Weighted Norm Inequalities of (1, q)-Type for Integral and Fractional Maximal Operators

Abstract

We study weighted norm inequalities of (1, q)- type for 0 < q < 1,

$$\displaystyle{\Vert \mathbf{G}\nu \Vert _{L^{q}(\Omega,d\sigma )} \leq C\,\Vert \nu \Vert,\quad \text{for all positive measures}\,\nu \text{ in }\Omega,}$$

along with their weak-type counterparts, where $\Vert \nu \Vert =\nu (\Omega )$, and G is an integral operator with nonnegative kernel,

$$\displaystyle{\mathbf{G}\nu (x) =\int _{\Omega }G(x,y)d\nu (y).}$$

These problems are motivated by sublinear elliptic equations in a domain $\Omega \subset \mathbb{R}^{n}$ with non-trivial Green’s function G(x, y) associated with the Laplacian, fractional Laplacian, or more general elliptic operator. We also treat fractional maximal operators M _α (0 ≤ α < n) on $\mathbb{R}^{n}$, and characterize strong- and weak-type (1, q)-inequalities for M _α and more general maximal operators, as well as (1, q)-Carleson measure inequalities for Poisson integrals.

Dedicated to Richard L. Wheeden

1 Introduction

In this paper, we discuss recent results on weighted norm inequalities of (1, q)- type in the case 0 < q < 1,

$$\displaystyle{ \Vert \mathbf{G}\nu \Vert _{L^{q}(\Omega,d\sigma )} \leq C\,\Vert \nu \Vert, }$$

(1)

for all positive measures ν in $\Omega$, where $\Vert \nu \Vert =\nu (\Omega )$, and G is an integral operator with nonnegative kernel,

$$\displaystyle{\mathbf{G}\nu (x) =\int _{\Omega }G(x,y)d\nu (y).}$$

Such problems are motivated by sublinear elliptic equations of the type

$$\displaystyle{ \left \{\begin{array}{@{}l@{\quad }l@{}} -\Delta u =\sigma u^{q}\,\,\text{ in }\Omega,\quad \\ \;u = 0\text{ on }\partial \Omega, \quad \end{array} \right. }$$

in the case 0 < q < 1, where $\Omega$ is an open set in $\mathbb{R}^{n}$ with non-trivial Green’s function G(x, y), and σ ≥ 0 is an arbitrary locally integrable function, or locally finite measure in $\Omega$.

The only restrictions imposed on the kernel G are that it is quasi-symmetric and satisfies a weak maximum principle. In particular, G can be a Green operator associated with the Laplacian, a more general elliptic operator (including the fractional Laplacian), or a convolution operator on $\mathbb{R}^{n}$ with radially symmetric decreasing kernel G(x, y) = k( | x − y | ) (see [1, 12]).

As an example, we consider in detail the one-dimensional case where $\Omega = \mathbb{R}_{+}$ and G(x, y) = min(x, y). We deduce explicit characterizations of the corresponding (1, q)-weighted norm inequalities, give explicit necessary and sufficient conditions for the existence of weak solutions, and obtain sharp two-sided pointwise estimates of solutions.

We also characterize weak-type counterparts of (1), namely,

$$\displaystyle{ \Vert \mathbf{G}\nu \Vert _{L^{q,\infty }(\Omega,d\sigma )} \leq C\,\Vert \nu \Vert. }$$

(2)

Along with integral operators, we treat fractional maximal operators M _α with 0 ≤ α < n on $\mathbb{R}^{n}$, and characterize both strong- and weak-type (1, q)-inequalities for M _α , and more general maximal operators. Similar problems for Riesz potentials were studied earlier in [6–8]. Finally, we apply our results to the Poisson kernel to characterize (1, q)-Carleson measure inequalities.

2 Integral Operators

2.1 Strong-Type (1, q)-Inequality for Integral Operators

Let $\Omega \subseteq \mathbb{R}^{n}$ be a connected open set. By $\mathcal{M}^{+}(\Omega )$ we denote the class of all nonnegative locally finite Borel measures in $\Omega$. Let $G: \Omega \times \Omega \rightarrow [0,+\infty ]$ be a nonnegative lower-semicontinuous kernel. We will assume throughout this paper that G is quasi-symmetric, i.e., there exists a constant a > 0 such that

$$\displaystyle{ a^{-1}\,G(x,y) \leq G(y,x) \leq a\,G(x,y),\quad x,y \in \Omega. }$$

(3)

If $\nu \in \mathcal{M}^{+}(\Omega )$, then by G ν and G ^∗ ν we denote the integral operators (potentials) defined respectively by

$$\displaystyle{ \mathbf{G}\nu (x) =\int _{\Omega }G(x,y)\,d\nu (y),\quad \mathbf{G}^{{\ast}}\nu (x) =\int _{ \Omega }G(y,x)\,d\nu (y),\quad x \in \Omega. }$$

(4)

We say that the kernel G satisfies the weak maximum principle if, for any constant M > 0, the inequality

$$\displaystyle{\mathbf{G}\nu (x) \leq M\quad \text{for all}\,\,\,x \in S(\nu )}$$

implies

$$\displaystyle{\mathbf{G}\nu (x) \leq hM\quad \text{for all}\,\,\,x \in \Omega,}$$

where h ≥ 1 is a constant, and $S(\nu ) \mathop{:}= \mathrm{supp}\,\nu$. When h = 1, we say that G ν satisfies the strong maximum principle.

It is well-known that Green’s kernels associated with many partial differential operators are quasi-symmetric, and satisfy the weak maximum principle (see, e.g., [2, 3, 12]).

The kernel G is said to be degenerate with respect to $\sigma \in \mathcal{M}^{+}(\Omega )$ provided there exists a set $A \subset \Omega$ with σ(A) > 0 and

$$\displaystyle{G(\cdot,y) = 0\quad d\sigma -\text{a.e. for }y \in A.}$$

Otherwise, we will say that G is non-degenerate with respect to σ. (This notion was introduced in [19] in the context of (p, q)-inequalities for positive operators T: L ^p → L ^q in the case 1 < q < p.)

Let 0 < q < 1, and let G be a kernel on $\Omega \times \Omega$. For $\sigma \in \mathcal{M}^{+}(\Omega )$, we consider the problem of the existence of a positive solution u to the integral equation

$$\displaystyle{ u = \mathbf{G}(u^{q}d\sigma )\quad \text{in}\,\,\,\Omega,\quad 0 <u <+\infty \,\,\,d\sigma \mathrm{-a.e.},\quad u \in L_{\mathrm{ loc}}^{q}(\Omega ). }$$

(5)

We call u a positive supersolution if

$$\displaystyle{ u \geq \mathbf{G}(u^{q}d\sigma )\quad \text{in}\,\,\,\Omega,\quad 0 <u <+\infty \,\,\,d\sigma \mathrm{-a.e.},\quad u \in L_{\mathrm{ loc}}^{q}(\Omega ). }$$

(6)

This is a generalization of the sublinear elliptic problem (see, e.g., [4, 5], and the literature cited there):

$$\displaystyle{ \left \{\begin{array}{@{}l@{\quad }l@{}} -\Delta u =\sigma u^{q}\quad \text{in}\,\,\Omega,\quad \\ \;u = 0\quad \text{on}\,\,\partial \Omega, \quad \end{array} \right. }$$

(7)

where σ is a nonnegative locally integrable function, or measure, in $\Omega$.

If $\Omega$ is a bounded C ²-domain then solutions to (7) can be understood in the “very weak” sense (see, e.g., [13]). For general domains $\Omega$ with a nontrivial Green function G associated with the Dirichlet Laplacian $\Delta$ in $\Omega$, solutions u are understood as in (5).

Remark 2.1

In this paper, for the sake of simplicity, we sometimes consider positive solutions and supersolutions $u \in L^{q}(\Omega,d\sigma )$. In other words, we replace the natural local condition $u \in L_{\mathrm{loc}}^{q}(\Omega,d\sigma )$ with its global counterpart. Notice that the local condition is necessary for solutions (or supersolutions) to be properly defined.

