Reduced measures for semilinear elliptic equations involving Dirichlet operators

We consider elliptic equations of the form (E) $-Au=f(x,u)+\mu$, where $A$ is a negative definite self-adjoint Dirichlet operator, $f$ is a function which is continuous and nonincreasing with respect to $u$ and $\mu$ is a Borel measure of finite potential. We introduce a probabilistic definition of a solution of (E), develop the theory of good and reduced measures introduced by H. Brezis, M. Marcus and A.C. Ponce in the case where $A=\Delta$ and show basic properties of solutions of (E). We also prove Kato's type inequality. Finally, we characterize the set of good measures in case $f(u)=-u^p$ for some $p>1$.


Introduction
Let E be a separable locally compact metric space and let m be a Radon measure on E such that supp[m] = E. In the present paper we study semilinear equations of the form −Au = f (x, u) + µ, (1.1) where µ is a Borel measure on E, f : E × R → R is a measurable function such that f (·, u) = 0, u ≤ 0, and f is nonincreasing and continuous with respect to u. As for the operator A, we assume that it is a negative definite self-adjoint Dirichlet operator on L 2 (E; m). Saying that A is a Dirichlet operator we mean that (Au, (u − 1) + ) ≤ 0, u ∈ D(A). (see [12,23].) The class of such operators is quite large. It contains many local as well as nonlocal operators. The model examples are Laplace operator ∆ (or uniformly elliptic divergence form operator) and the fractional Laplacian ∆ α with α ∈ (0, 1).

