{n}}}\right]}^{\frac{1}{r-1}}{\chi }_Q(x),x\in {\mathbb{R}}^n,$$

$${\mathbf{W}}_{\alpha ,r}\sigma (x)\ge c(\alpha ,r,n){\mathbf{W}}_{\alpha ,r}^d\sigma (x).$$

$${\mathcal{E}}_{\alpha ,r}[\sigma ]:={\int }_{{\mathbb{R}}^n}({\mathbf{I}}_{\alpha }\sigma {)}^{r^{\prime }}dx\le C(\alpha ,r){\int }_{{\mathbb{R}}^n}{\mathbf{W}}_{\alpha ,r}\sigma d\sigma ,$$

$${\int }_{{\mathbb{R}}^n}({\mathbf{I}}_{\alpha }\sigma {)}^{r^{\prime }}dx={\int }_{{\mathbb{R}}^n}{\mathbf{V}}_{\alpha ,r}\sigma d\sigma ,$$

$${\mathbf{V}}_{\alpha ,r}\sigma (x):={\mathbf{I}}_{\alpha }[({\mathbf{I}}_{\alpha }\sigma ){]}^{r^{\prime }-1}(x),x\in {\mathbb{R}}^n,$$

$${\mathbf{V}}_{\alpha ,r}\sigma (x)\ge c(\alpha ,r,n){\mathbf{W}}_{\alpha ,r}(x).$$

$$c{\int }_{{\mathbb{R}}^n}{\mathbf{W}}_{\alpha ,r}\sigma d\sigma \le {\mathcal{E}}_{\alpha ,r}[\sigma ]\le C{\int }_{{\mathbb{R}}^n}{\mathbf{W}}_{\alpha ,r}\sigma d\sigma ,$$

$${\mathcal{E}}_{\alpha ,r}[\sigma ]\approx \sum _{Q\in \mathcal{Q}}\frac{[\sigma (Q){]}^{r^{\prime }}}{|Q{|}^{\frac{n-\alpha r}{(r-1)n}}},$$

$$\Vert {\mathbf{W}}_{\alpha ,p}\sigma {\Vert }_{L^r({\mathbb{R}}^n)}^{p-1}\approx \Vert {\mathbf{I}}_{\alpha p}\sigma {\Vert }_{L^{\frac{r}{p-1}}({\mathbb{R}}^n)}.$$

$${\int }_{R^n}({\mathbf{W}}_{\alpha ,p}\sigma {)}^{r}dx\approx {\int }_{R^n}({\mathbf{W}}_{\alpha ,p}^d\sigma {)}^{r}dx\approx \sum _{Q\in \mathcal{Q}}\frac{[\sigma (Q){]}^{\frac{r}{p-1}}}{|Q{|}^{\frac{(n-\alpha p)r-n(p-1)}{(p-1)n}}},$$

$$\{\begin{array}{l}-{\Delta }_{p}u=\sigma \text{in}{\mathbb{R}}^n,\\ \underset{|x|\to \infty }{\mathrm{liminf}}u(x)=0,\end{array}$$

$$K^{-1}{\mathbf{W}}_{1,p}\sigma (x)\le u(x)\le K{\mathbf{W}}_{1,p}\sigma (x).$$

$${\int }_1^{\infty }{\left[\frac{\sigma (B(0,\rho ))}{s^{n-\alpha r}}\right]}^{\frac{1}{r-1}}<\infty .$$

$$\{\begin{array}{l}-{\Delta }_{p}u=\sigma u^q,u\ge 0\text{in}{\mathbb{R}}^n,\\ \underset{x\to \infty }{\mathrm{liminf}}u(x)=0,\end{array}$$

$$c^{-1}[({\mathbf{W}}_{1,p}\sigma {)}^{\frac{p-1}{p-1-q}}+{\mathbf{K}}_{1,p,q}\sigma ]\le u\le c[({\mathbf{W}}_{1,p}\sigma {)}^{\frac{p-1}{p-1-q}}+{\mathbf{K}}_{1,p,q}\sigma ],$$

$${\left({\int }_B|\phi {|}^{q}d\sigma \right)}^{\frac{1}{q}}\le \varkappa (B)||{\Delta }_p\phi |{|}_{L^1({\mathbb{R}}^n)}^{\frac{1}{p-1}},$$

$${\mathbf{K}}_{1,p,q}\sigma (x)={\int }_0^{\infty }{\left[\frac{\varkappa (B(x,r){)}^{\frac{q(p-1)}{p-1-q}}}{r^{n-p}}\right]}^{\frac{1}{p-1}}\frac{dr}{r},x\in {\mathbb{R}}^n.$$

$${\int }_1^{\infty }{\left[\frac{\varkappa (B(0,r){)}^{\frac{q(p-1)}{p-1-q}}}{r^{n-p}}\right]}^{\frac{1}{p-1}}\frac{dr}{r}<\infty .$$

$${\mathbf{K}}_{1,p,q}^d\sigma (x)=\sum _{Q\in \mathcal{Q}}{\left[\frac{\varkappa (Q{)}^{\frac{q(p-1)}{p-1-q}}}{|Q{|}^{1-\frac{p}{n}}}\right]}^{\frac{1}{p-1}}{\chi }_Q(x),x\in {\mathbb{R}}^n.$$