To pass from solutions u which are globally in $L^{q}(\Omega,d\sigma )$ to all solutions $u \in L_{\mathrm{loc}}^{q}(\Omega,d\sigma )$ (for instance, very weak solutions to (7)), one can use either a localization method developed in [8] (in the case of Riesz kernels on $\mathbb{R}^{n}$), or modified kernels $\tilde{G}(x,y) = \frac{G(x,y)} {m(x)\,m(y)}$, where the modifier $m(x) =\min \Big (1,G(x,x_{0})\Big)$ (with a fixed pole $x_{0} \in \Omega$) plays the role of a regularized distance to the boundary $\partial \Omega$. One also needs to consider the corresponding (1, q)-inequalities with a weight m (see [16]). See the next section in the one-dimensional case where $\Omega = (0,+\infty )$.

Remark 2.2

Finite energy solutions, for instance, solutions $u \in W_{0}^{1,2}(\Omega )$ to (7), require the global condition $u \in L^{1+q}(\Omega,d\sigma )$, and are easier to characterize (see [6]).

The following theorem is proved in [16]. (The case where $\Omega = \mathbb{R}^{n}$ and $\mathbf{G} = (-\Delta )^{-\frac{\alpha }{ 2} }$ is the Riesz potential of order α ∈ (0, n) was considered earlier in [8].)

Theorem 2.3

Let $\sigma \in \mathcal{M}^{+}(\Omega )$ , and 0 < q < 1. Suppose G is a quasi-symmetric kernel which satisfies the weak maximum principle. Then the following statements are equivalent:

• (1) There exists a positive constant ϰ = ϰ(σ) such that

  $$\displaystyle{\Vert \mathbf{G}\nu \Vert _{L^{q}(\sigma )} \leq \varkappa \Vert \nu \Vert \quad \text{for all}\,\,\nu \in \mathcal{M}^{+}(\Omega ).}$$

• (2) There exists a positive supersolution $u \in L^{q}(\Omega,d\sigma )$ to  (6) .

• (3) There exists a positive solution $u \in L^{q}(\Omega,d\sigma )$ to  (5) , provided additionally that G is non-degenerate with respect to σ.

Remark 2.4

The implication (1) ⇒ (2) in Theorem 2.3 holds for any nonnegative kernel G, without assuming that it is either quasi-symmetric, or satisfies the weak maximum principle. This is a consequence of Gagliardo’s lemma [10, 21]; see details in [16].

Remark 2.5

The implication (3) ⇒ (1) generally fails for kernels G which do not satisfy the weak maximum principle (see examples in [16]).

The following corollary of Theorem 2.3 is obtained in [16].

Corollary 2.6

Under the assumptions of Theorem  2.3 , if there exists a positive supersolution $u \in L^{q}(\Omega,\sigma )$ to  (6) , then $\mathbf{G}\sigma \in L^{ \frac{q} {1-q} }(\Omega,d\sigma )$ .

Conversely, if $\mathbf{G}\sigma \in L^{ \frac{q} {1-q},1}(\Omega,d\sigma )$ , then there exists a non-trivial supersolution $u \in L^{q}(\Omega,\sigma )$ to  (6) (respectively, a solution u, provided G is non-degenerate with respect to σ).

2.2 The One-Dimensional Case

In this section, we consider positive weak solutions to sublinear ODEs of the type (7) on the semi-axis $\mathbb{R}_{+} = (0,+\infty )$. It is instructive to consider the one-dimensional case where elementary characterizations of (1, q)-weighed norm inequalities, along with the corresponding existence theorems and explicit global pointwise estimates of solutions are available. Similar results hold for sublinear equations on any interval $(a,b) \subset \mathbb{R}$.

Let 0 < q < 1, and let $\sigma \in \mathcal{M}^{+}(\mathbb{R}_{+})$. Suppose u is a positive weak solution to the equation

$$\displaystyle{ -u^{{\prime\prime}} =\sigma u^{q}\quad \text{on}\,\,\,\mathbb{R}_{ +},\quad u(0) = 0, }$$

(8)

such that $\lim _{x\rightarrow +\infty }\frac{u(x)} {x} = 0$. This condition at infinity ensures that u does not contain a linear component. Notice that we assume that u is concave and increasing on [0, +∞), and $\lim _{x\rightarrow 0^{+}}u(x) = 0$.

In terms of integral equations, we have $\Omega = \mathbb{R}_{+}$, and G(x, y) = min(x, y) is the Green function associated with the Sturm-Liouville operator $\Delta u = u^{{\prime\prime}}$ with zero boundary condition at x = 0. Thus, (8) is equivalent to the equation

$$\displaystyle{ u(x) = \mathbf{G}(u^{q}d\sigma )(x) \mathop{:}=\int _{ 0}^{+\infty }\min (x,y)u(y)^{q}d\sigma (y),\quad x> 0, }$$

(9)

where σ is a locally finite measure on $\mathbb{R}_{+}$, and

$$\displaystyle{ \int _{0}^{a}y\,u(y)^{q}d\sigma (y) <+\infty,\quad \int _{ a}^{+\infty }u(y)^{q}d\sigma (y) <+\infty,\quad \text{for every}\,\,a> 0. }$$

(10)

This “local integrability” condition ensures that the right-hand side of (9) is well defined. Here intervals (a, +∞) are used in place of balls B(x, r) in $\mathbb{R}^{n}$.

Notice that

$$\displaystyle{ u^{{\prime}}(x) =\int _{ x}^{+\infty }u(y)^{q}d\sigma (y),\quad x> 0. }$$

(11)

Hence, u satisfies the global integrability condition

$$\displaystyle{ \int _{0}^{+\infty }u(y)^{q}d\sigma (y) <+\infty }$$

(12)

if and only if u ^′(0) < +∞.

The corresponding (1, q)-weighted norm inequality is given by

$$\displaystyle{ \Vert \mathbf{G}\nu \Vert _{L^{q}(\sigma )} \leq \varkappa \Vert \nu \Vert, }$$

(13)

where ϰ = ϰ(σ) is a positive constant which does not depend on $\nu \in \mathcal{M}^{+}(\mathbb{R}_{+})$. Obviously, (13) is equivalent to

$$\displaystyle{ \Vert H_{+}\nu + H_{-}\nu \Vert _{L^{q}(\sigma )} \leq \varkappa \Vert \nu \Vert \quad \text{for all}\,\,\nu \in \mathcal{M}^{+}(\mathbb{R}_{ +}), }$$

(14)

where H _± is a pair of Hardy operators,

$$\displaystyle{ H_{+}\nu (x) =\int _{ 0}^{x}y\,d\nu (y),\quad H_{ -}\nu (x) = x\int _{x}^{+\infty }d\nu (y). }$$

The following proposition can be deduced from the known results on two-weight Hardy inequalities in the case p = 1 and 0 < q < 1 (see, e.g., [20]). We give here a simple independent proof.

Proposition 2.7

Let $\sigma \in \mathcal{M}^{+}(\mathbb{R}_{+})$ , and let 0 < q < 1. Then  (13) holds if and only if

$$\displaystyle{ \varkappa (\sigma )^{q} =\int _{ 0}^{+\infty }x^{q}d\sigma (x) <+\infty, }$$

(15)

where ϰ(σ) is the best constant in  (13) .

Proof

Clearly,

$$\displaystyle{ H_{+}\nu (x) + H_{-}\nu (x) \leq x\,\Vert \nu \Vert,\quad x> 0. }$$

Hence,

$$\displaystyle{ \Vert H_{+}\nu + H_{-}\nu \Vert _{L^{q}(\sigma )} \leq \Big (\int _{0}^{+\infty }x^{q}d\sigma (x)\Big)^{\frac{1} {q} }\Vert \nu \Vert, }$$

which proves (14), and hence (13), with $\varkappa =\Big (\int _{0}^{+\infty }x^{q}d\sigma (x)\Big)^{\frac{1} {q} }$.