Equivalently, operator
Many other examples are to be found in [12,23].
T. Klimsiak Let Cap denote the capacity determined by (E, D[E]) (see Section 2). It is known (see [13]) that any Borel signed measure µ on E admits a decomposition µ = µ c + µ d into the singular (concentrated) part µ c with respect to Cap and the absolutely continuous (diffuse, smooth) part µ d with respect to Cap. The smooth part µ d is fully characterized in [20].
The study of semilinear equations of the form (1.1) in case µ is smooth, i.e. when µ c = 0, goes back to the papers by Brezis and Strauss [7] and Konishi [21] In [7,21] the existence of a solution of (1.1) is proved for µ ∈ L 1 (E; m). At present existence, uniqueness and regularity results are available for equation (1.1) involving general bounded smooth measure µ and operator corresponding to Dirichlet form (see Klimsiak and Rozkosz [17] for the case of symmetric regular Dirichlet form and [19] for the case of quasi-regular, possibly non-symmetric Dirichlet form). The case µ c = 0 is much more involved. Ph. Bénilan and H. Brezis [2] has observed that in such a case equation (1.1) need not have a solution even if A = ∆. In [5] (see also [4]) H. Brezis, M. Marcus and A.C. Ponce introduced the concept of good measure, i.e. a bounded measure for which (1.1) has a solution, and the concept of reduced measure, i.e. the largest good measure, which is less then or equal to µ. In case A = ∆ these concepts are by now quite well investigated (see [2,5]). The situation is entirely different in case of more general local operators or nonlocal operators. There are known, however, some existence and uniqueness results for (1.1) in case A is a diffusion operator (see Véron [28]) and in case A = ∆ α with α ∈ (0, 1) (see Chen and Véron [8]).
The main purpose of the paper is to present a new approach to (1.1) that provides a unified way of treating (1.1) for the whole class of negative defined self-adjoint Dirichlet operators A and for µ from some class of measures M including the class M b of bounded signed Borel measures on E. In particular, we give a new definition of a solution of (1.1) and investigate the structure of good and reduced measures relative to (1.1). In case A = ∆ our definition is equivalent to the definition of a solution adopted in [2,5], so our results generalize the results of [2,5] to wide class of operators. In fact, they generalize the existing results even in case A = ∆, because in this case M b M and M contains important in applications unbounded measures. The second purpose of our paper is to give a probabilistic interpretation for solutions of (1.1).
First, some remarks concerning our definition of a solution and the class M are in order. Suppose we want to consider problem (1.1) for some class of measures M including L 1 (E; m). Considering f ≡ 0 in (1.1) we see that then G := −A −1 should be well defined on L 1 (E; m), i.e. the following condition should be satisfied: T t g dt < ∞, m-a.e., g ∈ L 1,+ (E; m). (1.3) Condition (1.3) is nothing but the statement that the semigroup {T t , t ≥ 0} generated by A (or, equivalently, the Dirichlet form (E, D[E])) is transient (see [12,Section 1.5]). It is well known that then there exists a kernel {R(x, dy), x ∈ E} such that for every g ∈ L 1,+ (E; m), E g(y)R(·, dy) = Gg, m-a.e.
If u is a solution of (1.1) with f ≡ 0 then where R • µ is a Borel measure defined as E g(x)(R • µ)(dx) = E E g(y)R(x, dy)µ(dx), g ∈ B + (E).
Therefore R • µ must be absolutely continuous with respect to the measure m for every bounded Borel measure µ. This condition is known in the literature as the Meyer hypothesis (L) (see [3]) or the condition of absolute continuity of the resolvent {G α , α > 0} (see [12]). For the reasons explained above in the paper we assume that {T t , t ≥ 0} is transient and hypothesis (L) is satisfied. It is known that under these assumptions there exists a Borel function r : Using the kernel r we can give our first, purely analytical definition of a solution of (1.1). Namely, we say that a Borel function u on E is a solution of (1.1) if for m-a.e. x ∈ E. Of course, to make this definition correct we have to assume that the integrals in (1.4) exist. Therefore the class M we consider consists of Borel measures µ on E such that E r(x, y) |µ|(dy) < ∞ for m-a.e. x ∈ E. We will show that M b (E) ⊂ M. In general, the inclusion is strict. For instance, if A = ∆ α , α ∈ (0, 1], on an open set D ⊂ R d , then M includes the set of all Borel measures µ on E such that δ α · µ ∈ M b , where δ(x) = dist(x, ∂D). We also show that in case µ ∈ M b and A is a uniformly elliptic divergence form operator on a bounded domain in R d definition (1.4) is equivalent to Stampacchia's definition by duality (see [27]). Unfortunately, definition (1.4) is rather inconvenient for studying (1.1). One of the main results of the paper says that (1.4) is equivalent to our second, probabilistic in nature definition of a solution. At first glance the probabilistic definition seems to be more complicated than (1.4), but as a matter of fact suits much better to the purposes of the present paper. Let X = ({X t , t ≥ 0}, {P x , x ∈ E}) be a Hunt process with life time ζ associated with the form (E, D[E]). We say that u is a probabilistic solution of (1.1) if (a) f (·, u) · m ∈ M and there exists a local martingale additive functional M of X such that for quasi every (q.e. for short) x ∈ E (Here A µ d denotes a continuous additive functional of X of finite variation in the Revuz correspondence with µ d ), (b) for every polar set N ⊂ E, every stopping time T ≥ ζ and every sequence of stopping times {τ k } such that τ k ր T and E x sup t≤τ k |u(X t )| < ∞ for x ∈ E \ N and k ≥ 1 we have where E x denotes the integration with respect to probability P x and The above probabilistic definition allows us to develop a general theory of equations of the form (1.1). Moreover, in our opinion, the theory based on the probabilistic definition is elegant and simple. We first prove some regularity results. We show that if u is a solution of (1.1) and is an extension of the domain of the form E such that the pair (E, D e [E]) is a Hilbert space (see [12]). We also prove Stampacchia's type inequality which says that for every strictly positive excessive function ρ (for ρ ≡ 1 for instance) and We next study the structure of the set G of good measures and the set of reduced measures relative to A, f . Let us recall that the reduced measure is the largest measure µ * ∈ M such that µ * ≤ µ and there exists a solution of (1.1) with µ replaced by µ * . A measure µ ∈ M is good, if µ * = µ. By results of [17,19], if µ c = 0, then µ is good. In the present paper we first show that µ − µ * ⊥Cap.
Then we show that, as in the case of Laplace operator, the set G is convex and closed under the operation of taking maximum of two measures. We also show that µ ∈ G if and only if µ = g − Av for some functions g, v on E such that g · m, f (·, v) · m ∈ M and Av ∈ M. From this characterization of G we deduce that for every strictly positive excessive function ρ, where We also show that under some additional assumption on the growth of f (it is satisfied for instance if |f (x, u)| ≤ c 1 + c 2 e u 2 ), for every strictly positive excessive function ρ, where the closure is taken in the space (M ρ , · T V,ρ ). In Section 6 we prove the so-called inverse maximum principle and Kato's type inequality. In our context Kato's inequality says that if u is a solution of (1.1) then This form of Kato's inequality for Laplace operator was proved by H. Brezis and A.C. Ponce in [6].
In the last section we study the set of good measures G for problem (1.1) with f having at most polynomial growth, i.e. for f satisfying |f (x, u)| ≤ c|u| p , x ∈ E, u ≥ 0 for some p > 1. For this purpose, we introduce a new capacity Cap A,p , which in the special case, when A = ∆ α on an open bounded set D ⊂ R d with zero boundary condition is equivalent to the Bessel capacity defined as for compact sets K ⊂ D. We prove that if µ ∈ M and µ + is absolutely continuous with respect to Cap A,p ′ , where p ′ denotes the Hölder conjugate to p, then a solution of (1.1) exists, i.e. µ ∈ G. For f of the form we fully characterize the set G. Namely, we prove that the absolute continuity of µ + with respect to Cap A,p ′ is also necessary for the existence of a solution of (1.1). Thus, in case f is given by (1.6), Moreover, Cap A,p ′ denotes the absolutely continuous part of µ + with respect to Cap A,p ′ .