$${\mathbf{W}}_{1,p}\sigma \in L^{\frac{r(p-1)}{p-1-q}}({\mathbb{R}}^n)\text{and}{\mathbf{K}}_{1,p,q}\sigma \in L^r({\mathbb{R}}^n).$$

$$\{\begin{array}{l}{\left(-\Delta \right)}^{\alpha }u=\sigma u^q,u\ge 0\text{in}{\mathbb{R}}^n,\\ \underset{x\to \infty }{\mathrm{liminf}}u(x)=0,\end{array}$$

$$\sum _{Q\in \mathcal{Q}}\frac{[\varkappa (Q){]}^{\frac{qr}{p-1-q}}}{|Q{|}^{\frac{(n-p)r-n(p-1)}{n(p-1)}}}<\infty .$$

$$\Vert {\mathbf{K}}_{1,p,q}\sigma {\Vert }_{L^r({\mathbb{R}}^n)}^r\approx \sum _{Q\in \mathcal{Q}}\frac{[\varkappa (Q){]}^{\frac{qr}{p-1-q}}}{|Q{|}^{\frac{(n-p)r-n(p-1)}{n(p-1)}}},$$

$${\int }_{{\mathbb{R}}^n}{\int }_0^{\infty }\frac{[\varkappa (B(x,\rho )){]}^{\frac{qr}{p-1-q}}}{{\rho }^{\frac{(n-p)r}{p-1}}}\frac{d\rho }{\rho }dx<\infty .$$

$${\mathbf{W}}_{1,p}\sigma \in L^{\frac{r(p-1)}{p-1-q}}({\mathbb{R}}^n).$$

$$\sum _{Q\in \mathcal{Q}}\frac{[\sigma (Q){]}^{\frac{r}{p-1-q}}}{|Q{|}^{\frac{(n-p)r-n(p-1-q)}{n(p-1-q)}}}<\infty .$$

$$u={\mathbf{W}}_{\alpha ,p}(u^{q}d\sigma )\text{in}{\mathbb{R}}^n.$$

Full text

[TABLE]

in the sub-natural growth case $0<q<p-1$, where $\Delta_{p}u=\text{div}(|\nabla u|^{p-2}\nabla u)$ is the $p$-Laplacian, and $\sigma$ is a nonnegative measurable function (or measure) on $\mathbb{R}^{n}$.

As an application, we give a necessary and sufficient condition for the existence of a positive solution $u\in L^{r}(\mathbb{R}^{n})$ ($0<r<\infty$) to this problem, which was open even in the case $p=2$.

Our version of Wolff’s inequality for intrinsic nonlinear potentials relies on a new characterization of discrete Littlewood-Paley spaces $f^{p,q}(\sigma)$ defined in terms of characteristic functions of dyadic cubes in $\mathbb{R}^{n}$.

1. Introduction

Let $\mathcal{M}^{+}(\mathbb{R}^{n})$ denote the class of all locally finite Borel measures on $\mathbb{R}^{n}$. For $1<r<\infty$ and $0<\alpha<\frac{n}{r}$, the Wolff potential, or, more precisely, Havin-Maz’ya-Wolff potential (see [AH], [HM], [HW], [KM], [Maz]) ${\bf{W}}_{\alpha,r}\sigma$ of a measure $\sigma\in\mathcal{M}^{+}(\mathbb{R}^{n})$ is defined by

[TABLE]

where $B(x,\rho)=\{y\in\mathbb{R}^{n}:|x-y|<\rho\}$ is a ball centered at $x\in\mathbb{R}^{n}$ of radius $\rho>0$.

In the linear case $r=2$, the potential ${\bf{W}}_{\alpha,r}\sigma$ reduces (up to a constant multiple) to the Riesz potential ${\bf{I}}_{2\alpha}\sigma$, where

[TABLE]

is the Riesz potential of order $\beta\in(0,n)$.

In [HW], a useful dyadic version of ${\bf{W}}_{\alpha,r}$ was introduced:

[TABLE]

where the sum is taken over all dyadic cubes $Q\in\mathcal{Q}$.

Clearly,

[TABLE]

The converse inequality can be recovered, as usual, by replacing $\mathcal{Q}$ in (1.2) with a shifted dyadic lattice $\mathcal{Q}_{t}=\{Q+t\}$ ($t\in\mathbb{R}^{n}$), and then averaging over all $t\in B(0,1)$ (see, for instance, [COV2], [COV3], [V]). Wolff’s inequality obtained in [HW] says that the $(\alpha,r)$-energy

[TABLE]

where $1<r<\infty$ and $r^{\prime}=\frac{r}{r-1}$. The converse inequality holds as well, since obviously,

[TABLE]

where

[TABLE]

is the Havin-Maz’ya potential introduced in [HM]. It is easy to see (see [HM], [Maz]) that

[TABLE]

Hence, Wolff’s inequality demonstrates that

[TABLE]

where positive constants $c,C$ depend only on $\alpha,r,n$.

One can also use dyadic potentials ${\bf{W}}^{d}_{\alpha,r}\sigma$ in place of ${\bf{W}}_{\alpha,r}\sigma$ in (1.4), which yields the following discrete form of Wolff’s inequality ([HW]):

[TABLE]

where the constants of equivalence depend only on $\alpha,r,$ and $n$.