Conversely, suppose that (14) holds. Then, for every a > 0, and $\nu \in \mathcal{M}^{+}(\mathbb{R}_{+})$,

$$\displaystyle\begin{array}{rcl} & & \Big(\int _{0}^{a}x^{q}d\sigma (x)\Big)\Big(\int _{ a}^{+\infty }d\nu (y)\Big)^{q} {}\\ & & \leq \int _{0}^{a}\Big(x\int _{ x}^{+\infty }d\nu (y)\Big)^{q}d\sigma (x) {}\\ & & \leq \int _{0}^{+\infty }(H_{ -}\nu )^{q}d\sigma \leq \varkappa ^{q}\Vert \nu \Vert ^{q}. {}\\ \end{array}$$

For $\nu =\delta _{x_{0}}$ with x ₀ > a, we get

$$\displaystyle{ \int _{0}^{a}x^{q}d\sigma (x) \leq \varkappa ^{q}. }$$

Letting a → +∞, we deduce (15). □ 

Clearly, the Green kernel G(x, y) = min(x, y) is symmetric, and satisfies the strong maximum principle. Hence, by Theorem 2.3, Eqs. (8) and (9) have a non-trivial (super)solution $u \in L^{q}(\mathbb{R}_{+},\sigma )$ if and only if (15) holds.

From Proposition 2.7, we deduce that, for “localized” measures d σ _a  = χ _(a, +∞) d σ (a > 0), we have

$$\displaystyle{ \varkappa (\sigma _{a}) =\Big (\int _{a}^{+\infty }x^{q}d\sigma (x)\Big)^{\frac{1} {q} }. }$$

(16)

Using this observation and the localization method developed in [8], we obtain the following existence theorem for general weak solutions to (7), along with sharp pointwise estimates of solutions.

We introduce a new potential

$$\displaystyle{ \mathbf{K}\sigma (x) \mathop{:}= x\Big(\int _{x}^{+\infty }y^{q}d\sigma (y)\Big)^{ \frac{1} {1-q} },\quad x> 0. }$$

(17)

We observe that K σ is a one-dimensional analogue of the potential introduced recently in [8] in the framework of intrinsic Wolff potentials in $\mathbb{R}^{n}$ (see also [7] in the radial case). Matching upper and lower pointwise bounds of solutions are obtained below by combining G σ with K σ.

Theorem 2.8

Let $\sigma \in \mathcal{M}^{+}(\mathbb{R}_{+})$ , and let 0 < q < 1. Then Eq.  (7) , or equivalently  (8) has a nontrivial solution if and only if, for every a > 0,

$$\displaystyle{ \int _{0}^{a}x\,d\sigma (x) +\int _{ a}^{+\infty }x^{q}d\sigma (x) <+\infty. }$$

(18)

Moreover, if  (18) holds, then there exists a positive solution u to  (7) such that

$$\displaystyle\begin{array}{rcl} C^{-1}\Big[\Big(\int _{ 0}^{x}y\,d\sigma (y)\Big)^{ \frac{1} {1-q} } + \mathbf{K}\sigma (x)\Big]& &{}\end{array}$$

(19)

$$\displaystyle\begin{array}{rcl} \leq u(x) \leq C\,\Big[\Big(\int _{0}^{x}y\,d\sigma (y)\Big)^{ \frac{1} {1-q} } + \mathbf{K}\sigma (x)\Big].& &{}\end{array}$$

(20)

The lower bound in  (19) holds for any non-trivial supersolution u.

Remark 2.9

The lower bound

$$\displaystyle{ u(x) \geq (1 - q)^{ \frac{1} {1-q} }\Big[\mathbf{G}\sigma (x)\Big]^{ \frac{1} {1-q} },\quad x> 0, }$$

(21)

is known for a general kernel G which satisfies the strong maximum principle (see [11], Theorem 3.3; [16]), and the constant $(1 - q)^{ \frac{1} {1-q} }$ here is sharp. However, the second term on the left-hand side of (19) makes the lower estimate stronger, so that it matches the upper estimate.

Proof

The lower bound

$$\displaystyle{ u(x) \geq (1 - q)^{ \frac{1} {1-q} }\Big[\int _{0}^{x}y\,d\sigma (y)\Big]^{ \frac{1} {1-q} },\quad x> 0, }$$

(22)

is immediate from (21).

Applying Lemma 4.2 in [8], with the interval (a, +∞) in place of a ball B, and combining it with (16), for any a > 0 we have

$$\displaystyle{ \int _{a}^{+\infty }u(y)^{q}d\sigma (y) \geq c(q)\varkappa (\sigma _{ a})^{ \frac{q} {1-q} } = c(q)\Big[\int _{a}^{+\infty }y^{q}d\sigma (y)\Big]^{ \frac{1} {1-q} }. }$$

Hence,

$$\displaystyle{ u(x) \geq \mathbf{G}(u^{q}d\sigma ) \geq x\int _{ x}^{+\infty }u(y)^{q}d\sigma (y) \geq c(q)\,x\Big[\int _{ x}^{+\infty }y^{q}d\sigma (y)\Big]^{ \frac{1} {1-q} }. }$$

Combining the preceding estimate with (22), we obtain the lower bound in (19) for any non-trivial supersolution u. This also proves that (18) is necessary for the existence of a non-trivial positive supersolution.

Conversely, suppose that (18) holds. Let

$$\displaystyle{ v(x) \mathop{:}= c\,\Big[\Big(\int _{0}^{x}y\,d\sigma (y)\Big)^{ \frac{1} {1-q} } + \mathbf{K}\sigma (x)\Big],\quad x> 0, }$$

(23)

where c is a positive constant. It is not difficult to see that v is a supersolution, so that v ≥ G(v ^q d σ), if c = c(q) is picked large enough. (See a similar argument in the proof of Theorem 5.1 in [7].)

Also, it is easy to see that $v_{0} = c_{0}(\mathbf{G}\sigma )^{ \frac{1} {1-q} }$ is a subsolution, i.e., v ₀ ≤ G(v ₀ ^q d σ), provided c ₀ > 0 is a small enough constant. Moreover, we can ensure that v ₀ ≤ v if c ₀ = c ₀(q) is picked sufficiently small. (See details in [7] in the case of radially symmetric solutions in $\mathbb{R}^{n}$.) Hence, there exists a solution which can be constructed by iterations, starting from u ₀ = v ₀, and letting

$$\displaystyle{ u_{j+1} = \mathbf{G}(u_{j}^{q}d\sigma ),\quad j = 0,1,\ldots. }$$

Then by induction u _j  ≤ u _j+1 ≤ v, and consequently u = lim _{j → +∞} u _j is a solution to (9) by the Monotone Convergence Theorem. Clearly, u ≤ v, which proves the upper bound in (19). □ 

2.3 Weak-Type (1, q)-Inequality for Integral Operators

In this section, we characterize weak-type analogues of (1, q)-weighted norm inequalities considered above. We will use some elements of potential theory for general positive kernels G, including the notion of inner capacity, $\mathop{\mathrm{cap}}\nolimits (\cdot )$, and the associated equilibrium (extremal) measure (see [9]).

Theorem 2.10

Let $\sigma \in \mathcal{M}^{+}(\Omega )$ , and 0 < q < 1. Suppose G satisfies the weak maximum principle. Then the following statements are equivalent:

• (1) There exists a positive constant ϰ _w such that

  $$\displaystyle{ \Vert \mathbf{G}\nu \Vert _{L^{q,\infty }(\sigma )} \leq \varkappa _{w}\Vert \nu \Vert \quad \text{for all}\,\,\nu \in \mathcal{M}^{+}(\Omega ). }$$

• (2) There exists a positive constant c such that

  $$\displaystyle{ \sigma (K) \leq c\Big(\mathop{\mathrm{cap}}\nolimits (K)\Big)^{q}\quad \text{for all compact sets}\,\,K \subset \Omega. }$$

• (3) $\mathbf{G}\sigma \in L^{ \frac{q} {1-q},\infty }(\sigma )$ .

Proof

(1) ⇒ (2) Without loss of generality we may assume that the kernel G is strictly positive, that is, G(x, x) > 0 for all $x \in \Omega$. Otherwise, we can consider the kernel G on the set $\Omega \setminus A$, where $A \mathop{:}=\{ x \in \Omega: G(x,x)\not =0\}$, since A is negligible for the corresponding (1, q)-inequality in statement (1). (See details in [16] in the case of the corresponding strong-type inequalities.)