Preliminaries
In the paper E is a locally compact separable metric space and m is a positive Radon measure on E such that supp[m] = E. By (E, D[E]) we denote a symmetric regular Dirichlet form on L 2 (E; m) (see [12] or [23] for the definitions). We will always assume that (E, D[E]) is transient, i.e. there exists a strictly positive function g on E such that where u E = E(u, u), u ∈ D[E]. As usual, for α > 0 we set , where (·, ·) is the usual inner product in L 2 (E; m). By Riesz's theorem, for every α > 0 and f ∈ L 2 (E; m) there exists a unique function G α f ∈ L 2 (E; m) such that It is an elementary check that {G α , α > 0} is a strongly continuous contraction resolvent on L 2 (E; m). By {T t , t ≥ 0} we denote the associated semigroup and by (A, D(A)) the self-adjoint negative definite Dirichlet operator generated by {T t }. It is well known that A satisfies (1.2) (see [12,Section 1.3]). Conversely, one can prove (see [23, page 39 and then for arbitrary A ⊂ E we set We say that some property P holds quasi everywhere (q.e. for short) if a set for which it does not hold is exceptional.
We say that a function u on E is quasi-continuous if there exists a nest {F n } such that u |Fn is continuous for every n ≥ 1. It is known that each function u ∈ D[E] has a quasi-continuous m-version.
A Borel measure µ on E is called smooth if it does not charge exceptional sets and there exists a nest {F n } such that |µ|(F n ) < ∞, n ≥ 1. By S we denote the set of all smooth measures on E.
By S By u · µ w denote the Borel measure on E defined as whenever the integrals exist.
With a regular symmetric Dirichlet form (E, D[E]) one can associate uniquely a symmetric Hunt process X = ((X t ) t≥0 , (P x ) x∈E , (F t ) t≥0 , ζ) (see [12,Section 7.2]). It is related to (E, D[E]) by the formula where E x stands for the expectation with respect to the measure P x . For α, t ≥ 0 and f ∈ B + (E) we write Observe that for α, t > 0 and f ∈ L 2 (E; m), For simplicity we denote R 0 by R. We say that some function on E is measurable if it is universally measurable, i.e. measurable with respect to the σ-algebra where P(E) is the set of all probability measures on E and B µ (E) is the completion of B(E) with respect to the measure µ.
A positive measurable function u on E is called α-excessive if for every β > 0, (α + β)R α+β u ≤ u and αR α u ր u as α → ∞. By S α we denote the set of α-excessive functions. We put S = S 0 .
By S (0) 00 we denote the set of all µ ∈ S (0) 0 such that |µ|(E) < ∞ and R|µ| is bounded. For a Borel set B we set i.e. σ B is the first hitting time of B, D A is the first debut time of B and τ B is the first exit time of B.
By T we denote the set of all stopping times with respect to the filtration (F t ) t≥0 and by D the set of all measurable functions u on E for which the family For a Borel measure µ on E and α ≥ 0 we denote by µ • R α the measure defined as and by P µ we denote the measure In the whole paper we assume that m is the reference measure for X, i.e. for all x ∈ E and α > 0 we have R α (x, ·) ≪ m. It is well known (see [12,Lemma 4.2.4]) that in this case for every α ≥ 0 there exists a B(E) ⊗ B(E) measurable function It is also clear that by symmetry of X, r α (x, y) = r α (y, x) for x, y ∈ E, α ≥ 0. In what follows we put r(x, y) = r 0 (x, y), x, y ∈ E. Thanks to the existence of r α we may define R α µ for arbitrary positive Borel measure µ by putting It is well known (see [12,Section 5.1] and [3, Theorem V.2.1] that for each µ ∈ S there exists a unique perfect positive continuous additive functional A µ in the Revuz duality with µ, and moreover,

Linear equations
In this section we give some definitions of a solution of the linear problem where µ is a Borel measure such that R|µ|(x) < ∞ for q.e. x ∈ E. The class of such measures will be denoted by M.
In the whole paper we adopt the convention that E r(x, y) dµ(y) = 0 for every Borel measure µ on E such that E r(x, y) dµ + (y) = E r(x, y) dµ − (y) = ∞. We call u : E → R ∪ {−∞, ∞} a numerical function on E.