There are similar Wolff’s inequalities for potentials ${\bf{W}}_{\alpha,p}\sigma$, since for any $r>0$, $1<p<\infty$ and $0<\alpha<\frac{n}{p}$, we have

[TABLE]

Consequently,

[TABLE]

where the constants of equivalence depend only on $\alpha,p,r,$ and $n$.

Several proofs of (1.4) and its variations are known; in particular, it can be deduced from a weighted norm inequality of Muckenhoupt and Wheeden for fractional integrals [MW] (see also [AH], [HJ], [JPW], [V]). However, the original proof [HW] via dyadic potentials ${\bf{W}}^{d}_{\alpha,r}\sigma$ is most direct, and useful in more general situations. (See, for instance, a two-weight version and its applications in [COV3], [HV1], [HV2].)

The following important result due to T. Kilpeläinen and J. Malý [KiMa] gives precise pointwise estimates of $p$-superharmonic solutions $u\geq 0$ to the equation

[TABLE]

in terms of potentials ${\bf{W}}_{1,p}\sigma$: Let $\sigma\in\mathcal{M}^{+}(\mathbb{R}^{n})$ and $p>1$. Suppose $u$ is a $p$-superharmonic function in $\mathbb{R}^{n}$ satisfying (1.7). Then there exists a positive constant $K=K(n,p)$ such that

[TABLE]

Moreover, such a solution to (1.7) exists if and only if $1<p<n$, and ${\bf{W}}_{1,p}\sigma(x)<\infty$ for some $x\in\mathbb{R}^{n}$, or equivalently,

[TABLE]

If (1.9) holds, then ${\bf{W}}_{1,p}\sigma(x)<\infty$, $dx$-a.e. (and quasi-everywhere).

Throughout this paper, we use $p$-superharmonic solutions, or equivalently, locally renormalized solutions to equations involving the $p$-Laplace operator. We refer to [HKM], [KKT] for the corresponding definitions and properties of such solutions.

Let us now consider the quasilinear elliptic problem

[TABLE]

in the sub-natural growth case $0<q<p-1$, where $\sigma\in\mathcal{M}^{+}(\mathbb{R}^{n})$. We assume here that $u\in L^{q}_{loc}(\Omega,d\sigma)$, so that the right-hand side of (1.10) is a Radon measure, and we can use $p$-superharmonic, or locally renormalized solutions $u$, as in the case of (1.7).

The existence of solutions $u\in L^{\infty}(\mathbb{R}^{n})$ to (1.10) was characterized by Brezis and Kamin [BK] in the case $p=2$. They also proved uniqueness of bounded solutions. In fact, for all $1<p<\infty$, a solution $u\in L^{\infty}(\mathbb{R}^{n})$ to (1.10) exists if and only if ${\bf{W}}_{1,p}\sigma\in L^{\infty}(\mathbb{R}^{n})$ (see [CV3]). However, a similar problem for solutions $u\in L^{r}(\mathbb{R}^{n})$ with $r<\infty$ turned out to be more complicated. Some sharp sufficient conditions for that were established recently in [SV3] (see also [SV1], [SV2] where finite energy solutions and their generalizations are treated).

In this paper, we give a necessary and sufficient condition on $\sigma$, in terms of integrability of nonlinear potentials, for the existence of a positive solution $u\in L^{r}(\mathbb{R}^{n})$, $0<r<\infty$, to problem (1.10).

The following bilateral pointwise estimates of nontrivial (minimal) solutions $u$ to (1.10) in the case $0<q<p-1$ were obtained in [CV2]:

[TABLE]

where $c>0$ is a constant which depends only on $p$, $q$, and $n$.

Here $\mathbf{K}_{1,p,q}$ is the so-called intrinsic Wolff potential associated with (1.10), which was introduced in [CV2]. It is defined in terms of the localized weighted norm inequalities,

[TABLE]

for all test functions $\varphi$ such that $-\Delta_{p}\varphi\geq 0$, $\displaystyle{\liminf_{x\to\infty}}\,\varphi(x)=0$. Here $\varkappa(B)$ denotes the least constant in (1.12) associated with the measure $\sigma_{B}=\sigma|_{B}$ restricted to a ball $B$. Then the intrinsic potential $\mathbf{K}_{1,p,q}$ is defined by

[TABLE]

As was shown in [CV2], $\mathbf{K}_{1,p,q}\sigma\not\equiv+\infty$ if and only if

[TABLE]

In a similar way, we define constants $\varkappa(Q)$ for cubes $Q$ in place of $B$, and the dyadic potentials

[TABLE]

More general fractional potentials $\mathbf{K}_{\alpha,p,q}$, along with their dyadic analogues, are defined in Sec. 2.

Thus, a necessary and sufficient condition for the existence of a solution $u\in L^{r}(\mathbb{R}^{n})$ to (1.10) is given by:

[TABLE]

In fact, as we will show below, the first condition in (1.16) is a consequence of the second one, despite differences in pointwise behavior of $({\bf{W}}_{1,p}\sigma)^{\frac{p-1}{p-1-q}}$ and $\mathbf{K}_{1,p,q}\sigma$.

Moreover, we will simplify to some degree the second condition in (1.16) by proving an analogue of Wolff’s inequality (1.6) for potentials $\mathbf{K}_{1,p,q}\sigma$.

Similar results hold for the fractional Laplace problem

[TABLE]

where $0<q<1$ and $0<\alpha<\frac{n}{2}$. They are new even in the classical case $\alpha=1$, or if $\sigma$ is a locally integrable function on $\mathbb{R}^{n}$.