We remark that the kernel G is known to be strictly positive if and only if, for any compact set $K \subset \Omega$, the inner capacity cap(K) is finite [9]. In this case there exists an equilibrium measure λ on K such that

$$\displaystyle{ \mathbf{G}\lambda \geq 1\,\,\text{n.e.}\,\,\text{on}\,\,K,\quad \mathbf{G}\lambda \leq 1\,\,\text{on}\,\,S(\lambda ),\quad \Vert \lambda \Vert = \mathrm{cap}(K). }$$

(24)

Here n.e. stands for nearly everywhere, which means that the inequality holds on a given set except for a subset of zero capacity [9].

Next, we remark that condition (1) yields that σ is absolutely continuous with respect to capacity, i.e., σ(K) = 0 if cap(K) = 0. (See a similar argument in [16] in the case of strong-type inequalities.) Consequently, G λ ≥ 1 d σ-a.e. on K. Hence, by applying condition (1) with ν = λ, we obtain (2).

(2) ⇒ (3) We denote by σ _E the restriction of σ to a Borel set $E \subset \Omega$. Without loss of generality we may assume that σ is a finite measure on $\Omega$. Otherwise we can replace σ with σ _F where F is a compact subset of $\Omega$. We then deduce the estimate

$$\displaystyle{ \Vert \mathbf{G}\sigma _{F}\Vert _{ L^{ \frac{q} {1-q},\infty }(\sigma _{F})} \leq C <\infty, }$$

where C does not depend on F, and use the exhaustion of $\Omega$ by an increasing sequence of compact subsets $F_{n} \uparrow \Omega$ to conclude that $\mathbf{G}\sigma \in L^{ \frac{q} {1-q},\infty }(\sigma )$ by the Monotone Convergence Theorem.

Set $E_{t} \mathop{:}=\{ x \in \Omega: \,\mathbf{G}\sigma (x)> t\}$, where t > 0. Notice that, for all x ∈ (E _t )^c,

$$\displaystyle{ \mathbf{G}\sigma _{(E_{t})^{c}}(x) \leq \mathbf{G}\sigma (x) \leq t. }$$

The set (E _t )^c is closed, and hence the preceding inequality holds on $S(\sigma _{(E_{t})^{c}})$. It follows by the weak maximum principle that, for all $x \in \Omega$,

$$\displaystyle{ \mathbf{G}\sigma _{(E_{t})^{c}}(x) \leq \mathbf{G}\sigma (x) \leq h\,t. }$$

Hence,

$$\displaystyle{ \{x \in \Omega: \mathbf{G}\sigma (x)> (h + 1)t\} \subset \{ x \in \Omega: \mathbf{G}\sigma _{E_{t}}(x)> t\}. }$$

(25)

Denote by $K \subset \Omega$ a compact subset of $\{x \in \Omega: \mathbf{G}\sigma _{E_{t}}(x)> t\}$. By (2), we have

$$\displaystyle{\sigma (K) \leq c\,\Big(\mathop{\mathrm{cap}}\nolimits (K)\Big)^{q}}$$

If λ is the equilibrium measure on K, then G λ ≤ 1 on S(λ), and $\lambda (K) =\mathop{ \mathrm{cap}}\nolimits (K)$ by (24). Hence by the weak maximum principle G λ ≤ h on $\Omega$. Using quasi-symmetry of the kernel G and Fubini’s theorem, we have

$$\displaystyle\begin{array}{rcl} \mathop{\mathrm{cap}}\nolimits (K)& =& \int _{K}\,d\lambda {}\\ & \leq & \frac{1} {t}\int _{K}\mathbf{G}\sigma _{E_{t}}\,d\lambda {}\\ & \leq & \frac{a} {t} \int _{E_{t}}\mathbf{G}\lambda d\sigma {}\\ & \leq & \frac{ah} {t} \sigma (E_{t}). {}\\ \end{array}$$

This shows that

$$\displaystyle{ \sigma (K) \leq \frac{c(ah)^{q}} {t^{q}} \,\Big(\sigma (E_{t})\Big)^{q}. }$$

Taking the supremum over all K ⊂ E _t , we deduce

$$\displaystyle{ \Big(\sigma (E_{t})\Big)^{1-q} \leq \frac{c(ah)^{q}} {t^{q}}. }$$

It follows from the preceding estimate and (25) that, for all t > 0,

$$\displaystyle{ t^{ \frac{q} {1-q} }\sigma \Big(\left \{x \in \Omega: \mathbf{G}\sigma (x)> (h + 1)t\right \}\Big) \leq t^{ \frac{q} {1-q} }\sigma (E_{t}) \leq c^{ \frac{1} {1-q} }(ah)^{ \frac{q} {1-q} }. }$$

Thus, (3) holds.

(3) ⇒ (2) By Hölder’s inequality for weak L ^q spaces, we have

$$\displaystyle\begin{array}{rcl} \Vert \mathbf{G}\nu \Vert _{L^{q,\infty }(\sigma )}& =& \left \Vert \frac{\mathbf{G}\nu } {\mathbf{G}\sigma }\mathbf{G}\sigma \right \Vert _{L^{q,\infty }(\sigma )} {}\\ & \leq & \left \Vert \frac{\mathbf{G}\nu } {\mathbf{G}\sigma }\right \Vert _{L^{1,\infty }(\sigma )}\left \Vert \mathbf{G}\sigma \right \Vert _{ L^{ \frac{q} {1-q},\infty }(\sigma )} {}\\ & \leq & C\left \Vert \mathbf{G}\sigma \right \Vert _{ L^{ \frac{q} {1-q},\infty }(\sigma )}\Vert \nu \Vert, {}\\ \end{array}$$

where the final inequality,

$$\displaystyle{ \left \Vert \frac{\mathbf{G}\nu } {\mathbf{G}\sigma }\right \Vert _{L^{1,\infty }(\sigma )} \leq C\,\Vert \nu \Vert, }$$

with a constant C = C(h, a), was obtained in [16], for quasi-symmetric kernels G satisfying the weak maximum principle. □ 

3 Fractional Maximal Operators

Let 0 ≤ α < n, and let $\nu \in \mathcal{M}^{+}(\mathbb{R}^{n})$. The fractional maximal function M _α ν is defined by

$$\displaystyle{ M_{\alpha }\nu (x) \mathop{:}=\sup _{Q\ni x} \frac{\vert Q\vert _{\nu }} {\vert Q\vert ^{1- \frac{\alpha }{ n} }},\quad x \in \mathbb{R}^{n}, }$$

(26)

where Q is a cube, $\vert Q\vert _{\nu }\mathop{:}=\nu (Q)$, and | Q | is the Lebesgue measure of Q. If $f \in L_{\mathrm{loc}}^{1}(\mathbb{R}^{n},d\mu )$ where $\mu \in \mathcal{M}^{+}(\mathbb{R}^{n})$, we set M _α (fd μ) = M _α ν where d ν = | f | d μ, i.e.,

$$\displaystyle{ M_{\alpha }(fd\mu )(x) \mathop{:}=\sup _{Q\ni x} \frac{1} {\vert Q\vert ^{1- \frac{\alpha }{ n} }}\int _{Q}\vert f\vert \,d\mu,\quad x \in \mathbb{R}^{n}. }$$

(27)

For $\sigma \in \mathcal{M}^{+}(\mathbb{R}^{n})$, it was shown in [22] that in the case 0 < q < p,

$$\displaystyle{ M_{\alpha }: L^{p}(dx) \rightarrow L^{q}(d\sigma )\Longleftrightarrow M_{\alpha }\sigma \in L^{ \frac{q} {p-q} }(d\sigma ), }$$

(28)

$$\displaystyle{ M_{\alpha }: L^{p}(dx) \rightarrow L^{q,\infty }(d\sigma )\Longleftrightarrow M_{\alpha }\sigma \in L^{ \frac{q} {p-q},\infty }(d\sigma ), }$$

(29)

provided p > 1.