Solutions defined via the resolvent kernel and regularity results
Definition 3.1. We say that a measurable numerical function u on E is a solution of (3.1) if Let us note that by [3, Proposition V.1.4], if the above equality holds for every x ∈ E, then u is Borel measurable. Since µ ∈ M, u is finite q.e.
Proof. Since the form E is assumed to be transient, there exists a strictly positive Borel function f on E such that Rf < ∞, q.e. From this we conclude that f · m is a smooth measure. Hence, by [12,Theorem 2.2.4], there exists an increasing sequence {F n } of closed subsets of E such that n≥1 F n = E, q.e. and sup x∈E R(1 Fn f )(x) < ∞ (see also comments following [12, Corollary 2.2.2]). As a matter of fact, in [12] in the last condition sup is replaced by ess sup with respect to m, however in view of [3, Proposition II.3.2], it holds true also with supremum norm. We have Hence R|µ| is finite q.e., i.e. µ ∈ M. ✷ Using Definition 3.1 we can easily prove some regularity result for solutions of (3.1). For this purpose, for k ≥ 0 set and for every k ≥ 0, Proof. For α ≥ 0 and measurable functions u, v on E set whenever the integral exists. By the definition of a solution of (3 Therefore On the other hand, since αR α is Markovian, we have

Probabilistic solutions
In this subsection we give an equivalent definition of solution of (3.1) using stochastic equations involving a Hunt process X associated with the Dirichlet operator A. We begin with the following lemma.
Proof. Let u α = αR α u, α > 0. Then By the Markov property, for every t ≥ 0 and x ∈ E we have and let N α,x denote a cádlág modification of the martingaleN α,x . Then for every By the integration by parts formula applied to the processes e αt and e −αt u α (X) we get Since u is an excessive function, A α is an increasing process and u α (x) ր u(x) for every x ∈ E as α ր ∞. Hence denote the quadratic variations of processes u α (X) and u(X), respectively. By [12, Theorem 4.2.2] there exists an exceptional set N ⊂ E such that for every x ∈ E \ N , Let ζ i , ζ p denote the totally inaccessible and the predictable part of ζ, respectively. From [12,Theorem 4.2.2] it also follows that while by the fact that u α , u are potentials, By what has already been proved, By the generalized Dini theorem (see [10, p. 185]), u α (X) ր u(X) uniformly on compact subsets of [0, ∞). Observe that for every t ≥ 0 and q.e. x ∈ E, Hence u(X) is a supermartingale and lim t→∞ E x u(X t ) < ∞. Therefore by [25,Theorem III.13], for q.e. x ∈ E there exists an increasing predictable process Since the filtration is quasi-left continuous, M x has no predictable jumps. Since X is quasi-left continuous, it also has no predictable jumps, which implies that u(X) has no predictable jumps, because u is quasi-continuous. Thus C x is continuous. Since u(X) is a special semimartingale, there exists a localizing sequence {τ x n } ⊂ T such that for every n ≥ 1, By [12,Lemma A.3.3] there exists a process A such that A = C x for q.e. x ∈ E. Of course, A is a positive continuous additive functional. Putting we see that M is an additive functional and M x = M , P x -a.s. for q.e. x ∈ E. Thus M is a local martingale additive functional. By [12, Theorem 5.1.4] there exists ν ∈ S such that A = A ν . In particular, for every α ≥ 0, Observe that by the resolvent identity, for every α ≥ 0 we have On the other hand, by (3.2) and the integration by parts formula applied to the processes e −αt and u(X t ), as k → ∞. From this, (3.4) and (3.5) we conclude that for q.e x ∈ E, By this and [3, Proposition II.3.2], R α (µ − ν) ≥ 0. Since α ≥ 0 was arbitrary, applying Lemma 3.5 shows that µ ≥ ν. Since µ⊥Cap, it follows that ν ≡ 0 or, equivalently, that A ν ≡ 0. Therefore from (3.3) it follows that u(X) is a local martingale. ✷ Let us recall that a process M is called a local martingale additive functional (MAF) if it is an additive functional and M is an (F, P x )-local martingale for q.e. x ∈ E.
for q.e. x ∈ E. Moreover, for every polar set N ⊂ E, every stopping time T ≥ ζ and Proof. Let w = Rµ c and v = Rµ d . It is well known (see [17,Lemma 4.3]) that v is quasi-continuous and that there exists a uniformly integrable MAF for q.e. x ∈ E. By Theorem 3.6, w is quasi-continuous and there exists a local MAF M w such that for q.e. x ∈ E. Let N ⊂ E be a polar set such that (3.8), (3.9) hold for x ∈ E \ N . Let {τ k } be as in the formulation of the theorem. Then M v,τ k , M w,τ k are both uniformly integrable and by (3.8) and (3.9), Letting k → ∞ in the above equation yields which proves (3.7). Adding (3.8) to (3.9) gives (3.6). ✷ Remark 3.8. Under the assumptions of Theorem 3.7, for every α > 0, To see this we use (3.4) and arguments following it.
We are now ready to introduce the second definition of a solution of (3.1) making use of the Hunt process X associated with operator A. Solutions of (3.1) in the sense of this definition will be called probabilistic solutions or simply solutions, because we will show that our second definition is equivalent to the definition via the resolvent kernel. Definition 3.9. We say that a measurable numerical function u on E is a probabilistic solution of (3.1) if (a) there exists a local MAF M such that for q.e. x ∈ E, (b) for every polar set N ⊂ E, every stopping time T ≥ ζ and every sequence Any sequence {τ k } with the properties listed in (b) will be called the reducing sequence for u, and we will say that {τ k } reduces u.
x ∈ E. Therefore letting k → ∞ and using (b) we see that for q.e. x ∈ E, Note that if A is a uniformly elliptic divergence form operator then by [17,Proposition 5.3], u is also a solution of (3.1) in the sense of Stampacchia (see [27]). In the sequel we will show that this holds true for general Borel measures and wider class of operators. Proof. Assume that u is a solution of (3.1) in the sense of Definition 3.1. Then by Theorem 3.7, u is a probabilistic solution. Now suppose that u is a probabilistic solution of (3.1). Then using (a) and (b) of the definition of a probabilistic solution of (3.1) we obtain for q.e. x ∈ E. ✷