Theorem 1.1**.**

Let $1<p<n$, $0<q<p-1$, $\frac{n(p-1)}{n-p}<r<\infty$, and $\sigma\in\mathcal{M}^{+}(\mathbb{R}^{n})$ with $\sigma\not\equiv 0$. Then the following conditions are equivalent.

(i) There exists a positive $p$-superharmonic (super) solution $u\in L^{r}(\mathbb{R}^{n})$ to (1.10).

(ii) $\mathbf{K}_{1,p,q}\sigma\in L^{r}(\mathbb{R}^{n})$.

(iii)

[TABLE]

Moreover,

[TABLE]

where the constants of equivalence depend only on $p,q,r$, and $n$.

Remark 1.2**.**

It is easy to see that if $n\leq p<\infty$, or $1<p<n$ and $0<r\leq\frac{n(p-1)}{n-p}$, then there is only a trivial nonnegative supersolution $u\in L^{r}(\mathbb{R}^{n})$ to (1.10). Simpler sufficient conditions for (1.18) in the case $1<p<n$, $\frac{n(p-1)}{n-p}<r<\infty$, are given in [SV3]*Theorem 1.1.

Remark 1.3**.**

A condition equivalent to (1.18) can be stated in terms of $\varkappa(B)$ for balls $B=B(x,\rho)$ in place of dyadic cubes $Q$,

[TABLE]

A necessary (but generally not sufficient) condition for the existence of a nontrivial solution $u\in L^{r}(\mathbb{R}^{n})$ to (1.10) follows from (1.16),

[TABLE]

By Wolff’s inequality, (1.21) is equivalent to the condition

[TABLE]

Remark 1.4**.**

Theorem 1.1 holds for the $\mathcal{A}$-Laplacian in place of $\Delta_{p}$, under the standard structural assumptions on $\mathcal{A}$ (see [CV2], [HKM], [MZ]).

Our methods are applicable to intrinsic nonlinear potentials of fractional order and nonlinear integral equations of the type

[TABLE]

Here, a solution $u\geq 0$ is understood in the sense that $u\in L^{q}_{loc}(\mathbb{R}^{n},\sigma)$ satisfies (1.22). In the special case $p=2$, this integral equation, namely $u={\bf{I}}_{2\alpha}(u^{q}d\sigma)$, is equivalent to the corresponding problem for the fractional Laplacian (1.17).

Theorem 1.5**.**

Let $0<q<p-1$, $1<p<\infty$, $0<\alpha<\frac{n}{p}$ and $\sigma\in\mathcal{M}^{+}(\mathbb{R}^{n})$ with $\sigma\not\equiv 0$. Suppose that $\frac{n(p-1)}{n-\alpha p}<r<\infty$. Then there exists a positive solution $u\in L^{r}(\mathbb{R}^{n})$ to (1.22) if and only if $\mathbf{K}_{\alpha,p,q}\sigma\in L^{r}(\mathbb{R}^{n})$. Moreover,

[TABLE]

where the constants of equivalence depend only on $\alpha,p,q,r$, and $n$.

If $0<r\leq\frac{n(p-1)}{n-\alpha p}$, then there is only a trivial supersolution to (1.17).

In (1.23), we employ the localized embedding constants $\kappa(Q)$ associated with certain weighted norm inequalities for potentials ${\bf{W}}_{\alpha,p}$. They are used to define the intrinsic potentials $\mathbf{K}_{\alpha,p,q}\sigma$ and their dyadic analogues in the same manner as constants $\varkappa(Q)$ in the case $\alpha=1$ above (see Sec. 2).

A simple necessary, but not sufficient, condition for (1.23) is given by

[TABLE]

This paper is organized as follows. In Sec. 2, we give definitions of nonlinear potentials $\mathbf{K}_{\alpha,p,q}$ and discuss some of their properties. New expressions for norms of sequences in discrete Littlewood–Paley spaces $\mathbf{f}^{p,q}(\sigma)$ are discussed in Sec. 3. They are used in Sec. 4, where we prove Theorem 1.1 and Theorem 1.5.

2. Nonlinear potentials

Havin-Maz’ya potentials ${\bf{V}}_{\alpha,p}\sigma$ are known to satisfy the weak maximum (or boundedness) principle (see [AH]*Theorem 2.6.3). A similar weak maximum principle holds for Wolff potentials: If $\sigma\in\mathcal{M}^{+}(\mathbb{R}^{n})$, then

[TABLE]

Indeed, let $K=\textrm{supp}(\sigma)$. Suppose $x\not\in K$, and $x_{0}\in K$ minimizes the distance from $x$ to $K$. Then, clearly, $B(x,r)\subset B(x_{0},2r)$, for any $r>0$. Consequently,

[TABLE]

Let $1<p<\infty$, $0<\alpha<\frac{n}{p}$, and $0<q<p-1$. Let $\sigma\in\mathcal{M}^{+}(\mathbb{R}^{n})$. We denote by $\kappa$ the least constant in the weighted norm inequality

[TABLE]

We will also need a localized version of (2.2) for $\sigma_{E}=\sigma|_{E}$, where $E$ is a Borel subset of $\mathbb{R}^{n}$, and $\kappa(E)$ is the least constant in

[TABLE]

In applications, it will be enough to use $\kappa(E)$ where $E=B$ is a dyadic cube $Q$, or a ball in $\mathbb{R}^{n}$.