More general two-weight maximal inequalities

$$\displaystyle{ \Vert M_{\alpha }(fd\mu )\Vert _{L^{q}(\sigma )} \leq \varkappa \,\Vert f\Vert _{L^{p}(\mu )},\quad \text{for all}\,\,f \in L^{p}(\mu ), }$$

(30)

where characterized by E.T. Sawyer [18] in the case p = q > 1, R.L. Wheeden [24] in the case q > p > 1, and the second author [22] in the case 0 < q < p and p > 1, along with their weak-type counterparts,

$$\displaystyle{ \Vert M_{\alpha }(fd\mu )\Vert _{L^{q,\infty }(\sigma )} \leq \varkappa _{w}\,\Vert f\Vert _{L^{p}(\mu )},\quad \text{for all}\,\,f \in L^{p}(\mu ), }$$

(31)

where $\sigma,\mu \in \mathcal{M}^{+}(\mathbb{R}^{n})$, and ϰ, ϰ _w are positive constants which do not depend on f.

However, some of the methods used in [22] for 0 < q < p and p > 1 are not directly applicable in the case p = 1, although there are analogues of these results for real Hardy spaces, i.e., when the norm $\Vert f\Vert _{L^{p}(\mu )}$ on the right-hand side of (30) or (31) is replaced with $\Vert M_{\mu }f\Vert _{L^{p}(\mu )}$, where

$$\displaystyle{ M_{\mu }f(x) \mathop{:}=\sup _{Q\ni x} \frac{1} {\vert Q\vert _{\mu }}\int _{Q}\vert f\vert d\mu. }$$

(32)

We would like to understand similar problems in the case 0 < q < 1 and p = 1, in particular, when $M_{\alpha }: \mathcal{M}^{+}(\mathbb{R}^{n}) \rightarrow L^{q}(d\sigma )$, or equivalently, there exists a constant ϰ > 0 such that the inequality

$$\displaystyle{ \Vert M_{\alpha }\nu \Vert _{L^{q}(\sigma )} \leq \varkappa \,\Vert \nu \Vert }$$

(33)

holds for all $\nu \in \mathcal{M}^{+}(\mathbb{R}^{n})$.

In the case α = 0, Rozin [17] showed that the condition

$$\displaystyle{\sigma \in L^{ \frac{1} {1-q},1}(\mathbb{R}^{n},dx)}$$

is sufficient for the Hardy-Littlewood operator M = M ₀: L ¹(dx) → L ^q(σ) to be bounded; moreover, when σ is radially symmetric and decreasing, this is also a necessary condition. We will generalize this result and provide necessary and sufficient conditions for the range 0 ≤ α < n. We also obtain analogous results for the weak-type inequality

$$\displaystyle{ \Vert M_{\alpha }\nu \Vert _{L^{q,\infty }(\sigma )} \leq \varkappa _{w}\,\Vert \nu \Vert,\quad \text{for all}\,\,\nu \in \mathcal{M}^{+}(\mathbb{R}^{n}), }$$

(34)

where ϰ _w is a positive constant which does not depend on ν.

We treat more general maximal operators as well, in particular, dyadic maximal operators

$$\displaystyle{ M_{\rho }\nu (x) \mathop{:}=\sup _{Q\in \mathcal{Q}: Q\ni x}\rho _{Q}\,\vert Q\vert _{\nu }, }$$

(35)

where $\mathcal{Q}$ is the family of all dyadic cubes in $\mathbb{R}^{n}$, and $\{\rho _{Q}\}_{Q\in \mathcal{Q}}$ is a fixed sequence of nonnegative reals associated with $Q \in \mathcal{Q}$. The corresponding weak-type maximal inequality is given by

$$\displaystyle{ \Vert M_{\rho }\nu \Vert _{L^{q,\infty }(\sigma )} \leq \varkappa _{w}\,\Vert \nu \Vert,\quad \text{for all}\,\,\nu \in \mathcal{M}^{+}(\mathbb{R}^{n}). }$$

(36)

3.1 Strong-Type Inequality

Theorem 3.1

Let $\sigma \in M^{+}(\mathbb{R}^{n})$ , 0 < q < 1, and 0 ≤α < n. The inequality ( 33 ) holds if and only if there exists a function u≢0 such that

$$\displaystyle{ u \in L^{q}(\sigma ),\quad \text{and}\quad u \geq M_{\alpha }(u^{q}\sigma ). }$$

Moreover, u can be constructed as follows: u = lim_j→∞ u _j , where $u_{0} \mathop{:}= (M_{\alpha }\sigma )^{ \frac{1} {1-q} }$ , u _j+1 ≥ u _j , and

$$\displaystyle{ u_{j+1} \mathop{:}= M_{\alpha }(u_{j}^{q}\sigma ),\quad j = 0,1,\ldots. }$$

(37)

In particular, $u \geq (M_{\alpha }\sigma )^{ \frac{1} {1-q} }$ .

Proof

( ⇒ ) We let $u_{0} \mathop{:}= (M_{\alpha }\sigma )^{ \frac{1} {1-q} }$. Notice that, for all x ∈ Q, we have $u_{0}(x) \geq \Big ( \frac{\vert Q\vert _{\sigma }} {\vert Q\vert ^{1- \frac{\alpha }{n} }}\Big)^{ \frac{1} {1-q} }$. Hence,

$$\displaystyle{ u_{1}(x) \mathop{:}= M_{\alpha }(u_{0}^{q}d\sigma )(x) =\sup _{ Q\ni x} \frac{1} {\vert Q\vert ^{1- \frac{\alpha }{ n} }}\int _{Q}u_{0}^{q}d\sigma \geq \sup _{ Q\ni x}\Big( \frac{\vert Q\vert _{\sigma }} {\vert Q\vert ^{1- \frac{\alpha }{ n} }}\Big)^{ \frac{1} {1-q} } = u_{0}(x). }$$

By induction, we see that

$$\displaystyle{ u_{j+1} \mathop{:}= M_{\alpha }(u_{j}^{q}d\sigma ) \geq M_{\alpha }(u_{ j-1}^{q}d\sigma ) = u_{ j},\quad j = 1,2,\ldots. }$$

Let u = limu _j . By (33), we have

$$\displaystyle\begin{array}{rcl} \Vert u_{j+1}\Vert _{L^{q}(\sigma )}& =& \Vert M_{\alpha }(u_{j}^{q}\sigma )\Vert _{ L^{q}(\sigma )} {}\\ & \leq & \varkappa \Vert u_{j}\Vert _{L^{q}(\sigma )}^{q} {}\\ & \leq & \varkappa \Vert u_{j+1}\Vert _{L^{q}(\sigma )}^{q}. {}\\ \end{array}$$

From this we deduce that $\Vert u_{j+1}\Vert _{L^{q}(\sigma )} \leq \varkappa ^{ \frac{1} {1-q} }$ for j = 0, 1, …. Since the norms $\Vert u_{j}\Vert _{L^{q}(\sigma )}^{q}$ are uniformly bounded, by the Monotone Convergence Theorem, we have for $u \mathop{:}=\lim _{j\rightarrow \infty }u_{j}$ that u ∈ L ^q(σ). Note that by construction u = M _α (u ^q d σ).

( ⇐ ) We can assume here that M _α ν is defined, for $\nu \in \mathcal{M}^{+}(\mathbb{R}^{n})$, as the centered fractional maximal function,

$$\displaystyle{ M_{\alpha }\nu (x) \mathop{:}=\sup _{r>0} \frac{\nu (B(x,r))} {\vert B(x,r)\vert ^{1- \frac{\alpha }{ n} }}, }$$

since it is equivalent to its uncentered analogue used above. Suppose that there exists u ∈ L ^q(σ) (u ≢ 0) such that u ≥ M _α (u ^q d σ). Set $\omega \mathop{:}= u^{q}d\sigma$. Let $\nu \in \mathcal{M}^{+}(\mathbb{R}^{n})$.