Semilinear equations
In what follows µ ∈ M and f : E × R → R is a function satisfying the following conditions: R ∋ y → f (x, y) is continuous for every x ∈ E and E ∋ x → f (x, y) is measurable for every y ∈ R.
In this section we consider semilinear equation of the form Definition 4.1. We say that a measurable numerical function u on E is a solution of (4.1) if f (·, u) · m ∈ M and u is a solution of (3.1) with µ replaced by f (·, u) · m + µ.

Comparison results, a priori estimates and regularity of solutions
In the sequel, for a given real function u on E we write y ∈ R and f 1 or f 2 satisfies (H1). Then u 1 ≤ u 2 q.e., where u 1 (resp. u 2 ) is a solution of (4.1) with data f 1 , µ 1 (resp. f 2 , µ 2 ).
Proof. Let {τ k } be a common reducing sequence for u 1 and u 2 . We assume that f 1 satisfies (H1). By the Tanaka-Meyer formula (see [25, Theorem IV.66]), for every k ≥ 1, for q.e. x ∈ E. From the assumption µ 1 ≤ µ 2 and properties of the Revuz duality it follows that dA µ 1 d ≤ dA µ 2 d , P x -a.s. for q.e. x ∈ E. By (H1) and the assumptions on f 1 and f 2 , for q.e. x ∈ E, which proves the proposition. ✷ Proof. Let {τ k } be a common reducing sequence for u 1 and u 2 . By the Tanaka-Meyer formula, for q.e. x ∈ E. By (H1) the second term on the right-hand side of (4.2) is nonpositive. Therefore from (4.2) it follows that Proof. We apply Proposition 4.4 to u 1 = u, u 2 = 0, µ 1 = µ, µ 2 = −f (·, 0).

✷
Given a positive function ρ ∈ S, we denote by M ρ the set of all measures µ ∈ M such that µ ρ < ∞, where µ ρ = ρ · µ T V . Important examples of positive ρ ∈ S are ρ = 1 and ρ = Rη, where η is a positive Borel function on E. Let us also note that if A = ∆ α (with α ∈ (0, 1]) on an open bounded set D ⊂ R d (see Remark 4.13) then for ρ = R1 we have M ρ = {µ ∈ M : δ α · µ ∈ M b }, where δ(x) = dist(x, ∂D), because by [22] there exists c, C > 0 such that In the rest of the paper we assume that ρ ∈ S and ρ is strictly positive.