It is easy to see using estimates (1.8) that embedding constants $\kappa(B)$ in the case $\alpha=1$ are equivalent to the constants $\varkappa(B)$ in (1.12).

We define the intrinsic potential of Wolff type in terms of $\kappa(B(x,s))$, the least constant in (2.3) with $E=B(x,s)$:

[TABLE]

It is easy to see that $\mathbf{K}_{\alpha,p,q}\sigma(x)\not\equiv\infty$ if and only if

[TABLE]

for any (all) $a>0$.

As in the case of Wolff potentials ${\bf{W}}_{\alpha,p}$, sometimes a more convenient dyadic version of $\mathbf{K}_{\alpha,p,q}$ is useful:

[TABLE]

Similarly to (2.5), $\mathbf{K}_{\alpha,p,q}\not\equiv\infty$ if and only if

[TABLE]

for $P\in\mathcal{Q}$.

3. Equivalent norms on discrete Littlewood-Paley spaces

In this section, we give some new equivalent norms for discrete Littlewood-Paley spaces with respect to an arbitrary measure $\sigma\in\mathcal{M}^{+}(\mathbb{R}^{n})$ (see [CV], [FJ]), [HV1]). In this paper, we will need them only in the case of Lebesgue measure, but a more general setup is useful in various applications in harmonic analysis and PDE ([COV1]–[COV3], [HV1], [HV2]). In particular, they give new characterizations of the discrete Carleson embedding theorem in the case $0<q<1<p$ (see Corollary 3.3 below).

Let $\sigma\in\mathcal{M}^{+}(\mathbb{R}^{n})$. We use the notation $|E|_{\sigma}=\sigma(E)$, for Borel sets $E\subset\mathbb{R}^{n}$; $|E|$ stands for Lebesgue measure of $E$.

Let $\Lambda=(\lambda_{Q})_{Q\in{\mathcal{Q}}}$ be a sequence of nonnegative reals. We denote by $\mathcal{D}$ the collection of all dyadic cubes $Q\in\mathcal{Q}$ such that $|Q|_{\sigma}\not=0$.

For $0<\tau<\infty$, $0<r<\infty$, and $-\infty<q\leq\infty$ ($q\not=0$), we set

[TABLE]

[TABLE]

For $0<r<1$, we set

[TABLE]

where the supremum is taken over all sequences of nonnegative reals $\nu=(\nu_{R})$ such that $\nu_{Q}=0$ if $|Q|_{\sigma}=0$, and

[TABLE]

In other words, the supremum on the right-hand side of (3.3) is taken over all Carleson sequences $\nu=(\nu_{R})$ such that $||\nu||_{f^{\infty,\frac{1}{1-r}}_{0}(\sigma)}\leq 1$.

For $-\infty<r<\infty$ and $-\infty<q<\infty$ ($q\not=0$), we set

[TABLE]

We observe that $a_{4}(\Lambda)$ coincides with $a_{2}(\Lambda)$ in the special case $\tau=1$, $r>0$.

The following duality lemma is known in the case $0<r<1$ (see [CV], [HV1]).

Lemma 3.1**.**

Suppose $0<r<1$. There exists a positive constant $C$ depending only on $r$, $q$, and $\tau$ such that

[TABLE]

Theorem 3.2**.**

Let $0<q<r<\infty$, and $0<\tau<\infty$. Then there exists a positive constant $C$ depending only on $r$, $q$, and $\tau$ such that

[TABLE]

Theorem 3.2 is a consequence of the lemmas proved below.

The following corollary is immediate from Theorem 3.2.

Corollary 3.3**.**

Let $\sigma\in M^{+}(\mathbb{R}^{n})$. Suppose $0<q<1<p<\infty$. Then the following statements are equivalent.

(i) The “one-weight” inequality holds,

[TABLE]

for all $f\in L^{p}(\sigma)$.

(ii) $\sigma$ satisfies the condition

[TABLE]

where $r=\frac{pq}{p-q}$.

(iii) $\sigma$ satisfies the condition

[TABLE]

where $\tau=\frac{q(p-1)}{p-q}$, and

[TABLE]

We now prove a series of lemmas used in the proof of Theorem 3.2. Some of them might be of independent interest.

Lemma 3.4**.**

Let $\sigma\in M^{+}(\mathbb{R}^{n})$, and let $-\infty<q\leq\infty$.

(i) If either $0<r<1$ and $-(1-r)\leq q<\infty$, or $r\geq 1$ and $0<q<r<\infty$, then

[TABLE]

where $C$ is a positive constant depending only on $r$.

(ii) If either $-\infty<q<r<1$, or $r\geq 1$ and $0<q<\infty$, then the converse inequality holds:

[TABLE]

where $C$ is a positive constant depending only on $r$ and $q$.

Proof.

We first prove statement (i) in the case $0<r<1$. Since $q\geq-(1-r)$, we estimate using Jensen’s inequality,

[TABLE]

Hence,

[TABLE]

where we used summation by parts in the last line.