We note that we have

$$\displaystyle\begin{array}{rcl} \frac{M_{\alpha }\nu (x)} {M_{\alpha }\omega (x)}& =& \frac{\sup _{r>0} \frac{\vert B(x,r)\vert _{\nu }} {\vert B(x,r)\vert ^{1- \frac{\alpha }{n} }}} {\sup _{\rho>0} \frac{\vert B(x,\rho )\vert _{\omega }} {\vert B(x,\rho )\vert ^{1- \frac{\alpha }{n} }}} {}\\ & \leq & \sup _{r>0}\frac{\vert B(x,r)\vert _{\nu }} {\vert B(x,r)\vert _{\omega }} {}\\ & =:& M_{\omega }\nu (x). {}\\ \end{array}$$

Thus,

$$\displaystyle\begin{array}{rcl} \Vert M_{\alpha }\nu \Vert _{L^{q}(\sigma )}& =& \left \Vert \frac{M_{\alpha }\nu } {M_{\alpha }\omega }\right \Vert _{L^{q}((M_{\alpha }\omega )^{q}d\sigma )} {}\\ & \leq & \left \Vert \frac{M_{\alpha }\nu } {M_{\alpha }\omega }\right \Vert _{L^{q}(d\omega )} {}\\ & \leq & \left \Vert M_{\omega }\nu \right \Vert _{L^{q}(d\omega )} {}\\ & \leq & C\left \Vert M_{\omega }\nu \right \Vert _{L^{1,\infty }(\omega )} \leq C\Vert \nu \Vert, {}\\ \end{array}$$

by Jensen’s inequality and the (1, 1)-weak-type maximal function inequality for M _ω ν. This establishes (33). □ 

3.2 Weak-Type Inequality

For 0 ≤ α < n, we define the Hausdorff content on a set $E \subset \mathbb{R}^{n}$ to be

$$\displaystyle{ H^{n-\alpha }(E) \mathop{:}=\inf \left \{\sum _{ i=1}^{\infty }r_{ i}^{n-\alpha }: E \subset \bigcup _{ i=1}^{\infty }B(x_{ i},r_{i}),\right \} }$$

(38)

where the collection of balls {B(x _i , r _i )} forms a countable covering of E (see [1, 15]).

Theorem 3.2

Let $\sigma \in M^{+}(\mathbb{R}^{n})$ , 0 < q < 1, and 0 ≤α < n. Then the following conditions are equivalent:

• (1) There exists a positive constant ϰ _w such that

  $$\displaystyle{ \Vert M_{\alpha }\nu \Vert _{L^{q,\infty }(\sigma )} \leq \varkappa _{w}\,\Vert \nu \Vert \quad \text{for all}\,\,\nu \in \mathcal{M}(\mathbb{R}^{n}). }$$

• (2) There exists a positive constant C > 0 such that

  $$\displaystyle{ \sigma (E) \leq C\,(H^{n-\alpha }(E))^{q}\quad \text{for all Borel sets}\,\,E \subset \mathbb{R}^{n}. }$$

• (3) $M_{\alpha }\sigma \in L^{ \frac{q} {1-q},\infty }(\sigma )$ .

Remark 3.3

In the case α = 0 each of the conditions (1)–(3) is equivalent to $\sigma \in L^{ \frac{1} {1-q},\infty }(dx)$.

Proof

(1) ⇒ (2) Let K ⊂ E be a compact set in $\mathbb{R}^{n}$ such that H ^n−α(K) > 0. It follows from Frostman’s theorem (see the proof of Theorem 5.1.12 in [1]) that there exists a measure ν supported on K such that ν(K) ≤ H ^n−α(K), and, for every x ∈ K there exists a cube Q such that x ∈ Q and $\vert Q\vert _{\nu }\geq c\,\vert Q\vert ^{1- \frac{\alpha }{ n} }$, where c depends only on n and α. Hence,

$$\displaystyle{ M_{\alpha }\nu (x) \geq \sup _{Q\ni x} \frac{\vert Q\vert _{\nu }} {\vert Q\vert ^{1- \frac{\alpha }{ n} }} \geq c\quad \text{for all}\,\,x \in K, }$$

where c depends only on n and α. Consequently,

$$\displaystyle{ c^{q}\,\sigma (K) \leq \Vert M_{\alpha }\nu \Vert _{ L^{q,\infty }(\sigma )}^{q} \leq \varkappa _{ w}^{q}\Big(H^{n-\alpha }(K)\Big)^{q}. }$$

If H ^n−α(E) = 0, then H ^n−α(K) = 0 for every compact set K ⊂ E, and consequently σ(E) = 0. Otherwise,

$$\displaystyle{ \sigma (K) \leq \varkappa _{w}^{q}\Big(H^{n-\alpha }(K)\Big)^{q} \leq \varkappa _{ w}^{q}\Big(H^{n-\alpha }(K)\Big)^{q}, }$$

for every compact set K ⊂ E, which proves (2) with C = c ^−q ϰ _w ^q.

(2) ⇒ (3) Let $E_{t} \mathop{:}=\{ x: M_{\alpha }\sigma (x)> t\}$, where t > 0. Let K ⊂ E _t be a compact set. Then for each x ∈ K there exists Q _x  ∋ x such that

$$\displaystyle{ \frac{\sigma (Q_{x})} {\vert Q_{x}\vert ^{1- \frac{\alpha }{ n} }}> t.}$$

Now consider the collection {Q _x } _x ∈ K , which forms a cover of K. By the Besicovitch covering lemma, we can find a subcover {Q _i } _i ∈ I , where I is a countable index set, such that $K \subset \bigcup _{i\in I}Q_{i}$ and x ∈ K is contained in at most b _n sets in {Q _i }. By (2), we have

$$\displaystyle{\sigma (K) \leq [H^{n-\alpha }(K)]^{q},}$$

and by the definition of the Hausdorff content we have

$$\displaystyle{H^{n-\alpha }(K) \leq \sum \vert Q_{ i}\vert ^{1-\alpha /n}.}$$

Since {Q _i } have bounded overlap, we have

$$\displaystyle{\sum _{i\in I}\sigma (Q_{i}) \leq b_{n}\sigma (K).}$$

Thus,

$$\displaystyle{\sigma (K) \leq \left (b_{n}\frac{\sigma (K)} {t} \right )^{q},}$$

which shows that

$$\displaystyle{t^{ \frac{q} {1-q} }\sigma (K) \leq (b_{n})^{ \frac{1} {1-q} } <+\infty.}$$

Taking the supremum over all K ⊂ E _t in the preceding inequality, we deduce $M_{\alpha }\sigma \in L^{ \frac{q} {1-q},\infty }(\sigma )$.

(3) ⇒ (1). We can assume again that M _α is the centered fractional maximal function, since it is equivalent to the uncentered version. Suppose that $M_{\alpha }\sigma \in L^{ \frac{q} {1-q},\infty }(\sigma )$. Let $\nu \in \mathcal{M}(\mathbb{R}^{n})$. Then, as in the case of the strong-type inequality,

$$\displaystyle\begin{array}{rcl} \frac{M_{\alpha }\nu (x)} {M_{\alpha }\sigma (x)}& =& \frac{\sup _{r>0} \frac{\vert B(x,r)\vert _{\nu }} {\vert B(x,r)\vert ^{1- \frac{\alpha }{n} }}} {\sup _{\rho>0} \frac{\vert B(x,\rho )\vert _{\sigma }} {\vert B(x,\rho )\vert ^{1- \frac{\alpha }{n} }}} {}\\ & \leq & \sup _{r>0}\frac{\vert B(x,r)\vert _{\nu }} {\vert B(x,r)\vert _{\sigma }} =: M_{\sigma }\nu (x). {}\\ \end{array}$$

Thus, by Hölder’s inequality for weak L ^p-spaces,

$$\displaystyle\begin{array}{rcl} \Vert M_{\alpha }\nu \Vert _{L^{q,\infty }(\sigma )}& \leq & \Vert (M_{\alpha }\sigma )\,(M_{\sigma }\nu )\Vert _{L^{q,\infty }(\sigma )} {}\\ & \leq & \Vert M_{\alpha }\sigma \Vert _{ L^{ \frac{q} {1-q},\infty }(\sigma )}\,\Vert M_{\sigma }\nu \Vert _{L^{1,\infty }(\sigma )} {}\\ & \leq & c\Vert M_{\alpha }\sigma \Vert _{ L^{ \frac{q} {1-q},\infty }(\sigma )}\,\Vert \nu \Vert, {}\\ \end{array}$$

where in the last line we have used the (1, 1)-weak-type maximal function inequality for the centered maximal function M _σ ν. □ 

We now characterize weak-type (1, q)-inequalities (36) for the generalized dyadic maximal operator M _ρ defined by (35). The corresponding (p, q)-inequalities in the case 0 < q < p and p > 1 were characterized in [22]. The results obtained in [22] for weak-type inequalities remain valid in the case p = 1, but some elements of the proofs must be modified as indicated below.