Stampacchia's definition by duality
In [27] Stampacchia introduced a definition of a solution of (3.1) in case µ ∈ M b and A is uniformly elliptic operator of the form on a bounded open set D ⊂ R d . According to this definition, now called Stampacchia's definition by duality, a measurable function u ∈ L 1 (D; m), where m is the Lebesgue measure on R d , is a solution of (3.1) if (u, η) = (Gη, µ), η ∈ L ∞ (D; m).
The above definition has sense, because it is well known that for A as above Gη has a bounded continuous version. In the general case considered in the paper the original Stampacchia's definition has to be modified, because the measure µ is not assumed to be bounded, Gη may be not continuous for η ∈ L ∞ (E; m) and moreover, the solution of (3.1) may be not locally integrable (see [17,Example 5.7]). In [17] we introduced a generalized Stampacchia's definition for solutions of (4.1) with Dirichlet operator A and bounded measure µ such that µ ≪Cap. Here we give a definition for general measures of the class M.  Proof. Let u be a solution of (4.1) in the sense od Definition 3.1. Then by Proposition 4.5, |u| + R|f u | ≤ R|µ|, it is clear that u is a solution of (4.1) in the sense of Stampacchia. Now assume that u is a solution of (4.1) in the sense of Stampacchia. By Lemma 4.10 there exists a strictly positive ρ ∈ S such that µ ∈ M ρ . In fact, from the proof of Lemma 4.10 it follows that we may take ρ = Rg for some strictly positive Borel function g on E. We have Remark 4.14. In [18] renormalized solutions of (4.1) are defined in case µ is a bounded smooth measure. It is also proved there that u is a renormalized solution of (4.1) if and only it is a probabilistic solution. Thus, in case µ is smooth, all the definitions (renormalized, Stampacchia's by duality, probabilistic, via the resolvent kernel) are equivalent.
Remark 4.15. In case A is the Laplace operator on an open bounded set D ⊂ R d , also the so-called weak solutions of (4.1) are considered in the literature (see, e.g., [5]). A weak solution of (4.1) is a function u ∈ L 1 (D; dx) such that f u ∈ L 1 (D; dx) and for It is clear that the definition of weak solution is equivalent to Stampacchia's definition by duality. It is worth pointing out that in fact the concept of weak solutions is also due to Stampacchia (see [27,Definition 9.1]).

Existence of solutions
In [17] (see also [19] for the case of operator corresponding to general nonsymmetric quasi-regular form) it is proved that if µ is smooth then under conditions (H1)-(H3) there exists a solution of (4.1). It is well known that if A = ∆ and µ is not smooth, i.e. µ c = 0, then in general assumptions (H1)-(H3) are not sufficient for the existence of a solution of (4.1). In this section we give an existence result for (4.1) under the following additional hypothesis: (H4) there exists a positive Borel measurable function g on E such that g · m ∈ M and |f (x, y)| ≤ g(x), x ∈ E, y ∈ R.
Hypothesis (H4) imposes rather restrictive assumption on the growth of f but allows us to prove the existence of solutions for any µ ∈ M and any Dirichlet operator A. Proof. Let ̺ be a strictly positive Borel function on E such that Let us define Φ : L 1 (E; ̺ · m) → L 1 (E; ̺ · m) by Φ(u) = Rf (·, u) + Rµ.