In the case $r\geq 1$ and $0<q<r<\infty$, we use the maximal function inequality for the dyadic maximal operator $M_{\sigma}:\,L^{\frac{r}{q}}(\sigma)\to L^{\frac{r}{q}}(\sigma)$, $\frac{r}{q}>1$. Letting $\phi=(\sum_{R\in\mathcal{D}}\lambda_{R}\,\chi_{R})^{q}$, we estimate

[TABLE]

To prove statement (ii), by Jensen’s inequality it suffices to assume $q>0$. The case $r=1$ is trivial. Suppose $0<q<r<1$. Then

[TABLE]

Let $\phi=(\sum_{R\in\mathcal{D}}\lambda_{R}\,\chi_{R})^{q}$. To complete the proof of (3.10), it remains to show that, for $r>q$,

[TABLE]

The preceding inequality is proved using Hölder’s inequality with exponents $\frac{1}{r}$ and $\frac{1}{1-r}$, together with the maximal function inequality in $L^{\frac{r}{q}}(\sigma)$:

[TABLE]

which yields (3.11). Thus, $a_{4}(\Lambda)\geq C\,a_{1}(\Lambda)$.

We now consider the case $r>1$. By Jensen’s inequality, it suffices to consider $q>0$ small enough, so without loss of generality we will assume $0<q<\min(1,r-1)$.

If $1<r<\infty$, then by (3.13),

[TABLE]

We will also need the elementary summation by parts inequality, for $r\geq 1$ (see [COV2]),

[TABLE]

We consider separately two subcases, $1+q<r\leq 2$, and $r>2$.

Suppose first that $1+q<r\leq 2$. Then by Hölder’s inequality with exponents $t=\frac{1-q}{2-r}>1$ and $t^{\prime}=\frac{1-q}{r-1-q}$,

[TABLE]

Substituting this estimate into (3.12), we obtain

[TABLE]

Using Hölder’s inequality for sums with exponents $s=\frac{(1-q)(r-1)}{(2-r)q}$ and $s^{\prime}=\frac{(1-q)(r-1)}{r-1-q}$, we estimate

[TABLE]

By the known estimate for $r\geq 1$ (see [COV2]),

[TABLE]

where $C>0$ is a constant which depends only on $r$. Hence,

[TABLE]

which yields $a_{1}(\Lambda)\leq C\,a_{4}(\Lambda)$.

It the second subcase $r>2$, assuming as above that $0<q<1$, we estimate by Hölder’s inequality with exponents $t=\frac{r-1-q}{1-q}>1$ and $t^{\prime}=\frac{r-1-q}{r-2}$,

[TABLE]

By (3.14) and the preceding estimate,

[TABLE]

Using now Hölder’s inequality with exponents $s=\frac{r-1-q}{q(r-2)}>1$ and $s^{\prime}=\frac{r-1-q}{(r-1)(1-q)}$ for sums, so that $\frac{(r-1)s}{t^{\prime}}=\frac{r-1}{q}$ and $\frac{(r-1)s^{\prime}}{t}=1$, we estimate,

[TABLE]

where we used (3.12) again in the last line. This completes the proof of statement (ii). ∎

Lemma 3.5**.**

Let $\sigma\in M^{+}(\mathbb{R}^{n})$, and let $0<q\leq\infty$.

(i) If either $0<\tau\leq\min(r,1)$, or $q<r<\tau<\infty$, then

[TABLE]

where $C$ is a positive constant depending only on $r$, $q$, and $\tau$.

(ii) If $\max(q,\tau)<r<\infty$, then the converse inequality holds:

[TABLE]

where $C$ is a positive constant depending only on $r$, $q$, and $\tau$.

Proof.

We first prove statement (i). Let $0<r<1$.

Suppose $0<\tau\leq r$. Set

[TABLE]

Suppose $\nu=(\nu_{R})$ is a Carleson sequence such that $||\nu||_{f^{\infty,\frac{1}{1-r}}_{0}(\sigma)}\leq 1$ as in (3.3). Let $s=\frac{1-\tau}{1-r}\geq 1$. By Hölder’s inequality with exponents $s$ and $s^{\prime}$,

[TABLE]

Note that $(r-\frac{\tau}{s})s^{\prime}=1$, and $\gamma=\frac{r-\tau}{q}$, so that

[TABLE]

Letting

[TABLE]

we see that $||\mu||_{f^{\frac{1}{1-\tau},\infty}_{0}(\sigma)}\leq 1$, that is,

[TABLE]

It follows from (3.6) and (3.3) with the exponent $\tau$ in place of $r$, and $\mu_{R}$ in place of $\nu_{R}$,

[TABLE]

Lemma 3.4 (i) yields

[TABLE]

Combining the preceding estimates, we obtain

[TABLE]

which completes the proof of (3.15) in the case $0<\tau\leq r<1$.

In the case $0<\tau<1$, $r\geq 1$, we set

[TABLE]

where $\rho_{R}$ is defined by (3.8). Using summation by parts, we estimate

[TABLE]

We denote by $M_{\sigma}^{R}$ the dyadic maximal operator scaled to a cube $R$:

[TABLE]

Clearly,

[TABLE]

Hence, by Jensen’s inequality and the maximal inequality for $M_{\sigma}^{R}$,

[TABLE]

where in the last line we used Kolmogorov’s maximal inequality for $M^{R}_{\sigma}:\,L^{1}(\omega)\to L^{\epsilon}(\omega)$, $0<\epsilon<1$, with the probability measure $d\omega=\frac{1}{|R|_{\sigma}}\chi_{R}d\sigma$, applied to $f=\rho_{R}^{q}$ .