Theorem 3.4

Let $\sigma \in \mathcal{M}^{+}(\mathbb{R}^{n})$ , 0 < q < 1, and 0 ≤α < n. Then the following conditions are equivalent:

• (1) There exists a positive constant ϰ _w such that  (36) holds.

• (2) $M_{\rho }\sigma \in L^{ \frac{q} {1-q},\infty }(\sigma )$ .

Proof

(2) ⇒ (1) The proof of this implication is similar to the case of fractional maximal operators. Let $\nu \in \mathcal{M}(\mathbb{R}^{n})$. Denoting by $Q,P \in \mathcal{Q}$ dyadic cubes in $\mathbb{R}^{n}$, we estimate

$$\displaystyle\begin{array}{rcl} \frac{M_{\rho }\nu (x)} {M_{\rho }\sigma (x)}& =& \frac{\sup _{Q\ni x}(\rho _{Q}\,\vert Q\vert _{\nu })} {\sup _{P\ni x}(\rho _{P}\,\vert P\vert _{\sigma })} {}\\ & \leq & \sup _{Q\ni x}\frac{\vert Q\vert _{\nu }} {\vert Q\vert _{\sigma }} =: M_{\sigma }\nu (x). {}\\ \end{array}$$

By Hölder’s inequality for weak L ^p-spaces,

$$\displaystyle\begin{array}{rcl} \Vert M_{\rho }\nu \Vert _{L^{q,\infty }(\sigma )}& \leq & \Vert (M_{\rho }\sigma )\,(M_{\sigma }\nu )\Vert _{L^{q,\infty }(\sigma )} {}\\ & \leq & \Vert M_{\rho }\sigma \Vert _{ L^{ \frac{q} {1-q},\infty }(\sigma )}\,\Vert M_{\sigma }\nu \Vert _{L^{1,\infty }(\sigma )} {}\\ & \leq & c\Vert M_{\rho }\sigma \Vert _{ L^{ \frac{q} {1-q},\infty }(\sigma )}\,\Vert \nu \Vert, {}\\ \end{array}$$

by the (1, 1)-weak-type maximal function inequality for the dyadic maximal function M _σ .

(1) ⇒ (2) We set f = sup _Q (λ _Q χ _Q ) and d ν = f d σ, where $\{\lambda _{Q}\}_{Q\in \mathcal{Q}}$ is a finite sequence of non-negative reals. Then obviously

$$\displaystyle{ M_{\rho }\nu (x) \geq \sup _{Q}(\lambda _{Q}\rho _{Q}\chi _{Q}),\quad \text{and}\quad \Vert \nu \Vert \leq \sum _{Q}\lambda _{Q}\,\vert Q\vert _{\sigma }. }$$

By (1), for all $\{\lambda _{Q}\}_{Q\in \mathcal{Q}}$,

$$\displaystyle{ \Vert \sup _{Q}(\lambda _{Q}\rho _{Q}\chi _{Q})\Vert _{L^{q,\infty }(\sigma )} \leq \varkappa _{v}\,\sum _{Q}\lambda _{Q}\,\vert Q\vert _{\sigma }. }$$

Hence, by Theorem 1.1 and Remark 1.2 in [22], it follows that (2) holds. □ 

4 Carleson Measures for Poisson Integrals

In this section we treat (1, q)-Carleson measure inequalities for Poisson integrals with respect to Carleson measures $\sigma \in \mathcal{M}^{+}(\mathbb{R}_{+}^{n+1})$ in the upper half-space $\mathbb{R}_{+}^{n+1} =\{ (x,y): x \in \mathbb{R}^{n},y> 0\}$. The corresponding weak-type (p, q)-inequalities for all 0 < q < p as well as strong-type (p, q)-inequalities for 0 < q < p and p > 1, were characterized in [23]. Here we consider strong-type inequalities of the type

$$\displaystyle{ \Vert \mathbf{P}\nu \Vert _{L^{q}(\mathbb{R}_{+}^{n+1},\sigma )} \leq \varkappa \,\Vert \nu \Vert _{\mathcal{M}^{+}(\mathbb{R}^{n})},\quad \text{for all}\,\,\nu \in \mathcal{M}^{+}(\mathbb{R}^{n}), }$$

(39)

for some constant ϰ > 0, where P ν is the Poisson integral of $\nu \in \mathcal{M}^{+}(\mathbb{R}^{n})$ defined by

$$\displaystyle{ \mathbf{P}\nu (x,y) \mathop{:}=\int _{\mathbb{R}^{n}}P(x - t,y)d\nu (t),\quad (x,y) \in \mathbb{R}_{+}^{n+1}. }$$

Here P(x, y) denotes the Poisson kernel associated with $\mathbb{R}_{+}^{n+1}$.

By P ^∗ μ we denote the formal adjoint (balayage) operator defined, for $\mu \in \mathcal{M}^{+}(\mathbb{R}_{+}^{n+1})$, by

$$\displaystyle{ \mathbf{P}^{{\ast}}\mu (t) \mathop{:}=\int _{ \mathbb{R}_{+}^{n+1}}P(x - t,y)d\mu (x,y),\quad t \in \mathbb{R}^{n}. }$$

We will also need the symmetrized potential defined, for $\mu \in \mathcal{M}^{+}(\mathbb{R}_{+}^{n+1})$, by

$$\displaystyle{ \mathbf{P}\mathbf{P}^{{\ast}}\mu (x,y) \mathop{:}= \mathbf{P}\Big[(\mathbf{P}^{{\ast}}\mu )dt\Big] =\int _{ \mathbb{R}_{+}^{n+1}}P(x -\tilde{ x},y +\tilde{ y})d\mu (\tilde{x},\tilde{y}),\quad (x,y) \in \mathbb{R}_{+}^{n+1}. }$$

As we will demonstrate below, the kernel of PP ^∗ μ satisfies the weak maximum principle with constant h = 2ⁿ⁺¹.

Theorem 4.1

Let $\sigma \in \mathcal{M}^{+}(\mathbb{R}_{+}^{n+1})$ , and let 0 < q < 1. Then inequality ( 39 ) holds if and only if there exists a function u > 0 such that

$$\displaystyle{ u \in L^{q}(\mathbb{R}_{ +}^{n+1},\sigma ),\quad \text{and}\quad u \geq \mathbf{P}\mathbf{P}^{{\ast}}(u^{q}\sigma )\quad \mathrm{in}\,\,\mathbb{R}_{ +}^{n+1}. }$$

Moreover, if (39) holds, then a positive solution u = PP ^∗ (u ^q σ) such that $u \in L^{q}(\mathbb{R}_{+}^{n+1},\sigma )$ can be constructed as follows: u = lim_j→∞ u _j , where

$$\displaystyle{ u_{j+1} \mathop{:}= \mathbf{P}\mathbf{P}^{{\ast}}(u_{ j}^{q}\sigma ),\quad j = 0,1,\ldots,\quad u_{ 0} \mathop{:}= c_{0}(\mathbf{P}\mathbf{P}^{{\ast}}\sigma )^{ \frac{1} {1-q} }, }$$

(40)

for a small enough constant c ₀ > 0 (depending only on q and n), which ensures that u _j+1 ≥ u _j . In particular, $u \geq c_{0}\,(\mathbf{P}\mathbf{P}^{{\ast}}\sigma )^{ \frac{1} {1-q} }$ .

Proof

We first prove that (39) holds if and only if

$$\displaystyle{ \Vert \mathbf{P}\mathbf{P}^{{\ast}}\mu \Vert _{ L^{q}(\mathbb{R}_{+}^{n+1},\sigma )} \leq \varkappa \,\Vert \mu \Vert _{\mathcal{M}^{+}(\mathbb{R}_{+}^{n+1})},\quad \text{for all}\,\,\mu \in \mathcal{M}^{+}(\mathbb{R}_{ +}^{n+1}). }$$

(41)

Indeed, letting ν = P ^∗ μ in (39) yields (41) with the same embedding constant ϰ.