Good measures and reduced measures
In this section we develop the theory of reduced measures for (1.1) in case of general Dirichlet operator A and general measure µ of the class M. Our results generalize the corresponding results from H. Brezis, M. Marcus and A.C. Ponce [5] proved in the case where A is the Laplace operator on a bounded domain in R d and µ is a bounded measure. Also note that in [5] it is assumed that f does not depend on x.
In the whole section in addition to (H1)-(H3) we assume that f (x, y) = 0 for y ≤ 0.
Definition 5.1. We say that a measurable numerical function v on E is a subsolution of (4.1) if f v · m ∈ M and there exists a measure ν ∈ M such that ν ≤ µ and Then u n ց u * , where u * is a maximal subsolution of (4.1). Moreover, the measure µ * = −Au * − f (x, u * ) admits decomposition of the form µ * = µ d + ν with ν⊥Cap such that ν ≤ µ c .
Proof. Let {τ k } be a reducing sequence for u n . By the Tanaka-Meyer formula, x ∈ E. Letting k → ∞ in the above inequality we get for q.e. x ∈ E. Let v n , w n be solutions of the following equations −Av n = f + n (·, u n ) + µ + , −Aw n = f − n (·, u n ) + µ − .
Of course, v n , w n are excessive functions and by (5.1), v n = Rf + n (·, u n ) + Rµ + ≤ 2R|µ|, w n = Rf − n (·, u n ) + Rµ − ≤ 2R|µ|. (5.2) By [11, Lemma 94, page 306], from {v n } and {w n } one can choose subsequences convergent m-a.e. to excessive functions v and w, respectively. By (5.2) and [14], there exists ν 1 , ν 2 ∈ M + such that v = Rν 1 , w = Rν 2 . By Theorem 3.7 the function h = R|µ| is quasi-continuous. Therefore if we put δ 1 k = inf{t ≥ 0 : h(X t ) ≥ k} ∧ ζ, then δ 1 k ր ζ, P x -a.s. for q.e. x ∈ E. From Theorem 3.7 it also follows that h(X) is a special semimartingale. Therefore there exists a sequence {δ 2 k } ⊂ T such that δ 2 k ր ζ and for q.e. x ∈ E, E x sup We may assume that τ k = δ 1 k = δ 2 k . Since by Proposition 4.2, u n (x) ≥ u n+1 (x), n ≥ 1, for q.e. x ∈ E, there exists u * such that u n ց u * , q.e. Therefore letting n → ∞ in the equation and using (H1)-(H3), (5.1) (and the fact that for q.e. x ∈ E. By (5.1) and Fatou's lemma, f (·, u * ) · m ∈ M. Hence u * is a solution of (4.1) with µ replaced by µ * := µ d + ν c . What is left is to show that u * is the maximal subsolution of (4.1). By the construction of u * , u n ≥ u * . Therefore by condition (b) of the definition of a probabilistic solution of (4.1) and Lemma 3.5 (see also Remark 3.8) we have µ * c ≤ µ c , which when combined with the fact that µ * d = µ d shows that µ * ≤ µ, i.e. that u * is subsolution of (4.1). Suppose that v is another subsolution of (4.1). Then there exists β ∈ M such that β ≤ µ and v is a solution of (4.1) with µ replaced by β. Since β ≤ µ and f n ≥ f , applying Proposition 4.2 shows that u n ≥ v q.e., hence that u * ≥ v q.e., which completes the proof. ✷ Let µ ∈ M. From now on by µ * , u * we denote the objects constructed in Theorem 5.2. By Theorem 5.2, µ * ≤ µ. It is known (see [2]) that it may happen that µ * = µ, i.e. that there is no solution of (4.1) under assumptions (H1)-(H3). In what follows we denote by G the set of all good measures relative to A and f . Of course, µ * ∈ G.
x ∈ E. Of course, u is a subsolution of (4.1), so by Theorem 5.2, u = u * and u n ց u. By this and (5.4), for q.e. x ∈ E. Since f n (X, u n (X)) → f (X, u(X)), dt ⊗ dP x -a.e. for q.e.
x ∈ E and f n (X, u n (X)) ≤ 0, applying Vitali's theorem shows that the sequence {f n (X, u n (X))} is uniformly integrable under the measure dt ⊗ P x for q.e. x ∈ E, and hence for m-a.e.
Proof. Let {u n } be the sequence of functions of Theorem 5.2 associated with µ and let {v n } be a sequence constructed as {u n } but for µ replaced by ν. By Proposition 4.2, v n ≤ u n q.e. Consequently, f n (·, u n ) ≤ f (·, v n ) ≤ 0 q.e. Since µ ∈ G, we know from Proposition 5.5 that the sequence {f n (X, u n (X))} is uniformly integrable under the measure dt ⊗ P x for m-a.e. x ∈ E. Therefore {f n (X, v n (X))} has the same property. By Proposition 5.5, this implies that ν ∈ G. ✷ Corollary 5.7. If µ ∈ M and µ + ∈ G, then µ ∈ G.
Theorem 5.11. Let µ ∈ M. The following conditions are equivalent: (iv) µ = g − Av for some functions g, v on E such that g · m ∈ M and f (·, v) · m ∈ M.
Proof. That (i) is equivalent to (ii) follows from Corollary 5.7 and Corollary 5.8. That (ii) implies (iii) follows from the fact that µ c ≤ µ + and Proposition 5.6. Suppose that µ c ∈ G. Since µ d ∈ G and µ + = µ d ∨ µ c , it follows from Corollary 5.8 that µ + ∈ G. Thus (iii) implies (ii). Of course (i) implies (iv). Suppose now that (i) is satisfied.
Hence µ − g − f (·, v) ∈ G, and consequently (µ − g − f (·, v)) c = µ c ∈ G, because we already know that (i) implies (iii). Hence µ ∈ G, because we also know that (iii) implies Corollary 5.12. We have Let us consider the following hypothesis: Proof. First we show that G ρ is a closed subset of (M ρ , · ρ ). Let {µ n } ⊂ G ρ be a sequence such that µ n → µ in (M ρ , · ρ ) for some µ ∈ M ρ . Let u n denote a solution of (4.1) with µ replaced by µ n and let ψ be a strictly positive Borel function on E such that Rψ ≤ ρ, m-a.e. Let us observe that (R|µ n − µ m |, ψ) ≤ µ n − µ m ρ , n, m ≥ 1. Therefore there exists u ∈ L 1 (E; ψ · m) such that u n → u in L 1 (E; ψ · m). By the definition of a solution of (4.1), u n = Rf un + Rµ n , m-a.e.