Consequently,

[TABLE]

By Lemma 3.4, we have $a_{4}(\Lambda)\geq Ca_{1}(\Lambda)$, which completes the proof of (3.15) in the case $0<\tau<1$, $r\geq 1$.

Suppose now that $\tau>r>q$. Then

[TABLE]

The last inequality follows, as in the proof of Lemma 3.4, by letting \phi=\Big{(}\sum_{Q\in\mathcal{D}}\lambda_{Q}\,\chi_{Q}\Big{)}^{q}, and applying Hölder’s inequality with exponents $\frac{\tau}{r}$ and $(\frac{\tau}{r})^{\prime}$, together with the maximal function inequality in $L^{\frac{r}{q}}(d\sigma)$ for $\frac{r}{q}>1$:

[TABLE]

This proves the inequality $a_{1}(\Lambda)\leq C\,a_{2}(\Lambda)$.

The converse inequality for $\max(q,\tau)<r<\infty$ is immediate from the maximal function inequality in $L^{\frac{r}{q}}(\sigma)$: if \phi=\Big{(}\sum_{Q\in\mathcal{D}}\lambda_{Q}\,\chi_{Q}\Big{)}^{q}, then

[TABLE]

∎

Lemma 3.6**.**

Let $\sigma\in M^{+}(\mathbb{R}^{n})$, and let $0<q\leq\infty$.

(i) If $1\leq\tau<r$, then

[TABLE]

(ii) If $0<r<\tau$, then

[TABLE]

Proof.

Since $r\geq 1$, by Lemma 3.4 (ii), for every $s>0$,

[TABLE]

On the other hand, letting

[TABLE]

and applying Lemma 3.4 (i) with $r=\tau\geq 1$, $q=1$, and $\lambda_{R}\,a_{R}$ in place of $\lambda_{R}$, we obtain

[TABLE]

By Jensen’s inequality, it suffices to prove (3.15) for $q$ small, so that we may assume without loss of generality $r>\tau(1+q)$. Then $\gamma=\frac{1}{q}(\frac{r}{\tau}-1)>1$.

Notice that

[TABLE]

Next, we estimate using Jensen’s inequality for sums,

[TABLE]

We simplify using summation by parts,

[TABLE]

Hence, combining the preceding estimates and using the interpolation inequality,

[TABLE]

we obtain

[TABLE]

where in the last line we used Lemma 3.4 (ii) with $q=1$ in the expression for $a_{4}(\Lambda)$, and $r\geq 1$. This completes the proof of statement (i) of Lemma 3.6.

To prove statement (ii), we may assume without loss of generality that $q$ is small enough; in particular, $0<q\leq\frac{r}{\tau}$, where $r<\tau$. Let $\gamma=-\frac{1}{q}(\frac{r}{\tau}-1)>0$.

Consider first the case $\tau\geq 1$. By Lemma 3.4 (ii), with $\tau$ in place of $r$ and $q=1$, we estimate

[TABLE]

Note that

[TABLE]

for any $\epsilon>0$. Using the preceding inequality with $\epsilon=1-\frac{r}{\tau}$, we estimate

[TABLE]

where we used summation by parts in the last line.

Since we are assuming that $0<q\leq\frac{r}{\tau}<1$, it follows by Jensen’s inequality,

[TABLE]

where in the last inequality we used Lemma 3.4 (ii) with $\frac{r}{\tau}$ in place of $q$.

In the case $r<\tau<1$, we can assume again that $q$ is small enough; in particular, $0<q<r$. Using Lemma 3.4 (ii) again for the sequence $\lambda_{R}\,a_{R}$, with $r_{1}=\tau$ in place of $r$, and $q_{1}=s$ in place of $q$ where $0<s<\tau$, in the expression for $a_{4}(\Lambda)$, we have

[TABLE]

Let $\phi_{R}=(\sum_{Q\subseteq R}\lambda_{Q}\,\chi_{Q})^{q}$. Then

[TABLE]

where $M^{R}_{\sigma}$ is the localized maximal function (3.19). Hence,

[TABLE]

By Hölder’s inequality and the maximal function inequality (see (3.11)),

[TABLE]

where $p=\frac{s}{q}>1$ and $\delta=\frac{r}{\tau}<1$. Hence,

[TABLE]

Assuming without loss of generality that $q$ is small enough, so that $0<q\leq\frac{rs}{\tau}$, and using Jensen’s inequality we estimate

[TABLE]

Consequently, for $r<\tau<1$,

[TABLE]

which proves statement (ii).∎

The following corollary, which is merely a combination of Lemma 3.5 and Lemma 3.6, yields Theorem 3.2.

Corollary 3.7**.**

Let $\sigma\in M^{+}(\mathbb{R}^{n})$, and let $0<q\leq\infty$.

(i) If either $0\leq\tau\leq r$, or $r<q<\tau$, then

[TABLE]

(ii) If $0<q<r$ and $0<\tau\leq r$, then

[TABLE]

Remark 3.8**.**

Statement (i) of Corollary 3.7 fails if $0<r<\min(\tau,q)$; statement (ii) fails if $q>r>\tau$.