Conversely, suppose that (41) holds. Then by Maurey’s factorization theorem (see [14]), there exists $F \in L^{1}(\mathbb{R}_{+}^{n+1},\sigma )$ such that F > 0 d σ-a.e., and

$$\displaystyle{ \Vert F\Vert _{L^{1}(\mathbb{R}_{+}^{n+1},\sigma )} \leq 1,\quad \sup _{(x,y)\in \mathbb{R}_{+}^{n+1}}\mathbf{P}\mathbf{P}^{{\ast}}(F^{1-\frac{1} {q} }d\sigma )(x,y) \leq \varkappa. }$$

(42)

By letting y ↓ 0 in (42) and using the Monotone Convergence Theorem, we deduce

$$\displaystyle{ \sup _{x\in \mathbb{R}^{n}}\mathbf{P}^{{\ast}}(F^{1-\frac{1} {q} }d\sigma )(x) \leq \varkappa. }$$

(43)

Hence, by Jensen’s inequality and (43), for any $\nu \in \mathcal{M}^{+}(\mathbb{R}^{n})$, we have

$$\displaystyle{ \Vert \mathbf{P}\nu \Vert _{L^{q}(\mathbb{R}_{+}^{n+1},\sigma )} \leq \Vert \mathbf{P}\nu \Vert _{ L^{1}(\mathbb{R}_{+}^{n+1},\,F^{1-\frac{1} {q} }d\sigma )} =\Vert \mathbf{P}^{{\ast}}(F^{1-\frac{1} {q} }d\sigma )\Vert _{L^{1 }(\mathbb{R}^{n},\,d\nu )} \leq \varkappa \,\Vert \nu \Vert _{\mathcal{M}^{+}(\mathbb{R}^{n})}. }$$

We next show that the kernel of PP ^∗ satisfies the weak maximum principle with constant h = 2ⁿ⁺¹. Indeed, suppose $\mu \in \mathcal{M}^{+}(\mathbb{R}_{+}^{n+1})$, and

$$\displaystyle{ \mathbf{P}\mathbf{P}^{{\ast}}\mu (\tilde{x},\tilde{y}) \leq M,\quad \text{for all}\,\,(\tilde{x},\tilde{y}) \in S(\mu ). }$$

Without loss of generality we may assume that $S(\mu ) \Subset \mathbb{R}_{+}^{n+1}$ is a compact set. For $t \in \mathbb{R}^{n}$, let (x ₀, y ₀) ∈ S(μ) be a point such that

$$\displaystyle{ \vert (t,0) - (x_{0},y_{0})\vert = \text{dist}\Big((t,0),S(\mu )\Big). }$$

Then by the triangle inequality, for any $(\tilde{x},\tilde{y}) \in S(\mu )$,

$$\displaystyle{ \vert (x_{0},y_{0}) - (\tilde{x},-\tilde{y})\vert \leq \vert (x_{0},y_{0}) - (t,0)\vert + \vert (t,0) - (\tilde{x},-\tilde{y})\vert \leq 2\vert (t,0) - (\tilde{x},\tilde{y})\vert. }$$

Hence,

$$\displaystyle{ \sqrt{\vert t -\tilde{ x}\vert ^{2 } +\tilde{ y}^{2}} \geq \frac{1} {2}\,\sqrt{\Big[\vert x_{0 } -\tilde{ x}\vert ^{2 } + (y_{0 } +\tilde{ y})^{2 } \Big]}. }$$

It follows that, for all $t \in \mathbb{R}^{n}$ and $(\tilde{x},\tilde{y}) \in S(\mu )$, we have

$$\displaystyle{ P(t -\tilde{ x},\tilde{y}) \leq 2^{n+1}P(x_{ 0} -\tilde{ x},y_{0} +\tilde{ y}). }$$

Consequently, for all $t \in \mathbb{R}^{n}$,

$$\displaystyle{ \mathbf{P}^{{\ast}}\mu (t) \leq 2^{n+1}\mathbf{P}\mathbf{P}^{{\ast}}\mu (x_{ 0},y_{0}) \leq 2^{n+1}M. }$$

Applying the Poisson integral P[dt] to both sides of the preceding inequality, we obtain

$$\displaystyle{ \mathbf{P}\mathbf{P}^{{\ast}}\mu (x,y) \leq 2^{n+1}M\quad \text{for all}\,\,(x,y) \in \mathbb{R}_{ +}^{n+1}. }$$

This proves that the weak maximum principle holds for PP ^∗ with h = 2ⁿ⁺¹. It follows from Theorem 2.3 that (39) holds if and only if there exists a non-trivial $u \in L^{q}(\mathbb{R}_{+}^{n+1},\sigma )$ such that u ≥ PP ^∗(u ^q d σ). Moreover, a positive solution u = PP ^∗(u ^q σ) can be constructed as in the statement of Theorem 4.1 (see details in [16]). □ 

Corollary 4.2

Under the assumptions of Theorem  4.1 , inequality ( 39 ) holds if and only if there exists a function $\phi \in L^{1}(\mathbb{R}^{n})$ , ϕ > 0 a.e., such that

$$\displaystyle{ \phi \geq \mathbf{P}^{{\ast}}\Big[(\mathbf{P}\phi )^{q}d\sigma \Big]\quad \text{a.e. in}\,\,\mathbb{R}^{n}. }$$

Moreover, if ( 39 ) holds, then there exists a positive solution $\phi \in L^{1}(\mathbb{R}^{n})$ to the equation $\phi = \mathbf{P}^{{\ast}}\Big[(\mathbf{P}\phi )^{q}d\sigma \Big]$ .

Proof

If (39) holds then by Theorem 4.1 there exists u = PP ^∗(u ^q d σ) such that u > 0 and $u \in L^{q}(\mathbb{R}_{+}^{n+1},\sigma )$. Setting ϕ = P ^∗(u ^q d σ), we see that

$$\displaystyle{ \mathbf{P}\phi = \mathbf{P}\mathbf{P}^{{\ast}}(u^{q}d\sigma ) = u, }$$

so that ϕ = P ^∗[(P ϕ)^q d σ], and consequently

$$\displaystyle{ \Vert \phi \Vert _{L^{1}(\mathbb{R}^{n})} =\Vert u\Vert _{L^{q}(\mathbb{R}_{+}^{n+1},\sigma )}^{q} =\int _{ \mathbb{R}^{n}}u(x,y)dx <\infty. }$$

Conversely, if there exists ϕ > 0, $\phi \in L^{1}(\mathbb{R}^{n})$ such that $\phi \geq \mathbf{P}^{{\ast}}\Big[(\mathbf{P}\phi )^{q}d\sigma \Big]$, then letting u = P ϕ, we see that u is a positive harmonic function in $\mathbb{R}_{+}^{n+1}$ so that

$$\displaystyle{ u(x,y) = \mathbf{P}\phi (x,y) \geq \mathbf{P}\mathbf{P}^{{\ast}}(u^{q}d\sigma )(x,y),\quad (x,y) \in \mathbb{R}_{ +}^{n+1}. }$$

Notice that the kernel $P(x -\tilde{ x},y +\tilde{ y})$ of the operator PP ^∗ has the property

$$\displaystyle{ \int _{\mathbb{R}^{n}}P(x -\tilde{ x},y +\tilde{ y})dx = 1,\quad y> 0,\,\,(\tilde{x},\tilde{y}) \in \mathbb{R}_{+}^{n+1}, }$$

and consequently, for all y > 0,

$$\displaystyle{ \int _{\mathbb{R}^{n}}\int _{\mathbb{R}_{+}^{n+1}}P(x -\tilde{ x},y +\tilde{ y})u(\tilde{x},\tilde{y})^{q}d\sigma (\tilde{x},\tilde{y})\,dx =\int _{ \mathbb{R}_{+}^{n+1}}u(\tilde{x},\tilde{y})^{q}d\sigma (\tilde{x},\tilde{y}), }$$

Hence,

$$\displaystyle{ \Vert u\Vert _{L^{q}(\mathbb{R}_{+}^{n+1},\sigma )}^{q} =\int _{ \mathbb{R}^{n}}\Big[\mathbf{P}\mathbf{P}^{{\ast}}(u^{q}d\sigma )\Big](x,y)\,dx \leq \int _{ \mathbb{R}^{n}}u(x,y)dx =\Vert \phi \Vert _{L^{1}(\mathbb{R}^{n})} <\infty. }$$

Thus, inequality (39) holds by Theorem 4.1. □