Inverse maximum principle and Kato's inequality
In this section we consider the linear equation (3.1). The following theorem generalizes the inverse maximum principle proved by H. Brezis and A.C. Ponce in [6] in case A is the Laplace operator on a bounded domain in R d .
Proof. Assume that u ≥ 0. Let {τ k } be a reducing sequence for u. By the definition of a solution of (3.1), for every α ≥ 0, for q.e. x ∈ E. In particular, R α µ c (x) ≥ 0 for q.e. x ∈ E, and hence, by [3, Proposition II.3.2], R α µ c ≥ 0 everywhere. That µ c ≥ 0 now follows from Lemma 3.5. ✷ Proposition 6.2. Assume that µ ∈ M. Let u be a solution of (3.1) and let ϕ be a positive convex Lipschitz continuous function on R such that ϕ(0) = 0. Then Aϕ(u) ∈ M. Moreover, Proof. Let {τ k } be a reducing sequence for u. By the definition of a probabilistic solution of (3.1), for some local MAF M . By the Itô-Meyer formula, for some increasing process A, where ϕ ′ is the left derivative of ϕ. Let A p denote the dual predictable projection of A (one can find a version of A p which is independent of x; see [9]). Since A p is predictable, it is continuous, because the filtration (F t ) is quasi-left continuous. Therefore there exists a positive smooth measure ν such that and observe that ϕ(u)(X t ) + v 1 (X t ) + v 2 (X t ) = ϕ(u)(x) for some local MAFM . Set w = ϕ(u) + v 1 + v 2 . From the above equation and the fact that w ≥ 0 it follows that w(X) is a supermartingale. Therefore w is an excessive function. On the other hand, w ≤ |ϕ(u)| + v 1 + v 2 ≤ Lip(ϕ)|u| + Rν + Rµ − d ≤ Lip(ϕ)R|µ| + Rν + Rµ d . Therefore by [14,Proposition 3.9] there exists a positive β ∈ M such that w = Rβ. This implies that Aϕ(u) = β − ν − µ − d ∈ M. By (6.1) and the assumptions on ϕ, for q.e. x ∈ E. Letting k → ∞ and applying Lemma 4.6 we get the desired result. ✷ The following version of Kato's inequality was proved by H. Brezis and A.C. Ponce [6] (see also H. Brezis, M. Marcus and A.C. Ponce [5]) in case A is the Laplace operator on a bounded domain in R d ). By the resolvent identity, for every α ≥ 0 we have u = R α (µ + αu), u + = R α (ν + 1 {u>0} µ d − l + αu + ).

Equations with polynomial nonlinearity
In this section we give a necessary and sufficient condition on µ ensuring the existence of a solution of (4.1) with f satisfying the condition |f (x, u)| ≤ cu p , x ∈ E, u ≥ 0 (7.1) for some constants c ≥ 0, p > 1. We also calculate the reduced measure in the case where f (x, u) = −u p . In our study a primary role will be played by a new capacity Cap A,p , which we define below. Let p ≥ 1. By the Riesz-Thorin interpolation theorem one can extend the semigroup {T t , t ≥ 0} from L 2 (E; m) ∩ L p (E; m) to L p (E; m). We denote the extended semigroup by {T p t , t ≥ 0}, whereas by {R p α , α > 0} we denote its resolvent. Let (A p , D(A p )) be the operator generated by {T p }. It is well known that D(A p ) = R p 1 (L p (E; m)). We set D + (A p ) = R p 1 (L p,+ (E; m)). Each element of D + (A p ) is defined pointwise via the resolvent kernel. Let V p denote the space D(A p ) equipped with the norm u Vp = A p u L p (E;m) + u L p (E;m) .
We define the capacity of B ⊂ E as It is an elementary check that Cap A,p is subadditive and increasing (see, e.g., [1, Proposition 2.3.6]). We say that µ ∈ V ′ p ∩ M + if for every η ∈ V + p , (η, µ) ≤ c η Vp .
In the rest of the section we assume that p > 1. By p ′ we denote the Hölder conjugate to p. Then u ∈ L p ′ (E; m) ∩ L(E; m). Indeed, the fact that u ∈ L(E; m) follows from the inequality Ru ≤ Rµ. Now, for f ∈ L p,+ (E; m) set η = R p 1 f . Then which shows that u ∈ L p ′ (E; m). That µ is a good measure relative to f (u) = −|u| p ′ now follows from Theorem 5.11. ✷ Lemma 7.2. Let u ∈ D + (A p ). Then for every λ > 0, Cap A,p (u ≥ λ) ≤ λ −p u p Vp .
Corollary 7.8. Let the assumptions of Corollary 7.7 hold. Let µ + Cap A,p ′ denote the absolutely continuous part, with respect to Cap A,p ′ , of the measure µ + . Then Proof. It suffices to repeat step by step the proof of [5,Theorem 16]. ✷ Remark 7.9. Let us note that from [1, Proposition 2.3.13] (see also [15]) it follows that for all p > 1, α ∈ (0, 1] and open bounded set D ⊂ R d , where A = ∆ α on D with zero boundary condition (see Remark 4.13) and for a compact K ⊂ D the capacity Cap D α,p (K) is defined by (1.5).