4. Proofs of Theorems 1.1 and 1.5

It is shown in [CV2] that (1.10) has a positive (super) solution if and only if the same is true for (1.22) in the case $\alpha=1$. Moreover, the conditions in Theorems 1.1 and 1.5 are equivalent, since one can use embedding constants $\kappa(B)$ in place of $\varkappa(B)$ if $\alpha=1$ (see Sec. 2). Thus, it suffices to prove only Theorem 1.5.

Let $u\in L^{q}_{{\rm loc}}(\sigma)$ ($u\geq 0$) be a solution to (1.22). In [CV2], the following analogue of the bilateral pointwise estimates (1.11) was obtained for nontrivial (minimal) solutions $u$ to (1.22) in the case $0<q<p-1$:

[TABLE]

where $c>0$ is a constant which depends only on $\alpha$, $p$, $q$, and $n$. Moreover a nontrivial (super) solution exists if and only if both ${\bf{W}}_{\alpha,p}\sigma\not\equiv\infty$ and $\mathbf{K}_{\alpha,p,q}\not\equiv\infty$.

It follows that $u\in L^{r}(\mathbb{R}^{n})$ ($r>0$) exists if and only the following analogue of (1.16) holds:

[TABLE]

The first condition here actually follows from the second one, both in (1.16) (in the case $\alpha=1$), and in (4.2) that is,

[TABLE]

Indeed, suppose that $\mathbf{K}_{\alpha,p,q}\sigma\in L^{r}(\mathbb{R}^{n})$. Using the following trivial estimate for balls $B=B(x,\rho)$,

[TABLE]

we see that

[TABLE]

Hence,

[TABLE]

Estimates in [HJ], [JPW] yield that the preceding condition is equivalent to ${\bf{W}}_{\alpha,p}\sigma\in L^{\frac{r(p-1)}{p-1-q}}(\mathbb{R}^{n})$. This proves (4.3).

It remains to show that $\mathbf{K}_{\alpha,p,q}\sigma\in L^{r}(\mathbb{R}^{n})$ is equivalent to (1.18).

Suppose that $\mathbf{K}_{\alpha,p,q}\sigma\in L^{r}(\mathbb{R}^{n})$. Then by (4.3), there exists a nontrivial (super) solution $u\in L^{r}(\mathbb{R}^{n})$ to either (1.10) or (1.22). We set $d\omega=u^{q}d\sigma$. Then ${\bf{W}}_{\alpha,p}\omega\leq c\,u$, and hence ${\bf{W}}_{\alpha,p}\omega\in L^{r}(\mathbb{R}^{n})$. By the estimates in [HJ], [JPW] again, this is equivalent to the condition

[TABLE]

Using the estimate (see [SV2]*Lemma 4.2)

[TABLE]

where $C=C(\alpha,p,q,n)$, for $B=B(x,\rho)$, we obtain the following inequality (see (1.20) in the case $\alpha=1$),

[TABLE]

This condition obviously implies its dyadic version (see (1.23)),

[TABLE]

which also can be deduced independently using the pointwise estimate ${\bf{W}}_{\alpha,p}\omega\geq C\,{\bf{W}}^{d}_{\alpha,p}\omega$ (see [HW]) and a version of (4.5) for cubes $Q$ in place of balls $B$.

Let us next prove that, conversely, (4.6) yields $\mathbf{K}_{\alpha,p,q}\sigma\in L^{r}(\mathbb{R}^{n})$. It is enough to show this for the dyadic version, that is, $\mathbf{K}^{d}_{\alpha,p,q}\sigma\in L^{r}(\mathbb{R}^{n})$.

We first consider the case $0<r\leq 1$. Then, clearly,

[TABLE]

Integrating both sides of the preceding inequality over $\mathbb{R}^{n}$ with respect to $dx$ shows that (4.6) yields $\mathbf{K}^{d}_{\alpha,p,q}\sigma\in L^{r}(\mathbb{R}^{n})$.

We now treat the more difficult case $1<r<\infty$. Let

[TABLE]

Notice that by Theorem 3.2 with $\tau=1$, $q=s$ and $d\sigma=dx$, we have that $\mathbf{K}^{d}_{\alpha,p,q}\sigma\in L^{r}(\mathbb{R}^{n})$ if and only if, for some $0<s<r$,

[TABLE]

Let us fix a dyadic cube $R$, and denote by $u_{R}$ is a solution to the equation

[TABLE]

where $d\sigma_{R}=\chi_{R}d\sigma$ is the restriction of $\sigma$ to $R$. Such a solution exists since $\kappa(R)<\infty$; moreover, by [CV2]*Lemma 4.2 and Corollary 4.3 for every $Q\subseteq R$, we have

[TABLE]

By the first estimate in (4.8), we have

[TABLE]

Hence,

[TABLE]

Let $s=\min(1,p-1)<r$. Then by [SV2, Lemma 3.1], we can estimate the average value of \Big{[}{\bf{W}}_{\alpha,p}(u^{q}d\sigma_{R})\Big{]}^{s} over $R$:

[TABLE]

Further, by the second part of estimate (4.8),

[TABLE]

Combining these estimates, we deduce

[TABLE]

Consequently, for our choice of $\lambda_{R}$, we have

[TABLE]

Thus, (4.7) holds, which proves that $\mathbf{K}_{\alpha,p,q}\sigma\in L^{r}(\mathbb{R}^{n})$. This completes the proofs of Theorems 1.1 and 1.5.∎